2
Contents Preface
3
1 Classical Relativity 1.1 Frames of Reference 1.2 The Galilean-Newtonian Relativity Principle 1.2.1 1.2.1 Galil Galilean ean Veloc Velocity ity Addit Addition ion 1.2.2 The Galilean Transformation
4 4 6 6 7
2 Special Relativity 2.1 Ein Einsteins Pos Postt!lat lates 2.1.1 Einstein’s Two ost!lates of "#ecial $elati%ity 2.1. 2.1.22 The The &ore &orent nt'' Tran Transf sfor orma mati tion on 2.1.3 $elati%istic Addition of Velocities 2.1. 2.1.44 The The )ich )ichel elso son* n*)o )orl rley ey E+#e E+#eri rime ment nt 2.2 Sim!ltaneity 2." Time #ilation 2.3. 2.3.11 Time Time -ila -ilati tion on 2.3. 2.3.22 The The Twin Twin ara arado do++ 2.3. 2.3.33 -eca -ecayy of the the )! )!on on 2.$ %en&th Co Contraction 2.4.1 &en/th Contraction 2.' Relat lativis vistic (o (omen ment!m 2.) Relativistic (ass 2.* Relat lativis vistic Ener Ener&&y 2.+ 2.+ Rela Relati tivi vist stic ic #opp #opple lerr Ef Effect ect
11 11 12 14 1( 23 2, 31 31 4 4 43 43 0 02 04 6
". Gene General ral Relat Relativi ivity ty ".1 ".1 ,ner ,nerti tial al (ass (ass an an Gra Gravi vita tati tion onal al (as (asss ".2 ".2 The Princ Principl iples es of E!i E!ival valen ence ce an an Cova Covari rian ance ce "." "." Test Testss of the the Gene Genera rall Theo Theory ry of of Rel Relat ativ ivit ityy ".$ Geometry of Spacetime ".' Einsteins C!rvat!re of Spacetime ".) /lac0 /lac0 oles oles
63 63 64 66 67 6( 6(
$. Prolems
73
References
,1
,nternet Reso!rces
,2
2
Contents Preface
3
1 Classical Relativity 1.1 Frames of Reference 1.2 The Galilean-Newtonian Relativity Principle 1.2.1 1.2.1 Galil Galilean ean Veloc Velocity ity Addit Addition ion 1.2.2 The Galilean Transformation
4 4 6 6 7
2 Special Relativity 2.1 Ein Einsteins Pos Postt!lat lates 2.1.1 Einstein’s Two ost!lates of "#ecial $elati%ity 2.1. 2.1.22 The The &ore &orent nt'' Tran Transf sfor orma mati tion on 2.1.3 $elati%istic Addition of Velocities 2.1. 2.1.44 The The )ich )ichel elso son* n*)o )orl rley ey E+#e E+#eri rime ment nt 2.2 Sim!ltaneity 2." Time #ilation 2.3. 2.3.11 Time Time -ila -ilati tion on 2.3. 2.3.22 The The Twin Twin ara arado do++ 2.3. 2.3.33 -eca -ecayy of the the )! )!on on 2.$ %en&th Co Contraction 2.4.1 &en/th Contraction 2.' Relat lativis vistic (o (omen ment!m 2.) Relativistic (ass 2.* Relat lativis vistic Ener Ener&&y 2.+ 2.+ Rela Relati tivi vist stic ic #opp #opple lerr Ef Effect ect
11 11 12 14 1( 23 2, 31 31 4 4 43 43 0 02 04 6
". Gene General ral Relat Relativi ivity ty ".1 ".1 ,ner ,nerti tial al (ass (ass an an Gra Gravi vita tati tion onal al (as (asss ".2 ".2 The Princ Principl iples es of E!i E!ival valen ence ce an an Cova Covari rian ance ce "." "." Test Testss of the the Gene Genera rall Theo Theory ry of of Rel Relat ativ ivit ityy ".$ Geometry of Spacetime ".' Einsteins C!rvat!re of Spacetime ".) /lac0 /lac0 oles oles
63 63 64 66 67 6( 6(
$. Prolems
73
References
,1
,nternet Reso!rces
,2
reface
3
t is called annus mirabilis mirabilis the year 1(0. 1(0. An !nnown !nnown cler in the ern atent atent 5ffice 5ffice in "wit'erla "wit'erland nd #!lished a s!ccession of fo!r #a#ers in the #resti/io!s #resti/io! s German o!rnal Annalen o!rnal Annalen der Physik and the world of #hysics was chan/ed fore%er. fore%e r. -!rin/ this mirac!lo!s year Alert Alert Einstein re%ol!tioni'ed re%ol!tioni'ed the field of #hysics with his s#ecial theory of relati%ity relati%ity which alon/ with the wor of other scientists scientists in the emer/in/ emer/in/ field of 8!ant!m mechanics mechanics s!ch as 9iels ohr :erner ;eisener/ :olf/an/ a!li and Erwin "chr #hysics was no lon/er e8!i##ed to deal shatterin/ the ill!sion that after electroma/netis electroma/netism m o#tics and mechanics mechanics were !nderstood at the end of the nineteenth cent!ry cent!ry there was not m!ch else left for #hysicists #hysicists to do !t refine their theories theories and tie !# the loose ends. The d!al wa%e*#article wa%e*#article nat!re of matter the im#ossiility of definin/ oth moment!m moment!m and #osition of a #article sim!ltaneo!sly sim!ltaneo!sly the s!#er#osition of 8!ant!m states and other 8!ant!m #henomena soon e+#osed the inade8!acy of classical #hysics and called for fo r a radical %iew. :hile 8!ant!m mechanics was raisin/ to!/h 8!estions Einstein introd!ced a conce#t of s#ace and time that dee#ened o!r !nderstandin/ of the eha%ior of macrostr!ct!res s!ch as the stars the /ala+ies and the !ni%erse. !ni%erse. 5f the fo!r #a#ers Einstein Einstein s!mitted one descried descried the eection of electrons electrons y #hotons now called the #hotoelectric effect effect which hel#ed !ild the fo!ndation for 8!ant!m theory and which earned him the 9oel #ri'e for #hysics in 1(21. Another #a#er determined determined the si'es of molec!les molec!les from a st!dy of s!/ar molec!les in a water sol!tion and the n!mer of molec!les in a /i%en mass of a s!stance. The ne+t #a#er showed how the irre/!lar 'i/'a//in/ mo%ements called rownian motions of #articles of smoe #ro%ide e%idence of the e+istence e+istence of molec!les and atoms. !t the most im#ortant is his #a#er on s#ecial relati%ity. relati%ity. This seminal wor from a #atent cler who had to hide his o!tside in%esti/ation in his office drawer whene%er he heard footste#s is nothin/ short of mirac!lo!s. mirac!lo!s. 9ow we now that neither time nor s#ace is asol!te and that we can only s#ea of time and s#ace in relation to some frame of reference and that they differ de#endin/ on the frames of reference. "#ecial relati%ity as its name im#lies is only a s#ecial case of the /eneral theory of relati%ity and deals with #henomena that that occ!r in inertial reference reference frames. frames. The /eneral theory theory in%esti/ates eha%ior eha%ior in non* inertial frames. :e will e+amine the theory of relati%ity in its totality. The tri#artite di%ision of this mono/ra#h aims at treatin/ the conce#t of relati%ity with the thoro!/hness that will satisfy the c!rio!s as well as the st!dent of relati%ity. To fi+ the theory firmly in the reader’s mind ha%e incl!ded co#io!s e+am#les and #rolems for #ractice. "ome st!dents of #hysics may find them !sef!l. As this year mars the one h!ndredth anni%ersary of the s#ecial theory of relati%ity a re%iew of the theory of relati%ity is a fittin/ tri!te to the /eni!s who dominated science and #o#!lar ima/ination for m!ch of the twentieth cent!ry. Thomas -. &e 31 -ecemer 20
1. Clas Classi sica call $ela $elati ti%i %ity ty 1.1
4
Frame of Reference
Einstein’s s#ecial theory of relati%ity relati%ity introd!ced in 1(0 descries the world as it is. 5!r conce#tion of s#ace and time radically chan/es y this new insi/ht and we /ain a dee#er !nderstandin/ of #hysical reality as a res!lt. res!lt. ;owe%er ;owe%er the idea of relati%ity had e+isted since Galileo’s Galileo’s time time aleit as a constricted one that was ade8!ate eno!/h to acco!nt for e%eryday #henomena #henomena within the framewor of classical classical #hysics. Einstein’s Einstein’s contri!tion was a #aradi/m shift not only in that it e+tended relati%ity to a wider ran/e of #henomena as it refines the theory !t also in that it re%eals the !ni%erse as %astly different from one concei%ed with the classical #hysics #aradi/m. The f!ndamental 8!estion of s#ecial relati%ity deals with the differences etween the meas!rements of a sin/le #henomenon made in two different frames of reference that mo%e with a !niform s#eed relati%e to each other. These meas!rements are of s#ace and time since all nat!ral #henomena occ!r within reference frames frames that are defined in terms of s#ace and time. "#ecifically a phenomenon =or an event> event> occ!rs within the three dimensio dimensions ns of s#ace s#ace and one dimensio dimensionn of time time all of which delimit delimit a frame frame of refere reference. nce. Altho!/ Altho!/hh int!iti%ely we may feel that time is somehow different from s#ace as we now more ao!t Einstein’s theory of relati%ity we will e con%inced that s#ace and time are !st #art of a contin!!m #art of the faric of spacetime. &et !s now e/in o!r in8!iry with the frame of reference. reference. :hile ridin/ in a car tra%elin/ tra%elin/ at a constant s#eed of 6 miles #er #er ho!r yo! throw a all !# in the air. :here does it land? land? "trai/ht "trai/ht down into yo!r hand if yo! do not mo%e. This oser%ation oser%ation wo!ld e identical if yo! carried o!t the same e+#eriment e+#eriment standin/ standin/ in yo!r li%in/ room. room. The law of /ra%ity /ra%ity wors the same same way in either either case. @et yo! were were in two different different frames of reference reference a car mo%in/ with !niform s#eed and a motionless room. There is no e+#eriment yo! can thin of that shows from the eha%ior of the tossed all whether yo! are in a mo%in/ car or in a room at rest. :e call them inertial reference frames eca!se frames eca!se they are either at rest or mo%in/ at a constant %elocity. 9ow thin thin of an oser%er oser% er standin/ y the t he roadside while yo!r yo !r car was dri%in/ #ast. ;ow did she see the all dro#? d ro#? To her the all al l did not fall down %ertically % ertically as a s yo! had e+#erienced. e+ #erienced. "he saw it fall forward with the %elocity of the car in a #araolic c!r%e. The #araolic #ath of the all is in accordance with the laws of mechanics. Clearly the all traectory differs de#endin/ on the frames of reference a/ain i.e. whether the oser%er oser%er is mo%in/ or stationary. ;owe%er ;owe%er the stationary oser%er oser%er is in a different different inertial frame than the #assen/er in the t he car. The oser%er’s reference frame is the earth. Altho!/h the t he earth re%ol%es re%ol% es aro!nd its i ts a+is and orits the s!n for #ractical #!r#oses and for the time ein/ we can re/ard its acceleration and rotation as ne/li/ile ne/li/ile and the earth as as an inertial inertial frame. frame. This shows shows that motion is not asol!te !t relative to the reference frame in which it occ!rs. occ!rs. There is e8!i%alence e8!i%alence of the laws of mechanics mechanics =e./. =e./. law of inertia inertia of !ni%ersal /ra%itation and so forth> in different inertial frames of reference. And no inertial frame is #referredB #referredB y the laws of mechanics o%er any other inertial inertial frame since they are all e8!i%alent. @o! co!ld indifferently indifferently say in the e+am#le ao%e that the car is mo%in/ and the earth is at rest or the car is at rest and the earth is mo%in/ which is what the #assen/er in the car #ercei%es anyway. The all*throwin/ e+#eriment leads to the same res!lt whether yo! are tra%elin/ in a train on a oat or in an air#lane mo%in/ mo%in/ at a constant =non*acceler =non*accelerated> ated> %elocity. %elocity. The laws of mechanics mechanics a##ly e8!ally to the #assen/er as well as to the oser%er fi+ed on o n Earth. or e+am#le when whe n a fli/ht attendant in an air#lane #o!rs #o !rs coffee it fills the c!# in the the same way that it does when when yo! #o!r it in yo!r itchen. E%erythin/ in an air#lane flyin/ at a !niform s#eed v’ of 1 mDh oeys oeys the same laws laws of mechanics mechanics as it does on earth. earth. To an oser%er in the stationary reference frame of the earth a fli/ht attendant who wals toward the front of the air#lane with the !niform s#eed u’ of of 3 mDh adds her s#eed to that of the air#lane. ;ence to an oser%er on earth she tra%els at u = u’ + v’ or u = 3 mDh F 1 mDh 13 mDh altho!/h in her own reference frame of the air#lane she wals 3 mDh. n /eneral if a reference frame S’ =e./. =e./. the air#lane> mo%es with a %elocity of v’ with res#ect to another reference frame S =e./. =e./. the earth> then the %elocity u of an oect relati%e to the frame S is is e8!al to the s!m of the two %elocities v’ and and u’ the latter ein/ the %elocity of the oect in the mo%in/ reference frame S’ . 5f co!rse if the air#lane accelerates as at taeoff or flies flies thro!/h a dist!rance it is no lon/er an inertial reference frame and the laws of mechanics do not a##ly in the same way as efore. &et !s tae the e+am#le of the air#lane =which we call reference frame S’ > tra%elin/ with a !niform =i.e. non*accelerated> %elocity of 1 mDh a/ain. All oects in the air#lane are tra%elin/ with the same %elocity relati%e to the earth. @et when a #assen/er in her seat #ics !# a oo to read they are oth at rest relati%e to
0 the air#lane. :hen she lets /o of the oo it falls strai/ht down !st as it does in her li%in/ room in accordance with the law of /ra%ity. There is nothin/ inside the air#lane other than the r!mlin/ of the en/ines and the occasional dis#lay on the monitor in the seat in front of her that tells her she is tra%elin/ at 1 mDh. The oo she is readin/ tra%els with the same %elocity as she is so to her it is stationary. To an earth*o!nd oser%er =which we call reference frame S > the oo is not motionless. :ho is correct the #assen/er or the oser%er? oth are correct. The difference in their e+#erience comes from the frames of reference in which they find themsel%es. The #assen/er and her oo are at rest with res#ect to the reference frame of the #lane and are mo%in/ at 1 mDh with res#ect to the reference frame of the earth. 5r yo! can e%en correctly say that the flyin/ air#lane is at rest and the earth is mo%in/. Any air#lane tra%eler thins this is the case. The classical laws of mechanics s!ch as the law of !ni%ersal /ra%itation a##ly e8!ally in oth reference frames. A/ain there is no one inertial frame that is etterB or #referred o%er any other inertial frame. 9ow s!##ose that yo! are in a cain of a shi# mo%in/ in a strai/ht line with constant s#eed. f yo! dro# a oo from the ceilin/ it falls strai/ht down in e+actly the same way it wo!ld in yo!r ho!se. 9ot only that it accelerates at (., mDs2 as it does on shore. "ince yo!r reference frame is an inertial frame accordin/ to Galileo the !niform mo%ement of the reference frame has no oser%ale effects. 9ewton’s first law of motion =the law of inertia> and second law =the time rate*of*chan/e of moment!m> a##ly in this case. n fact all laws of mechanics a##ly as well. These e+am#les and co!ntless others lie them show that there is no s!ch thin& as asol!te motion. There is motion only relati%e to a /i%en frame of reference. Galileo and 9ewton new this and called it relativity. ="ee "ection 1.2. elow.> As an ill!stration a reference frame S =the !n#rimed frame> is re#resented y a set of coordinates H x, y, z, t I called Cartesian coordinates the first three of which are /ra#hically re#resented y three lines that are #er#endic!lar to one another with the oser%er normally #laced at their common ori/in O as in i/!re 1*1 elow where x y and z are s#atial dimensions and t is the time dimension. The time dimension t does not ha%e a /ra#hical re#resentation. To meas!re an e%ent =an e%ent occ!rs in some reference frame> we m!st now not only where it occ!rred !t also when it did. ;ence a fo!rth dimension is needed. &iewise a reference frame S’ =the #rimed frame> is descried y the set of coordinates H x’, y’, z’, t’ I. y’
y
%t’
+’ direction of motion
5’ 5
"’
J
" +
'
+’
+
'’ Fi&!re 1-1. Frames of reference
i/!re 1*1 ill!strates two reference frames The =!n#rimed> frame S is at rest and the =#rimed> frame S’ is in motion =:e normally mae S’ the mo%in/ frame.>. :e can consider S’ as the inertial frame mo%in/ with a constant %elocity v for time t’ in the x direction for e+am#le a #assen/er in a train at O’ . n the frame S’ the #assen/er at O’ sits a distance x’ from the ac of the seat !st in front of her at P . The train is mo%in/ away from the stationary oser%er at O =not shown> on the /ro!nd in the frame S. The distance x from the /ro!nd oser%er to the seat in front of the onoard #assen/er which is e8!al to the distancevt’ F x’ will increase as the train mo%es away with %elocity v. Kee# this in mind for the %elocity addition disc!ssed in "ection 1.2.1 elow.
6 y now we can see when s#eain/ of s#ace and time that e%erythin/ e+ists in some frame of reference. ;ence the most im#ortant factor in sol%in/ relati%ity #rolems is to identify the reference frame. 1.2
The Galilean-Newtonian Relativity Principle
The conce#t of relati%e motion was nown to Galileo and 9ewton =sometimes calledGalilean or Newtonian relativity> in the se%enteenth cent!ry. Accordin/ to this #rinci#le if the laws of mechanics a##ly in one inertial reference frame they also a##ly in any other inertial reference frame mo%in/ with constant %elocity relati%e to the first one so that it is im#ossile to tell whether an inertial frame is at rest or in motion. n other words all inertial frames are e8!i%alent and there is no #referred inertial frame. @o! can %erify this with yo!r own e+#erience. :hile sittin/ in a train ready to lea%e the station yo! oser%e the train on the ne+t trac mo%in/. or a while yo! are not s!re if yo!r train or the other is #!llin/ o!t of the station. @o! can tell only y looin/ o!t the window !sin/ the ac/ro!nd scene as reference #oint. The same can e said of an oser%er in the other train. &iewise while looin/ o!t the window of a standin/ air#lane with its en/ines on if yo! see another #lane ta+iin/ neary yo! cannot at first tell which #lane mo%es witho!t referrin/ to a fi+ed oect on the /ro!nd. n classical #hysics the Galilean or Newtonian relativity principle holds that all 9ewtonian laws of mechanics eha%e the same way in all inertial frames. This is the asic ass!m#tion of classical mechanics. And it conforms to common sense as we do not e+#ect these laws to differ when we chan/e #laces. ndeed 9ewton’s laws of motion can #erfectly acco!nt for motions of macrostr!ct!res at s#eeds far less than the s#eed of li/ht. ;owe%er the Galilean relati%ity #rinci#le failed to a##ly to )a+well’s theory of electroma/netism which had een 8!ite s!ccessf!l in descriin/ electroma/netic #henomena of which li/ht was elie%ed to e one. :hile the electroma/netic e8!ations #redicted the s#eed of li/ht to e c 3 + 1, mDs the Galilean transformations allowed a mo%in/ ody to e+ceed the s#eed of li/ht ="ee "ection 1.2.1>. This contradiction and st!dies of motions with %elocities a##roachin/ that of li/ht soon re%ealed the inade8!acies of 9ewton’s laws and the Galilean relati%ity #rinci#le. The remedy comes from Einstein’s s#ecial theory of relati%ity which can acco!nt for oects mo%in/ at any s#eed from 'ero !# to c th!s main/ the Galilean relati%ity #rinci#le a s#ecial case. 1.2.1
Galilean Velocity Addition
Common sense tells !s that if a mo%in/ #erson throws a all forward the s#eed of the all is e8!al to the #erson’s s#eed #l!s the all’s s#eed. This addition of %elocity is a nat!ral res!lt oser%ed in o!r daily e+#erience and we ne%er thin twice ao!t its conse8!ences for o!r int!ition a/rees with the #hysics of the #henomenon. 9othin/ we e+#erience contradicts this oser%ation. :e can formali'e the oser%ation with an e+am#le. "!##ose a #assen/er wals at the s#eed u’ of 3 mDs toward the front of a train tra%elin/ at a constant s#eed v of 3 mDs. :hat is the s#eed of the #assen/er as meas!red y an oser%er standin/ on the #latform? Tain/ the train as the S’ frame and the oser%er on the #latform as the S frame we ha%e u = u’ + v 3 mDs F 3 mDs 33 mDs where u is the #assen/er’s s#eed relati%e to the earth =i.e. the stationary oser%er> u’ is the s#eed of the #assen/er as meas!red in the " frame and v’ the s#eed of the train. This e8!ation re#resents the Galilean velocity aition. As far as the train #assen/er is concerned he mo%es only 3mDs. 9ow if the same #assen/er wals toward the rear of the train with the same s#eed he is mo%in/ in the direction o##osite to that of the train or u’ L3 mDs. )ore formally if the train mo%es alon/ the x*a+is in the x*direction then the #assen/er mo%es in the L x*direction. "!stit!tin/ this %al!e of u’ in the form!la we otain the #assen/er’s s#eed with res#ect to the stationary oser%er in S u = u’ + v – 3 mDs F 3 mDs 27 mDs. A/ain as far as the train #assen/er is concerned he mo%es only 3 mDs. Tho!/h oth #assen/er and oser%er oser%ed the same #henomenon they arri%ed at different concl!sions. @et they are oth ri/ht eca!se this relati%ity is the nat!re of the #hysical world.
7 9ote that o!r e+am#le ill!strates e%eryday s#eeds which are far less than the s#eed of li/ht orv c. And since we ne%er e+#erience any s#eeds close to c in daily life we ne%er s!s#ect anythin/ remiss ao!t classical relati%ity. The addition of %elocity seems to e worin/ correctly. To see how this %elocity addition within the Galilean =classical> relati%ity does not wor at a s#eed close to the s#eed of li/ht consider a Gedanen =tho!/ht> e+#eriment. nstead of a #assen/er in a train thin of an astrona!t tra%elin/ in a s#aceshi# with the s#eed close to that of li/ht v = .( c and he shoots a rocet forward in the direction of motion of the s#aceshi# at u’ = .10 c with res#ect to the s#aceshi#. :hat is the rocet’s s#eed relati%e to an oser%er on the earth? A##lyin/ the Galilean %elocity addition ao%e we /et u = u’ + v
314
u .10 c F .( c 1.0 c. where c is the s#eed of li/ht. "ince u is /reater than the s#eed of li/ht either the Galilean %elocity addition is wron/ or )a+well’s electroma/netic theory is wron/. )a+well’s electroma/netic theory #redicts that the s#eed of li/ht is inde#endent and has a definite %al!e c ! 3 + 1,mDs. ;owe%er )a+well’s theory which was #ro%en correct in n!mero!s e+#eriments cond!cted !nder all sorts of conditions wo!ld e contradicted eca!se then li/ht wo!ld #ro#a/ate at different s#eeds in different reference frames that mo%ed with res#ect to the ether which was elie%ed at the end of the nineteenth cent!ry to e the hy#othetical medi!m in which li/ht mo%ed. The )ichelson*)orley e+#eriment in "ection 2.1.4 elow shows the asence of the ether as well as the in%ariance of c. esides as we will see in "ection 2.1.1 elow the Galilean %elocity addition %iolates Einstein’s second #ost!late of the s#ecial theory of relati%ity. efore introd!cin/ Einstein’s s#ecial theory which will sol%e this #rolem and re%ol!tioni'e o!r !nderstandin/ of s#ace and time we e+amine how the Galilean transformation e+#resses the relationshi# etween an e%ent as meas!red in the S frame and the same e%ent as meas!red in the S’ frame. 1.2.2
The Galilean Transformation
The Galilean transformation is a set of relations that allows the meas!rements of an e%ent otained in one inertial frame S with coordinates H x, y, z, t I to e con%erted into meas!rements of the same e%ent in another inertial frame S’ defined y H x’, y’, z’, t’ I. These meas!rements are called the Galilean transformation. x = x’ + vt’
x’ = x – vt
y = y’
y’ = y
z = z’
z’ = z
t = t’
t’ = t
y
5
y’
%t "
+
5’
+’ "’
324
direction of motion J +’
+ Fi&!re 1-2. Ty#ical relati%ity sit!ation stationary frame S frame S’ mo%in/ at v relati%e to S e%ent P in S’. After time t P has mo%ed a total of x = vt + x’ with res#ect to S.
, &et !s see how this transformation came ao!t. i/!re 1*2 shows a ty#ical sit!ation a frame at rest S a mo%in/ frame S’ and an e%ent =oect #article s#acecraft> P s#acecraft> P mo%in/ mo%in/ in S’ . :e ass!me ass!me that inertial inertial referenc referencee frames frames S and and S’ ha%e ha%e their x their x*a+es *a+es alon/ the same line. &et the ori/in O’ of of frame S’ mo%e mo%e with constant %elocity v in the direction of the common x common x*a+is *a+is i.e. to the the ri/ht. "!##ose for for sim#licity sim#licity that at at time t = t’ = " the " the two #oints of ori/in coincide so that at some later time t’ they are se#arated y the distance meas!red y the #rod!ct of the %elocity v and the time ela#sed t’ or or vt’ . An e%ent denoted P denoted P mo%in/ mo%in/ with frame S’ co!ld then e descried y either the coordinates of frame S #x, y, z$ z$ or those of frame frame S’ #x’, y’, z’$. z’$. ts x ts x coordinate coordinate in the S frame frame is then x = x’ + vt’ and its x’ its x’ coordinate coordinate in the S’ frame frame is sim#ly the in%erse x’ = x – vt The min!s si/n maes sense eca!se relati%e to an e%ent P e%ent P in frame S’ which mo%es ri/htward frame S mo%es to the left. Gi%en that only the distance alon/ the x the x*a+is *a+is chan/es and there is no chan/e alon/ the y% the y% a+is or the z% the z%a+is a+is P P ’s y ’s y and and z z %al!es %al!es are !nchan/ed in oth frames y = y’ z = z’ And since in classical classical #hysics oth time and s#ace are considered considered asol!te clocs in the two inertial inertial frames show the same time. ;ence t = t’ 9ote that in the first set of relations all the %al!es on the left*hand side are !n#rimed and the %al!es on the ri/ht*hand side are #rimed. M!st the re%erse re%erse is tr!e in the second set of relations. The e8!ations in the second set are directly deri%ale from the first set. These corres#ondences show the !nderlyin/ ass!m#tions of classical mechanics namely that oth time and s#ace as said ao%e ha%e !ni%ersal and asol!te %al!es. or e+am#le the e8!ation t = t’ im#lies im#lies that time is the same re/ardless of reference frames. As we saw earlier these these transfo transforma rmation tion e8!ations e8!ations hold tr!e tr!e only only at %elociti %elocities es m!ch less less than the s#eed s#eed of li/ht li/ht v c. ;owe%er at s#eeds a##roachin/ the s#eed of li/ht they fail as we ha%e seen in "ection 1.2.1 ao%e. To see how the e8!ations of the Galilean transformation =2> are related to the Galilean %elocity addition =1> =1> consider consider the #oint P #oint P mo%in/ mo%in/ in the x*direction with constant %elocity v relati%e to frame S a/ain. ts %elocity %elocity u relati%e relati%e to frame frame S is u = N x & Nt and and its %eloc %elocity ity u’ relati% relati%ee to frame S’ is u’ = N x’ & Nt’ . nt!iti%ely these are related to e8!ation =1> u = u’ + v’ of the Galilean %elocity addition. :e co!ld deri%e the e8!ation =1> from the e8!ation e8!ation =2>. "!##ose the #article P #article P is is at coordinate coordinate x x' or x x '’ at at time t ' and at coordinate x coordinate x( or x x(’ at at time t (. The ela#sed time is then Nt Nt = t ( – t '. rom E8!ation =2> we /et N x x( L x x' = )x( * L x L x'’ + v)t ( – t ' = N x’ + vN vNt N x & Nt = N x’ & Nt’ + v u = u’ + v
=1>
As seen ao%e this last e8!ation e8!ation of %elocity addition addition #oses a serio!s #rolem. #rolem. t allows the s#eed of li/ht to e e+ceeded contrary to the s#ecial theory of relati%ity’s second #ost!late. The Galilean %elocity addition allows u to e /reater than c if u’ is is %ery close to c. ;owe%er for e%eryday e%ents where s#eeds are m!ch less than c the Galilean*9ewtonian framewor wors #erfectly well. 9ow that we ha%e disc!ssed disc!sse d the Galilean Galil ean #rinci#le of relati%ity and its shortcomin/s sh ortcomin/s at s#eeds close to the s#eed of li/ht we will t!rn to Einstein’s Einstein’s theory of relati%ity and see how it re%ol!tioni'ed o!r %iew of s#ace and time. As a re%iew let !s a##ly the e8!ations ao%e to a few e+am#les.
Examle 1. A #assen/er on a train mo%in/ at 1 mDh is walin/ toward the front at a s#eed of 0 mDh. :hat is the s#eed of the #assen/er with res#ect to an oser%er standin/ on the /ro!nd as the train #asses?
(
!olution. This is a classical %elocity addition. The train is the mo%in/ frameS’ frameS’ whose whose s#eed is v = 1 mDh carryin/ a #assen/er whose s#eed is u’ 0 mDh , and the oser%er is at rest in frame S. A##lyin/ the Galilean %elocity addition form!la yields the #assen/er’s s#eed relati%e to the stationary /ro!nd oser%er u = u’ + v u = 0 mDh F 1 mDh '" km&h. Examle 2 A train tra%elin/ at 12 mDh carryin/ a #assen/er who is walin/ toward its rear at 4 mDh. To a stationary oser%er on the /ro!nd how fast is the #assen/er mo%in/? !olution. This e+am#le is analo/o!s to E+am#le 1. ;owe%er instead of mo%in/ in the direction of the train the #assen/er mo%es in the o##osite direction main/ his s#eed a ne/ati%e %al!e. The same %elocity addition a##lies u = u’ + v u = L = L 4 mDh F 12 mDh = mDh = ''- km&h. Examle " A /ro!nd oser%er at rest meas!res the s#eed of an airline #assen/er in fli/ht walin/ toward the front front of the air#lan air#lanee to e 126 mDh. Knowin/ Knowin/ the air#lane’ air#lane’ss s#eed to e 12 mDh how fast is the #assen/er mo%in/? !olution. This e+am#le e+am#le is analo/o!s analo/o!s to the the other e+am#les ao%e. ao%e. A##lyin/ the the %elocity %elocity addition e8!ation e8!ation we /et u = u’ + v 126 mDh = u’ + 12 mDh u’ = 126 mDh L mDh L 12 12 mDh mDh = - km&h. Examle # Consider the frame S and and S’ at at t = t’ when O and O’ coincide coincide =$efer to i/!re 1*2>. rame S’ mo%es mo%es with the s#eed v = 40 mDs with res#ect to the frame S. A S. A #article P #article P in in S’ finally finally comes to rest at coordinates x’ coordinates x’ 3 m y’ m y’ 20m and z’ . Calc!late Calc!late the #osition of P of P with with res#ect to S = x, = x, y, z at z at =a> t = 2. s and => t = (. s. !olution. !olution. =a> :e !se the Galilean transformation e8!ation =2> to /et the x the x coordinate coordinate at t = 2. s x’ = x L x L vt vt
=2>
y’ = y z’ = z 3 m x L x L 40 mDs + 2 s x L x L ( m x x 3 m F ( m '("m. "ince the motion of P of P is is only in the x the x*direction *direction its y its y and z and z coordinates coordinates remain !nchan/ed y’ !nchan/ed y’ 20m and z’ . => A##lyin/ the Galilean Galilean transforma transformation tion a/ain a/ain we /et /et the x the x coordinate coordinate t = (. s
3 m x L x L 40 mDs + ( s x x L L 40 m x x 3 m F 40 m / m. A/ain P A/ain P ’s y ’s y and z and z coordinates coordinates remain !nchan/ed y’ !nchan/ed y’ 20m and z’ and z’ .
1
2. "#ec "#ecial ial $ela $elati ti%i %ity ty 2.1
11
Einsteins Post!lates
n a #a#er of 1(0 Einstein introd!ced an ele/ant and sim#le theory to show that time and s#ace are not asol!te !t %ary de#endin/ on the reference frames. ;e called itthe it the special theory of relativity eca!se relativity eca!se it a##lies only to a s#ecial set of reference frames called inertial5 i.e.5 frames in which everythin& is at rest or movin& at a constant5 non-accelerate velocity. velocity. :e will see in Cha#ter 3 that the s#ecial theory is only a s#ecial case of the /eneral theory which deals with all sorts of motions incl!din/ accelerated motions. The s#ecial theory theory of relati%ity is ased on two sim#le #ost!lates. #ost!lates. Taen se#arately se#arately the two #ost!lates #ost!lates that form the fo!ndation fo!ndation of the s#ecial theory seem innoc!o!s. innoc!o!s. The first #ost!late #ost!late is a /enerali'ation /enerali'ation of the classic classical al relati relati%ity %ity #rinci# #rinci#le le merely merely e+tendin e+tendin// its a##licat a##lication ion to all of #hysics #hysics incl!din incl!din// mechani mechanics cs electrodynamics electrodynamics o#tics o#tics and thermodynamics. thermodynamics. E%en the second #ost!late #ost!late #ositin/ the s#eed of li/ht as constant re/ardless re/ardless of the motion of any frames frames of reference reference tho!/h re8!irin/ ad!stment of o!r %iew of the #hysical world does not ta+ o!r cred!lity e+cessi%ely. And altho!/h the mathematics in%ol%ed chan/e accordin/ly they are remaraly sim#le. t is only when a##lied to/ether that the #ost!lates #rod!ce startlin/ res!lts that re%ol!tioni'e o!r !nderstandin/ of s#ace time and the !ni%erse. :e will see that instead of ein/ asol!te and !nchan/in/ time chan/es with reference frames. A min!te in yo!r li%in/ li%in/ room does not ha%e ha%e the same d!ratio d!rationB nB as a min!te min!te in a et #lane #lane flyin/ flyin/ at )ach 1. t is eca!se a tra%elin/ cloc r!ns r! ns slower than a stationary one a #henomenon calledtime called time ilation. ilation. 9ot only do clocs r!n slower in fast*mo%in/ fast*mo%in/ frames of reference reference all #hysical #rocesses #rocesses also r!n slower. slower. A #erson’s hearteat metaolism a/in/ #rocess slow down accordin/ly. s this an ill!sion or is it the nat!re of time? t is not an ill!sion. ill!sion. 5ne conse8!en conse8!ence ce of time time dilatio dilationn is that /i%en /i%en a##ro#r a##ro#riate iately ly ad%anced ad%anced technolo technolo/y /y astrona!ts astrona!ts in the distant f!t!re co!ld tra%el thro!/h inter/alactic inter/alactic s#ace within their lifetime and come ac to earth yo!n/er than the friends they had left ehind. This certainly is a mind*o//lin/ mind*o//lin/ and tantali'in/ #ros#ect for manind and fertile /ro!nd for science*fiction ima/inin/s in the meantime. Another Another conse8!enc conse8!encee is that there is no s!ch thin/ as asol!t asol!tee sim!lta sim!ltaneo! neo!ss e%ents. e%ents. E%ents E%ents that are #ercei%ed as sim!ltaneo!s in one reference frame are not sim!ltaneo!s in another. !t first what are sim!ltaneo!s sim!ltaneo!s e%ents? Einstein /a%e /a%e this definition definition f say for e+am#le e+am#le the train arri%es arri%es here at 7B this means the coincidence of the small hand of my watch with the n!mer 7 and the arri%al of the train are sim!ltaneo!s e%ents.0 e%ents.0 =Calle 22 #. 46> Then he clarified that s!ch a definition is not satisfactory when e%ents occ!rrin/ occ!rrin/ at widely widely dis#ersed dis#ersed #laces #laces are considered. considered. &et !s tae a tri%ial tri%ial case. case. ;ow can we say say that an e%ent occ!rrin/ occ!rrin/ in Toyo and another e%ent occ!rrin/ occ!rrin/ in 9ew @or are sim!ltaneo!s? sim!ltaneo!s? "!##ose a #erson in Toyo and her friend in 9ew @or wanted to li/ht !# their Christmas trees at the same time and comm!nicated comm!nicated %ia tele#hone to synchroni'e their actions. actions. They a/reed that at the co!nt of three oth wo!ld fli# the switch on instantly. And they did. Can we say that these two e%ents were were sim!ltaneo!s? 9o eca!se the tele#hone con%ersation had to tae time to tra%el across tho!sands of miles and y the time friend heard the co!nt of three from friend A the latter’s cloc had shown a later time and she had already thrown the switch e%en if their clocs had een synchroni'ed. ;ence the e%ents cannot e descried as sim!ltaneo!s. ;owe%er this e+am#le is a rather wea one for oth friends are still considered to e within the same inertial reference reference frame of the earth. earth. n o!r tho!/ht it is #ossile to %is!ali'e %is!ali'e them synchroni'in/ synchroni'in/ their clocs clocs at a distance and e+#eriencin/ e+#eriencin/ sim!ltaneo!s e%ents e%ents es#ecially when hi/her s#eeds are in%ol%ed. The sit!ation is 8!ite different when they are in different reference frames e./. one on Earth and the other tra%elin/ in a s#aceshi#. :e will see that in these casessim!ltaneity cases sim!ltaneity is relative. relative. @et another another conse8!enc conse8!encee of s#ecial s#ecial relati% relati%ity ity a##ears a##ears in the contrac contraction tion of distanc distance e called called len&th contraction. contraction. A mo%in/ meter stic is shorter shorter than a meter stic at rest. The 2*cm*diameter 2*cm*diameter #late on which yo!r friend’s dinner is ser%ed on oard a rocet shi# flyin/ #ast yo!r s#ace station shrins relati%ely to yo!. n this res#ect time and s#ace seem to e a contin!!m so that it maes sense to tal ao!t s#acetime as the fo!r dimensions dimensions of the !ni%erse the three three s#atial dimensions dimensions and the fo!rth dimension dimension which is time. Time dilation and len/th contraction are !st manifestations of the same conse8!ence of s#ecial relati%ity yo! /ain in time what yo! lose in s#ace when seen from different reference frames. @et in each frame of reference all laws of #hysics hold tr!e eca!se all inertial reference frames are e8!i%alent. or most of !s time is so different different from s#ace that callin/ it the fo!rth dimension /oes a/ainst common sense or int!ition. int!ition. :e do not e%en meas!re meas!re time in the same way we meas!re len/th. @et time and s#ace im#in/e on o!r e%eryday e%eryday life in th ine+tricale ine+tricale ways. "!##ose yo!r friend friend and yo! ha%e a date to meet on the 1 floor of the Em#ire "tate !ildin/ in 9ew @or City at the corner of 34th "treet and ifth A%en!e at 3 #m "at!rday "e#temer 24
12 20. The e%ent is defined y the three dimensions of s#ace one %ertical and two hori'ontal with time main/ !# the fo!rth. This e%ent cannot #ossily tae #lace e+ce#t in these fo!r dimensions. This is why we re#resent it as a system of fo!r coordinates H z, y, z, t I for mathematical disc!ssion. There is no com#lete descri#tion of motion witho!t time. n order to /ain a com#lete descri#tion we m!st s#ecify how a ody chan/es its #osition o%er time. To con%ince yo!rself !st wal from yo!r edroom to the itchen and see if the cloc shows the same time at the e/innin/ as at the end of yo!r tri#. f the cloc shows the same time yo! don’t need t to descrie yo!r motion. !t the cloc does show a different time and yo! are mo%in/ oth in s#ace and in time. 9ow ima/ine yo!rself sittin/ comfortaly in an armchair in yo!r li%in/ room readin/ a oo from , to ( one e%enin/. To yo! nothin/ in yo!r li%in/ room mo%es. @o!r #osition is defined s#atially y z, H y, zI and yo! may thin time has nothin/ at all to do with yo!r #osition. n reality yo! are tra%elin/ not only thro!/h s#ace !t also thro!/h time. The e%ent of yo!r readin/ taes #lace in the same s#ace in which the earth is walt'in/ aro!nd the s!n for one ho!r. @o! may e at restB with res#ect to the reference frame of the earth !t the earth is mo%in/ from the reference frame of an oser%er on an asteroid flyin/ y or from the s!n’s #oint of %iew. @o!r #osition at restB act!ally alters with time eca!se the earth mo%es at v 3. + 14 mDs. @o! are in a different location from where yo! were a second a/o. There is nothin/ yo! can do to e+tricate time from the s#atial dimensions. ;owe%er in o!r e%eryday e+#erience when s#eeds are m!ch smaller than the s#eed of li/htv c the effects of time dilation and len/th contraction are too small to e #erce#tile in the order of nanoseconds and nanometers. E%en o!r fastest rocets seem to crawl when com#ared to li/ht. "omeday when h!man ci%ili'ation is ca#ale of desi/nin/ machines that can accelerate contin!o!sly to %ery hi/h s#eeds e./. s#eeds a##roachin/ li/ht %elocity all of the #redictions of the s#ecial theory will e common#lace !st as the classical laws of inertia and of !ni%ersal /ra%itation are common#lace today. !t efore we /ot carried away y the tho!/ht of s#ace tra%el at hi/h %elocity we m!st reali'e that in order to accelerate to nearc we m!st #rod!ce infinite ener/y. 2.1.1
Einstein’s T$o %ostulates of !ecial &elativity
!ildin/ on the achie%ements of Galileo Galilei and 9ewton Einstein /ro!nds his s#ecial theory on two sim#le #ost!lates. :ith these #rinci#les the conflict etween 9ewtonian mechanics and )a+well’s electroma/netic theory is ele/antly resol%ed. The 1irst 23stulate the principle of relativity states The laws of physics are the same in all inertial frames of reference. This #ost!late is easy to acce#t. After all it conforms to o!r e%eryday e+#eriences and e+#ectations. or instance we wo!ld e+#ect the laws of #hysics =incl!din/ mechanics electroma/netism thermodynamics o#tics> to wor similarly in an en%ironment at rest as well as in one mo%in/ !niformly. :e wo!ld not e+#ect to see coffee flyin/ o!t of o!r c!# when we fly in an air#lane =ass!min/ the #lane is not /oin/ thro!/h a weather dist!rance or acceleratin/> any more than we wo!ld e+#ect to see it do the same in o!r itchen. Also it is im#ossile to distin/!ish one inertial reference frame from any other eca!se the laws of #hysics a##ly e8!ally well in any of them. Th!s no inertial reference frame whether at rest or in motion is #referred o%er another. The notion of asol!te motion and asol!te rest loses all si/nificance. :e concl!de that motion is relative. ;owe%er it is the sec3nd 23stulate called the principle of the constancy of the spee of li&ht that is harder to acce#t. %i&ht propa&ates thro!&h empty space with a efinite spee c inepenent of the motion of the so!rce or of the oserver. Accordin/ to )a+well’s electroma/netic e8!ations the s#eed of li/htc deri%es from the form!la OOOOO c 1 & 4 5" 6"
13 where the constant 5" is the permittivity of vac!!m =or free s#ace> and the constant 6" is the permeaility of vac!!m oth ein/ !sed in electric calc!lations. :hat is remarale is that this form!la !ses electrical methods to calc!late the s#eed of li/ht instead of a direct meas!rement. c == 5" 6">
L1D2
P = ,.,042 + 1
= 2.((7( + 1, mDs.
7 (
L12 OOOOOOOO
8 9 m(
8 9 s(
L7 OOOOOOO
> = 4: + 1
7 (
> Q L1D2
The asto!ndin/ feat achie%ed y !sin/ electroma/netic methods to estalish the s#eed of li/ht so #recisely leads to the acce#tance of li/ht as an electroma/netic wa%e. This s!ccess rans )a+well’s electroma/netic theory as a /reat theoretical achie%ement in #hysics alon/side 9ewton’s laws of motion and Einstein’s theory of relati%ity. n 1(,2 !sin/ a hi/hly staili'ed laser scientists deri%ed the s#eed of li/ht from the form!la %elocity wa%elen/th + fre8!ency v = ;1 as c = 2665*625$'+ 7 1.2 m8s. or most #!r#oses we !se c 3. + 1, mDs. As scientists are constantly tinerin/ with li/ht they are ale to slow it down ="hirer 24> sto# it or #!sh it eyond c. !t this is another story. n 1(,3 the "e%enteenth General Conference on :ei/hts and )eas!res ado#ted a definition of the meter ased on the ao%e s#eed of li/ht The meter is the len&th of path travele y li&ht in vac!!m !rin& a time interval of 182665*625$'+ of a secon. -es#ite all the con%incin/ calc!lations of the s#eed of li/ht the second #ost!late still r!ns co!nter to o!r common sense notions. That the s#eed of li/ht is inde#endent of the motion of its so!rce is not #rolematic. n this res#ect li/ht eha%es !st as so!nd does. 5nce emitted li/ht and so!nd ecome inde#endent of their so!rce or its motion. "o!nd s#eed then de#ends on the medi!m in which it #ro#a/ates. And efore the twentieth cent!ry scientists elie%ed that li/ht wo!ld e similarly affected y the hy#othetical ether wind called the l!minifero!s ether in which it tra%eled. :e will see how this hy#othesis was shattered y the )ichelson*)orley e+#eriment in "ection 2.1.4 elow. @et we find it hard to elie%e that a #erson tra%elin/ toward or away from a li/ht so!rce or !st stayin/ motionless meas!res the s#eed of li/ht in %ac!!m always to e c 3. + 1, mDs. 5!r int!ition wo!ld re8!ire !s to add or s!tract the s#eed of the oser%er de#endin/ on whether the oser%er is mo%in/ in the same direction as the li/ht so!rce or in the o##osite direction. 9ow ima/ine two s#aceshi#s "hi# 1 is at rest with res#ect to an o!tside so!rce of li/ht "hi# 2 is mo%in/ with %elocity v toward the same o!tside eacon. oth shi#s meas!re the li/ht from the eacon as well as the li/ht from their own internal so!rces. "hi# 1 meas!res the s#eed of internal li/ht as well as that of the eacon as c as is e+#ected from the first #ost!late. :hat ao!t "hi# 2? t meas!res the s#eed of the internal li/ht asc =accordin/ to the second #ost!late> !t it m!st meas!re the eacon’s li/ht as ha%in/ s#eed c also not c – v< for otherwise oth so!rces of li/ht wo!ld e !sed as a motion detector to determine that "hi# 2 is mo%in/ in %iolation of the first #ost!late. $ecall that accordin/ to the first #ost!late all inertial reference frames are e8!i%alent i.e. no reference frame is #referred o%er any other reference frame. :e concl!de that re&arless of the spee of an inertial oservers motion towar or away from the so!rce of li&ht5 an re&arless of the spee of a li&ht so!rces motion towar or away from the inertial oserver5 the spee of li&ht in vac!!m is always the constant c. Examle '. Astrona!t A #ilotin/ a s#aceshi# tra%elin/ eastward with %elocity .,c sends a li/ht eam forward. "ome distance away Astrona!t in another s#aceshi# tra%els westward toward the first s#aceshi# with %elocity .,0 c. :ith what s#eed does Astrona!t see the li/ht eam from Astrona!t A #ass y? !olution. The s#eed of li/ht in free s#ace is always c no matter how fast or slow an inertial oser%er =Astrona!t > mo%es toward or away from it and no matter how fast or slow its so!rce =Astrona!t A> mo%es toward or away from the inertial oser%er. This res!lt is also shown mathematically in E+am#le , elow.
14 The two #ost!lates at the heart of Einstein’s s#ecial theory of relati%ity tho!/h sim#le in each form!lation re%ol!tioni'e o!r conce#t of time and s#ace. y /i%in/ !# the asic ass!m#tion of 9ewton’s laws that s#ace and time are asol!te and inde#endent of anythin/ e+ternal to them the s#ecial theory reconciles )a+well’s electroma/netism and 9ewtonian mechanics in a sim#le way. n the remainin/ sections of this cha#ter we e+amine the ar/!ments and e+#eriments that corroorate the s#ecial theory and disc!ss its ama'in/ conse8!ences. !t first since we ha%e seen that the Galilean transformation wors only for s#eeds m!ch less than the s#eed of li/ht how do we con%ert meas!rements from one inertial system to meas!rements in another one or oth of which mo%e at s#eed close to the s#eed of li/ht? This leads to a set of e8!ations called the &orent' transformation which Einstein deri%ed inde#endently. 2.1.2
The (orent) Transformation
:ithin the framewor of classical #hysics where s#eed is m!ch smaller than the s#eed of li/htv c the Galilean %elocity addition and transformation e8!ations are #erfectly ade8!ate. ;owe%er they will not wor with s#eeds close to c. or hi/h s#eed the Galilean transformation will e re#laced y a new set of transformation e8!ations called the %orent9 transformation. f we e+amine the familiar two inertial frame systems S and S’ we can ass!me that the new e8!ations will also e linear. &et !s ass!me that the new e8!ations differ from the Galilean e8!ations y a constant factor and ha%e the forms x = x’ + vt’ > ,
y = y’,
z = z’
The constant factor is to e determinedR y and z do not chan/e and t will e deri%ed. 5f co!rse the con%erse set will e x’ = x – vt > ,
y’ = y,
z’ = z
Consider a li/ht eam that lea%es the common ori/in of S and S’ at time t = t’ = . After a time t it will ha%e tra%eled a distance x = ct 3r x’ = ct’ alon/ the x*a+is. "!stit!tin/ these %al!es in the first e8!ations ao%e we /et ct = =ct’ + vt’ > = =c + v>t’ ct’ = =ct – vt > = =c – v>t t’ = =c – v>t & c y s!stit!tin/ t’ in the first e8!ation we otain ct = =c + v>t’ = =c + v> =c – v> t & c ct = ( =c( – v(> t & c from which we deri%e after cancelin/ t ( = c & =c( – v(> & c ( = c( & =c( – v(> ( = 1 & =1 – v( & c(> OOOOOOOOOO * = 1 , 1 – v2 c2 Ssin/ we now find the relation etween t and t’ . :e s!stit!te x in x’ x’ = x – vt >= P = x’ + vt’ > – vt Q = ( = x’ + vt’ > – vt
3"4
10 rom nown e8!ations ao%e ct = =ct’ + vt’ > and x’ = ct’ or t’ = x’ & c we deri%e
ct = =ct’ + vx’ & c> t = =t’ + vx’ & c(> OOOOOOOO O t = 1 & 4 1 – =v( & c(> =t’ + vx’ & c(>
"imilarly we deri%e t’ and otain OOOOOOOOO t’ = 1 & 4 1 – =v( & c(> =t – vx & c(> The %orent9 transformation e!ations are then x = * 3 x’ + vt’ 4
x’ = * 3 x – vt 4
y = y’
y’ = y
z = z’
z’ = z
t = * 3t’ + vx’ c24
t’ = * 3t – vx c24
3$4
These e8!ations are sli/htly different from the ori/inal transformation #ro#osed y &orent' to acco!nt for the n!ll res!lt of the )ichelson*)orley e+#eriment ="ection 2.1.4 elow>. Einstein =1(61 ##. 110*122> arri%ed at his &orent' transformation e8!ations y a totally different deri%ation. >he ?3rentz trans13rmati3n e@uati3ns are n3thin but the relativistic eneralizati3n 31 the Balilean trans13rmati3n e@uati3ns. As can e s!rmised the factor * plays a cr!cial role in relativistic meas!rements. OOOOOOOOOO * = 1 , 1 – -v2 c2 in s 3"4 or since C = v & c OOOOOO * = 1 , 1 – / 2
in s
3"4
The spreasheet form!la for * is * : 3103v8c4;24;-31824
3'4
"!stit!te act!al %al!es for v and c in the form!la =0> to /et the res!lt. f v is e+#ressed in terms of c e./. .(0 c then this v %al!e taes the #lace of %Dc in the form!la. And the spreasheet form!la for the reciprocal 18 * or * – 1 of * is * – 1 : 310 3v8c4;24;31824
3)4
$ememer that an E+cel form!la e/ins with an e8!al si/n. M!st s!stit!te the real %al!es forv and c. Tale 2*3 elow is constr!cted !sin/ this s#readsheet form!la.
16 To sim#lify the calc!lations always try to red!ce v to a fraction of c. or e+am#le if v 3 + 17 mDs mae it v .1 c. This ins!res that c( will not enter into the calc!lations at all. And the s#readsheet form!las for * and * – 1 red!ce to * : 310v;24;-31824
3'a4
* – 1 : 310 v;24;31824
3)a4
and
res#ecti%ely. * in &elativistic easurements "ince * is the factor that affects all relati%istic meas!rements s!ch as relati%istic len/th time ener/y mass and moment!m let !s e+amine in some detail. ts reci#rocal * – 1 is easily deri%ed. M!st ear in mind that * 1 and * – 1 U 1. =1> n ’s form!lation v cannot e /reater than c for otherwise the %al!e !nder the radical si/n wo!ld e ne/ati%e res!ltin/ in an ima/inary n!mer. ;owe%er in the 1(6’s some scientists #ointed o!t that there is nothin/ in where%er it may e !sed that #re%ents v ein/ /reater than c. They hy#othesi'ed the e+istence of a fast #article called tachyonB =meanin/ fastB> whose s#eed v c. f %ery fast #articles e+isted their rest mass m" wo!ld ha%e to e an ima/inary n!mer in the mass form!la m = m" in order for their mass m to e real. "o far the hy#othetical tachyons ha%e not een fo!nd. Th!s the s#eed of li/ht is still the !ltimate s#eed of the !ni%erse. =2> :hen v = c the %al!e !nder the radical is e8!al to and is not defined. Therefore v cannot e8!al c. =3> f v or v c 1. All oects and e%ents are at rest. Therefore there are no relati%istic effects. =4> The ratio v & c also called C controls the %al!e of . f v /oes to infinity =an im#ossiility y the second #ost!late> is an ima/inary n!mer as the %al!e !nder the radical is ne/ati%e. :e saw this in =1> ao%e. f c ecomes infinity =which it can ne%er e> 1 and there are no relati%istic effects. =0> :hen sol%in/ relati%istic #rolems con%ert v into a fraction of c to sim#lify calc!lations. Th!s C red!ces to v in terms of c. "ince the ratio v & c determines the %al!e of let !s e+amine a few e+am#les of this ratio and its reci#rocal all of which occ!r in relati%istic meas!rements. =;echt 1((, #. (64>
/ = v c .
Relations etween / 5 1 *5 an * OOOOOOOOOO OOOOOOOOOO 1 * = , 1 – -v2 c2 * = 1 , 1 – -v2 c2
. .1 .2 .3 .4 .0 .6 .7 ., .( .(( .((( .((( ( .((( (( .((( (((
1. .((4 (,7 .(7( 7(6 .(03 (3( .(16 010 .,66 20 ., .714 143 .6 .430 ,( .141 07 .44 71 .14 142 .4 472 .1 414
1.
.
1. 1.0 3, 1.2 621 1.4, 2,0 1.(1 ,( 1.104 71 1.20 1.4 2, 1.666 667 2.2(4 107 7.,, ,12 22.366 27 7.712 40 223.67 77.17 di%ision y 'ero
17 Tale 2-1. Relations etween / 5 1 *5 an *. These im#ortant relations rec!r where%er relati%ity meas!rements are made e they relati%istic len/th time mass ener/y or moment!m. "ince C = v & c the ratio of the s#eed of a mo%in/ oect to the s#eed of li/ht sho!ld ne%er reach 1 enforces so to s#ea the !##er limit for all s#eeds which can ne%er e reached e+ce#t for li/ht. The ao%e tale also shows that at e%eryday s#eed which is %ery small com#ared to c v c the &orent' transformation red!ces to the Galilean transformation eca!se has no effect ein/ %ery close to 1. n other words the Galilean relati%ity #rinci#le is a s#ecial case of the first #ost!late of the s#ecial theory of relati%ity. Gi%en that the %al!es C and are !i8!ito!s in relati%ity #rolems let !s rememer that * < 1 and its reci#rocal 1 * = 1 since v in the ratio C = v & c is ne%er 'ero. To a%oid to tedio!sness of calc!latin/ when v is %ery small i.e. v c it is !sef!l to wor o!t an a##ro+imate e+#ression for . :e !se a inomial form!la to this end. A inomial form!la has this form n
n
=a F b> a F
n
OO
1
a
n–1
b F
n=n L1>
OOOOOOOOOO
1X 2
a
n–2 2
n=n L1> =n L2>W=n L r F 1>
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOO
1X 2 X 3 Y Y Y Y Y r
b F
n=n L1> =n L2>
OOOOOOOOOOOOOOOOOOO
1X 2 X 3
an – 3b3 F W F
an L r br F W F bn
"ince the form!la for = =1 L C 2> L1D2 is a inomial we a##ly the inomial e+#ansion ao%e main/ a 1 b L C ( and n = L Z retainin/ only the first two terms and treatin/ the rest of the terms as ne/li/ile =1 F b>n [ 1 F n)an – 1b> [ 1 F n=1 – 1D2 – 1>=b> [ 1 F nb ;ence [ 1 F nb "!stit!tin/ the %al!es for n and b in the ao%e relation yields the shortene inomial form!la of * > =1 L C 2> 1 F =L1D2> = L C 2> * > 1 ? / 2 2
3*4
The spreasheet form!la of relation 3*4 is * : 1?33 @;24824
3+4
"!stit!te an act!al %al!e for C in the form!la =,> to /et the res!lt. "ince C v&c if v is e+#ressed in terms of c e./. .(0 c then this v %al!e taes the #lace of C in the form!la. This a##ro+imate %al!e of comes in handy when we need to 8!icly chec o!r calc!lation res!lts when v c e./. !# to aro!nd v = .6 c. or v closer to c, v > c or C [ 1 the followin/ approAimation sho!ld e !sed =1 L C (> =1 L C > =1 F C > [ 2=1 L C > * > 231 B / 4 B182
364
The spreasheet form!la of relation 364 is * : 231-@4;3-1824
31D4
1, "!stit!te an act!al %al!e for C in the form!la 1 to /et the res!lt. "ince C v&c if v is e+#ressed in terms of c e./. .(0 c then this v %al!e taes the #lace of C in the form!la. n other words if v is eApresse as a fraction of c5 then / : v. Gi%en the im#ortance of and its com#onent C let !s list a few nown s#eeds and their corres#ondin/ and C %al!es in Tale 2*2. ="li/htly modified from ;echt 1((, #. (0,> Spees of oects an their corresponin& * an / val!es ect
Spee v
/ : v c =v in terms of c>
* = 31 – 3v2 c244 B182
;!man walin/ 1*yard dash =ma+.> Commercial a!tomoile =ma+.> "o!nd "$*71 reconnaissance et )oon aro!nd Earth "#ace sh!ttle A#ollo 1 =re*entry> Esca#e s#eed =Earth> ioneer 1 Earth aro!nd "!n )erc!ry orital s#eed ;elios solar #roe Earth*"!n aro!nd /ala+y Electrons in a TV t!e )!ons at CE$9 Electrons at "tanford &inear Accelerator ="&AC>
, mDh 1. mDs 62 mDs 333 mDs (, mDs 1 mDs 70(( mDs 11.1 mDs 11.2 mDs 14.4 mDs 2(.6 mDs 47.( mDs 66.7 mDs 2.1 + 10 mDs ( + 17 mDs 2.((6 + 1, mDs
. 7 . 33 . 21 . 1 11 . 3 27 . 3 33 . 20 33 . 37 . 37 . 4, . (( . 16 . 22 . 7 .3 .((( 4
1 1. 1 1 1. 1. 1. 1. 1 1. 1 1. 1 1. 0 1. 13 1. 20 1. 240 1.0 2,.,7
2.((7 ( + 1, mDs
.((( ((1 , ((0
246.(0
Tale 2-2. Spees of oects an their corresponin& * an / val!es 9ow let !s constr!ct a more com#rehensi%e tale with !# to 10 decimal #laces with v as close to c as the E+cel s#readsheet allows. t starts with the s#eed near that of the s#ace sh!ttle the hi/hest s#eed achie%ed y a manned flyin/ machine. E%en with the s#eed of a s#ace sh!ttle of 17 miDh 2730, mDh or 7.0(( mDs =olded in the followin/ tale> ! 1. This hi/h s#eed is only ao!t 1D3(041th the s#eed of li/ht. As the s#eed v of an oect /ets %ery close to c its %al!e increases dramatically ="ee Tale 2*3 elow>. aution3 >he bin3mial ex2ansi3n 31 is, 13r 2ractical 2ur23ses, 1airly reliable 3nly 13r v c e.. 13r C = v & c ! ".-" because 31 the retenti3n 31 3nly the 1irst tw3 terms 31 the ex2ansi3n. Dey3nd this C value, the sec3nd a22r3ximati3n 13rmula E ives results that are remarkably similar t3 th3se 3btained with the 1ull 13rmula, as the table bel3w )>able (%/ clearly sh3ws. Fhen in d3ubt, use the 1ull 13rmula 3r its s2readsheet e@uivalent. G3wever, i1 y3u have a calculat3r and want 3nly a @uick, reas3nably acce2table result, use the bin3mial 13rmula 13r v = ".-" c and the sec3nd a22r3ximati3n 13r v H ".-" c. Iust bear in mind that even the calculat3r will tackle the 1ull 13rmula with3ut undue tr3uble. or con%enience a s#readsheet %ersion of the f!ll * e+#ression a##ears in orm!la =0> and a s#readsheet %ersion of its reci#rocal * B1 is orm!la =6> ao%e. Tale 2*3 elow lists a n!mer of C = v & c %al!es with !# to 10 decimal #laces and their corres#ondin/ %al!es deri%ed y the shortened inomial form!la =orm!la 7 rows 1*12> the second a##ro+imation =orm!la ( row 13 to end> and the f!ll form!la =orm!la 3 in the third col!mn>. 9ote that from row 24 to the end of the tale the second a##ro+imation and the f!ll * form!la yield identical res!lts. This tale re#resents to est a##lication of the inomial and a##ro+imation form!las. #iscrepancies etween the shortene inomial eApansion of * an the * form!la / : v c
* 4 1 + 3v2 c24 8 2
* = 31 – 3v2 c244 B182
=v in terms of c>
=inomial e+#ansion for v c> * > 231 B / 4 B182 =for v > c rows 13 and followin/>
.2 1.2 5.55552'""5555555 1.555555555"256555 .2 1.2 .2 1.2 .2 1.2 .1 1.0 .2 1.2 .3 1.40 .37 1.6,40 .4 1., .0 1.120 .6 1.1, D.*DDDDDDDDDDDDDDD 1.26D66$$$+*"'+1DD ., 1.0,113,,3,41( .( 2.23667(774((7( .(( 7.7167,11,6047 .((( 22.3667(774((7( .(((4 2,.,6701340(4,2 .(((( 7.7167,11,60,6 .((((( 223.667(7704,, .(((((( 77.167,11763,1 .((((((( 2 236.67(7,,,27 .(((((((( 7 71.677(4127 D.666666666DDDDDDD 225")D.)+DD61166'DDDDDD .(((((((((7 4,24.,273074007 .(((((((((( 771.6701(33411 .((((((((((( 22366.7,,4((320 .(((((((((((( 77114.6202046( .((((((((((((( 223072.41126022 .(((((((((((((( 773,(0.3,,,26 .((((((((((((((( 2236(621.3333333
=in seconds>
1(
1.2 1.555555555"256555 1.2 1.26 1.2621 1.037,1020(21 1.26272610(66 1.4,2,4,36721(2 1.763,(472,6771 1.(1,(40117((6 1.104703,37(20 1.20 1.$DD2+DD+$D2+D1DD 1.66666666666667 2.2(410733,7062 7.,,,120,330 22.3662724212(4 2,.,71,44061217 7.712440(01(140 223.6730676(620 77.16(07(4(232 2236.6,33(402( 771.67,1372642 225")D.)+DD61166'DDDDDD 4,24.,273074007 771.6701(33411 22366.7,,4((320 77114.6202046( 223072.41126022 773,(0.3,,,26 2236(621.3333333
Tale 2-". "hortened inomial form!la and the second a##ro+imation of %ers!s f!ll form!la for %ario!s C v & c %al!es. The shortened inomial form!la 7 /i%es fairly /ood a##ro+imations of !# to the %al!e of C .6. rom that #oint on the second a##ro+imation form!la ( taes o%er and #rod!ces remaraly close %al!es of !# to C .((((((((. eyond this %al!e as v f!rther a##roaches c the second a##ro+imation /i%es the same %al!es of * as the f!ll form!la. The /ra#h of this tale =shown in i/!re 2*( "ection 2.3 elow> shows how dramatic the ra#id rise of is as v a##roaches c. 9ote that C v & c can ne%er reach c y the second #ost!late i.e. v \ c so that C and C U 1 and 1and L1 U 1. n addition the f!ll form!la shows that wo!ld e !ndefined if v c eca!se of the di%ision y 'ero. 9ote howe%er that the tr!ncated inomial form!la of wo!ld not show this effect of v c which #oints to another of its !ndesirale as#ects. 2.1." &elativistic Addition of Velocities
2 Ssin/ the res!lts otained in the &orent' transformation let !s determine relativistic velocity aition. As seen ao%e the classical %elocity addition u = u’ + v, is %alid only at %ery small s#eeds com#ared to c. Consider o!r familiar #air of inertial reference frames S and S’ .
J
!’ =relati%e to "’>
!’ =relati%e to "’> J % =relati%e to ">
% =relati%e to "> "’
"’
5’
5’ +’ "
"
5
5 =a>
+ =>
Fi&!re 2-1. )o%in/ e%ent P within mo%in/ S’ with res#ect to stationary S . "!##ose at time t = t’ = " the ori/ins O and O’ coincide =i/!re 2*1a> and the #article P #asses the ori/in at s#eed u’ = x’ & t’ with res#ect to frame S’ in the F x direction. The S’ frame itself mo%es with constant s#eed v relati%e to the S frame. At a later time the #article is at x in the S frame where its s#eed is u = x & t and at x’ in the S’ frame where its s#eed is u’ = x’ & t’ =i/!re 2*1>. Ssin/ the &orent' transformation e8!ation 4 s!stit!tin/ x’ cancelin/ and factori'in/ t’ we /et x’ + vt’ JJJJJJJ x 4 1 – =v( & c(> t’ =u’ + v> u = J = JJJJJJJ = JJJJJJJ t t’ + =vx’ & c(> t’ =1 + u’v & c(> JJJJJJJ 4 1 – =v( & c(> y cancelin/ t’ we otain the relativistic velocity aition e!ation u’ + v u = 77777 1 + u’v c2
3114
where u is the %elocity of a mo%in/ oect relati%e to the reference frame of a stationary oser%er u’ is the %elocity of the oect with res#ect to its own reference frame and v is the %elocity of mo%in/ frame relati%e to the stationary frame. The same a##lies if the oser%er is also mo%in/ alon/ with the oect. The #rimed co!nter#art of u has a ne/ati%e v !st lie in the Galilean %elocity addition u– v u’ = 77777 1 – uv c2
3124
9ote that the n!merator of the relati%istic %elocity transformation e8!ation is the same as the n!merator of the Galilean %elocity addition. The difference etween these transformations resides in the denominator which is
21 res#onsile for main/ s!re that no oect can tra%el at c. And that incl!des #rotons electrons "tarshi# Enter#rise or any other machine we can de%ise in the f!t!re. :hen oth u and v are e8!al to c the denominator e8!als 'ero. This is why c is the !ltimate spee. &et !s now e+#lore the a##lications of the &orent' transformation e8!ations and the relati%istic %elocity transformation. The most im#ortant thin/ to rememer when worin/ with relati%ity #rolems is to identify the reference frame !nder consideration. "#ecial relati%ity deals only with inertial frames i.e. frames that are at rest or in constant motion. There are three entities in%ol%ed =1> the stationary frameS =relati%e to the other> =2> the mo%in/ frame S’ and =3> an e%ent a #article or an oect P mo%in/ in S’. n a##lyin/ the %elocity addition form!la ee# in mind that u’ is the %elocity of an oect in a frame =S’ > mo%in/ with s#eed v with res#ect to the frame =S > in which s#eed u of P is meas!red. Examle 8. An astrona!t in a s#aceshi# mo%in/ away from the earth with a constant s#eed of .(c fires a rocet in the direction of tra%el with a s#eed of .0 c relati%e to the s#aceshi#. :hat is the s#eed of the rocet as oser%ed y an eartho!nd oser%er? !olution. &et S e the earth’s reference frame and S’ e the reference frame of the s#aceshi#. The s#aceshi# tra%els at v = .( c and the rocet at u’ = .0 c. The 8!estion is to determine the s#eed of the rocet relati%e to earth i.e. frame S . As we cannot a##ly the Galilean %elocity addition since it wo!ld %iolate the second #ost!late .( c + .0 c = 1.4 c we a##ly the relati%istic %elocity transformation u = =u’ + v> & P1 + =u’v & c(>Q
=11>
u =.0 c F .( c> D P1 F =.0 c> =.( c> D c(Q 1.4 c D P1 F =.40 c( D c(>Q u 1.4 c D 1.40 ".E-K c. Examle 9. Two stars A and mo%e away from Earth in o##osite directions with %elocity .(c and .,c res#ecti%ely. ind the s#eed of star with res#ect to star A. !olution. irst we mae star A the S frame =the frame at rest> with res#ect to which star ’s s#eed is /oin/ to e meas!red. :e mae Earth the mo%in/ frame S’ in which Earth is at rest while oth stars are mo%in/ relati%e to Earth. ;owe%er in frame S star A is at rest and Earth is mo%in/ with %elocity v = .( c relati%e to star A and star is mo%in/ at %elocity u’ = ., c relati%e to Earth. :e are to find the s#eed of star relati%e to star A. :e a##ly the relati%istic %elocity addition e8!ation u = =u’ + v> & P1 + =u’v & c(>Q
=11>
u =., c F .( c> D =1 F =., c>=.( c> D c(> u 1.7 c D 1.72 ".ELL c. Examle 6. "how that the relati%istic %elocity addition e8!ation ins!res that no mo%in/ oect can e+ceed the s#eed of li/ht c. !olution. ma/ine an astrona!t in a s#aceshi# mo%in/ at s#eed v relati%e to an oser%er sends a li/ht eam forward. The astrona!t meas!res the li/ht’s s#eed as c. The oser%er meas!res the s#eed of the li/ht eam and fo!nd the res!lt consistent with Einstein’s second #ost!late. "!stit!tin/ c for u’ in the relati%istic %elocity transformation we /et u = =u’ + v> & P1 + =u’v & c(>Q and after m!lti#lyin/ oth n!merator and denominator yc u = =c + v> & P1 + =cv & c(>Q = c=c + v> & =c + v> = c.
=11>
22 Examle :. A s#aceshi# tra%els away from )ars with %elocity .00 c and fires a lander ac at )ars at the s#eed of .6c relati%e to the s#aceshi#. :hat is the s#eed of the lander as seen from )ars? !olution. Tain/ the direction of the s#aceshi# as #ositi%e we a##ly the relati%istic %elocity e8!ation. The s#aceshi# mo%es at v .00 c and the lander’s s#eed u’ – .6 c =ne/ati%e meanin/ in the o##osite direction>. The lander’s %elocity with res#ect to )ars is u = =u’ + v> & P1 + =u’v & c(>Q
=11>
u = = – .6 c + .00 c> & P1 + = – .6 c>=.00 c> & c(Q u = = – .0 c> & P1 – =.33 c( & c( >Q – .0 c D .67 – "."K c. Examle 15. $ocet shi# A and rocet shi# a##roach their s#ace station from the same direction shi# A with the s#eed of .660 c and shi# with the s#eed of .0,0 c. :hat is the s#eed of shi# A with res#ect to shi# ? !olution. The first ste# is to identify the reference frames. &et !s choose the s#ace station as the frame at rest S. Then oth rocet shi#s are in the mo%in/ frame S’ . $ocet shi# A has %elocity u .660c and rocet shi# has %elocity v = .0,0 c. :e a##ly %elocity addition e8!ation =12> u’ = =u – v> & P1 – =uv & c(>Q
=12>
u’ = =.660 c – .0,0 c> & P1 L =.660 c> =.0,0 c> & c(>Q u’ = =., c> & P1 L =.3,( c( & c(>Q ., c & .611 ".'/' c. Examle 11. A rocet shi# lea%es its s#ace station at the s#eed of .4 c and emits a li/ht #!lse ac toward the s#ace station. At what s#eed does an oser%er in the s#ace station see the li/ht #!lse comin/ to her? !olution. The rocet shi# is mo%in/ in frame S’ in the F x direction =i.e. away from the s#ace station> at v = .4c, and the s#ace station is at rest in frame S. The na%i/ator aoard the rocet shi# meas!res the li/ht #!lse and finds its s#eed to e u’ – c the ne/ati%e si/n desi/natin/ mo%ement in the –x direction =toward the station>. A##lyin/ the relati%istic %elocity transformation we /et u = =u’ + v> & P1 + =u’v & c(>Q
=11>
u = = – c + .4 c> & P1 + = – c>=.4 c> & c(Q u = .6 c D =1 L .4> c. This is another confirmation of the second #ost!late. 9o matter whether the so!rce of li/ht mo%es away from or toward the oser%er the s#eed of li/ht remains the constant c. Examle 12. Gala+y A and Gala+y mo%e away from )ars in o##osite directions at a s#eed of .,c with res#ect to )ars. ind the s#eed at which the /ala+ies more a#art relati%e to each other. !olution. A/ain we ha%e the familiar S and S’ frames and the #oint P the three oects that re8!ire the addition of %elocities. &et Gala+y A mo%in/ left e theS frame )ars the S’ frame then Gala+y mo%in/ ri/ht is P. A strai/ht a##lication of the relati%istic %elocity transformation yields u = =u’ + v> & P1 + =u’v & c(>Q u =., c F ., c> P1 F =., c>= ., c > D c(Q u 1.6 c D 1.64 ".EK c.
=11>
23 The two /ala+ies can ne%er se#arate with a s#eed e8!al to or /reater than the s#eed of li/ht. This im#lies that comm!nication %ia li/ht and /ra%itational and electroma/netic interactions that #ro#a/ate atc are always #ossile etween any two inter/alactic odies eca!se they can ne%er o!tr!n li/ht. 2.1.# The ichelson0orley Exeriment At the end of the nineteenth cent!ry a /reat #!''le left ehind y )a+well remained to e sol%ed. Galilean relati%ity #redicted that the s#eed of li/ht wo!ld %ary de#endin/ on the inertial reference frames. "!ch was the dominance of 9ewtonian mechanics that scientists elie%ed that li/ht m!st tra%el in some sort of medi!m !st lie so!nd wa%es and water wa%es. They #ost!lated the l!minifero!s ether a hy#othetical in%isile massless thin yet ri/id s!stance that filled s#ace and acted as a medi!m thro!/h which electroma/netic wa%es m!st #ro#a/ate. "tran/e s!stance indeed. @et e%en )a+well himself elie%ed there was s!ch an ethereal medi!m fillin/ s#ace and #ermeatin/ e%erythin/. The earth too m!st rotate aro!nd its a+is and re%ol%e aro!nd the s!n thro!/h the ether. )a+well’s e8!ations hold and li/ht tra%el at s#eedc only in an asol!te reference frame which is at rest with res#ect to the ether. This is called theasol!te frame. n other frames of reference the e8!ations /a%e contradictory res!lts and needed to e modified to wor. This seems to e at odds with the classical relati%ity #rinci#le wherey no inertial frame is #ri%ile/ed o%er any other. Consider for e+am#le that a s#aceshi# tra%elin/ at 1. + 1, mDs t!rns on a li/ht eacon and meas!res its s#eed to e 3. + 1, mDs. To a stationary oser%er the li/ht tra%els at s#eed 1. + 1, mDs F 3. + 1, mDs 4. + 1, mDs. Th!s the same electroma/netic effect has different %al!es when meas!red in different reference frames. !t )a+well’s theory does not #ro%ide for relati%e s#eed. t #redicted the s#eed of li/ht to ec = 3. + 1 , mDs. t seemed intri/!in/ that )a+well’s electroma/netic e8!ations which wored so s!ccessf!lly and were corroorated y n!mero!s e+#eriments sho!ld ha%e to e an e+ce#tion to the Galilean relati%ity #rinci#le which a##lied to 9ewtonian mechanics !t not to )a+well’s electroma/netic theory. Clearly this dilemma called for in%esti/ation to reconcile the classical relati%ity #rinci#le and the electroma/netic theory. n 1,7( the year Einstein was orn Mames Cler )a+well =1,31*1,7(> wrote a letter efore he died in which he disc!ssed a #otential e+#eriment to meas!re the s#eed v of the earth as it tra%els thro!/h the ether wind. )any e+#eriments were cond!cted !sin/ the interferometer !t yielded !ncon%incin/ res!lts. 9one was as famo!s as the one #erformed y the American #hysicist Alert A. )ichelson. =1,02*1(31> and the American chemist Edward :. )orley =1,3,*1(23>. orn in "trelno r!ssia =now "tr'elno oland> )ichelson at a/e 4 immi/rated with his #arents to America. $eected y the 9a%al Academy he mana/ed to /ain admission thro!/h a chanceB meetin/ with resident Grant who was im#ressed with the yo!n/ man. After /rad!ation )ichelson stayed at the Academy as an instr!ctor of #hysics and chemistry. n 1,7, he was thinin/ ao!t how to meas!re the s#eed of li/ht etter !t new he needed more formal trainin/. "o in 1,, he too a lea%e of asence to #!rs!e ad%anced st!dies in rance and Germany. :hile in E!ro#e he came across )a+well’s last letter ao!t meas!rin/ the s#eed of the earth thro!/h the ether. ast e+#erimenters had !sed the interferometer to meas!re the effects of motion on the transmission of li/ht thro!/h the ether. Altho!/h they failed to detect the ether the limited #recision of their instr!ments made their findin/s less than con%incin/. )ichelson desi/ned a new instr!ment with a #recision that wo!ld allow him to settle the 8!estion definiti%ely. ;e ho#ed to detect the ether wind that carried li/ht wa%es with his new and im#ro%ed interferometer. f the ether e+isted so went the reasonin/ it wo!ld act lie any wind or stream im#edin/ or aidin/ the #ro/ress of an oect mo%in/ in it. &et !s tae an e+am#le to ill!strate the #rolem. :e !se the analo/y of oats tra%elin/ on a ri%er. n i/!re 2*2 a oat tra%els at a fi+ed s#eed of c = 0 mDh in the ri%er which flows to the ri/ht with a s#eed of v = 3 mDh.
c
2
C v’
v
OOOOOO 1 v’ = ] c( – v(
c
downstream 2
C
v
A
v %elocity of ri%er c+v
c–v 1
A
!#stream 1
24 Fi&!re 2-2. The downstream tri# made y oat 1 at %elocity c + v is faster than its !#stream tri# at c – v. 5n the cross*stream tri# oat 2 starts from A heads for C to land at =trian/le 1>. 5n the ret!rn tri# oat 2 starts from headin/ for C and is carried ac y the stream to A =trian/le 2>. &et !s com#are the tra%el time of two tri#s 4 m downstream and ac %ers!s crossin/ the ri%er and ac. A oat =oat 1> headin/ downstream tra%els at %elocity c =not the s#eed of li/ht> a!/mented y the ri%er’s %elocity v mo%in/ a total of c + v 0 mDh F 3 mDh. , mDh. The tri# downstream taes 4 m D , mDh .0 h. 5n the way ac the oat tra%elin/ !#stream maes slower #ro/ress e8!i%alent to c – v or 2 mDh. The tri# ac therefore taes 4 m D 2 mDh 2 h. The total transit time !# and down stream taes .0 h F 2 h 2.0 h. 9ow the oat =oat 2> crosses the ri%er from #ier A to #ier . :ith the ri%er c!rrent flowin/ to the ri/ht if oat 2 heads strai/ht across from #ier A it will e dra//ed downstream and miss #ier . To com#ensate oat 2 has to head into the c!rrent =!#stream> at a certain an/le so that the comined effect ofc and v wo!ld allow it to reach ier . This an/le de#ends on c and v. n i/!re 2*2 the width of the ri%er A the distance AC tra%eled y oat 2 and the distance C of the effect of the ri%er flow on oat 2 form the ri/ht trian/le 1. &iewise on the ret!rn tri# oat 2 starts from ier heads into the stream for C and reaches A dra//ed y the downstream c!rrent =as de#icted y trian/le 2>. Ssin/ the ytha/orean theorem we can deri%e the len/th of any of its side /i%en those of the other two y !sin/ the form!la c( = v( + v’ (. This /eometry shows that y main/ the tri# as de#icted oat 2 e11ectively tra%els v’ =c( – v(>1D2 =20 L (>1D2 4 m and taes !st 1 h =eca!se it tra%els 0 m at 0 mDh> to mae the tri# across and 1 h to come ac a total of 2 h. ;ence in this case for the same total distance of , m the oat taes less time crossin/ the ri%er than /oin/ !# and down stream. n /eneral the times for the ro!nd tri#s will differ. The )ichelson*)orley e+#eriment =in fact many were cond!cted at %ario!s #laces and times> !ilds on the oat tri# analo/y !st descried. t !ses a #recision interferometer re#resented in schematic form in i/!re 2*3. This e+#eriment aimed to meas!re the ether wind with res#ect to the earth or e8!i%alently the earth’s %elocity relati%e to the ether. :e will !se the analo/y of the oat’s o!rney on the ri%er to inter#ret the res!lt of the e+#eriment. The ether wind is analo/o!s to the %elocity of the ri%er flowR and the two oats /oin/ !# and down stream and across the ri%er and ac are analo/o!s to the li/ht eams 1 and 2 on the #er#endic!lar arms of the interferometer. :e wo!ld e+#ect that the total tra%el times of the eams wo!ld differ in the same way as the oats’ total transit times differed. )o%ale mirror )1 Ether wind v l '
eam 1
&i/ht so!rce
;alf*sil%ered mirror )s
eam 2
)2 i+ed mirror
l (
Telesco#e
Fi&!re 2-". The (ichelson-(orley eAperiment. n this interferometer the li/ht eam tra%els from its so!rce to the half*sil%ered mirror =eam s#litter> )s an/led at 40^ which s#lits the li/ht eam in two. 5ne eam contin!es its #ath to mirror )2 is reflected ac toward )s where it is a/ain reflected to the telesco#e.
20 The other eam is reflected to mirror ) 1 where it is sent ac thro!/h )s strai/ht to the telesco#e. The two eams recomine and form an interference #attern that is oser%ed thro!/h the telesco#e. At the laoratory of ;ermann %on ;elmholt' in Germany )ichelson in%ented the interferometer ased on a )a+well idea #ro#osed in 1,70 to meas!re the earth’s %elocity relati%e to the ether. Accordin/ to )a+well as the earth mo%es thro!/h s#ace the ether #ermeatin/ s#ace wo!ld create a wind. f we meas!re the %elocity of li/ht in the direction of the earth’s mo%ement aro!nd the s!n i.e. in the direction o##osite the ether wind lie the oat tra%elin/ !#stream =eam 2> the s#eed of li/ht with res#ect to the ether wo!ld e e8!al to the %al!e meas!red min!s the s#eed of the ether wind c – v. f we meas!re the s#eed of li/ht in the direction of the ether =eam 2> i.e. downstream its %al!e wo!ld e the meas!rement otained #l!s the s#eed of the ether c + v. And if we meas!re the s#eed of li/ht in the direction at ri/ht an/les with the ether wind =eam 1> the sit!ation wo!ld e analo/o!s to the oat crossin/ the ri%er. i/!re 2*2 ao%e ill!strates these scenarios. n the interferometer e+#eriment a li/ht eam is sent to a eam s#litter the half*sil%ered mirror )s an/led at 40_ de/rees which s#lits it in two one eam is transmitted to the fi+ed mirror )2 which reflects it ac to )s where it is reflected toward the telesco#e. The other eam is reflected to the mo%ale mirror )1 which ret!rns it to )s where it is transmitted on to the telesco#e. E%ent!ally the two eams !nite and interfere. As in the oat analo/y the two e8!al*len/th ro!nd tri#s of eam 1 =from )s to )1 and ac to )s> and eam 2 =from )s to )2 and ac to )s> sho!ld tae different amo!nts of time which can e detected in the interference #attern oser%ale with the telesco#e. n his first attem#t(ichelson fo!n no ifference in the spee of li&ht which to him was an !ns!ccessf!l res!lt. n 1,,7 )ichelson now at Case :estern $eser%e Sni%ersity tried the e+#eriment a/ain with his collea/!e Edward :. )orley a #rofessor of chemistry. They set !# their interferometer on a s8!are sla of stone floatin/ on merc!ry to minimi'e %irations when the de%ice was rotated in different directions. -ifferent interference #atterns wo!ld res!lt from orientin/ the instr!ment alon/ the direction of the earth’s motion or across it. :hen the interferometer was rotated (^ the arm alon/ with the li/ht eam that was #arallel to the earth’s %elocity wo!ld ecome #er#endic!lar to it and the other arm and eam that were #er#endic!lar wo!ld ecome #arallel. Th!s a shift in the interference #attern wo!ld e e+#ected. irst let !s consider eam 2 which is #arallel to the ether wind in i/!re 2*3. Ssin/ the oat analo/y of i/!re 2*2 as eam 2 tra%els the distance l ( downstream from )s to )2 its s#eed is c + v and the time taen is t = l ( & =c + v>. 5n its ret!rn tri# from )2 to ) s its s#eed is c – v and the time t = l ( & =c – v>. Th!s eam 2’s total time is t ( l ( & =c + v> F l ( & =c – v> t ( 2l ( & =c( – v(> 2l ( & c=1 – v( & c(>. 9ow let !s consider eam 1 which tra%els across the ether wind from )s to )1 and ac for a total distance of 2 l '. Ssin/ the oat analo/y in i/!re 2*2 eam 1’s motion forms a ri/ht trian/le of sidesc v and v’ . rom the ytha/orean theorem we /et eam 1’s s#eed v’ ] c( – v(. eam 1`s total time is th!s OOOOOOOO OOOOOOOOO ( ( t ' 2l ' & ] c – v 2l ' & c ] 1 – v( & c( Ass!me that l ' and l ( are e8!al. Then eam 1la/s ehind eam 2 y an amo!nt of time OOOOOOOOO 2 2 t : t 2 – t 1 : 2l 2 c 31 – v c 4 – 2l 1 c H 1 – v2 c2 OOOOOOOOO ( ( Nt 2l & c P1 D =1 – v & c > – 1 D ] 1 – v( & c( Q
31"4
This relation can e sim#lified if we !se the inomial e+#ansions ee#in/ only the first two terms and
1 – =v( & c(> – 1 [ 1 F v( & c( 1 – =v( & c(> L1D2 [ 1 F Z =v( & c(> Nt 2l & c P=1 F v( & c(> – =1 F Z =v( & c(>>Q l v( & c/.
26 rom this Nt if v and the two eams will e in #hase !st as they were at the e/innin/. fv \ the two eams will e o!t of #hase and v co!ld e determined. ;owe%er the earth cannot e sto##ed nor can l ' and l ( e inde#endently ass!med e8!al. At this #oint eam 1 is ali/ned to the arml ' and eam 2 is ali/ned to the arm l ( of the a##arat!s which is #arallel to the ether wind. To detect the difference in #hase )ichelson and )orley rotated the interferometer thro!/h (^ so that eam 1 ecame #arallel and eam 2 ecame #er#endic!lar to the ether wind. n this rotated #osition the roles of the eams were re%ersed and the times =desi/nated y #rimes> wo!ld e OOOOOOOOO ( ( t’ ' 2l ' & c =1 – v & c > and t’ ( 2l ( & c ] 1 – v( & c( . The time difference in the new #osition th!s ecomes OOOOOOOOO t’ : t’ 2 – t’ 1 : 2l 2 cH 1 – v2 c2 – 2l 1 c 31 – v2 c24 OOOOOOOOO Nt’ 2l & c P1 D ] 1 – v( & c( – 1 D =1 – v( & c(>Q
31$4
:ith the arms of the interferometer th!s rotated !sin/ relations =(> and =1> the frin/e #attern will shift y OOOOOOOOO OOOOOOOOO ( ( ( ( Nt – Nt’ 2l ( & c=1 – v & c > – 2l ' & c] 1 – v & c – 2l ( & c] 1 – v( & c( – 2l ' & c=1 – v( & c(> OOOOOOOOO ( ( Nt – Nt’ 2 D c =l ' F l (> P1 D =1 – v & c > – 1 & ] 1 – v( & c( Q A/ain if we ass!me v & c UU 1 we can !se inomial e+#ansions to /et Nt – Nt’ [ 2 D c =l ' F l (> P1 F =v( & c(> – 1 – Z =v( & c(>Q t – t’ > 3l 1 ? l 24 3v2 c"4
31'4
:e tae the s#eed of the earth aro!nd the s!n =or e8!i%alently the s#eed of the ether wind>v 3. + 14 mDs and the s#eed of li/ht c 3. + 1, mDs. n the early )ichelson*)orley e+#eriments each li/ht eam was reflected many times y mirrors res!ltin/ in an effecti%e len/th for each arm of ao!t 11 m. "!stit!tin/ these %al!es in the relation =10> we otain the time difference Nt – Nt’ [ 22 m =3. + 14 mDs>2 D =3. + 1, mDs>3
=10>
Nt – Nt’ [ 22 m =3. + 1 L16 mDs> Nt – Nt’ [ 7. + 1 L16 s. :ith the center of the %isile li/ht s#ectr!m at wa%elen/th ; 0.0 + 1 L7m its fre8!ency wo!ld e 1 = c & ;M 1 = =3. + 1, mDs> D =0.0 + 1 L7m> 1 [ 1., + 110s. 1 = 0.0 + 114 ;'. Th!s li/ht wa%e crests wo!ld #ass a #oint e%ery 1 D =0.0 + 114 ;'> [ 1., + 1 L10s. )ichelson and )orley wo!ld ha%e fo!nd the time difference of 7. + 1 L16 s to ca!se the interference #attern to shift y less than half a frin/e 7. + 1 L16 s D 1., + 1 L10 s [ .4 frin/e. "ince their a##arat!s co!ld detect frin/e shifts as small as .1 frin/e the ao%e shift sho!ld ha%e een easily detected. @et they 13und n3 sini1icant shi1t in 1rines. They made oser%ations with different orientations relati%e to the s!n e+#erimented at different times of day and ni/ht and in different seasons and at different ele%ations. The res!lt was still the same. There was no shift in the frin/e #attern. 5ther e+#erimenters also
27 fo!nd the same ne/ati%e res!lt. "cientists scramled to find an e+#lanation. 9e%er efore had a n!ll res!lt of e+#eriments ca!sed so m!ch concern and interest. 5ne e+#lanation #ro#osed that the ether is at rest with res#ect to the earth and not to the s!n and stars. !t then it wo!ld mean the earth is somehow #referredB o%er the s!n and stars contrary to the relati%ity #rinci#le. Another idea was that the ether was #!lled alon/ y the earth and other stars and therefore had 'ero s#eed on their s!rfaces. !t e+#eriments with hi/h*flyin/ alloons at altit!des where the ether mi/ht e detected also fo!nd no traces of the ether wind. The most notale hy#othesis was ad%anced y the rish #hysicist G. . it'Gerald =1,01*1(1> in 1,,2. ;e #ro#osed that the ether wind com#resses all odies that mo%ed with it at the s#eed v incl!din/ the interferometer arm the li/ht eam a meter stic or any other oect. ;ence this contraction maes it im#ossile to inde#endently demonstrate the contraction. This shortenin/ is y a factor e+actly e8!al to the difference etween the s#eed of li/ht alon/ the ether’s direction and the s#eed of li/ht across it calc!lated to e =1 L v2 D c2> 1D2 or since C v & c =1 L C (>1D2 which is the reci#rocal of . n 1,(0 the -!tch #hysicist ;endri A. &orent' =1,03*1(2,> inde#endently #ro#osed the same contraction in terms of chan/es in the electroma/netic forces etween the atoms !t witho!t #ro%idin/ em#irical e%idence. 9ow nown as the %orent9-Fit9Geral contraction the ad hoc #ro#osal ad%anced merely to e+#lain the )ichelson*)orley n!ll res!lts is !st a conse8!ence of the s#ecial theory of relati%ity. The contraction howe%er is real as we will see in "ection 2.4.1 elow. or his wors with li/ht )ichelson ecame the first American to win the 9oel #ri'e in #hysics in 1(7. The concl!sion drawn from the n!ll res!lts of the )ichelson*)orley e+#eriments is that there is no ether to mae a difference in the time of tra%el of the two li/ht eams alon/ and across the ether. The s#eed of li/ht is not in the least affected y the earth’s motion. A 1(7( %ersion of the )ichelson*)orley e+#eriment !sin/ %ery stale lasers to im#ro%e the meas!rement #recision y a factor of 4 still fo!nd no traces of the ether wind. Examle 1"; n a )ichelson*)orley e+#eriment s!##ose each of the li/ht eams has len/th l = 2, m. Gi%en v = 3. + 14 mDs and c 3. + 1, mDs =a> what is the time difference ca!sed y rotatin/ the arms of the interferometer thro!/h (^? => Ass!min/ the wa%elen/th of the li/ht !sed is ; 0 nm what is the e+#ected frin/e shift? !olution. =a> A##lyin/ the time difference form!la =10> we ha%e Nt – Nt’ [ =l ' F l (> =v( & c/>
=10>
Nt – Nt’ [ =2, + 2> m =3. + 14 mDs>2 D =3. + 1, mDs>3 Nt – Nt’ [ 06 m =3. + 1 L16 mDs> Nt – Nt’ [ 'E." x '" –'- s. => irst let !s find the fre8!ency of the li/ht !sed 1 = c & ; 1 = =3. + 1, mDs> D =0. + 1 L7m> 1 [ 1.7 + 110 s. ;ence the frin/e shift is the time difference di%ided y the reci#rocal of the fre8!ency of the li/ht eam 1(. + 1 L16 s D 1.7 + 1 L10 s '.' 1rines. :e now now that e%idence of the hy#othesi'ed ether wind does not e+ist and Einstein’s theory of relati%ity dis#oses of the conce#t entirely. Examle 1#; =a> f a #article is mo%in/ at a s#eed of .2 c find the %al!e of 1 D =1 L v2 D c2> 1D2. => Calc!late the of a #article mo%in/ at a s#eed of .2 c. =c> ind the of a #article mo%in/ at a s#eed of .(7, c =ro!/hly the s#eed of a commercial etliner>. !olution. =a> :e s!stit!te the #article’s s#eed v in the form!la
2, 1 D =1 L v2 D c2> 1D2 1 D P1 L =.2 c>2 D =c2 >Q1D2 1 D =1 L .4> 1D2 1 D .(7(, '."('. => :e a##ly the same #roced!re 1 D =1 L v2 D c2> 1D2 1 D P1 L =.2 c>2 D =c2 >Q1D2 1 D =1 L .4> 1D2 =1 D .(((((6> 1D2 '."""""(. =c> The same #roced!re yields 1 D =1 L v2 D c2> 1D2 1 D P1 L =.(7, c>2 D =c2 >Q1D2 1 D =1 L .(06> 1D2 =1 D .((((((430> 1D2 '.""""""KL. 5r !sin/ the a##ro+imation of deri%ed y inomial e+#ansion we /et [ 1 F C ( & (
=7>
[ 1 F =.(7, c2 D c2>2 D 2 1 F .47, '.""""""KL. f we com#are the res!lts of =a> and =c> we can see the effects of s#eed on relati%istic len/th and e%eryday len/th in the meas!rement of which is the only factor that se#arates them. $ecall that in the Galilean transformation x x’ F vt’ and in the &orent' transformation x = x’ F vt’ >. At a hi/h s#eed of .2 c is fairly si/nificant at 1.21 while at a slower et #lane s#eed a of 1.47, is so close to 1 that its effect on len/th is ne/li/ile. Tale 2*1 shows that as v /ets m!ch closer to c for e+am#le v .((( ((( c ecomes h!/e 77.17. The factor * is important in relativity meas!rements. As shown in Tale 2*1 ao%e since C v & c is ne%er 'ero =e+ce#t when the oect is at rest> altho!/h it is %ery small in e%eryday e+#erience 1 whereas its reci#rocal is less than 1 1 D U 1. 2.2
Sim!ltaneity
asic to 9ewtonian mechanics is the #remise of !ni%ersal time which is the same for all oser%ers inde#endent y nat!re of anythin/ e+ternal to it. This conforms to o!r e%eryday common sense e+#erience. :e are !sed to elie%in/ that an ho!r is e8!al to an ho!r whether we are in California or on the )oon. ;owe%er we now that Einstein had to aandon the notion of fi+ed time in his theory of relati%ity. Accordin/ to this theory meas!rements of time chan/e de#endin/ on reference frames. Einstein #ro#osed a tho!/ht e+#eriment to #ro%e his #oint. A o+car tra%elin/ to the ri/ht at constant %elocity is str!c y two olts of li/htnin/ that lea%e mars on each end of the o+car and on the /ro!nd =i/!re 2*4>. The mars on the o+car are A’ and ’ and the corres#ondin/ mars on the /ro!nd are A and . n the mo%in/ o+car =frame S’ > an oser%er 5’ stands halfway etween A’ and ’. 5n the /ro!nd the stationary oser%er 5 =frame S > is halfway etween A and . "!##ose the li/ht #!lses from the li/htnin/ olts reach the midway /ro!nd oser%er 5 sim!ltaneo!sly. =i/!re 2*4>. The oser%er 5 says that she sees the two olts striin/ at A and at the same time and concl!des that the tw3 events are simultane3us. !t with the mo%in/ oser%er 5’ thin/s are different. y the time the li/ht #!lse has reached the oser%er 5 the o+car has mo%ed a distance to the ri/ht toward the front olt carryin/ the oser%er 5’ with it and th!s shortenin/ the distance etween her and the front li/ht #!lse. The li/ht #!lse from ’ has already #assed 5’ while the #!lse from A’ has not yet arri%ed. And this is eca!se as the o+car mo%es to the ri/ht it /ets closer to the li/htnin/ olt at the front and recedes from the one in the rear. "ince the s#eed of li/ht is constant li/ht taes less time tra%elin/ a shorter distance. The oser%er 5’ therefore sees the li/htnin/ olt at the front of the car strie first and the li/htnin/ olt at the rear strie ne+t. Th!s t3 O’ the tw3 events are n3t simultane3us. :e concl!de that events that are meas!re to e sim!ltaneo!s in one frame of reference are not sim!ltaneo!s in another frame.
2(
"’
"
A’
J 5’
’
A
J5
=a> &i/ht #!lses "’ A’ "
A
J5
J 5’
’
=> Fi&!re 2-$. =a> To a midway stationary oser%er 5 on the /ro!nd =frame "> the two li/htnin/ olts a##ear t3 strike simultane3usly at 23ints A and D. => To the midway oser%er 5’ in the train mo%in/ to the ri/ht =frame "’> the li/htnin/ olt at the front of the car stries first efore the li/htnin/ olt in the ac stries. >he tw3 events A’ and D’ are n3t simultane3us. This tho!/ht e+#eriment shows that events that are sim!ltaneo!s in one reference frame are not sim!ltaneo!s in another reference frame even tho!&h oth are inertial frames. ;ence !st as there is no s!ch thin/ as asol!te motion or asol!te rest there is no s!ch thin& as asol!te sim!ltaneity. !t there is relative sim!ltaneity i.e. relati%e to reference frames. This is a direct conse8!ence of the second #ost!late. Alon/ this same line of in%esti/ation let !s tae another e+am#le. t shows that if two e%ents are sim!ltaneo!s for an inertial oser%er located halfway etween them they may not e sim!ltaneo!s for all midway inertial oser%ers re/ardless of their motion. n i/!re 2*0 a stationary s#ace station is commanded y Commander Achilles while a rocet shi# mo%in/ in the o##osite direction is !nder Commander ;elen. :e will arran/e at the instant that Commander Achilles and Commander ;elen are o##osite each other =i/!re 2*0a> for two flares to /o off at the same time. lare A maes a dent mar A on Achilles’ station and A’ on ;elen’s rocet shi#. lare maes a dent mar on Achilles’ station and ’ on ;elen’s shi#. As a midway oser%er Achilles saw the flashes from lare A and lare occ!r in the same instant. ;ence he concl!ded that the two flashes were sim!ltaneo!s e%ents. To him the len/th A was e8!al to the len/th A’’. As he looed o!t his window he saw ;elen’s shi# slidin/ off to his station’s rear mo%in/ toward lare A and away from lare while his station was at rest. ;e th!s saw that ;elen reached the #!lse from lare A first efore the #!lse from lare reached her. t was o%io!s to Achilles that ;elen sho!ld see what he saw i.e. lare A fired first and lare fired ne+t. The flares did not /o off at the same time. or her #art as a midway oser%er ;elen considered herself ein/ stationary and saw Achilles’ station mo%e off to the rear of her shi#. ;elen oser%ed the flash from lare A occ!r first and lare fired second as the station was mo%in/ toward the latter. Th!s she es#ied that the #oints A and A’ coincided first efore the #oints and ’ did. To her two e%ents were ne%er sim!ltaneo!s and the len/th A was not e8!al to A’’.
3 Achilles’ s#ace station A
J Achilles
lare A
lare A’
J ;elen
’
;elen’s rocet shi# =a> &i/ht #!lses
A
J Achilles
lare A A’
lare
J ;elen
’ =>
Fi&!re 2-'. =a> Achilles a midway oser%er in his stationary s#ace station oser%es that lares A and /o off at the same time and A A’’. !t he sees the flash from lare A reach ;elen’s rocet shi# first as it mo%es toward the station’s rear. => As a midway oser%er ;elen sees Achilles’ station slidin/ off toward her shi#’s rear and oser%es that the #!lse from lare A occ!rs efore that from lare . To ;elen A \ A’’. oth oser%ers are correct y the first #ost!late of the s#ecial theory of relati%ity. 9either Achilles nor ;elen wo!ld a/ree on len/ths times or sim!ltaneity. @et y the first #ost!late of the s#ecial theory of relati%ity they are oth correct. There is no #referred reference frame. s there any way to reconcile their different oser%ations? 9o. t is in the nat!re of thin/s the way the !ni%erse is. t is reality. !ynchroni)ation of loc.
31 y t
t=x&c
1
2
t = (x & c "
" 5
+ +
2+
' Fi&!re 2B). dentical clocs are e%enly s#aced at nown distances x from 5 in stationary frame S . A flash of li/ht is set off at 5. At that instant the oser%er at 5 sets her cloc 1 to t . :hen the li/ht #!lse reaches #oint + the oser%er at + sets her cloc 2 to t = x & c. &iewise the oser%er at #oint 2+ sets her cloc 3 tot = (x & c at the #!lse’s arri%al. A flare flashed y oser%er 5 at t tra%els at li/ht %elocity. 5ser%er 5 sets her cloc 1 tot . :hen the li/ht #!lse arri%es at cloc 2 at #oint + the second oser%er sets it to t = x & c this ein/ the time li/ht too to tra%el the distance x. A third cloc located at 2+ is set in the same manner. y !sin/ this #roced!re identical clocs in the frame S can e s!ccessi%ely set and they are all synchroni'ed. The same #rocess can also e a##lied to synchroni'e identical clocs in a mo%in/ frame S’ . 9ow s!##ose we want to set an array of clocs in a mo%in/ frame S’ to the ori/inal cloc in frame S. At time t = " oth frames coincide so that their ori/ins 5 and 5’ are s!#er#osed. :e synchroni'e the clocs at the ori/ins to that time. :hen the oser%er in frame S sets her cloc to time t = " her co!nter#art in frame S’ sets her cloc to t’ = ". Then when an e%ent occ!rs at x in frame S the oser%er in S’ records the same e%ent at #oint +’ and time t’ = x’ & c. 9ow a #rolem arises since the time of a cloc in S’ de#ends not only on time t of a cloc in S !t also on its #osition in that frame. The /reater is the distance of this cloc from its ori/in in S the /reater is the discre#ancy etween t and t’. This is a conse8!ence of the constancy of the s#eed of li/ht =the second #ost!late>. Time is relati%e and not fi+ed as 9ewton wo!ld ha%e it. "o too is sim!ltaneity relati%e. f in e%eryday life we are not aware of all this relati%ity it is eca!se its effects are noticeale only when the relati%e s#eed etween reference frames is %ery lar/e =close to c> or when the frames are se#arated y %ast distances. 2." Time #ilation :e ha%e seen that one of the conse8!ences of the s#ecial theory of relati%ity is thatsim!ltaneity of events is relative to the oservers frame of reference. E%ents that occ!r in the same #lace and are sim!ltaneo!s in one inertial frame are not sim!ltaneo!s in another. @et another conse8!ence of the theory is a %ery co!nterint!iti%e #henomenon of time dilation time that is not fi+ed or asol!te !t r!ns slow de#endin/ on frames of reference. 2.".1
Time ilation
n 9ewtonian #hysics time is asol!te and e+ists inde#endently of anythin/ o!tside it. This corres#onds to o!r conce#t of time which we consider distinct from s#ace and not ordinarily a dimension.B esides we do not easily acce#t the notion of time as ein/ relati%e eca!se we ha%e ne%er e+#erienced it. !t !st lie any re%ol!tionary ad%ances in science the theory of relati%ity !#sets old notions and introd!ces new #ers#ecti%es ao!t the reality. To see what effects this theory has on o!r meas!rement of time we will as !s!al consider a tho!/ht e+#eriment with a li/ht cloc as de#icted in i/!re 2*7. Tra%elin/ in a s#aceshi# =frame S’ > mo%in/ at hi/h s#eed v to the ri/ht oser%er 5’ flashes a li/ht. The li/ht #!lse tra%els to the mirror fi+ed on the shi#’s ceilin/ at a distanced from the shi#’s floor. 5ser%er 5’
32 meas!res the ro!nd tri# of the li/ht #!lse =to the mirror and ac to the detector> with an acc!rate cloc !sin/ the definition of the %elocity =i/!re 2*7a>. "ince li/ht tra%els atc the ro!nd tri# taes a time Nt’ distance tra%eled D %elocity 2d & c. The distance tra%eled is d = cNt’ & 2. v "’ d 5’
=a> &i/ht #!lse sent y mo%in/ oser%er 5’ tra%els strai/ht !# and down co%erin/ the distanced twice. Time tra%eled to mirror and ac Nt’ 2d & c. "
v
cNt D2
d vNt D2 5 vNt
=> "tationary oser%er 5 in frame S on Earth sees the same li/ht #!lse tae a dia/onal #ath not the %ertical distance d and th!s tain/ a lon/er time Nt to com#lete the ro!nd tri#. Fi&!re 2-*. =a> A flash !l set off in a s#aceshi# tra%elin/ at the s#eed v in frame S’ is reflected off a mirror fi+ed on the ceilin/ at a distance d to reach a detector y the oser%er 5’ at the li/ht so!rce. The li/ht’s !#* and*down ro!nd tri# taes time Nt’ 2d & c. => To a stationary oser%er 5 on earth =frame S > howe%er the li/ht’s #ath is a 'i/'a/ as shown i.e. a lon/er distance than d . Therefore the ro!nd tri# time tra%eled Nt in frame S is /reater than Nt’ in frame S’. The time interval t etween two events that occ!r at the same location meas!re y a sin&le cloc0 as in frame S’ is called the proper time. Proper means characteristic of or elon/in/ to. The #ro#er time is !s!ally denoted as Nt which is the notation we !se from here on. >he 2r32er time is the sh3rtest time interval measured with a sinle cl3ck by any 3bserver . :e ha%e the #ro#er time form!lation t D : 2d c from which we deri%e the istance travele d = cNt & 2.
31)4
33 n the conte+t of a li/ht cloc Nt maes !# one tic. )eanwhile the stationary oser%er 5 in frameS of the earth e8!i##ed with an acc!rate cloc oser%es the same #rocess and notices that the li/ht’s #ath is not strai/ht !# and down. y the time the li/ht reaches the mirror on the ceilin/ the s#aceshi# has mo%ed a certain distance. As the li/ht is reflected down to reach the detector the s#aceshi# has a/ain mo%ed a com#arale distance as shown in i/!re 2*7 co%erin/ a total distance of vNt =from the definition of %elocity>. or its #art the li/ht #!lse will ha%e tra%eled d!rin/ the same time a distance of cNt . As can e seen in i/!re 2*7 the distance co%ered y li/ht to the mirror is the hy#oten!se of a ri/ht trian/le whose other sides are the distance tra%eled y the s#acecraft and the distanced from the mirror to the detector. A##lyin/ the ytha/orean theorem yields =cNt & 2>2 d 2 F =vNt & 2>2 "ince d cNt & 2 =cNt & 2>2 =cNt & 2>2 F =vNt & 2>2 =c2Nt 2 & 4> L =v2Nt 2 & 4> c2Nt 2 & 4 c2Nt 2 L v2Nt 2 c2Nt 2 "ol%in/ for Nt we /et Nt 2 =c2 L v2> c2Nt 2 Nt 2 c2Nt 2 D =c2 L v2> Nt cNt D =c2 L v2> 1D2 Nt cNt D c=1 L v2 D c2> 1D2 Time ilation form!la t : t D 8 31 B v2 8 c24 182
31*4
Nt Nt
=1,>
The spreasheet form!la of time ilation relation 31*4 is t : tD831 L 3v8c4;24 ;31824
3164
"!stit!te act!al %al!es for Nt " v and c in the form!la 1( to /et the res!lt. f v is e+#ressed in terms of c e./. .(0 c then this v %al!e taes the #lace of %Dc in the form!la. The proper time form!la is Nt Nt &
=2>
t D : t 31 B v2 8 c24 182
3214
The spreasheet form!la of the proper time relation 3214 is tD : t31 L 3v8c4;24 ;31824
3224
"!stit!te act!al %al!es for Nt v and c in the form!la 22 to /et the res!lt. f v is e+#ressed in terms of c e./. .(0 c then this v %al!e taes the #lace of %Dc in the form!la. ;ence the factor * in terms of the two time inter%als Nt and Nt * : t 8 t D
32"4
34 To find the spee of travel v in terms of Nt and Nt we start from the ao%e form!la Nt D Nt
=23>
=Nt D Nt > 2 =1 L v2 D c2> L1 Nt 2 =1 L v2 D c2> L1 Nt 2 Nt 2 =1 L v2 D c2> Nt 2 Nt 2 D Nt 2 =1 L v2 D c2> v2 D c2 1 L Nt 2 D Nt 2 v : 31 B t D2 8 t 24 182 c
32$4
n the time dilation form!la =17> ao%e note that the denominator is the reci#rocal of the factor which rec!rs in relati%istic meas!rements. "ince and its reci#rocal 1 D are !i8!ito!s in relati%istic meas!rements we need to learn to mani#!late them with ease. The time dilation form!la indicates that the time in the mo%in/ frame S’ or the 2r32er time Nt is shorter than the time Nt in the stationary frame S eca!se 1 =Tale 2*1 ao%e>. n other words time r!ns more slowly in a movin& oy than in a oy at rest y eAactly the factor *. This #henomenon is not d!e to any cloc’s acc!racyR it is sim#ly the nat!re of time i.e.time is relative a reality that is ca#t!red ele/antly y the s#ecial theory of relati%ity. :e call this slowin/ down of time time ilation. :hat is the meanin/ of ? rom one #ers#ecti%e is the factor y which the #ro#er time differs from the time meas!red of two e%ents in two different reference frames. Th!s is the ratio Nt & Nt " of the time inter%al etween two e%ents occ!rrin/ in ifferent frames as meas!red y a cloc at rest to the #ro#er time i.e. the time inter%al meas!red of two e%ents occ!rrin/ at the same location y an inertial oser%er !sin/ a sin/le cloc. The factor is in effect the amo!nt of time Nt corres#ondin/ to e%ery !nit of #ro#er time Nt ". or instance if Nt " 1 s Nt s. The factor is s#eed*sensiti%e. n Tale 2*1 we see that as the s#eed ratio C v & c increases increases. or e+am#le when v .4 c 1.( and when v .0 c 1.10. Time ilation as a conce#t is not easy to /ras# y o!r common sense eca!se we are so !sed to thinin/ of time as asol!te. @et it has een corroorated y e+#erimental e%idence. &et !s e+amine the time dilation form!la in f!rther detail. f the s#eed of a mo%in/ oect v is then oth times are e8!al Nt Nt as e+#ected. n other words when nothin/ mo%es there is no time dilation. f v !t v UU c as in e%eryday e+#erience then [ 1 and time dilation has no oser%ale effect i.e. it is not meas!rale y ordinary clocs. 5nly when v is closer to c as in certain e+#eriments already cond!cted in f!t!re inter/alactic tra%els or in #article accelerators does time dilation ha%e the meas!rale effects s#elled o!t in form!la =17>. Time dilation can e meas!red y !sin/ clocs that are acc!rate to the nanoseconds. E+#erimental confirmation of time dilation was achie%ed in 5ctoer 1(71 the #hysicists M. C ;afele and $. E. Keatin/ #laced fo!r hi/h*#recision cesi!m*eam atomic clocs =acc!rate to the nanoseconds> on commercial air#lanes flown aro!nd the world twice in o##osite directions and identical clocs at the S.". 9a%al 5ser%atory. They fo!nd that after flyin/ aro!nd the world at hi/h s#eed the flyin/ clocs ran slower than the stationary cloc in the la. rom time dilation calc!lations the eastward clocs were e+#ected to ha%e lost 4 b 23 ns in the tri# whereas the westward clocs sho!ld ha%e /ained 270 b 21 ns. The e+#erimental res!lts were in remarale a/reement with theory The eastward clocs lost 0( b 1 ns and the westward clocs /ained 273 b 7 ns -!e to the relati%istic effects of time dilation atomic clocs that are mo%ed from one location to another m!st now e ad!sted for these effects. ;owe%er in e%eryday acti%ities we ha%e no clocs that are acc!rate eno!/h to meas!re time dilation. Knowin/ that =1 L v2 D c 2> L1D2 =orm!la 3> and notin/ also that Nt & Nt =orm!la 23> as deri%ed from the time dilation e8!ation we wor o!t a few data #oints that relate C and for the /ra#h in i/!re 2*, elow which shows time dilation as a f!nction of s#eedv i.e. as a f!nction of C Time ilation as a f!nction of spee v
30
Nt & Nt "
7 6 0 4 3 2 1
/ : v c
* = 31 B v2 8 c24 B182
.1 .2 .3 .4 .0 .6 .7 ., .( .((
1.037,10 1.262726 1.4,2,4,37 1.(1,(401 1.104703, 1.20 1.42,,4 1.666666667 2.2(410733( 7.,,,120
.2
.4
.6
.,
1.
"#eed C v & c
Fi&!re 2-+. Relationship etween * an / or time ilation as a f!nction of spee. The data ao%e shows a %ery /rad!al rise in the %al!e of Nt & Nt " startin/ from a v of 1 of the s#eed of li/ht. "o far we do not ha%e technolo/y to #rod!ce any machine that can mo%e that fast. The time c!r%e => hardly chan/es for s#eeds v !# to .2 c =most e%eryday s#eeds are well elow this> and rises moderately !ntil v reaches .( c when Nt & Nt " more than do!les its %al!e. The last two data #oints /i%e a hint of the ra#id rise of as C v & c increases in si/nificant decimal #laces. rom this #oint on the rise of is dramatic. "till the c!r%e will ne%er reach 1. on the x*a+is eca!se c is the !nreachale !##er limit of all s#eeds. The /ra#h in i/!re 2*, can e /enerali'ed as as a f!nction of s#eed v & c. This f!nction rec!rs in all relati%istic calc!lations. n a time dilation #rolem three im#ortant %al!es are in%ol%ed =1> the 2r32er time t 5 etween two e%ents occ!rrin/ at the same #hysical location mo%in/ at =2> a s2eed v relative t3 a stati3nary re1erence 1rame and =3> the time t )s3metimes called stati3nary time> etween these e%ents as meas!red in the stationary frame. f any two of these %al!es are nown the third one can e deri%ed in a strai/htforward manner from the time dilation e8!ation. The fo!rth %al!e is a /i%en the s#eed of li/htc. &et !s ill!strate time dilation with a few e+am#les. Examle 1'; A li/ht cloc tra%els in a s#aceshi# with a s#eed of .6 c relati%e to a stationary oser%er on the earth. ;ow lon/ does it tae for the tra%elin/ cloc to ad%ance y 1 second accordin/ to this oser%er? !olution. irst we determine the #ro#er time Nt " the time ela#sed etween two e%ents that tae #lace in the same location. n this case it is the 1. second recorded y the mo%in/ li/ht cloc in the s#aceshi#. :e are to find the time Nt as meas!red y the oser%er on the earth. A##lyin/ the time dilation form!la =17> we /et Nt Nt D =1 L v2 D c2> 1D2 Nt 1. s D P1 L =.6 c>2 D c2Q 1D2 Nt 1. s D =.64> 1D2 1. s D ., '.( s.
=17>
36 Examle 18; ind the factor y which a cloc on the s#ace sh!ttle tra%elin/ at 70(( mDs r!ns slower than an identical cloc on the earth. !olution. $ecall that is the factor y which Nt differs from the #ro#er time Nt as indicated in the form!la 14 Nt Nt . :e find as follows = =1 L v2 D c2> L1D2
=3>
P1 L =70(( mDs>2 D =3. + 1, mDs>2Q L1D2 '."""""""""/("L"". or e%ery second recorded y the cloc on the s#ace sh!ttle the cloc on the earth records 1.32, seconds. 5%io!sly no ordinary clocs can meas!re this time. ;encethe time dilati3n e11ect is t3tally neliible at the shuttle s2eed, as it is in all s2eeds we enc3unter in everyday ex2erience. Examle 19; The starshi# Enco!nter tra%els with the s#eed of .7 c relati%e to the earth for 10 years accordin/ to the shi#’s cloc. To an oser%er on the earth how many years ela#se d!rin/ the tri#? !olution. :e determine that the #ro#er time Nt the time meas!red y a cloc at rest on oard the shi# is 10 years. y a##lyin/ the time dilation form!la Nt Nt D =1 L v2 D c2> 1D2 we find the earth d!ration to e Nt Nt D =1 L v2 D c2> 1D2
=17>
Nt 10 y D P1 L =.72 c>2 D c2Q1D2 10 y D =1L .72> 1D2 Nt 10 y D .011D2 ('."" y. Examle 16; Astrona!t -aedal!s tra%els with a s#eed of .((7 c to Gala+y )inos 2 ly =li/ht*years> away then t!rns ac to Earth. ;is son car!s a/e 2 when he left comes to "tar#ort to /reet him. =a> ;ow old is -aed!l!s on his ret!rn to Earth if he was 40 years old when he set o!t on his o!rney? => ;ow old is car!s when he sees his father a/ain? !olution. As in all time dilation #rolems we first determine two e%ents and the #ro#er time. The two e%ents are -aedal!s’ de#art!re from and ret!rn to Earth. The #ro#er time is the time -aedal!s meas!res d!rin/ his 2*ly %oya/e. irst we find the d!ration of -aelal!s’ tra%el from the car!s’ #oint of %iew. Knowin/ that the distance -aelal!s tra%eled is 4 ly =ro!nd tri#> i.e. a distance co%ered y li/ht for 4 years or 4 c =1 y> at the %elocity of .((7 c the time it too -aelal!s as meas!red from Earth is Nt d & v Nt 4 c =1 y> D .((7 c 3( y =a> -aedal!s howe%er too the #ro#er time Nt to mae the o!rney Nt Nt =1 L v2 D c2> 1D2
=21>
Nt 3( y P1L =.((72 c>2 D c2Q 1D2 Nt 3( y =.0((1> 1D2 3.2 y rom his #oint of %iew -aedal!s had tra%eled only 3.2 years. Therefore at the end of the o!rney he is 40 y F 3.2 y L."( years 3ld.
37
=> ;is son car!s who stayed ac on Earth is 3( years older 2 y F 3( y E years 3ld. >hus the 1ather is y3uner than the s3n a1ter /E years 31 absenceN
Examle 1:; Ca#tain Valiant emared at %ery hi/h s#eed on an interstellar odyssey that lasted 12 years y Earth’s cloc. :hen he came home he had a/ed 2 years. :hat was his a%era/e s#eed? !olution. :e now the two e%ents his de#art!re and his ret!rn. The time meas!red with res#ect to Earth is Nt = 12 y. !t Ca#tain Valiant’s #ro#er time is only Nt 2 y. :e a##ly the s#eed form!la v =1 L Nt 2 D Nt 2> 1D2 c
=24>
v =1 L 22 y D 122 y> 1D2 c ".EL- c. Examle 25; )ycenaen Commander A/amemnon’s s#aceshi# taes him on a lon/ tri# to Gala+y Troy with %elocity of .70 c and rin/s him ac to Earth. :hen he checs the cloc and calendar in his ho!se he finds that a #eriod of se%en years has ela#sed. ;ow lon/ did his tri# tae? !olution. A/ain the two e%ents are his de#art!re and his ret!rn. Earth time d!ration is Nt 7 y. :e are to find the #ro#er time Nt y !sin/ the #ro#er time form!la =21> Nt Nt =1 L v2 D c2> 1D2
=21>
Nt 7 y =1 L .702 c2 D c2> 1D2 Nt 7 y =1 L .0620> 1D2 7 y =.4370> 1D2 .-/ y. Examle 21; A s#acecraft reenters Earth at the s#eed of 10 mDs and taes 3 seconds to land. :hat is the time dilation effect on the s#acecraft d!rin/ this time? !olution. The two e%ents are the s#acecraft’s reentry and to!chdown. The #ro#er time Nt for the s#acecraft is Nt 3 s. f we now we can deri%e the effect Nt y a##lyin/ the time dilation form!la Nt Nt D =1 L v2 D c2> 1D2
=17>
irst we con%ert v into a fraction of c for ease of handlin/ v 10 mDs D 3 + 16 mDs .0 c A##lyin/ the time dilation form!la yields Nt 3 s D =1 L .02 c2 D c2> 1D2 Nt 3s D =.((((((((70> 1D2 3 s + 1. 1 20 /".""" """ /L s. E%en at this !nrealistically hi/h s#eed the effect of time dilation is so tiny it is !ndetectale y an ordinary cloc. Examle 22; Cosmona!t "%ooda in a s#acecraft h!rtlin/ toward Gala+y Tranto )ir meas!res her hearteat to e one e%ery .( s while )ission Control on Earth meas!res her heart to eat once e%ery 1.0 s. ;ow fast is "%ooda tra%elin/ relati%ely to Earth? !olution. :e find the two e%ents in the e/innin/ and end of one of Cosmona!t "%ooda’s hearteat. "ince the hearteat e/ins and ends in the same location her heart the hearteat she meas!red is the #ro#er time Nt . Knowin/ the time Nt meas!red in the earth’s reference frame =)ission Control> we a##ly the s#eed relation deri%ed from the time dilation form!la
3, v =1 L Nt 2 D Nt 2> 1D2 c
=24>
v =1 L .(2 s D 1.02 s> 1D2 c ".L" c. Examle 2"; An e+#eriment cond!cted in a laoratory on Earth taes 3 s to com#lete. A )artian s#ace tra%eler mo%in/ with a constant %elocity of .6 c relati%e to Earth ha##ens to oser%e the e+#eriment. ;ow lon/ does the )artian meas!re the e+#eriment to last? !olution. The two e%ents are the e/innin/ and end of the la e+#eriment. "ince oth e%ents occ!r in the la the time meas!red is the #ro#er time Nt 3 s. :e find the time Nt meas!red y the )artian to e Nt Nt D =1 L v2 D c2> 1D2
=17>
Nt 3 s D =1 L .62 c2D c2> 1D2 Nt 3 s D =1 L .36 c2D c2> 1D2 3 s D ., /K s. >hat are the imlications of relativistic time dilation? The conce#t of time dilation im#lies that as Nt /oes to infinity one tic of the cloc =or any one !nit of time> taes an infinite amo!nt of time ass!min/ that Nt " is one s!ch tic. To #!t it another way as v a##roaches c the cloc /rad!ally /rinds to a halt. That is another sense ofc’s ein/ the u22er limit 31 all s2eeds. Another way to loo at time dilation is that the #ro#er time Nt " the time ela#sed etween two e%ents that occ!r at the same location e./. the time tra%eled y an astrona!t remains small while =1 L v2 D c2> L1D2 increases %ery ra#idly as the s#eed v a##roaches !t ne%er reaches c. The relation Nt Nt deri%ed from the time dilation e8!ation says that the time Nt ela#sed on Earth is e8!al to the #ro#er time Nt " m!lti#lied y . That is e%ery second of #ro#er time that an astrona!t tra%els near the s#eed of li/ht corres#onds to seconds as meas!red y an eartho!nd oser%er. As /rows y lea#s and o!nds e%en with tiny C v & c increases so also do the time dilation effects. Th!s as shown y Tale 2*3 in "ection 2.1.2 ao%e if the astrona!t tra%els with a %elocity of .(((4 c for e%ery year =second min!te ho!r and so on.> she clocs in her s#aceshi# her rother on the earth clocs 2,.,7 years =seconds min!tes ho!rs and so on>. And if she tra%els at %elocity .(((( c for a year her rother will a/e 7.71 years an e+tremely tantali'in/ consideration for attem#tin/ interstellar tra%el. Another e+am#le of s#ace tra%el made #ossile !t hi/hly im#la!sile y time dilation. A s#aceshi# that can accelerate at a constant and comfortale rate of =acceleration of Earth’s /ra%ity (.,660 mDs2> can tae tra%elers to Andromeda /ala+y 2 million li/ht*years away and ac in 0( years. -!rin/ this time the earth has a/ed more than 4 million years. And if the shi# tra%els 7, years with the same acceleration it can reach a /ala+y 0 million li/ht*years away to find !#on ret!rn that the earth has ecome one illion years older. This is the law of nat!re. !t as with other laws of nat!re we do not ha%e the technolo/y or the now*how yet to tae f!ll ad%anta/e of them. 5ne ca%eat ao!t s#ace tra%el howe%er. t taes an enormo!s amo!nt of ener/y in fact an infinite amo!nt of ener/y in order to !ild !# eno!/h acceleration to reach this near li/ht s#eed an en/ineerin/ feat that o!r technolo/y is not ca#ale of achie%in/ within the foreseeale f!t!re. To add a little #ers#ecti%e 9A"A’s s#ace sh!ttle which in 20 is the fastest manned flyin/ machine a%ailale needs 3 s#ace sh!ttle main en/ines ="")E’s !ilt into the oriter> which are f!eled y the an e+ternal tan =ET> of #ro#ellant and 2 re!sale solid rocet oosters ="$’s> !rnin/ solid #ro#ellant to lift it free of Earth’s /ra%itational #!ll for orit insertion. The 2((37*/ e+ternal tan with its 71(110*/ load of li8!id o+y/en and li8!id hydro/en wei/hs a total of 74(02 /. Each of the two ,6137*/ "$’s holds 4(464 / of rocet #ro#ellant wei/hin/ a total of 0,21 /. To/ether the 2 "$’s deli%er 71.4 =or 2((374 /> of the total thr!st to the sh!ttle at liftoff and d!rin/ first sta/e ascent. At liftoff the ET recei%es a total thr!st of 303,07 / from the 2 "$’s and 3 "")E’s. Each of the three "")E’s wei/hs 33( /. Alto/ether the #ro#!lsion system of the s#ace sh!ttle wei/hs 1(1,071 /. The s#ace sh!ttle at liftoff wei/hs ao!t 241166 / with a ma+im!m car/o ca#acity of 2,,3 /. After ,.0 min!tes of rocet acceleration the s#ace sh!ttle reaches an altit!de of 40 m the e+ha!sted ET is ettisoned and the 2 "$’s also se#arate lea%in/ the sh!ttle’s 3 main en/ines the tas of oostin/ the oriter from 4,2, mDh to 2730, mDh in 6 min!tes to reach orit. :itho!t the ET the "")E’s do not ha%e m!ch f!el to /o far. Th!s it taes the s#ace sh!ttle a total of 14.0 min!tes after la!nch
3( and a cons!m#tion of 177340 / of #ro#ellant !st to carry 2,,3 / of #ayload into orit 304 m =22 miles> ao%e Earth. As Tale 2*3 shows e%en this asto!ndin/ %elocity !nder/oes a tiny time dilation factor of 1.32, since it is only 1D3(041th the s#eed of li/ht. :e are a lon/ way from interstellar tra%el indeed. As a contin!ation of the /ra#h in i/!re 2*, we now tae the data from Tale 2*3 and start at a %ery hi/h s#eed v =in terms of c> or which amo!nts to the same thin/ C v & c. :e /ra#h the times corres#ondin/ to v .( c to v .((( ((( ((( ((( ((( c =10 decimal #laces> comin/ as !nrealistically close to c as we can /et within the limits of floatin/ #oint n!mers allowed y E+cel. 9ote that the to# of the /ra#h shows that if an astrona!t can mana/e to o!rney at %elocity v .((( ((( ((( ((( ((( c e%ery second she tra%els thro!/h s#ace is e8!i%alent to 2236(621 seconds on Earth Time Dilation as Speeds Approach c 25,000,000
20,000,000
s d n o c e 15,000,00 0 s n i a m m a g 10,000,000 s a e m i T 5,000,000
0 0.9
1
1
1
1
1
1
1
Spee d v / c fr om 0.9-0.999999999999999 c Time in seconds
Fi&!re 2-6. Time ilation &rowin& rapily at very hi&h spees approachin& c. n calc!latin/ time dilation it is im#ortant to identify the two e%ents =e./. the e/innin/ and end of a o!rney the e/innin/ and end of a li/ht #!lse the e/innin/ and end of a hearteat the creation and decay of a #article etc.> and the proper time t D which is the time inter%al etween two e%ents that occ!r at the same location for an oser%er e./. in a s#aceshi#. The proper time is the shortest time etween these two events. The time inter%al t meas!red y a stationary oser%er on Earth oser%in/ the same two e%ents is not the #ro#er time eca!se the e%ents occ!r in two different locations. A consideration of the time dilation form!la Nt Nt D =1 L v2 D c2> 1D2 shows that the hi/her the s#eed v is the hi/her ecomes and the /reater Nt ecomes. t is &reater than t D y the factor * at any spee v @ D. ;owe%er in v c the time difference etween Nt and Nt is so small it can e detected only y atomic clocs ca#ale to meas!rin/ nanoseconds. :e ha%e seen that time r!ns slower for a tra%elin/ oser%er. This does not mean that the cloc r!ns slowerR the cloc is acc!rate and f!nctions normally. nstead it is in the nat!re of time to slow down =to
4 dilate> at %ery hi/h s#eed com#ared to time as meas!red y a stationary oser%er. Time in an en%ironment mo%in/ with hi/h s#eed with res#ect to Earth marches to a different dr!meat than time on Earth. :e say a/ain that time is relative5 i.e.5 relative to the reference frame. The o%io!s 8!estion is :hat ao!t the normal #hysical and iolo/ical #rocesses that occ!r in a h!man tra%elin/ at hi/h s#eed? The answer is that e%ery #hysical #rocess from hearteat di/estion metaolism to reathin/ and a/in/ slows down accordin/ly y e+actly the same dilation factor . The !#shot is that e%erythin/ from the tics of the cloc to a h!man’s life #rocesses #roceed as they wo!ld normally do on Earth. The tra%elin/ astrona!ts wo!ld notice nothin/ different in themsel%es or in their clocs. To s!m !# oth the hi/h*%elocity s#ace tra%eler and the eartho!nd oser%er li%e their li%es normally in their res#ecti%e en%ironments. The effects of time dilation ecome meas!rale when the s#ace tra%eler comes ac to Earth. The traveler a&es less than the oserver on Earth. "#ace tra%el howe%er leads to a dilemma we will consider in the ne+t section. 2.".2 The T$in %aradox As ill!strated in E+am#le 1, ao%e if yo! tra%el with the s#eed of .((7 c yo! will mae a 4*ly tri# to a star and ac in only 3.2 years. ather -aedal!s made the 4*ly interstellar tri# in 3.2 years while his son car!s who stayed on Earth a/ed 3( years d!rin/ the same tri# makin him 3lder than his 1ather 3n the latter’s return a scenario rife with #romises for science fiction writers. This startlin/ #ossiility is #redicted y the s#ecial theory of relati%ity. To the s#ace tra%elers their shi# is at rest and the stars and e%erythin/ else mo%e #ast them. To oser%ers o!tside the s#aceshi# it is the s#aceshi# that mo%es and e%erythin/ else is at rest. :hether -aedal!s thins he is at rest and his son mo%es or %ice %ersa or whether he thins he mo%es and his son is at rest or %ice %ersa maes no difference in the res!lt. Each side is ri/ht in his claims. Each #oint of %iew =or reference frame> is e8!i%alent to the other. And in accordance with the first #ost!late of s#ecial relati%ity each inertial reference frame is treated the same y the laws of #hysics. There are no #referred reference systems. f all reference frames are e8!i%alent can the s#ace tra%eler -aedal!s claim that it was his son car!s that tra%eled while he was at rest and therefore sho!ld e the one that a/ed more slowly? Can car!s claim that his father a/ed more slowly eca!se he is the one that tra%eled? !t differently if there is com#lete symmetry etween reference frames are oth claims e8!ally %alid? The answer is no eca!se the tw3 re1erence systems are n3t e@uivalent . ather -aedal!s’ s#aceshi# had to accelerate many times to reach its cr!isin/ s#eed of .((7 c and had to accelerate a/ain on the ret!rn tri#. ;is s#aceshi# is not an inertial system. !t "on car!s was in an inertial frame that of the Earth. f we now s!stit!te two identical twins Tweedledee and Tweedled!m for -aedal!s and car!s the sit!ation does not chan/e. y the first #ost!late of the s#ecial theory of relati%ity Tweedledee the tra%eler can claim that he is stationary and it is his rother who tra%els. "o Tweedled!m sho!ld a/e more slowly than Tweedledee. Tweedled!m on his #art will #rotest that it is his rother Tweedledee that tra%els and therefore sho!ld a/e more slowly. n this case we ha%e a #!tati%e #arado+ nown as theTwin ParaoA. y the first #ost!late each twin can claim that it is his rother who a/es more slowly. @et oth twins sim#ly cannot a/e more slowly than each other at the same time. A/ain the resol!tion resides in the definition of the inertial frame. f we rememer that only Tweedledee has to accelerate in his interstellar %oya/e the a##arent #arado+ dissol%es. This is a clear case in%ol%in/ an inertial system =Tweedled!m> and an acceleratin/ system =Tweedledee>. The two systems are not e8!i%alent. ;ence the first #ost!late does not a##ly and there is no #arado+. :e will see acceleratin/ systems treated y the /eneral theory in Cha#ter 3. 2."."
ecay of the uon
Another e+#erimental confirmation of the s#ecial theory of relati%ity in%ol%es the eha%ior of the m!on an !nstale s!atomic #article created y cosmic radiation hi/h in the earth’s atmos#here that tra%els down to the earth at a s#eed a##roachin/ the s#eed of li/ht and has an a%era/e life of 2.2 s =2.2 microseconds or millionths of a second> efore it decays. Consider a m!on created at an altit!de of 0 m ao%e Earth’s s!rface. f the m!on tra%els down toward Earth at the s#eed of .((0 c it will decay after tra%elin/ only d = .((0 + 3 + 1, m + 2.2 + 1*6 s 607 m.
41 Another way to loo at this is from the #oint of %iew of the m!on itself. f an oser%er co!ld accom#any the m!on she wo!ld notice that the m!on was at rest and the earth was mo%in/ !# at the s#eed of .((0 c tra%elin/ 607 m efore the m!on decayed. "ince the m!on is created somewhere at the to# of the atmos#here and decays somewhere near the ottom of the atmos#here to the m!on this distance is the thicness of the atmos#here. &ar/e n!mers of m!ons ha%e een detected on the /ro!nd eca!se as they tra%el at near the s#eed of li/ht they a/e more slowly d!e to the effect of time dilation in their descent and th!s li%e lon/ eno!/h to reach the /ro!nd. Time dilation for radioacti%e #articles was first meas!red in 1(41 y r!no $ossi and -.. ;all. !t the more notale e+#eriment was cond!cted in 1(63 y the #hysicists -. ;. risch and M. ;. "mith. n the article )eas!rement of $elati%istic Time -ilation Ssin/ 6*)esonsB #!lished in American I3urnal 31 Physics )ay 1(63 they descried in detail an e+#eriment in which they co!nted the n!mer of m!ons near the #ea of )o!nt :ashin/ton 9ew ;am#shire and the n!mer of m!ons in their laoratory in Camrid/e )assach!setts. eca!se of the m!on’s short life it was e+#ected that there wo!ld e more of them s!r%i%in/ at hi/her altit!des than at sea le%el. risch and "mith set their detector to detect m!ons tra%elin/ with s#eeds v within the narrow ran/e of .((0 c and co!nt the n!mer of m!ons as they came down near the s!mmit of )o!nt :ashin/ton efore their decay. n one e+#eriment r!n their a##arat!s fo!nd a mean m!on life of 2.2 s and an a%era/e arri%al rate of 063 m!ons #er ho!r The m!ons’ o!rney thro!/h a %ertical distanced of 1(7 m from their set!# on )o!nt :ashin/ton to their laoratory in Camrid/e meas!red y a cloc at rest relati%e to the laoratory taes a time Nt d & v = 1(7 m D =.((0>=3 + 1, s> 6.4 + 1 L6 or 6.4 s Accordin/ to their calc!lations after the time Nt 6.4 s has ela#sed only 27 m!ons #er ho!r wo!ld e e+#ected to s!r%i%e the tri#. ;owe%er the detector at Camrid/e co!nted an a%era/e ho!rly s!r%i%al rate of 4, m!ons. This rate accordin/ to the e+#erimenters’ calc!lations corres#onds to a mean life of Nt .7 s. To find the %elocity with which the m!ons fell to Earth /i%en the #ro#er time Nt .7 s i.e. the mean life at rest and the stationary time Nt 6.4 s we !se the time dilation findin/s y a##lyin/ the s#eed form!la to otain v =1 L Nt 2 D Nt 2> 1D2 c
=24>
v =1 L .72 s D 6.42 s> 1D2 c .((4 c which is incredily close to the e+#erimental s#eed of .((0 c set for the detectors therey confirmin/ time dilation and the s#ecial theory of relati%ity. !rther #article e+#eriments in 1(,0 with fast*mo%in/ atoms of neon e+cited y a laser to otain hi/h* #recision fre8!ency also confirmed time dilation. &et !s tae a few more n!merical e+am#les to ill!strate. Examle 2#; f a cosmic ray*#rod!ced m!on’s mean life at rest =i.e. tra%elin/ with the m!on> is 2.2 s =a> how far does the m!on tra%el at this s#eed of decay? => what is its mean life when it tra%els with the s#eed of .(( c? =c> ;ow far does it tra%el at this s#eed efore decayin/? !olution. =a> n the m!on’s frame of reference tra%elin/ for 2.2 s the m!on co%ers a distanced mu3n of d mu3n = .((+ 3 + 1, m + 2.2 + 1*6 s -/ m. rom the m!on’s frame the earth’s atmos#here is only 603 m thic. => The two e%ents the irth and decay of the m!on occ!r in the same location only in a rest frame. Gi%en the m!on’s mean life at rest is the #ro#er time Nt we !se the time dilation form!la =17> to find the time Nt Nt Nt D =1 L v2 D c2> 1D2 Nt 2.2 s D =1 L .((2 c2D c2> 1D2 '.- 6s.
=17>
is
=c> Accordin/ to an oser%er on Earth the distance d arth tra%eled y the m!on d!rin/ this time
42
d arth Nt v = =10.6 + 1 L6 s> =.((> =3 + 1, mDs> -/( m. or the eartho!nd oser%er the m!on has tra%eled 4632 m to arri%e close to the /ro!nd le%el. Examle 2'; Ass!min/ a m!on is created at an altit!de of 00 m and has a mean life at rest of 2.2 s =a> what is its mean life when it tra%els with the s#eed of .(0 c? => what is the distance it tra%els to Earth efore decayin/? !olution. =a> "ince the m!on’s mean life at rest is its #ro#er time Nt we !se the time dilation form!la =17> to find its mean life when tra%elin/ with %elocity .(0 c. Nt Nt D =1 L v2 D c2> 1D2
=17>
Nt 2.2 s D =1 L .(02 c2 D c2> 1D2 K." 6s 3r K." x '" –- s. => The distance the m!on tra%els at the s#eedv .(0 + 3 + 1, mDs 2.,0 + 1, mDs d : t v d 7.0 + 1 L6 s + 2.,0 + 1, mDs (""E m. Examle 28; :hat is the mean life of a m!on meas!red in the laoratory frame of reference if its mean life at rest is 2.2 + 1 L6 s and its s#eed is 2. + 1, mDs relati%e to the la? !olution. The m!on’s mean life at rest is its #ro#er time Nt . A##lyin/ the time dilation form!la yields Nt Nt D =1 L v2 D c2> 1D2
=17>
Nt 2.2 + 1 L6 s D P1 L =2. + 1, mDs>2 D =3. + 1, mDs>2Q1D2 Nt 2.2 + 1 L6 s D P1 L =2 D3>2Q1D2 2.2 + 1 L6 s D P1 L =4 D (>Q1D2 (.E x '" –- s. Examle 29; A m!on’s a%era/e lifetime at rest is fo!nd to e 2.0 + 1 L6 s and the time it taes to reach the laoratory is 6., + 1 L6 s. ind =a> the s#eed with which it tra%eled and => the distance it tra%eled. !olution. =a> ;ere we ha%e Nt 2.0 + 1 L6 s and Nt 6., + 1 L6 s. A##lyin/ the s#eed e8!ation we otain v =1 L Nt 2 D Nt 2> 1D2 c
=24>
v P1 L =2.0 + 1 L6 s >2 D =6., + 1 L6 s>2Q 1D2 c ".E/ c. => The distance the m!on tra%els at the s#eed v .(3 c d Nt v d 6., + 1 L6 s + .(3 + 3 + 1, mDs 1.,(7 + 13 m 'LEK m. Examle 26; The Conseil e!ro#en #o!r la recherche n!claire =CE$9 E!ro#ean Co!ncil for 9!clear $esearch> straddlin/ rance and "wit'erland o!tside Gene%a is the #remier site of #article #hysics research center e8!i##ed with the world’s lar/est #article accelerators and detectors. t #rod!ced m!ons accelerated to hi/h s#eeds with a mean life 3 times /reater than the m!on’s mean lifetime at rest. :hat is the s#eed of a CE$9 m!on?
43 !olution. The mean life of a m!on has een determined to e Nt 2.2 + 1 L6 s. The CE$9 m!on’s mean life is Nt 2.2 + 1 L6 s + 3 66. + 1 L6 s. A##lyin/ the s#eed form!la yields v =1 L Nt 2 D Nt 2> 1D2 c
=24>
v P1 L =2.2 + 1 L6 s >2 D =66. + 1 L6 s>2Q 1D2 c ".EEE c. Examle 2:; rod!ced y cosmic rays a m!on tra%els 3 m to Earth with a mean life in its fall of 44 + 1 L6 s. :hat are its s#eed v, its and its #ro#er time? !olution. =a> Knowin/ the m!on’s mean life in fall and the distance tra%eled we !se the distance form!la to deri%e its %elocity d Nt v v d DNt v 3 m D 44 + 1 L6 s --.'L x '"- m&s 3r ".-- c. => The factor is deri%ed from = =1 L v2 D c2> L1D2
=3>
= =1 L .66 2 c2D c2> L1D2 './/. =c> The #ro#er time is deri%ed from Nt Nt =1 L v2 D c2> 1D2
=21>
Nt 44 + 1 L6 s =1 L .66 2 c2 D c2> 1D2 // x '" –- s. 2.$
%en&th Contraction
M!st as time and sim!ltaneity are relati%e to reference frames distance or len/th is also relati%e to reference frames as we will see shortly. f someone tells yo! that a starshi# flyin/ at close to the s#eed of li/ht =reference frame S’ > shrinks in lenth in the direction of motion with res#ect to an oser%er on Earth =reference frame S > what will yo! thin? 9ot only does the starshi# shrins e%erythin/ else in it incl!din/ h!mans also shrins as meas!red in the S frame. And if the starshi# mana/es to a##roach the s#eed of li/ht its len/th shrins to 'ero @o! will #roaly feel %ery !ncomfortale with the notion of a mo%in/ oect’s shrinin/ sim#ly y tra%elin/ with a s#eed close to c. @et this is what nat!re is lie and the #henomenon conforms to the s#ecial theory of relati%ity. 2.#.1
(enth ontraction
The shortenin/ of distances m!st y now e familiar to the reader /i%en what we ha%e seen in the &orent'* it'Gerald contraction as well as in time dilation. :e saw in "ection 2.1.4 how scientists were tryin/ to e+#lain the n!ll res!lt of the )ichelson*)orley e+#eriment y #ositin/ an ad*hoc factor nown as the &orent'*it'Gerald contraction which t!rns o!t to e the factor . $ecall E+am#le 1, ao%e in which an astrona!t father -aedal!s came home yo!n/er than his eartho!nd son car!s after a %oya/e to a star 2 ly away y a meas!rement made from Earth. -aedal!s co%ered the distance of 4 ly in !st Nt 3.2 years tra%elin/ at the s#eed of .((7 c whereas to his son the same tri# too Nt 3( years y a cloc on Earth. This distanced of 3.2 y + .((7 c amo!nts to only 3.1 ly as com#ared to the 4 ly meas!red from the Earth’s #oint of %iew. Clearly distance has shr!n for the s#ace tra%eler. This shortenin/ of distance co!#led with the slowin/ down of time maes s#ace tra%el a %ery
44 attracti%e #ro#osition. The one maor h!rdle aside from financial matters is the technolo/y of #ro#!lsion #owerf!l eno!/h to reach %ery hi/h s#eeds. 9ow tae E+am#le 24 ao!t the m!on #rod!ced in the !##er atmos#here y cosmic rays. rom the m!on’s reference frame it tra%els only 603 m while from the Earth’s frame it tra%els 4632 m at %elocity .(( c. Clearly distance contracts for the m!on. :hen the conce#t of asol!te sim!ltaneity was o%ert!rned it ro!/ht down with it the conce#t of asol!te time as well as that of asol!te len/th or distance. To meas!re the len/th of an oect e./. the len/th of a meter stic we need to see oth ends of it at the same time. !t since two e%ents that are sim!ltaneo!s in one frame of reference S’ are not necessarily sim!ltaneo!s in another reference frame S in motion with res#ect to the first we are faced with a meas!rement #rolem. The meas!rements of time and distance in these frames will differ. To e+amine this iss!e 8!antitati%ely let !s consider a tho!/ht e+#eriment. A s#aceshi# tra%els from star A to star with a s#eed v. An oser%er from Earth meas!res the distance etween the stars as ? =called the proper len&th> and accordin/ to this oser%er the tri# taes the time Nt ? D v or v ? D Nt or their #art the s#aceshi# astrona!ts maintain that they are at rest and the destination star is mo%in/ toward them at the s#eed v. Accordin/ to the astrona!ts the distance in the direction of motion etween the stars is ?, and the time it taes them to reach their destination is Nt ? D v or v ? D Nt "ince oser%er and astrona!ts a/ree that the s#acecraft’s s#eed is v in either case the ao%e relations can e rewritten to eliminate v ? D Nt ? D Nt ? ? Nt D Nt !t form!la =23> has estalished the relation etween Nt and Nt as Nt D Nt
=23>
;ence the form!la for relati%istic len/th contraction in the direction of motion is ( = (D * B1 or ( = (D 31 B v2 8 c24 182
32'4
In the spreasheet e!ivalent of the len&th contraction form!la isJ % : %D 31 L 3v;2 8 c;244;31824
32)4
"!stit!te a %al!e for the #ro#er len/th (D and the s#eed v to otain the contracted len/th (. Gi%en a contracted len/th ? and the s#eed v we erive the proper len&th (D as follows (D = * ( or (D = ( 8 31 B v2 8 c24 182
32*4
which5 when translate into a spreasheet form!la5 ecomesJ (D = ( 8 31 L 3vB2 8 cB244;3 182 4
32+4
f v is e+#ressed in terms of c !se the followin/ form!la (D = ( 8 31 L 3vB2 44;3 182 4
3264
"!stit!te the contracted len/th ? and v in terms of c in the ao%e form!la to otain the #ro#er len/th.
40 "ol%in/ for the s#eed v we /et ? = ? =1 L v2 D c2> 1D2 ?( = ?2 =1 L v2 D c2> ?( & ?2 = 1 L v2 D c2 v2 D c2 = 1 L ?( & ?2 v2 = =1 L ?( & ?2 > c2 v = c 31 B (2 (D2 4 182
3"D4
or if we want to find the s#eed v in terms of c v c = 31 B (2 (D2 4 182
3"14
Translatin& relation 3"D4 into a spreasheet form!la5 we &et the spee v v : c31 L %;2 8 %D;24;3 1824
3"24
or the spee v in terms of c v : 31 L %;2 8 %D;24;3 1824
3""4
"ince the reci#rocal of U 1 ? U ?. The #ro#er len/th is /reater than the len/th as meas!red y the tra%elin/ astrona!ts y the reci#rocal of the factor . This #henomenon is called len&th contraction. The istance (D5 calle the roer lenth5 is the len&th of an oect meas!re y an oserver at rest with respect to that oect. n o!r e+am#le the #ro#er len/th is the distance etween star A and star as meas!red y an eartho!nd oser%er. The distance ? is meas!red y the tra%elin/ astrona!ts. The #ro#er len/th ? is /reater than the contracted len/th ? of the same oect in motion. And the len/th contracted is shorter than the #ro#er len/th y the factor L1. NotesJ =1> The astrona!ts tra%el d!rin/ the #ro#er time inter%al Nt !t co%er the distance ?. n3t the 2r32er lenth ?. 5n the other hand the eartho!nd oser%er meas!res the distance etween the stars as the 2r32er lenth ? !t meas!res the time the astrona!ts tae to co%er this distance as the stationary time Nt . Therefore it sh3uld never be assumed that an 3bserver that measures the 2r32er time als3 measures the 2r32er lenth 3r vice versa. =2> &en/th contraction occ!rs only in the len/th that is #arallel to the direction of motion of the oect. t does not occ!r in the dimension #er#endic!lar to the direction of motion i.e. the oect’s hei/ht remains !nchan/ed ass!min/ the motion is hori'ontal. =3> The factor a##roaches 1 as the s#eed v a##roaches e%eryday s#eeds v c and len/th contraction ecomes !nnoticeale. :hen v i.e. when the oect is at rest 1 and there is no len/th contraction. =4> Gi%en that no oect tra%els faster than li/ht i.e.v no oect has len/th 'ero in any reference frame. "ee the /ra#h in i/!re 2*11 elow. To the 8!estion ;ow lon/ is a meter stic?B the answer is t de#ends on whether the stic mo%es and how fast it mo%es and whether the oser%er who meas!res it mo%es or not and at what s#eed.B g!antitati%ely len/th meas!rements de#end on the factor which in t!rn de#ends on the s#eed v of the system. :e concl!de that there is n3 such thin as abs3lute lenth )distance 3r s2ace. %en&th5 li0e motion5 time an sim!ltaneity5 is relative. Effects of (enth ontraction
46 &en/th contraction has startlin/ effects. A s#aceshi# that #asses yo! at near the s#eed of li/ht not only a##ears shorter !t also shows its rear s!rface e%en tho!/h yo! are directly alon/side it. This effect which is not a relati%istic effect occ!rs eca!se of the finite s#eed of li/ht. y the time li/ht from the side of the s#aceshi# reaches yo! the s#aceshi# has mo%ed. &i/ht from the rear of the s#aceshi# m!st ha%e left earlier than li/ht from the side in order to reach yo! at the same time. y the same toen if yo! fly #ast a !ildin/ at nearly the s#eed of li/ht yo! can see oth the front and side of a !ildin/ as yo! #ass it. This effect maes it wron/ for science*fiction writers to re#resent !ildin/s or #eo#le to e sinny and elon/ated in the %ertical direction. &en/th contraction allows the m!on created y cosmic radiation hi/h !# in the atmos#here to reach /ro!nd le%el in s#ite of its short life. Tae the e+am#le of a trian/!lar s#aceshi# in i/!re 2*1 that mo%es to the ri/ht with a s#eed ofv .,0 c. ts len/th x at rest =i.e. its #ro#er len/th> meas!res 70 m and its hei/ht y at rest is 20 m. At the s#eed v the shi#’s contracted len/th is /i%en y form!la =20> ? = ? =1 L v2 D c2> 1D2
=20>
? = 70 =1 L .,02 > 1D2 3(.0 m
y 20 m + 70 m =a> "#aceshi# at rest.
y 20 m
% .,0 c
+ 3(.0 m => "#aceshi# mo%in/ with % .,0 c. Fi&!re 2-1D. =a> &en/th contraction occ!rs only in the direction of motion affectin/ the s#aceshi#’s len/th x. => &en/th contraction does not occ!r in the direction #er#endic!lar to the direction of motion. Therefore the shi#’s hei/ht y remains !nchan/ed. g!antitati%e e+am#les of len/th contraction follow. Examle "5; A s#aceshi# meas!rin/ ( m on the r!nway tra%els at a s#eed of .60 c with res#ect to an eartho!nd oser%er. ;ow lon/ does the s#aceshi# in fli/ht a##ear to the same oser%er? !olution. Gi%en the #ro#er len/th ? and the s#eed v we find the contracted len/th ? of the s#aceshi# with ? = ? =1 L v2 D c2> 1D2
=20>
? = ( =1 L .602 > 1D2 -L./E m. Examle "1; A s#aceshi# tra%els #ast the earth at a s#eed of .70 c relati%e to an eartho!nd oser%er. ;ow lon/ does a meter stic in the s#aceshi# a##ear to the same oser%er? !olution. Gi%en the #ro#er len/th ? and the s#eed v we find the contracted len/th ? of the meter stic with
? = ? =1 L v2 D c2> 1D2
=20>
47
? = 1 =1 L .702 > 1D2 ".-- m. Examle "2; To an oser%er on Earth the 1m*lon/ rocet shi# meas!res 00.,0 m. ;ow fast does the rocet shi# fly #ast the oser%er? !olution. :e ha%e the rocet shi#’s #ro#er len/th ? and its contracted len/th ?. ts s#eed is /i%en y the form!la v D c = =1 L ?( & ?2 > 1D2
=31>
v D c = =1 L 1( m & 00.,02 m> 1D2 ".L/ c. Examle ""; f a s#aceshi# as meas!red y a stationary oser%er is 37.2 m lon/ as it mo%es #ast the oser%er at a s#eed of .60 c how lon/ is it when at rest on the /ro!nd? !olution. Gi%en the contracted len/th ? and the s#eed v we find the #ro#er len/th ? of the rocet shi# with ? = ? D =1 L v2 D c2> 1D2
=27>
? = 32.7 m D =1 L .602 > 1D2 /." m. Examle "#; At what s#eed does a s#aceshi# ha%e to tra%el #ast the earth for an eartho!nd oser%er to meas!re its len/th to e 10 m if it is 170 m lon/ when at rest? !olution. Gi%en the contracted len/th ? and the #ro#er len/th ? of the s#aceshi# we deri%e its s#eed with v D c = =1 L ?( & ?2 > 1D2
=31>
v D c = =1 L '"( & 'K2 > 1D2 ".( c. Examle "'; A rocet shi# h!rtles thro!/h s#ace at the s#eed of .30 c to an earth oser%er. =a> ind its . => ind its contracted len/th ? as oser%ed y the earth oser%er. =c> ind its N G, the chan/e in its hei/ht in the direction #er#endic!lar to its motion as oser%ed y the earth oser%er. =d> ind its contracted len/th ? as oser%ed y an onoard astrona!t. =e> Calc!late the chan/e N ? in its len/th in the direction of motion as oser%ed y the earth oser%er. !olution. =a> The contraction factor for s#eed v = .30 c is /i%en y = 1 & =1 – %2 D c2 1D2
=3>
= 1 & =1 – .302 c2D c2 1D2 '."K. => The rocet shi#’s contracted len/th ? as oser%ed y the earth oser%er is /i%en y ? = ? =1 L v2 D c2> 1D2
=20>
? = ? =1 L 302 c2D c2> 1D2 ".E ?". =c> "ince hei/ht is #er#endic!lar to the direction of motion there is no chan/e in the rocet shi#’s hei/ht. =d> "ince the onoard astrona!t mo%es with the rocet shi# there is no chan/e in the shi#’s len/th oser%ed y the astrona!t. =e> The chan/e N ? is meas!red y com#arin/ ? and ?
4, N ? ? L ? =1D L 1> .(4 ? L =.(367 L 1> ? L "."E ?". Examle "8; ;ow lon/ does an eartho!nd oser%er meas!re a 10*m*lon/ s#aceshi# to e when it flies #ast the oser%er =a> with a %elocity of .64 c? => with a %elocity of .1 c? !olution. Gi%en the #ro#er len/th ? of the s#aceshi# and its s#eed we deri%e its the contracted len/th ? =a> at s#eed .64 c
? = ? =1 L v2 D c2> 1D2
=20>
? = 10 m =1 L .642 c2D c2> 1D2 '' m. => at s#eed .1 c
? = ? =1 L v2 D c2> 1D2
=20>
? = =10 m> =1 L .12 c2D c2> 1D2 'E.EEEE m. Examle "9; rom an altit!de of 00 m a s#aceshi# mo%es downward toward Earth with a %elocity of .(( c. :hat is the altit!de of the s#aceshi# as meas!red y an astrona!t in the s#aceshi#? !olution. 9ote that the direction of motion is %ertically downward and therefore len/th contraction occ!rs. Gi%en the #ro#er len/th ? which is the s#aceshi#’s altit!de and its s#eed we deri%e the altit!de =contracted len/th> ? as meas!red y the astrona!t ? = ? =1 L v2 D c2> 1D2
=20>
? = =00 m> =1 L .((2 c2D c2> 1D2 KK.E m.
Length Contraction 1.2 1.0
L 0.8 h t g 0.6 n e L 0.4 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 0.9 0.99
1
1
1
1
1
Speed v/c
Fi&!re 2-11. Contraction of a meter stic0 of len&th ( as it moves at spee v c. As its s#eed a##roaches the s#eed of li/ht its len/th a##roaches 'ero. n #rinci#le atv = c ? = . ;owe%er since nothin/ mo%es as fast as li/ht v H c and ? H . Each of the x coordinates after .(( has an additional ( in the ne+t decimal #lace. i/!re 2*11 shows how the len/th of a mo%in/ oect =or distance> a##ears contracted to an oser%er at rest. Contraction increases with s#eed and ecomes e+treme =a##roaches 'ero> at %ery hi/h s#eeds. !t /i%en that
4( no oect can tra%el at the s#eed of li/ht no distance or len/th is red!ced to 'ero. This is another sense in which c is considered the !ltimate s#eed. 9ote that oects only a##ear contracted from the stationary oser%er’s reference frame. rom the reference frame of the mo%in/ oect howe%er nothin/ in that oect contracts. A meter stic on a s#ace shi# tra%elin/ at .( c meas!res 1 meter !st as it does when the shi# is at rest on the /ro!nd. y reason of symmetry the mo%in/ oser%er finds the stationary stic to e shortened. )ore im#ortantly len/th contraction is related to time dilation. $ecall in "ection 2.3.3 the m!on’s o!rney to Earth’s s!rface from its atmos#heric hei/ht of 4*0 m. rom its #oint of %iew the distance it tra%els is short =we call it contracted> eno!/h to co%er efore it decays in nanoseconds. Tho!/h the m!on thins it tra%els only ao!t 6 m in its rief life it act!ally arri%es near the /ro!nd. And the m!ons reach the /ro!nd in lar/e eno!/h 8!antity to e detected. Also to a stationary oser%er a diner on a s#aceshi# tra%elin/ at a hi/h s#eed a##ear to tae more time e./. 2 min!tes to finish a meal that is ser%ed on a #late whose diameter has shr!n to 10 cm. To the tra%elin/ diner the meal taes only 10 min!tes and his #late meas!res 2 cm in diameter. Visual Effects of (enth ontraction or some 0 years after the s#ecial theory had een #ro#osed it was tho!/ht that len/th contraction co!ld e oser%ed or #hoto/ra#hed directly. or instance an astrona!t mo%in/ at hi/h s#eed in a s#aceshi# co!ld tae a #hoto/ra#h of another identical shi# mo%in/ in the o##osite direction that shows its shortened len/th in the direction of motion. ;owe%er there is a difference etween 3bservin len/th contraction and seein len/th contraction. n fact $o/er enrose in 1(0, and Mames Terrell in 1(0( worin/ inde#endently fo!nd that the &orent' contraction of distant oects is more of a rotation than a shortenin/. The followin/ astract of the article n%isiility of the &orent' ContractionB y the American #hysicist Mames Terrell #!lished in the o!rnal Physical eview =116 ##.141L140 1(0(> #ro%ides an o%er%iew t is shown that if the a##arent directions of oects are #lotted as #oints on a s#here s!rro!ndin/ the oser%er the &orent' transformation corres#onds to a conformal transformation on the s!rface of this s#here. Th!s for s!fficiently small s!tended solid an/le an oect will a##earo#ticallythe same sha#e to all oser%ers. A s#here will #hoto/ra#h with #recisely the same circ!lar o!tline whether stationary or in motion with res#ect to the camera. An oect of less symmetry than a s#here s!ch as a meter stic will a##ear when in ra#id motion with res#ect to an oser%er to ha%e !nder/one rotation not contraction. The e+tent of this rotation is /i%en y the aerration an/le =theta L theta’> in which theta is the an/le at which the oect is seen y the oser%er and theta’ is the an/le at which the oect wo!ld e seen y another oser%er at the same #oint stationary with res#ect to the oect. 5ser%ers #hoto/ra#hin/ the meter stic sim!ltaneo!sly from the same #osition will otain #recisely the same #ict!re e+ce#t for a chan/e in scale /i%en y the -o##ler shift ratio irres#ecti%e of their %elocity relati%e to the meter stic. E%en if methods of meas!rin/ distance s!ch as stereosco#ic #hoto/ra#hy are !sed the &orent' contraction will not e %isile altho!/h correction for the finite %elocity of li/ht will re%eal it to e #resent. This effect is nown as the Terrell effect. n order to see the relati%istic effects of len/th contraction we need to slow down the s#eed of li/ht c and tra%el with it. This can e accom#lished y com#!ter sim!lation. n the article "eein/ relati%ity Vis!alisin/ s#ecial relati%ityB =1((7*1(((> Anthony "earle of the Sni%ersity of 9ew "o!th :ales A!stralia descried the %is!al effects in %ideos made from com#!ter sim!lation of reduced c scenes. )ain/ li/ht tra%el 0 mDs we will see the effects of slow li/ht. irst we oser%e the effects on a stationary frame. A streetlam# does not instantly flood the scene with li/ht !t emits s#herical alls of li/ht that /rad!ally e+#and in all directions !ntil they hit a s!rface and mae it %isile to !s. f we fli# the lam# on and off we see the lam# t!rn on first then #arts of the /ro!nd and finally the wall e%en tho!/h the wall is closer to the streetlam# than the /ro!nd. This is eca!se li/ht taes less time to reach o!r eyes %ia the /ro!nd =the shortest distance> than %ia the wall =a lon/er distance>. f we now consider a mo%in/ frame s!ch as a tram mo%in/ at .,66 c which is /amma 2 the effects are different. The tram loos distorted its colors chan/e and their intensity also chan/e. E%en its shadow slants at an !n!s!al an/le. :e can st!dy each effect se#arately distortion as an/!lar aerration color chan/e as the -o##ler effect and ri/htness as the headli/ht effect y eliminatin/ the other effects. f we st!dy distortion the tram not only a##ears shorter !t also sheared and sli/htly ent. "hear is the effect that res!lts in the tram’s sides de%iatin/ from the #er#endic!lar. irst as the tram mo%es at near*li/ht s#eed it !nder/oes the &orent' contraction. Then we see the ac side e%en efore we see the front s!rface
0 res!ltin/ in the shearin/ effect. This means li/ht comin/ from the ac which has a lon/er distance to tra%el to o!r eye m!st ha%e left earlier than li/ht comin/ from the front. :hat we are seein/ is the ima/e of the ac side at an earlier time. :e call this comination of contraction and shear theTerrell effect or the Penrose-Terrell rotation. This effect occ!rs with oects that are small and distant. At close 8!arters howe%er different #arts of the tram a##ear rotated y different amo!nts res!ltin/ in e+treme distortion. &ines #er#endic!lar to an oser%er’s direction of motion will a##ear c!r%ed while lines #arallel will a##ear contracted. The whole tram =or a c!e> will a##ear rotated. The oser%er’s field of %iew a##ears com#ressed in the forward direction and e+#anded in the re%erse direction. Also oects ehind the oser%er will e rotated into si/ht. 9ote that the enrose*Terrell rotation is only an o#tical effect for what really rotates is the ima/e and not the tram. f the oect is a s#here its len/th contraction is e+actly offset y its rotation so that it retains its sha#e to an oser%er. 2.'
Relativistic (oment!m
:e ha%e seen that in relati%istic terms there is no asol!te sim!ltaneity %elocity time or len/th. They are all relati%e to the frame of reference in which they are meas!red. :e now e+amine one of the most im#ortant 8!antities of mechanics which is the %ector called moment!m and determine whether moment!m eha%es in the same way classically and relati%istically. n 9ewtonian mechanics thelinear moment!m of an oect e8!als the #rod!ct of its mass and %elocity as denoted y 2 = mv
=34>
where 2 is the moment!m of an oect m its mass and v its %elocity e+#ressed in the " !nit /.mDs2. 9ote that this relation does not set the !##er limit of the %elocity. A f!ndamental law in 9ewtonian mechanics the conservation of moment!m states that when tw3 3bQects )e.., balls, cars, trains, and s3 3n c3llide, the t3tal m3mentum 31 the system remains c3nstant be13re and a1ter the c3llisi3n, alth3uh the m3mentum 31 each 3bQect may chane, 2r3vided n3 external 13rces c3me int3 2lay. n other words moment!m is conser%ed in isolate systems where the only forces e+erted are internal to the oects which may ran/e from #articles to /ala+ies or their indi%id!al stars. :e note that the notion of collision etween oects m!st e /enerali'ed to all inds of collisionB whether there is contact etween them or not. or e+am#le a #roton and an al#ha #article oth of which ha%e #ositi%e char/e and re#el each other still oey the law of conservation of 3linear4 moment!m 2 2’ whose /eneral e+#ression is m1v1 F m2v2 m1v’ 1 F m2v’ 2
=30>
where the s!scri#ts refer to the two oects in collision the !n#rimed 8!antities re#resent the %al!es efore the collision and the #rimed 8!antities the %al!es after the collision. The law holds for collisions of all inds of oects from hard odies soft odies to s!atomic #articles and /ala+ies. Conser%ation of moment!m m!st e distin/!ished from conser%ation of inetic ener/y or ener/y associated with motion. :hereas moment!m is always conser%ed 0inetic ener&y =KE Z mv2> may or may not. f ener/y is conser%ed as in collidin/ illiard alls we ha%e an elastic collision. After the collision the two odies mo%e off in different directions. f after the collision the two odies stic to/ether as may sometimes ha##en in a train collision we ha%e a perfectly inelastic collision. )ost of the collisions are inelastic collisions as they are neither elastic nor #erfectly inelastic. Consider a one*dimensional head*on collision of two cars in which they stic to/ether. "!ch a collision is #erfectly inelastic and their final %elocity v’ is therefore the same v’ = v’ 1 = v’ 2 as they now form a system. rom the relation =30> we determine the final %elocity in terms of the masses and the initial %elocities i.e. !sin/ moment!m. :e /et the conser%ation of moment!m in a #erfectly inelastic collision as follows m1v1 F m2v2 m1v’ 1 F m2v’ 2
=30>
m1v1 ? m2v2
3")4
: 3m1? m24 v’
;ence the final %elocity v’ of odies in%ol%ed in a #erfectly inelastic collision is v’ =m1v1 F m2v2 > D =m1 F m2>
01 v’ : 3m1v14 8 3m1 ? m24 ? 3m2v24 8 3m1 ? m24
3"*4
Examle "63 After a head*on collision etween a f!ll*si'ed car with a mass 17 / and an initial %elocity of ( mDh and a small car with a mass of ,0 / and a s#eed of 12 mDh mo%in/ in the o##osite direction they stic to/ether. =a> :hat is the final %elocity of the two cars immediately after the collision? => :hat is the acceleration of each car? !olution3 =a> This is an e+am#le of a #erfectly inelastic collision in which the two odies stic to/ether and mo%e as one. The final %elocity v’ of the system is v’ =m1v1> D =m1 F m2> F =m2v2> D =m1 F m2>
=37>
&et !s first note that the mass of the small car is only half that of the lar/e one. ;encem1 2m2 and v’ =2m2 + ( mDh> D =2m2 F m2> F =m2 + L 12 mDh> D =2m2 F m2> Eliminatin/ all m2 and the common denominator and notin/ that the small car mo%es in the ne/ati%e direction with res#ect to the i/ car we /et the final %elocity v’ 2 =( mDh> L 12 mDh + - km&h. The #ositi%e si/n indicates that the system is mo%in/ in the direction of motion of the i/ car. => The chan/e in s#eed for the i/ car is Nv' = F 6 mDh L ( mDh L / km&h. And the chan/e in s#eed for the small car is /i%en y Nv( = F 6 mDh L =L 12 mDh> + 'L km&h. Gi%en that acceleration is defined as Nv & Nt and that Nt is identical for oth cars it is clear that the i/ car has ac8!ired an acceleration e8!al to only one third the acceleration of the small car. assen/ers in the small car who e+#erience an acceleration three times that of the i/ car are s!ected to /reater forces and /reater #otential for in!ries. Consider now a case of elastic collision. f a all dro##ed from hei/ht h reo!nds to the same hei/ht h’ that is h = h’ we ha%e an elastic collisionR in which case oth the inetic ener/y and the moment!m are conser%ed. f h R h’ we ha%e an inelastic collision in which the moment!m is conser%ed !t the inetic ener/y is not. The loss of inetic ener/y is con%erted into other forms s!ch as thermal ener/y. Ssin/ an elastic collision etween two identical #articles =or alls> we can in%esti/ate the effect of relati%ity on moment!m. ma/ine two railroad flatcarsS and S’ r!nnin/ toward each other on #arallel tracs. As they a##roach each other all A is thrown with s#eed v from S toward S’ #er#endic!lar to its motion. At the same time an identical all is la!nched with the same s#eed v from S’ toward S also #er#endic!lar to its motion. ;a%in/ tra%eled the same distance y the two alls collide elastically and reo!nd efore ein/ ca!/ht y their res#ecti%e oser%ers. all A with mass m has th!s tra%eled 2 y in S frame and all with identical mass m has tra%eled 2 y’ in S’ frame. "ince there is no len/th contraction in the y direction the distances tra%eled y the alls are the same y = y’. Th!s symmetry e+ists etween the two frames of reference. ;owe%er if the sit!ation is analy'ed from the #oint of %iew of one of the mo%in/ frames e./. S’ time dilation taes effect. As %iewed from S’ the two e%ents of throwin/ and catchin/ the all occ!r in the same frame and is meas!red y the same cloc. They occ!r in the 2r32er time and no time dilation effect emer/es. !t in S these two e%ents occ!r in two different locations and the time meas!red in S is related to the #ro#er time y the time dilation form!la Nt Nt D =1 L v2 D c2> 1D2. ;ence altho!/h the alls tra%el the same distance y = y’ the times are different and the y com#onents of their s#eeds m!st also e different. The %elocity of A in S is /i%en y
02
v A 2 y & Nt v whereas the %elocity of in S’ is v D 2 y & Nt =1 L v2 D c2> 1D2 v=1 L v2 D c2> 1D2.
"ince the %elocities of A and are not the same their momenta m!st differ. @et conser%ation of moment!m is so f!ndamental in classical and relati%istic mechanics it m!st e #reser%ed. After all y the first #ost!late of s#ecial relati%ity all laws of #hysics are the same in all inertial frames. Conse8!ently moment!m m!st e redefined to tae into acco!nt the effect of relati%ity. "ince the y com#onents of the %elocities of A and differ y the factor =1 L v2 D c2> 1D2 we can /enerali'e the relati%istic moment!m definition as follows Relativistic moment!m pJ = mv 8 31 B v2 8 c24 182
or
*mv
3"+4
" !nit /.mDs As has een oser%ed in other relati%istic effects the factor indicates that at e%eryday s#eed v UU c = close to 1> relati%istic moment!m red!ces to classical moment!m. Also the moment!m 2 increases as the %elocity v increases !t v ne%er reaches c. As the moment!m increases an e%er lar/er force is needed to accelerate the #article f!rther. An infinite amo!nt of ener/y wo!ld e re8!ired to achie%e the s#eed of li/ht main/ it the !##er limit of all s#eeds. Examle ":3 :hat is the moment!m of an electron that tra%els at .(((((((((7c in terms of mc? !olution3 :e a##ly the form!la for relati%istic moment!m. 2 = mv D =1 L v2 D c2> 1D2
=3,>
=m> .(((((((((7 c D =1 L .(((((((((72 c2D c2> ."L x '" mc. Examle #53 :hat is the moment!m of an electron that tra%els at .((c if its mass is (.1(4 + 1*31 /? !olution3 :e a##ly the form!la for relati%istic moment!m. 2 = mv
=3,>
=7.,,,> =(.1(4 + 1*31 / > =.(( c> =7.,,,> =(.1(4 + 1*31 / > =2.(67( + 1, mDs> '.E x '"%(' k.m&s. 2.)
Relativistic (ass
5ne of the on*/oin/ contro%ersial iss!es ao!t s#ecial relati%ity is the inter#retation of the conce#t of relativistic mass. n the relati%istic moment!m form!la ao%e m is inter#reted as the mass of the oect as meas!red y an oser%er at rest i.e. the rest mass m". The rest mass is also called invariant mass. Th!s we can write the moment!m e8!ation as 2 = m"v or m"v D =1 L v2 D c2> 1D2
=3(>
!t the moment!m form!la 2 = mv is also inter#reted to mean that as the %elocity v increases the mass m increases i.e. mass is %elocity*de#endent as relati%istic mass is s!##osed to e as shown 2 = m"v D =1 L v2 D c2> 1D2 mv = m"v D =1 L v2 D c2> 1D2
03 -i%idin/ oth sides y v we otain the e+#ression for the relativistic mass m = m5 8 31 B v2 8 c24 182
3$D4
The form!la =4> im#lies that relati%istic mass is related to rest mass y the factor and increases with the s#eed v. This moti%ates some a!thors to associate the mass in 9ewton’s moment!m 2 = mv with relati%istic mass main/ it a /eneral relation for all masses. t m!st e noted that at v UU c mass increase is infinitesimally small and therefore im#erce#tile. Snder this inter#retation as the %elocityv a##roaches c the =relati%istic> mass m tends to infinity as indicated y relation =4>. A constant force a##lied to the mass of an oect mo%in/ with e%er*increasin/ s#eed #rod!ces e%er*decreasin/ acceleration as v j c. n other words if v c the denominator in relation =4> ecomes 'ero and the mass m wo!ld e infinite. Also it wo!ld tae infinite ener/y to /enerate the s#eed v c an im#ossile feat. This is another reason to consider li/ht as the !ltimate s#eed. Examle #13 :hat is the relati%istic mass of a all tra%elin/ at .7c if its rest mass is .4 /? !olution3 A##lyin/ e8!ation =4> we /et m = m" D =1 L v2 D c2> 1D2
=4>
.4 / D =1 L .72 D c2> 1D2 ".- k. )any e/innin/ #hysics te+toos as well as #o#!lar oos on s#ecial relati%ity incl!din/ wors y "te#hen ;awin/ = A brie1 hist3ry 31 time> , $ichard eynman = ?ectures 3n 2hysics> and rian Greene =>he eleant universe> introd!ce relati%istic mass increase at least as a he!ristic. )ichael owler =1((6> too defends relati%istic mass increase th!s -ecidin/ that masses of oects m!st de#end on s#eed lie this = = m&)'%v ( &c( '&( addition mine> seems a hea%y #rice to #ay to resc!e conser%ation of moment!m ;owe%er it is a #rediction that is not diffic!lt to chec y e+#eriment. The first confirmation came in 1(, meas!rin/ the mass of fast electrons in a %ac!!m t!e. n fact the electrons in a color TV t!e are ao!t half a #ercent hea%ier than electrons at rest and this m!st e allowed for in calc!latin/ the ma/netic fields !sed to /!ide them to the screen. )!ch more dramatically in modern #article accelerators %ery #owerf!l electric fields are !sed to accelerate electrons #rotons and other #articles. t is fo!nd in #ractice that these #articles ecome hea%ier and hea%ier as the s#eed of li/ht is a##roached and hence need /reater and /reater forces for f!rther acceleration. Conse8!ently the s#eed of li/ht is a nat!ral asol!te s#eed limit. articles are accelerated to s#eeds where their mass is tho!sands of times /reater than their mass meas!red at rest !s!ally called the krest mass.k =owler 1((6> Another te+too states that Wwhen we consider the motion of a system of #articles =s!ch as /as molec!les in a mo%in/ container> the total mass of the system m!st e taen to e the s!m of the relati%istic masses of the #articles rather than the s!m of their rest masses.B ="ears emansy and @o!n/ 1(,7 #. ,7> @et in the ne+t #ara/ra#h "ears et al. =1(,7> admits the #itfalls of !sin/ the relati%istic mass. or e+am#le 9ewton’s second law cannot e /enerali'ed y its relati%istic co!nter#art as T = mrel a nor can the relati%istic inetic ener/y of a #article e /enerali'ed as KE Z mrel v(. t is th!s etter to consider the moment!m e+#ression in E8!ation =3(> as a /enerali'ed definition of moment!m withm constr!ed as the rest mass which is a #ro#erty of the #article on the same #ar with char/e inde#endent of motion. An increasin/ n!mer of te+toos #!lished d!rin/ 2*20 ha%e not relied on the conce#t of relati%istic mass. The reasons are many. n 1(4, Einstein in a letter to &incoln arnett wrote t is not /ood to introd!ce the conce#t of the mass = m&)'%v( &c( '&( of a ody for which no clear definition can e /i%en. t is etter to introd!ce no other mass than the rest mass m. nstead of introd!cin/ it is etter to mention the e+#ression for the moment!m and ener/y of a ody in motion. Einstein himself was ami%alent ao!t the term relati%istic mass which he almost ne%er !sed altho!/h in a #a#er of 1(0 H-oes the inertia of a ody de#end on its ener/y content?B> he stated that the inertia of a ody
04 de#ended on its ener/y. This is in accord with what some scientists say when they #o#!lari'e the to#ic for the /eneral #!lic. n "te#hen ;awin/ in A brie1 hist3ry 31 time we find keca!se of the e8!i%alence of ener/y and mass the ener/y which an oect has d!e to its motion will add to its mass.k and $ichard eynman in >he character 31 2hysical law writes kThe ener/y associated with motion a##ears as an e+tra mass so thin/s /et hea%ier when they mo%e.k n the 1(2s #hysicists a!li Eddin/ton and orn freely !sed relati%istic mass in their te+toos. Gis =1((6> #roaly est s!mmari'es the conf!sin/ state of affairs ao!t relati%istic mass and rest mass in the followin/ #assa/e There is sometimes conf!sion s!rro!ndin/ the s!ect of mass in relati%ity. This is eca!se there are two se#arate !ses of the term. "ometimes #eo#le say kmassk when they mean krelati%istic massk mr !t at other times they say kmassk when they mean kin%ariant massk m". These two meanin/s are not the same. The in%ariant mass of a #article is inde#endent of its %elocity v whereas relati%istic mass increases with %elocity and tends to infinity as the %elocity a##roaches the s#eed of li/ht c. They can e defined as follows mr EDc2 m s8rt=E2Dc4 – #2Dc2> where is ener/y 2 is moment!m and c is the s#eed of li/ht in a %ac!!m. The %elocity de#endent relation etween the two is mr m Ds8rt=1 – %2Dc2> 5f the two the definition of in%ariant mass is m!ch #referred o%er the definition of relati%istic mass. These days when #hysicists tal ao!t mass in their research they always mean in%ariant mass. The symol m for in%ariant mass is !sed witho!t the s!scri#t . Altho!/h the idea of relati%istic mass is not wron/ it often leads to conf!sion and is less !sef!l in ad%anced a##lications s!ch as 8!ant!m field theory and /eneral relati%ity. Ssin/ the word kmassk !n8!alified to mean relati%istic massis wron/ eca!se the word on its own will !s!ally e taen to mean in%ariant mass. or e+am#le when #hysicists 8!ote a %al!e for kthe mass of the electronk they mean its in%ariant mass. y 1(0s with the rise of #article #hysics more and more scientists e/an to sh!n the term relati%istic mass and whene%er they referred to mass they meant in%ariant mass. Certainly it maes no sense to refer to the rest mass of a #hoton as 'ero since #hotons are ne%er at rest. hotons are sim#ly massless. Altho!/h #hotons ha%e moment!m which is related to total ener/y the conce#t of relati%istic mass has no si/nificance to #hotons. 2.*
Relativistic Ener&y
f there is one e8!ation that dominates the twentieth cent!ry amon/ #hysicists and in the minds of the #!lic it is the mass*ener/y e8!i%alence relation mc(. "im#le as it is Einstein’s form!la has far*reachin/ conse8!ences. t !shers in the atomic a/e y showin/ that a small amo!nt of mass is e8!i%alent to an enormo!s amo!nt of ener/y. The tric is to find the technolo/y to effect the con%ersion. 5ne #eacef!l a##lication of this insi/ht is e%ident in n!clear #ower #lants where small amo!nts of !rani!m #rod!ce y fission tremendo!s heat ener/y that is !sed to /enerate electricity !sin/ con%entional t!rines. The s!n too emits %ast 8!antities of ener/y y !rnin/ tiny amo!nts of its own mass. Con%ersely ener/y has een con%erted into electrons and #rotons in the laoratory. Two of the most f!ndamental conce#ts in #hysics are conser%ation of ener/y and conser%ation of moment!m. :e see these laws of conser%ation at wor in collisions and other chemical and #hysical #henomena. :e shall see the same laws a##lied in deri%in/ the relationshi# etween mass and ener/y. n what follows Mones and Childers =1((3 #. 7,> introd!ce Einstein’s famo!s e8!ation y way of moment!m. )a+well’s electroma/netic theory #redicts that li/ht e+erts #ress!re when it stries an oect. This #ress!re im#lies that li/ht rays carry moment!m which is #ro#ortional to the ener/y carried y the li/ht rays. The moment!m is e+#ressed y 2 = & c where is the ener/y and c the s#eed of li/ht.
=41>
00 ma/ine in a tho!/ht e+#eriment y Einstein a closed rectan/!lar o+ of len/th ? and mass s!s#ended y a wire to allow free frictionless motion. At an instant of time a series of flash!ls on one end of the o+ fires a flash of li/ht ca!sin/ the o+ to recoil y a certain distance x with an e8!al !t o##osite moment!m to that of the li/hts so that we ha%e 2b3x v 2liht & c
=42>
where v is the %elocity of the o+’s recoil and the ener/y of the li/ht. The li/ht tra%els d!rin/ time t ? & c to reach the o##osite end of the o+ where it is asored while the o+ mo%es a distance L x in its recoil. At the end of the li/ht’s tra%el the system ret!rns to its rest state th!s conser%in/ moment!m. -!rin/ the mo%ement of the o+ its center of mass m!st stay stationary. or this to e the case there m!st e an com#arale shift ca!sed y an e8!i%alent mass m associated with the li/ht s!ch that the mo%ement of m o%er the distance ? is com#ensated y the mo%ement of the o+’s mass o%er the distance L x as indicated y m? x
=43>
"ince the moment!m of the o+ and that of the li/ht are e8!al we can sol%e for the %elocity of the o+ y !sin/ relation =42> v & c
=44>
Gi%en that the ela#sed time t is the time d!rin/ which the li/ht tra%els the distance ? we can deri%e the distance tra%eled y the o+ as follows x = vt = = & c> = ? & c>
=40>
"!stit!tin/ the %al!e of x in e8!ation =43> in e8!ation =40> we /et m? & = = & c> = ? & c> Eliminatin/ ? and from oth sides we deri%e the mass-ener&y e!ivalence relation as E : mc2
3$)4
" !nit M. n the ao%e Mones and Childers #assa/e the #rinci#le of conser%ation of moment!m #lays a cr!cial role. The mass*ener/y relation e8!ation says that ener/y is e8!al to mass times the s#eed of li/ht s8!ared. irst it means that ener/y and mass are e8!i%alent i.e. they are !st two forms of the same thin/. "econd since the s#eed of li/ht is a h!/e n!mer a small amo!nt of mass can #rod!ce an enormo!s amo!nt of ener/y. :e ha%e seen this in n!clear reactions. Examle #23 f a 1*/ /ra#efr!it co!ld e con%erted directly into electroma/netic ener/y how m!ch ener/y is otained in M and in )eV? !olution3 :e note that no oect can e com#letely con%erted into ener/y. A##lyin/ the relation =46> we otain the ener/y that co!ld theoretically e released e%en tho!/h c!rrent technolo/y falls far short of the feat mc2 : =1. /> =3. + 1, mDs>2 E."" x '"'- I. or since 1 )eV 1.62 + 1*13 M
=46>
06
(. + 116 M D 1.62 + 1*13 M .-((E eU.
This is eno!/h ener/y to r!n a 1*: motor for 2.,0 million years. Gi%en that 1 ilowatt*ho!r is e8!al to 1 MDs + 6sDmin + 6 minDh 3.6 )M we /et = (. + 116 M D 3.6 + 1, 2.0 + 11 :Dh. "ince there are ro!/hly 24 hDday + 360 dayDy ,76 hDy this ener/y emitted at 1 :Dh will last 2.0 + 11 :Dh D ,76 h 2.,0 + 16 years. Cinetic Enery Kinetic ener/y is ener/y #rod!ced y an oect in motion. Classically it is V 1D2 mv( that is the inetic ener/y of an oect of mass m mo%in/ with s#eed v. This form!la wors only where v c. ;ence !st as Galilean relati%ity is a s#ecial case of s#ecial relati%ity so is classical inetic ener/y a s#ecial case of relati%istic inetic ener/y. :e will now redefine the conce#t of inetic ener/y in order to accommodate oects mo%in/ at slow as well as fast s#eeds. $ecall the relati%istic mass relation m = m" D =1 L v2 D c2> 1D2
=4>
y a##lyin/ the inomial theorem e+#andin/ we /et m m" =1 F Z v2 & c2 F 3D, =v2 & c(>2 F W L 1> )!lti#lyin/ oth sides y c( and ee#in/ only the first two terms of the series eca!se the remainin/ terms are too small to mae a difference yields mc2 m" c2 F Z m" v2 9otin/ that mc2 the left*hand term is total ener/y the first term of the ri/ht*hand side is ener/y related to the rest mass and the second term is its classical V we ha%e the total ener&y e8!ation E : m5 c2 ? CE
3$*4
$earran/in/ the terms we can say that the 0inetic ener&y of an oect is the ifference etween its total ener&y E an its rest ener&y m5 c2 CE = mc2 B m5 c2
or CE = E B m5 c2
3$+4
"ince m = m" D =1 L v2 D c2> 1D2 = e8!ation 4> we can also write the inetic relation as CE = m5c2 8 31 B v2 8 c24 182 B m5 c2
3$64
f the oect is at rest its inetic ener/y is 'ero and its total ener/y ecomes itsrest ener&y E 5 : m5 c2
3'D4
And if we !se the relati%istic mass e8!ation =4> the total ener/y can e e+#ressed y E = m5 c2 8 31 B v2 8 c24 182
or E = * m5 c2
3'14
This and #re%io!s form!las im#ly e8!i%alence of mass and ener/y. The intercon%ersion etween mass and ener/y has een am#ly demonstrated in the laoratory and in n!clear #ower #lants. Electricity is /enerated in n!clear #lants with ener/y released when the n!clei of !rani!m f!el s#lit into two n!clei and one or two ne!trons which in t!rn create more fission and more ne!trons in a chain reaction. The #rocess res!lts in a loss of small amo!nts of !rani!m mass and a release of immense inetic ener/y which t!rns water into steam which dri%es t!rines to /enerate electricity. n a re%erse #rocess electroma/netic radiation can e con%erted
07 into matter s!ch as electrons in the laoratory. Also in #air*#rod!ction a #hoton is con%erted into an electron and a #ositron with inetic ener/ies released. Examle #"3 ind the total ener/y and inetic ener/y in M and in )eV of an electron that mo%es with a s#eed of .7 c and whose mass is (.11 + 1*31 /. !olution3 A##lyin/ e8!ation =01> we calc!late the total ener/y = m" c2 D =1 L v2 D c2> 1D2
=01>
= =(.11 + 1*31 /> =3. + 1, mDs>2 D =1 L .72 c2 D c2> 1D2 ,.2 + 1*14 M + 1.4 ''.L x '"%' I. 11.4, + 1*14 M D 1.6 + 1*1 M ".K'K eU. The inetic ener/y is e8!al to the difference etween the total ener/y and the rest ener/y V = L m" c2
=4,>
11.4, + 1*14 M L ,.2 + 1*14 M /.(L x '"%' I . 3.2, + 1*14 M D 1.6 + 1*1 M ".(" eU. Ener/y is often e+#ressed in terms of )eV and mass is e+#ressed in )eVDc2 =witho!t carryin/ o!t the di%ision> or in /. "ince mass and ener/y are !st two faces of the same coin the intercon%ersion etween them maes it #ossile to e+#ress them in either !nit of meas!rement. or e+am#le 1 / 0.6( 0 + 12( )eV The followin/ tale lists the rest mass and rest ener/y of a n!mer of common #articles. (asses an Rest Ener&ies of Common Particles an Itoms Particle
(ass m 30&4
hoton 9e!trino Electron or #ositron e )!on 6W i meson =ne!tral> : " i meson =char/e> : W ion =F> Atomic mass !nit u roton 2 9e!tron n -e!teron d or 2 G Triton Al#ha #article X or 4 Ge ;ydro/en atom -e!teri!m atom
(.1( 3,( 7 + 1*31 1.,,3 333 + 1*2, 2.40 (0 + 1*2, 2.4,, 0 + 1*2, 2.416 0 + 1*2, 1.66 04 + 1*27 1.672 623 + 1*27 1.674 (2( + 1*27 3.343 0,4 + 1*27 0.7 307 + 1*27 6.644 72 + 1*27 1.673 034 + 1*27 3.344 4(7 + 1*27
Rest Ener&y mc2 3(eK4 .01 ((( 10.60, 134.(64 13(.06( 130.06 (31.4(4 (3,.272 (3(.060 1,07.612 2,,.(2 3727.41 (3,.7,3 1,76.12
Tale 2-$. The tale of masses and ener/ies shows the corres#ondence etween them A lar/er mass corres#onds to a lar/er rest ener/y. These %al!es facilitate the calc!lations of relati%istic ener/y.
0, n sit!ations where the moment!m and ener/y are nown !t not the %elocity it is con%enient to !se a total ener/y relation that does not de#end on s#eed. Gi%en mc2 and 2 mv where m = m " D =1 L v2 D c2> 1D2 we s8!are oth sides of the total ener/y e8!ation and write the total ener&y relation in terms of moment!m 2 m2c4 m2c2=c2 F v2 L v2> 2 m2c2v2 F m2c2=c2 L v2> "!stit!tin/ m" D =1 L v2 D c2> 1D2 for m in the second term we otain 2 22c2 F m"2c4=1 L v2 & c2> D =1 L v2 & c2> E 2 : m52c$ ? 2c2
3'24
E 2 : E 52 ? 3 c42
3'"4
or
The form!la =02> says that when the oect is at rest its moment!m is 'ero and the e8!ation ecomes the familiar m"c2. And when its total ener/y is m!ch /reater than its rest ener/y HH mc 2 the first term is ne/li/ile and can e omitted to yield thehi&h-ener&y e!ation E : c
3'$4
The relation =04> also shows that since the total ener/y de#ends solely on moment!m the #article m!st ha%e 'ero mass. "!ch is the case with #hotons ne!trinos and /ra%itons. 5f the last two ne!trinos ha%e een identified e+#erimentally whereas /ra%itons are hy#othesi'ed !t not confirmed. !rthermore e8!ation =02> shows that total ener/y %aries in direct #ro#ortion to moment!m. eca!se ofc a small chan/e in moment!m res!lts in an enormo!s chan/e in total ener/y. rom relation =03> we can deri%e a !sef!l form!la to find the moment!m in terms of total ener&y an mass 2 "2 F = 2c>2 : 3 E 2 B E 52 4182 8 c
=03>
3''4
Another !sef!l relation concerns %elocity which can e deri%ed from ener/y alone. "ince " " D =1 – v2 & c2> 1D2 we deri%e velocity in terms of c th!s " D =1 – v2 & c2> 1D2 "8!arin/ oth sides we /et 2 "2 D =1 – v2 & c2> 1 – v2 & c2 "2 D 2 1 – "2 D 2 v2 & c2 v2 c2 =1 – "2 D 2 > v : 31 – E 52 8 E 2 4 182 c
3')4
Examle ##3 :ith a mass of (3,.27 )eVDc2 a #roton has a inetic ener/y of 21 )eV. ind the #roton’s total ener/y moment!m and s#eed. !olution3 Knowin/ rest mass and inetic ener/y we !se e8!ation =47> to find the total ener/y
0( m" c2 F V
=47>
=(3,.27 )eVDc2 > c2 F 21 )eV ''L.(K eU . The moment!m may then e calc!lated from e8!ation =00> 2 = 2 L "2 >1D2 D c
=00>
2 = =114,.27 )eV>2 L =(3,.27 )eV>2 >1D2 D c --'.E eU & c. To find the s#eed we !se e8!ation =06> v =1 – "2 D 2 > 1D2 c
=06>
v =1 – =(3,.27 )eV>2 D =114,.27 )eV>2 > 1D2 c ".K- c. Examle #'3 A #roton has a inetic ener/y e8!al to its rest ener/y. ind the #roton’s moment!m and s#eed. !olution3 Knowin/ rest ener/y ="ee Tale 2*4 ao%e> and inetic ener/y are e8!al we !se e8!ation =47> to find the total ener/y m" c2 F V
=47>
= (3,.27 )eV F (3,.27 )eV 2 + (3,.27 )eV. The moment!m is 2 = 2 L "2 >1D2 D c
=00>
2 (3,.27 )eV =4 L 1>1D2 D c '-( eU & c. And the s#eed is v =1 – "2 D 2 > 1D2 c
=06>
v =1 – (3,.272 )eV D =2 + (3,.27>2 )eV> 1D2 c v =1 L 1 D 4> 1D2 c ".L-- c. Examle #83 f an electron mo%es with s#eed ., c find its total ener/y its inetic ener/y and its moment!m. !olution3 Ssin/ Tale 2*4 we /et m" c2 .01 ((( )eV. To find the total ener/y we !se e8!ation =01> = m" c2 D =1 L v2 D c2> 1D2
=01>
= .011 )eV D =1 L .,2 D c2> 1D2 ".L( eU. ts inetic ener/y is /i%en y e8!ation =4,> V = L m" c2 V = .,02 )eV L .011 )eV "./' eU . The moment!m is then /i%en y e8!ation =00>
=4,>
2 = 2 L "2 >1D2 D c
=00>
6
2 =.,022 )eV L .0112 )eV >1D2 D c ".-L' eU & c. :e can also otain the moment!m y !sin/ the relati%istic moment!m form!la =3,>. 2 = mv
=3,>
2 = mv D =1 L v2 D c2> 1D2
=3,>
=.011 )eV> = ., c> D =1 L .,2 c2D c2> ".-L' eU & c. 2.+
Relativistic #oppler Effect
-escried y the A!strian #hysicist Christian -o##ler =1,0*1,03> in 1,42 the -o##ler effect on so!nd wa%es occ!rs when we hear a hi/h #itch of a fire tr!c’s siren a##roachin/ !s and a lower #itch as the tr!c recedes from !s. This is eca!se the so!nd wa%es are com#ressed to a hi/her fre8!ency =or shorter wa%elen/th> in front of the tr!c than they are ehind it. The same effect is felt when the wa%e so!rce is stationary and the oser%er is mo%in/ toward or away from it or oth so!rce and oser%er are mo%in/ toward or away from each other. n 1,4( the rench #hysicist Armand ;i##olyte &o!is i'ea! =1,1(*1,(6> a##lied the shift to li/ht wa%es. The #oppler effect =also nown as #oppler-Fi9ea! effect> a##lies to all wa%es so!nd electroma/netic wa%es e%en water. t is !sed in radar to meas!re the s#eed of cars and air#lanes. "i/nificant differences e+ist etween the -o##ler effect for so!nd and that for li/ht. irst if an oect a##roaches an oser%er at /reater than the s#eed of so!nd she will not hear anythin/ !ntil the sonic oom hits. n contrast nothin/ tra%els faster than li/ht. Then the -o##ler effect for li/ht de#ends entirely on the %elocity of the emitter relati%e to that of the recei%er. The effect for so!nd de#ends on the %elocities of oth emitter and recei%er relati%e to air. inally the -o##ler effect occ!rs when the oect is in the line of si/ht only. f the li/ht si/nal tra%els alon/ a line #er#endic!lar to the line of si/ht we ha%e atransverse #oppler shift. "!##ose a train is tra%elin/ alon/ a strai/ht railroad trac far away from yo!. @o! wo!ld not hear any -o##ler shift for the so!nd. !t for li/ht yo! wo!ld see a fre8!ency decrease e8!al only to the time dilation factor !t no effect d!e to the motion of the train. n the case of li/ht emission when the oser%er mo%es toward the li/ht so!rce =for e+am#le a star or a /ala+y> or %ice %ersa alon/ the line of si/ht the li/ht’s fre8!ency intensifies and shifts toward the l!e =or shorter wa%elen/th> end of the s#ectr!m. This is called a l!eshift. :hen the star or /ala+y mo%es away from the oser%er the li/ht emitted shifts toward the lower fre8!ency =lon/er wa%elen/th> or red end of the s#ectr!m and we ha%e a reshift. As stated ao%e there is no -o##ler effect if the li/ht is #er#endic!lar to the line of si/ht. n this case there is only the time dilation effect !t no -o##ler effect. Astronomers !se the -o##ler effect as an im#ortant tool to determine the raial velocities =s#eeds of the celestial odies in their motion #arallel to o!r line of si/ht> and motions of #lanets stars and /ala+ies. "ince most /ala+ies and stars are nown to e redshifted the concl!sion is that they are mo%in/ away from the Earth and hence that the !ni%erse is e+#andin/. The e8!ation that e+#resses the nonrelativistic #oppler shift is sim#le D 8 DD : v 8 c
3'*4
where N ; N ; L N ; is the difference in wa%elen/th etween the nown rest wa%elen/th N ; and that oser%ed N ; and v the radial %elocity. This e8!ation wors only ifv c. Examle #93 n a st!dy of the s#ectr!m of the star Ve/a a hydro/en line ;p is oser%ed to ha%e a wa%elen/th of 606.200 nm. This line has a normal wa%elen/th of 606.2,0 nm. ind the star’s wa%elen/th shift and radial %elocity. !olution3 The star’s wa%elen/th shift is N ; L N ; 606.200 nm L 606.2,0 nm L "."/" nm. :e deri%e its radial %elocity from e8!ation =07>
61 N ; D ; v D c v =N ; D ; > c =L .3 nm D 606.2,0 nm> + 3. + 1, mDs L '/.K m&s. The min!s si/n indicates that Ve/a is mo%in/ toward !s. At %ery hi/h %elocity we enco!nter a more com#licated relativistic #oppler shift eca!se of time dilation. ma/ine a laser eam mo!nted on a s#aceshi# mo%in/ at s#eed v toward an Earth oser%er. The laser has fre8!ency 1 in the reference frame in which the shi# is at rest. -!rin/ each time #eriod> of the li/ht eam the s#aceshi# has tra%eled the distance v> and the li/ht has co%ered c> . -!e to time dilation > a##ears to r!n slower to the oser%er on Earth than to the astrona!ts in the s#aceshi# who meas!re the #ro#er #eriod> . The #eriod > etween emissions is > 1 D 1 . As the s#aceshi# a##roaches the oser%er its motion com#resses the wa%elen/th of the li/ht increasin/ the fre8!ency ein/ oser%ed. $ecallin/ that the #eriod> > the wa%elen/th is th!s ; c> L v> =c – v> > D =1 L v2 D c2> 1D2 The wa%e e8!ation /i%es the fre8!ency 1 meas!red y the Earth oser%er 1 c D ; or !sin/ 1 1 D > 1 1 =1 L v2 D c2> 1D2 D 1 L v D c E+#andin/ the s8!are root of the n!merator and rewritin/ the denominator we ha%e 1 1 P=1 F v D c> =1 L v D c>Q 1D2 D P=1 L v D c> =1 L v D c>Q 1D2 "im#lifyin/ we /et the fre8!ency of the li/ht a##roachin/ the oser%er f : f D L31 ? v 8 c4 8 31 B v 8 c4M 182
3approachin&4
3'+4
n this case as the so!rce is mo%in/ toward the oser%er its fre8!ency increases =or e8!i%alently its wa%elen/th decreases>. f the so!rce and the oser%er mo%e a#art the s#eed v ecomes ne/ati%e and the fre8!ency decreases as shown elow. f : f D L31 B v 8 c4 8 31 ? v 8 c4M 182
3recein&4
3'64
A/ain what matters in the relati%istic -o##ler effect for li/ht is the relati%e s#eed of the so!rce and oser%er. Examle #63 A rocet shi# is tra%elin/ toward a distant star that emits yellow li/ht at the fre8!ency of 0. + 114 ;' meas!red in the star’s rest frame. f the shi#’s %elocity relati%e to the star is .10c what fre8!ency does the rocet shi#’s crew oser%e for the star’s li/ht? !olution3 :e a##ly form!la =0,> 1 1 P=1 F v D c> D =1 L v D c>Q 1D2
=0,>
0. + 114 ;' P=1 F .10 c D c> D =1 L .10 c D c>Q 1D2 .L( x '"' Gz. Con%erted to wa%elen/th the ao%e fre8!ency is ; c D 1 ; 3. + 1, mDs D 0.,2 + 114 ;' '- nm.
62 This corres#onds to /reen li/ht. y meas!rin/ the -o##ler shift of the li/ht of stars and /ala+ies astronomers ha%e determined that its fre8!ency decreases as com#ared with line s#ectra maintained in the laoratory. That is it has redshifted to the lon/er wa%elen/th end of the s#ectr!m. Th!s stars and /ala+ies are mo%in/ away from Earth and the !ni%erse is e+#andin/. !rthermore the oser%ed red shift seems #ro#ortional to their distance from !s i.e. the farther away the celestial oects are the faster they recede from Earth in accord with ;!le’s &aw. or red shifts lar/er than a few tenths the -o##ler form!la =07> /i%es the incorrect answer. or e+am#le a redshift of 3. a##lied to a 8!asar wo!ld indicate that the 8!asar is mo%in/ away at three times of s#eed of li/ht in %iolation of the second #ost!late of s#ecial relati%ity. Therefore the redshift e+#ression has to e modified as follows ) : D 8 DD : L31 ? v 8 c4 8 31 B v 8 c4M 182 B 1
3)D4
where z is the symol for the redshift. As an e+am#le astronomers oser%ed a redshift of 3.2 in the series of hydro/en emissions from a 8!asar that were -o##ler shifted into the red re/ion of the s#ectr!m. A##lyin/ the form!la =6> we /et z N ; D ; P=1 F v D c> D =1 L v D c>Q 1D2 L 1 3.2 P=1 F v D c> D =1 L v D c>Q 1D2 L 1 =1 F v D c> D =1 L v D c> =3.2 F 1>2 17.6 =1 F v D c> =17.6> =1 L v D c> 1,.6 v D c 16.6 v D c .,(. Th!s a redshift of 3.2 means that the 8!asar is mo%in/ away from Earth at the s#eed of .,(c.
=6>
63
3. General $elati%ity
:e ha%e seen that s#ecial relati%ity a##lies to laws of #hysics in inertial or non*acceleratin/ reference frames. n 1(10 Einstein introd!ced the /eneral theory of relati%ity to accommodate all reference frames incl!din/ acceleratin/ ones. asically the /eneral theory of relati%ity is a theory of /ra%ity. :hereas 9ewtonian #hysics foc!ses on force /eneral relati%ity foc!ses on motion. Tho!/h oth a##roaches yield satisfactory res!lts at low e%eryday s#eeds relati%ity alone is ade8!ate to acco!nt for the eha%ior of oects mo%in/ at hi/h s#eeds and ha%in/ lar/e /ra%itational fields. :e will first disc!ss the classical dichotomy inertial mass* /ra%itational mass to see how it leads to the #rinci#le of e8!i%alence. :e will see the #rinci#le of co%ariance which to/ether with the e8!i%alence #rinci#le lays the fo!ndation of the /eneral theory of relati%ity. inally we will disc!ss the %ario!s e+#erimental e%idences that s!##ort the theory and the im#lications of /eneral relati%ity in astronomy and cosmolo/y. ".1
,nertial (ass an Gravitational (ass
n 9ewtonian mechanics a distinction e+ists etween inertial mass that resists acceleration and &ravitational mass that e+erts force etween oects. Altho!/h 9ewton elie%ed inertial mass and /ra%itational mass are the same he meas!red them differently. 9ewton’s first law that states that a ody at rest or in motion at a constant %elocity in a strai/ht line tends to ee# its rest #osition or its motion !nless acted on y an o!tside force. This law and thesecon law elow a##ly to inertial mass. = mi a
3)14
where T is the net force e+erted on an inertial mass mi to affect its inertia or motion and a is acceleration. !t wei&ht is a force of /ra%ity that acts on a ody as shown in > : m
3)24
where F is the force e+erted y /ra%ity on /ra%itational mass m and is the acceleration of /ra%ity on Earth with an a%era/e %al!e of ! (., mDs2. nertial mass mi is defined as a meas!re of the inertial #ro#erty of matter. n #ractice it is easier to determine mass not y oser%in/ its interaction with another oect of nown mass !t y meas!rin/ the force of /ra%ity on the oect. :e now that when the 3nly force aro!nd is /ra%ity all oects re/ardless of mass =or wei/htI fall with the same acceleration . f we now call the acceleration of free*fallin/ oects and : the force of /ra%ity =which is also called wei/ht> then we can deri%e the second e8!ation =62> from 9ewton’s second law =e8!ation 61>. rom e8!ations =61> and =62> we can write a : m mi
3)"4
f acceleration is constant for a /ra%itational field which it is then the ratio of /ra%itational to inertial mass is also constant. And if acceleration is chosen to e e8!al to the force of /ra%itation as it is ao%e then we can say that /ra%itational mass and initial mass are e8!al. Gi%en this we can also say that inertia and wei/ht are !st two manifestations of the same thin/. :e note in #assin/ that the second e8!ation maes clear the fact that wei/ht is not the same as mass. )ass in one definition is the `8!antity of matter’ in a ody or more #recisely the meas!re of the inertia of a ody. )ass is also re/arded as an inherent #ro#erty of matter so that an oect may e wei/htless =where > !t still has mass. 5n the moon where /ra%ity is one*si+th that of the earth oects wei/h one*si+th their wei/ht on earth e%en tho!/h their mass remains the same. There are a few e+ce#tions hotons ne!trinos and /ra%itons are massless #articles. 9ewton’s insi/ht is to see the same force actin/ in the fall of an a##le to the /ro!nd and ee#in/ the moon oritin/ aro!nd the earth. This !nified %iew is emodied in the form!lation of the law of !niversal &ravitation that /o%erns the attraction etween any two odies in the !ni%erse from #articles to /ala+ies : G m1 m2 8 r 2
3)$4
64 where T is the force of attraction B is the constant of #ro#ortionality called /ra%itational constant which is e8!al to 6.672 + 1 L11 9.m2D/2 m' is the /ra%itational mass of the first oect m2 that of the second oect and r the distance etween the centers of /ra%ity of the oects. oth the a##le and the moon are attracted to the center of the earth y the same /ra%itational force in this case a centri#etal one. f there were no centri#etal force to ee# the moon in its nearly circ!lar orit it wo!ld follow its strai/ht inertial #ath into s#ace. rom the law of /ra%itation we can say first that all odies in the !ni%erse attract =not re#el> all other odies. "econd the ma/nit!de of the /ra%itational force is directly #ro#ortional to the /ra%itational masses of the odies in%ol%ed e./. if the mass of one ody do!les the /ra%itational force do!les. Third the /ra%itational force is in%ersely #ro#ortional to the s8!are of the distance that se#arates them e./. the /ra%itational force is red!ced y half if the distance s8!ared =not the sim#le distance> is do!led. inally the &ravitational constant G is a !ni%ersal constant that had to e determined e+#erimentally. "ince B is so small the /ra%itational force etween any two oects enco!ntered in e%eryday life is tiny. As an e+am#le for two 3*/ tr!cs whose centers of /ra%ity are 3 m a#art their force of m!t!al attraction is only ao!t 67 9 =micronewton or millionth of a newton a newton ein/ a !nit of force 9 1/.mDs2>. Gravitational onstant G 5f the fo!r forces of nat!re the stron/ wea electroma/netic and /ra%itational forces the last one is the weaest. t is also %ery diffic!lt to meas!re. Amon/ the diffic!lties may e cited the errors inherent in the meas!rin/ a##arat!s s!ch as inconstancy of the torsional moment of s!s#ension in torsion alance and torsion #end!l!m de%ices and the inhomo/eneities of the masses !sed in the a##arat!s. 5thers incl!de en%ironmental factors s!ch as the dist!rances of the /ro!nd the effects of amient tem#erat!re the ma/netic and electric infl!ences the /radients of the /ra%itational field and the chan/es of the s!rro!ndin/ field. 5%er the last two h!ndred years a n!mer of scientists ha%e meas!red B amon/ whom the rench #hysicist Charles de Co!lom =1736*1,6> the En/lish #hysicist ;enry Ca%endish =1731*1,1> who !sed the torsion alance and others !# to modern times. "ome #hysics classes e%en ha%e st!dents cond!ct meas!rements of B as a #roect. )ost te+toos settle on the %al!e of B = 6.6720( + 1 L11 9.m2D/2 b .,0 recommended y the C5-ATA =Committee on -ata for "cience and Technolo/y of the nternational Co!ncil of "cientific Snions>. ;owe%er attem#ts ha%e een made in recent years to refine the meas!rement of B which some scientists thin is still incorrect. )ore modern a##arat!ses ha%e een !sed with %aryin/ res!lts so that it is reasonale to e+#ect the contro%ersy to contin!e. ".2
The Principles of E!ivalence an Covariance
:e ha%e seen that the two /ra%itational and inertial masses ha%e een shown e+#erimentally to e %irt!ally e8!al to within a few #arts in 111 5ne is the force of m!t!al /ra%itational attraction amon/ masses =m > and the other the resistance of a sin/le mass to acceleration =mi>. :ith his /eneral theory Einstein showed the f!ndamental connection etween them i.e. there is e8!i%alence etween them and their effects. :e will now see how Einstein !ses this corres#ondence to set forth the /eneral theory of relati%ity. n a tho!/ht e+#eriment #ro#osed y Einstein =#. 66> an oser%er is inside a closed chest the si'e of a room s!s#ended in s#ace which d!rin/ the ne+t ste# will e mo%ed !#ward y an o!tside force !nenownst to the oser%er. :e may moderni'e the e+#eriment y s!stit!tin/ an ele%ator or a rocet shi# for the chest if we wish !t the res!lt will e the same. efore #roceedin/ f!rther let !s !nderstand the conce#t of /ra%itational field. t is common nowled/e that two thin/s se#arated in s#ace cannot interact witho!t some sort of intermediate a/ent. There is no s!ch thin/ as action at a distance. An a##le in the tree does not fall to the /ro!nd witho!t what we now call /ra%ity. M!st as a ma/net attracts iron filin/s eca!se it is s!rro!nded y a ma/netic field so does the earth attract the a##le and the moon eca!se it is s!rro!nded y a /ra%itational field. The closer to earth an oect is the stron/er it is affected y the earth’s /ra%itational force. ar o!t in s#ace the earth’s /ra%itational field diminishes in ma/nit!de. Snder the infl!ence of the /ra%itational field all oects are s!ected to the same acceleration re/ardless of their mass material or #hysical state. A l!m# of lead and a feather fall with e+actly the same s#eed in the same /ra%itational field in %ac!!m. 9ow in the closed chest mentioned ao%e there is no /ra%itational field actin/ on the system. "!##ose that witho!t the oser%er’s nowled/e the chest is lifted y an o!tside force y means of a ro#e thro!/h a hoo fastened to the to# of the chest with an acceleration e8!al to (., mDs2. The acceleration of the chest mo%in/ !#ward transmits its force to her le/s which #ress down on the chest’s floor with an e8!al force so
60 that the net res!lt is that she stands on the chest floor e+actly as she wo!ld in her li%in/ room witho!t feelin/ any difference. f the oser%er dro#s a r!er all from her hand the acceleration of the chest is no lon/er transmitted to the all and it a##roaches the floor with accelerated motion. "he also notices that the acceleration of the all toward the floor is the same if she !sed a hea%y wei/ht. The circ!mstances !st descried ill!strate the effect of acceleration created y an e+ternal force =rocet firin/ an/els liftin/ or whate%er> on inertial mass. "!##ose now that the chest is #laced ac on earth a/ain witho!t the oser%er’s nowled/e. The earth’s /ra%itational force is transmitted to her le/s and she stands on the floor in e+actly the same way she wo!ld in her own li%in/ room. f she dro#s the all it falls to the /ro!nd with the same accelerated motion as oser%ed ao%e. ;ere is an e+am#le of the effect of a /ra%itational field on /ra%itational mass. n oth cases the oser%er oser%es the same eha%ior of the all. ;ence inertial mass and /ra%itational mass are e8!al in the sense that they eha%e in identical ways. !rthermore acceleration and /ra%itational field #rod!ce identical effects. n fact there are no e+#eriments mechanical or otherwise that allow the oser%er to distin/!ish acceleration d!e to /ra%itational field from acceleration d!e to forces of other inds. This leads !s to the principle of e!ivalence which is one of the #ost!lates of the /eneral theory of relati%ity. The rincile of eFuivalence3 EAperiments performe in an inertial frame containin& a &ravitational fiel an eAperiments performe in a !niformly acceleratin& frame yiel ientical res!lts. The other #ost!late is nown as the principle of covariance which states Ill physical laws can e form!late to e vali for any oserver in any spacetime reference frame whether accelerate or not. This #rinci#le calls for the form!lation of relati%ity e8!ations in any system of s#acetime coordinates. There is still on/oin/ contro%ersy o%er the co%ariance #rinci#le and o%er the consistency of co%ariance and e8!i%alence !t this is eyond the sco#e of this wor. "!ffice it say that the two #rinci#les of e8!i%alence and of co%ariance form the fo!ndation of /eneral relati%ity. As a theory of /ra%itation the /eneral theory of relati%ity does away with the notion of mass*de#endent force which 9ewtonian #hysics relies on to e+#lain /ra%itation. nstead relati%ity foc!ses on motion and maes !se of the conce#t of s#acetime and the /eometry of s#ace. :ith this new #ers#ecti%e /ra%itation is nothin/ !t a conse8!ence of the s#acetime c!r%at!re. :e will ne+t loo at the test of the /eneral theory of relati%ity. The urvature of !acetime efore /oin/ f!rther let !s riefly familiari'e o!rsel%es with s#acetime c!r%at!re which is one of the startlin/ insi/hts of /eneral relati%ity. Accordin/ to this theory we li%e in a world with fo!r dimensions three s#ace dimensions and one time dimension in one contin!!m !nlie the 9ewtonian world where s#ace and time are two se#arate entities. :e will treat this s!ect in more detail later !t for now a #reliminary introd!ction is needed to aid in com#rehendin/ the e%idence #resented elow. Einstein maintains that /ra%ity is !st a conse8!ence of the c!r%at!re of s#acetime. n other words it is the /eometry of s#ace that dictates the eha%ior of oects. "#ace is not flat !t c!r%ed aro!nd celestial odies. Almost all odies war# the s#ace aro!nd them y %irt!e of their mass. The more mass a ody has the more distortion of s!rro!ndin/ s#ace it #rod!ces. ma/ine a #iece of faric that is held !# and e#t ta!t y hoos attached to all its corners. f we #lace a owlin/ all on the faric it will roll to the center and mae a de#ression where it rests. The s!rface of the faric aro!nd the owlin/ all is c!r%ed inward lie a f!nnel. f we roll a marle on the faric toward the owlin/ all it will first follow a strai/ht #ath when still far from the all then starts to di# when it /ets closer to the all. The marle merely follows the c!r%at!re of the faric aro!nd the owlin/ all. &ie the owlin/ all the s!n war#s the s#ace aro!nd it and any li/ht that comes near it will eha%e in the same way as the marle does in the %icinity of the all. The li/ht nat!rally follows a /eodesic the shortest distance etween two #oints thro!/h s#ace. ar away from any mass li/ht #ro#a/ates in a strai/ht line !st lie in E!clidean =flat> s#ace. !t when it reaches the c!r%ed s#ace aro!nd a massi%e ody s!ch as the s!n a star or a /ala+y it taes the c!r%ed #ath i.e. it ends.
"."
66
Tests of the General Theory of Relativity
The first test is the enin& of li&ht. General relati%ity #redicts that li/ht ends in the %icinity of a ody with a lar/e mass eca!se of the c!r%at!re of s#acetime it #rod!ces. &et !s tae a tho!/ht e+#eriment in which an oser%er is inside an ele%ator at rest in the /ra%itational field of the earth. "he shines a li/ht hori'ontally and it stries the o##osite wall at the same le%el to her naed eye. The endin/ effect is im#erce#tile eca!se it is too small. @et in an analo/o!s sit!ation when she tosses a all it follows a c!r%ed #ath. "!##ose now that witho!t her nowled/e the ele%ator starts mo%in/ with acceleration e8!al to . As efore if she sends a li/ht eam across the ele%ator she will see a strai/ht eam with her own !naided eyes. y the #rinci#le of e8!i%alence etween /ra%itational field and acceleration we sho!ld /et the same e+#erimental res!lt. ;ence if the ele%ator’s width is 3 m the li/ht tra%erses it in 1 L, s. -!rin/ this time the ele%ator wo!ld ha%e mo%ed !# 0 + 1 L16 m to meet the li/ht and the li/ht eam sho!ld hit the wall elow the hori'ontal y the same distance. n either case the deflection of li/ht is im#erce#tile. :e need a lar/er scale and a stron/er mass to a!/ment the endin/ and mae it meas!rale. To that effect we !se the s!n for its /reat mass and the s#ace war# it creates aro!nd it. Consider a ray of li/ht from a distant star. As it a##roaches the s!n the c!r%at!re of s#ace near its s!rface deflects the li/ht ray from its strai/ht*line #ath toward Earth =i/!re 3*1>. t is as if the s!n acts as a &ravitational lens that refracts li/ht. An earth oser%er sees the star as if it is in the line of si/ht. Act!ally she merely sees the refraction of the star. The li/ht deflection is #redicted to e small howe%er only 1.70 arc*seconds =1 arc*second 1D36 of a de/ree> eca!se the "!n’s /ra%ity is wea y relati%istic standards. Act!al #osition of star
A##arent #osition of star
Tr!e line of si/ht
"!n
&i/ht ray ent
Earth
Fi&!re "-1. %i&ht #eflection #!e to Gravitational Fiel of the S!n The li/ht rays from the star are deflected from their strai/ht*line #aths y the "!n. "ince the "!n’s /ra%ity is wea the deflection is only 1.70 arc*seconds. The fi/!re is e+a//erated to show li/ht endin/. To confirm the #rediction of /eneral relati%ity scientists first too #hoto/ra#hs of the stars near the s!n when it was in total ecli#se y the moon to a%oid its lindin/ li/ht. "i+ months later when the s!n was on the other side of the earth scientists too #hoto/ra#hs of the same stars a/ain. They then com#ared the #ositions of the stars in the #hoto/ra#hs. n an e+#edition to Africa in 1(1( d!rin/ a total solar ecli#se the En/lish astronomer "ir Arth!r Eddin/ton =1,,2*1(44> confirmed the ao%e res!lts. The deflections meas!red less than 2 arc sec in a/reement with theoretical #redictions. This s!ccess leads to an entirely new %iew of the str!ct!re of s#acetime. n contrast 9ewtonian mechanics cannot #redict li/ht deflection eca!se li/ht has no mass and its calc!lations of /ra%itational force de#end on mass. n other tests with radio wa%es astronomers !sin/ interferometry acc!rately meas!re the deflection of radio wa%es to within 1 #ercent of the %al!e #redicted y the theory. n cases where a h!/e mass s!ch as a /ala+y act as a /ra%itational lens etween the earth and the distant star it ends the li/ht’s #ath so as to mae it %isile to an earth oser%er. f the mass has circ!lar symmetry the /ra%itational lens effect creates a rin/ of li/ht. An e+am#le of this is Einstein’s $in/ a do!le*loed radio /ala+y formed y a lensin/ /ala+y alon/ the line of si/ht. f the mass has irre/!lar sha#e the li/ht is roen into #atches s!ch as in 8!asar G2237F30 called Einstein’s Cross which has fo!r
67 #atches aro!nd a central /low. n /eneral since the c!r%at!re called for y /eneral relati%ity is so sli/ht in the solar system 9ewtonian mechanics yields !st as /ood res!lts as Einstein’s theory. Another test of which /eneral relati%ity has /i%en an e+#lanatory acco!nt is the #!''lin/ eAcess precession of the planet (erc!rys orit. or half a cent!ry astronomers ha%e nown that the maor a+is of )erc!ry’s orit shifts aro!nd the #lane of orit =a rotation called #recession> and its #erihelion =the #oint of closest a##roach to the "!n> ad%ances nearly 0 arc sec a cent!ry. The #recession of )erc!ry’s orit meas!res 06 seconds of arc #er cent!ry. 9ewtonian e8!ations are ale to #redict the #recessions of all #lanets in the solar system e+ce#t )erc!ry. E%en after all the effects of other #lanets the fact that Earth is not an inertial frame and the s!n’s sli/ht deformation d!e to its rotation ha%e een acco!nted for 9ewton’s e8!ations #redict a #recession of 0007 arc sec #er cent!ry a discre#ancy etween oser%ation and #rediction of 43 arc sec #er cent!ry. f s#ecial relati%ity considerations s!ch as mass chan/e and time dilations are taen into acco!nt the discre#ancy is 21 seconds of arc #er cent!ry. :hen /eneral relati%ity is a##lied the #redicted motion d!e to c!r%at!re of s#acetime yields the additional 43 arc sec #er cent!ry in line with oser%ations to within a few #ercent. This is a tri!m#h of /eneral relati%ity o%er 9ewtonian mechanics. A third test may e called &ravitational reshift not to e conf!sed with the -o##ler shift. General relati%ity #redicts that clocs r!n slower in stron/er /ra%itational fields than in weaer ones. Th!s a cloc in the asement r!ns sli/ht slower than one in the attic. Consider an e+#eriment in which a cloc #laced on the floor of an ele%ator risin/ from the /ro!nd. The cloc emits a li/ht #!lse toward a mirror at the to# of the ele%ator. As the ele%ator accelerates the mirror recedes !t e%ent!ally the li/ht #!lse reaches the mirror where it is reflected ac. 5n the tri# !# the distance co%ered is m!ch /reater than the hei/ht of the ele%ator eca!se it has to catch !# to the recedin/ mirror. 5n the tri# down the distance co%ered is shorter eca!se the floor of the ele%ator accelerates !#ward to meet it. Calc!lations show that the total distance tra%eled y the li/ht #!lse is /reater than twice the hei/ht of the ele%ator which is the distance that wo!ld e co%ered y a cloc at rest. "ince the s#eed of li/ht is constant it follows that the time taen for the ro!nd tri# is lon/er in an accelerated reference frame than in a frame at rest. y the e8!i%alence #rinci#le we can say that time r!ns slower in the #resence of a /ra%itational field. And this is in addition to the time dilation effect. &i/ht from a %ery com#act star with a stron/ /ra%itational field has a hi/her fre8!ency =i.e. shorter wa%elen/th> than li/ht comin/ from dee# s#ace. Th!s li/ht lea%in/ a star with stron/ /ra%ity shifts to the red =low fre8!ency> end of the s#ectr!m. "!##ose an oser%er sends a li/ht eam from a c!r%ed re/ion of s#ace near a massi%e oect to another oser%er located in a re/ion far away from a massi%e oect. The distant oser%er will meas!re a lower fre8!ency or lon/er wa%elen/th for the li/ht than the sender. :e call this chan/e of wa%elen/th /ra%itational redshift. "ince li/ht has oscillations that can e !sed as clocs /eneral relati%ity’s #redictions a/ree with oser%ations. inally the e+istence of o!le !asars #ro%ides f!rther confirmation of &ravitational lensin& #redicted ao%e. g!asars 8!asi stellar radio so!rces are %ery old starlie oects with h!/e redshifts and stron/ radio emission orn when the !ni%erse was %ery yo!n/. They are some 14 illion li/ht*years away and are recedin/ fast. ;ence their li/ht taes illions of years to reach !s and /i%es !s a /lim#se into the early days of the !ni%erse. n 1(7( astronomers -ennis :alsh $oert Carswell and $ay :eymann fo!nd to their s!r#rise two 8!asars called (07F061A and (07F061 which are only 6 arc sec a#art. "ince 8!asars are rarely close to/ether they st!died their emission line redshifts and fo!nd them %irt!ally identical. They concl!ded that the do!le ima/e came from the same so!rce for which an inter%enin/ #roaly elli#tical /ala+y acts as a /ra%itational lens. y 1(( nearly two do'ens do!le 8!asars had een re#orted of which si+ cases were con%incin/. E%ery new disco%ery of do!le 8!asars #ro%ides additional confirmation of /eneral relati%ity. All the ao%e tests cannot e considered asol!te #roof of the /eneral theory of relati%ity since they may e e+#lained y other theories as well. !t the %alidity of the #rinci#le of e8!i%alence and the s!ccess of /eneral relati%ity whose #redictions are consistent with oser%ations mae the theory the most remarale achie%ement in twentieth*cent!ry #hysics. ".$
Geometry of Spacetime
As a theory of /ra%itation the /eneral theory of relati%ity descries /ra%ity in terms of the /eometry of s#acetime. :e will first e+amine all the #ossile /eometries the !ni%erse can ha%e then disc!ss the asic #remise of /eneral relati%ity that /ra%ity c!r%es the faric of s#acetime. "ince it is /enerally hard for !s to %is!ali'e three*dimensional c!r%ed s#ace in what follows we will !se e+am#les of two*dimensional oects to ill!strate their #ro#erties.
lat or Euclidean Geometry – ero urvature
6,
E!clidean /eometry is familiar to !s as it is ta!/ht in school. :e ha%e learned that two e+tended #arallel lines will ne%er meet that the an/les of a trian/le total 1,^ and that the circ!mference of a circle is e8!al to 2:r . n this flat &eometry there are two dimensions. eca!se #arallel lines stretched to infinity stay a#art we call it open &eometry. "#ace has infinite area no o!ndary no ed/e and hence no center .as a center can only e determined from o!ndaries. &i/ht tra%els in a strai/ht line fore%er and ne%er ret!rns. n a flat !ni%erse there is 9ero c!rvat!re i.e. s#ace is flat. Also a flat !ni%erse contains an infinite n!mer of /ala+ies and hence infinite mass. Altho!/h it is hard to ima/ine an infinite flat s#ace /ettin/ lar/er e+#ansion at least of some #arts is #ossile. M!st ima/ine a sheet of r!er that can e stretched indefinitely. The /eodesic the shortest distance etween two #oints is a strai/ht line. A tra%eler headin/ o!t in this s#ace will ne%er ret!rn to the #oint of de#art!re. !herical Geometry – %ositive urvature or disc!ssion ima/ine an Earth witho!t any /eolo/ical feat!res. The s!rface of the Earth =or a all> is a /ood e+am#le of the spherical &eometry or close &eometry which has positive c!rvat!re. This ty#e of /eometry is closed since #arallel lines meet. :hile the Earth is a three*dimensional oect its s!rface is two* dimensional. "#ace in this /eometry is two*dimensional and there is neither o!ndary nor center. There is nowhere on the Earth its two*dimensional inhaitants can call the ed/e. E%erywhere on the Earth is a two* dimensional s!rface. And the area is finite. arallel lines if e+tended will e%ent!ally intersect. To satisfy yo!rself of this !st ima/ine yo! and yo!r friend lea%in/ the E8!ator for the 9orth ole each followin/ a line of lon/it!de #er#endic!lar to the E8!ator. eca!se of the c!r%at!re of the Earth yo! and yo!r friend will end !# meetin/ at the 9orth ole while thinin/ all alon/ that yo! ha%e een tain/ #arallel #aths. n this /eometry s#ace can e+#and !st lie a alloon aro!nd its center which is not reachale to two*dimensional inhaitants eca!se it re8!ires a third dimension which is not a%ailale. The n!mer of /ala+ies is finiteR so is the total mass. As the all e+#ands all #oints on its s!rface /et farther a#art. There are no strai/ht lines. The /eodesic is not a strai/ht line !t an arc of a circle. &i/ht tra%els alon/ a /eodesic not alon/ a strai/ht line. A trian/le can e drawn from three /eodesics and its an/les s!m to more than 1,^. The circ!mference of a small circle is still e8!al to 2:r !t as the circles /et lar/er its circ!mference will fall short of 2:r y an increasin/ amo!nt. E%ent!ally the circ!mference /ets %ery small for a circle with a radi!s reachin/ halfway across the !ni%erse f this s#ace is %ast com#ared to its inhaitants s!ch as #eo#le on the Earth its c!r%at!re is not easy to discern. ;owe%er thro!/h meas!rin/ instr!ments it is #ossile for them to descrie this ind of !ni%erse with acc!racy. HyerIolic Geometry – Jeative urvature This /eometry is est e+em#lified y the saddle. n hyperolic &eometry #arallel lines di%er/e and hence ne%er meet. or this reason it is called open &eometry. The entire two*dimensional ne&atively c!rve s#ace cannot e drawn !t we can a##ro+imate it y !sin/ the middle of the saddle*sha#e s!rface. The an/les of a trian/le on this s!rface total less than 1,^. The circ!mference of a circle is always /reater than 2:r tho!/h in small circles the difference is diffic!lt to see. &ie flat s#ace hy#erolic ne/ati%ely c!r%ed s#ace has infinite area that e+tends in all directions. The n!mer of /ala+ies is infinite and so is mass. t also has no o!ndaries and no center. &ast it can e e+#anded indefinitely y stretchin/ its two*dimensional s!rface. :hen e+#anded this !ni%erse #!shes its two*dimensional /ala+ies farther away from one another. The uture of the Kniverse At the #resent time it is not nown which of the three #ossile /eometries the !ni%erse has. @et the f!t!re of the !ni%erse cr!cially de#ends on the ty#e of /eometry and the density of the !ni%erse. General relati%istic models !se the same three models ao%e for o!r three*dimensional !ni%erse altho!/h it is im#ossile to %is!ali'e a !ni%erse made of s#ace that has #ositi%e 'ero or ne/ati%e c!r%at!re. f the s#ace is c!r%ed we need an e+tra dimension for it to c!r%e into. ;owe%er we do not need to ima/ine what a fo!r*dimensional c!r%ed !ni%erse loos lie. :e can still !nderstand its #ro#erties y e+tra#olatin/ from the two*dimensional models descried ao%e.
6( Accordin/ to a asic #remise of /eneral relati%ity matter determines the s#acetime c!r%at!re. ;ence the c!rvat!re of space and ensity of matter are interconnected. The term matterB here refers to all sorts of matter and ener/y in the !ni%erse %isile as well as in%isile. "ince it is im#ossile to determine the c!r%at!re of s#ace /eometrically scientists foc!s on determinin/ the density of matter in the !ni%erse. f the a%era/e density of matter is lar/e the !ni%erse has #ositi%e s#atial c!r%at!reR and if the density is small the c!r%at!re is ne/ati%e. :hen the a%era/e density of matter L is e+actly e8!al to the critical ensity symoli'ed y Lc the c!r%at!re of the !ni%erse is 'ero. 9ote that the critical density is %ery small Lc 1 – 26 /Dm2 or ao!t 1 hydro/en atoms #er c!ic meter. or con%enience a #arameter called 5me/a has een desi/ned to e+#ress the ratio of the a%era/e density of matter in the !ni%erse to the critical densityo : L D Lc. The s!scri#t oB refers to #resent ratio. f the density of matter is less than the critical density L U Lc then o = 1. This means the !ni%erse is infinite with ne/ati%e c!r%at!re and it will e+#and fore%er. f the a%era/e density is /reater than the critical density L Lc then o < 1 and the !ni%erse is finite and has #ositi%e c!r%at!re and it will e%ent!ally contract. f the a%era/e density is e8!al to the critical density L Lc then o : 1 the !ni%erse is infinite and flat and it will also e+#and fore%er. 5f these three scenarios which one is the most liely to e followed y the !ni%erse? :e now that at the #resent time the !ni%erse is e+#andin/. The 8!estion is whether the e+#ansion will contin!e fore%er or whether it will e%ent!ally sto# and the !ni%erse will e/in to contract. ;ere an analo/y with a tossed all will shed li/ht. ma/ine "!#erman throwin/ a all in the air. f the tossin/ force is small at first the all rises with a##ro#riate acceleration a/ainst /ra%ity which tends to slow it. Then at some #oint the all reaches its hi/hest #oint and finally it falls ac down to the Earth. f the all is thrown m!ch more ener/etically it /ains tremendo!s s#eed ee#s risin/ fore%er and ne%er ret!rns to Earth. The all has reached its esca#e %elocity. The cr!cial factors that determine whether the all will ret!rn are its s#eed and /ra%ity. or the !ni%erse these two factors are the ;!le’s constant which descries the e+#ansion of the !ni%erse and the a%era/e density of matter which characteri'es the self*/ra%itation and c!r%at!re of the !ni%erse. The !ni%erse faces two alternati%es. f the ;!le’s constant is small and the a%era/e density is lar/e e+#ansion will #roceed slowly !ntil it reaches its ma+im!m then the !ni%erse will sto# e+#andin/ and start colla#sin/. This is sometimes referred to as the i/ Cr!nch =a i/ an/ in re%erse> in which the !ni%erse ends in a /reat confla/ration. Alternati%ely if the ;!le’s constant is lar/e and the a%era/e density is small the !ni%erse ee#s e+#andin/ fore%er !ntil all stars and /ala+ies ha%e !rned !# all their f!el and die. :e ha%e the i/ Chill. The !ni%erse will meet its fate in fire or in ice. ". '
Einsteins C!rvat!re of Spacetime
The asic #ost!late of the /eneral theory of relati%ity is that matter determines the c!r%at!re of s#acetime. This means e%ery oect c!r%es or war#s the s#acetime faric aro!nd it. or a lar/er oect the war# is more #rono!nced than for a smaller oect. And this c!r%in/ of s#ace determines the #ath of any oect that comes near it. Einstein does not need the /ra%itational force to e+#lain the attraction that odies e+ert on one another. "ince all odies follow their nat!ral motion s#acetime c!r%at!re maes s!re they follow the a##ro#riate #aths. As an analo/y of s#acetime c!r%at!re ima/ine a sheet of r!er stretched hori'ontally in mid*air. 9ow ima/ine that yo! #lace a hea%y all in the middle of the r!er sheet. t #!lls the sheet down and creates a c!r%at!re in the r!er sheet. f yo! roll a marle on the r!er sheet from the ed/e it first follows a strai/ht #ath then as it a##roaches the all it mo%es alon/ a c!r%ed #ath created in the r!er sheet y the all. 9ewtonian mechanics e+#lains this #henomenon as the effect of /ra%itational force e+erted y the all on the marle. n Einstein’s theory there is no force affectin/ the #ath of the marle. t sim#ly follows the c!r%at!re of s#acetime aro!nd the all. &i/ht too follows the s#acetime c!r%at!re so that in the %icinity of a massi%e ody li/ht is deflected as we ha%e seen in li/ht endin/ and /ra%itational lensin/ )atter not only creates a c!r%at!re in the s#acetime faric it also slows time as we saw in /ra%itational shift ao%e. n this and other #henomena /eneral relati%ity has #ro%ided an e+#lanation that is oth sim#le and ele/ant. ".)
/lac0 oles
After a #eriod of e+citement the /eneral theory of relati%ity lost its #o#!larity. y the late 1(0’s /eneral relati%ity had ecome a d!ll !n#rod!cti%e s!ect too hard to learn and com#rehend. !t y the 1(7’s it !nderwent a re%i%al and commanded renewed interest es#ecially in astro#hysics cosmolo/y and elementary
7 #article #hysics. 9ow /eneral relati%ity is front and center in the st!dy of lac holes 8!asars and in other astronomical research. :e concl!de this wor with a rief loo at the lac holes. The ormation of a Mlac< Hole )any stars in the )ily :ay and other /ala+ies will die. f the star’s mass is /reater than 3 solar masses it cannot ecome a ne!tron star or a white dwarf. As the star is dyin/ its matter !rnin/ o!t of control creates s!ch intense inward #ress!re on all sides that it o%ercomes o!tward forces and shrin ra#idly. As the s#here ecomes com#ressed to tremendo!s densities /ra%ity on the s!rface increases /reatly. The distortions of the s#acetime faric ecome more and more se%ere as #redicted y /eneral relati%ity. As the star contin!es contractin/ li/ht from its s!rface /rad!ally ends inward and wra#s aro!nd the star. hotons on its s!rface no lon/er lea%e the star’s s!rface tra##ed y the intense /ra%itational field. The esca#e %elocity of li/ht on its s!rface e8!als the s#eed of li/ht with the res!lt that no li/ht esca#es. The star ecomes dar. At this #oint the s#ace aro!nd the now dimin!ti%e star ecomes so c!r%ed that it ri#s a hole in the faric of the !ni%erse. The re/ion of s#ace aro!nd the hole loos lie a f!nnel. The dyin/ star is dead in this hole in%isileR it has now ecome a lac0 hole. A lac hole is a re/ion of s#acetime from which nothin/ incl!din/ li/ht esca#es. The !tructure of the Mlac< Hole f we a##roach a lac hole we first reach a re/ion that is flat eca!se of the wea /ra%itational field there. 9ear the hole /ra%ity is stron/ and the s#ace c!r%at!re is #rono!nced. As we come closer immediately s!rro!ndin/ the lac hole is a s#here called event hori9on where the esca#e s#eed is e8!al to the s#eed of li/ht. t is also nown as the s!rface of the lac hole. E%ent hori'on
R Sch "in/!larity
Fi&!re "-2. The Str!ct!re of a /lac0 ole A non*rotatin/ lac hole has a sin/!larity at its center and the e%ent hori'on s!rro!ndin/ it. The distance etween the sin/!larity and the e%ent hori'on is the "chwar'schild radi!s =$ "ch>. nside the e%ent hori'on the esca#e %elocity e+ceeds the s#eed of li/ht so that nothin/ in it can esca#e. 5nce the dyin/ star falls into the e%ent hori'on it disa##ears com#letely. A non*rotatin/ lac hole has a sim#le str!ct!re =1> the e%ent hori'on which is its s!rfaceB and =2> the sin/!larity in its center. 5nce a dyin/ star has im#loded inside the e%ent hori'on no force in the !ni%erse can #re%ent it shrinin/ to a #oint. At this sta/e the star has infinite density and is called asin&!larity. The distance from the sin/!larity to the e%ent hori'on is the Schwar9schil rai!s =R Sch> after the German astronomer Karl "chwar'schild =1,73*1(16> who first sol%ed Einstein’s /eneral relati%ity e8!ations. The form!la for this radi!s is R Sch : G 8 c2
3)'4
71 where B is the /ra%itational constant is the mass of the lac hole. The solar mass of any oect /i%es its "chwar'schild radi!s directly The radi!s is 3 m for each solar mass. A lac hole of 1 solar masses has a radi!s of 3 m. "ince /ra%ity c!r%es s#ace and slows time if an astrona!t falls toward a lac hole where /ra%ity is stron/ time e/ins to slow down !ntil it sto#s clicin/ in the e%ent hori'on. !t the astrona!t accelerates with /reat s#eed and is stretched to a thin wire y /reat forces. @et to her time tics normally as she contin!es fallin/. The reason for the difference is that the c!r%at!re of s#acetime ca!ses time to tic more slowly for the astrona!t than for an o!tside oser%er. rom the astrona!t’s #oint of %iew she crosses the e%ent hori'on and accelerates 8!icly toward the sin/!larity witho!t noticin/ anythin/ s#ecial. To the o!tside oser%er howe%er she contin!es to slow down and finally ho%ers o%er the e%ent hori'on fore%er. That is not e+actly correct since the li/ht the astrona!t emits ecomes e%er more /ra%itationally redshifted !ntil she com#letely disa##ears from the oser%er’s %iew. The remarale thin/ is that oser%er and astrona!t co!ld not a/ree on whether the astrona!t e%er crosses the e%ent hori'on.
4. rolems
72
$.1 Frames of Reference 4.1.1
4.1.2
4.1.3
Two air#lanes A and are flyin/ side y side at the same altit!de. lane A is o%ertain/ #lane at 6 mDh. A #assen/er in #lane A is walin/ toward the front at 2 mDh while a fli/ht attendant is walin/ toward the rear at 3 mDh. To an oser%er in #lane what are the s#eeds of the #assen/er and the attendant? &ea%in/ directly from the an with the s#eed of 3 mDh a oat crosses a ri%er flowin/ at 4 mDh. f a #assen/er wals dia/onally toward the oat’s stern with a s#eed of 1.3( mDs his #ath formin/ an an/le of 03.1_ toward the !#stream direction from the len/th of the oat what is his s#eed with res#ect to the ri%eran? A train #assen/er is r!nnin/ 1 mDh down the aisle a/ainst the train’s %elocity of 1 mDh as it mo%es #ast the station. ind =a> the #assen/er’s s#eed relati%e to the stationR => the #assen/er’s s#eed relati%e to the station if he r!ns in the o##osite direction.
$.2 Relativistic Kelocity Iition 4.2.1 4.2.2 4.2.3 4.2.4 4.2.0
4.2.6 4.2.7 4.2., 4.2.( 4.2.1 4.2.11 4.2.12
4.2.13
Two /ala+ies are tra%elin/ away from the earth in o##osite directions oth with s#eed % .,0c. :hat wo!ld an oser%er from one /ala+y meas!re for the s#eed of the other /ala+y? A s#aceshi# a##roachin/ Ven!s at a s#eed of .,2 c la!nches a #roe toward the #lanet at a s#eed of .7 c relati%e to the s#aceshi#. :hat is the s#eed of the #roe as meas!red y an oser%er on Ven!s? A s#aceshi# a##roachin/ the )oon at .0 c sends a li/ht si/nal to an oser%er stationed in a l!nar station. =a> :hat is the s#eed of the li/ht si/nal as meas!red y the s#aceshi#’s #ilot? => ;ow fast does the li/ht si/nal tra%el relati%e to the l!nar oser%er? Two s#aceshi#s lea%e the earth in o##osite directions with the same s#eed of .0c with res#ect to the earth. =a> :hat is the %elocity of shi# 1 relati%e to shi# 2? => :hat is the %elocity of shi# 2 relati%e to shi# 1? A s#aceshi# tra%els with the s#eed of .712 c relati%e to the earth A #roe la!nched from the s#aceshi# tra%els at .,23 c with res#ect to the s#aceshi#. :hat is the s#eed of the #roe with res#ect to the s#aceshi# if it is fired =a> in the direction of tra%el of the s#aceshi#? => in the o##osite direction? The ca#tain of a s#ace station sees his rocet shi# Al#ha a##roachin/ with the %elocity of .04(c. ;e sends o!t rocet shi# eta with %elocity of .612 c to meet Al#ha. :hat is the %elocity of the Al#ha relati%e to the eta. Two asteroids head toward Ven!s. Asteroid Centa!r a##roaches Ven!s with the s#eed of .67(c and Asteroid Ante!s a##roaches Ven!s with the s#eed of .772 c. :hat is the s#eed of Centa!r with res#ect to Ante!s? "tarshi# Enter#rise and $ocet "hi# ntre#id are mo%in/ away from the )a/n!m "#ace "tation with s#eeds of .7,3 c and .6(, c res#ecti%ely. :hat is the s#eed of the Enter#rise relati%e to the ntre#id? A #roton mo%es to the left with a s#eed .7,( c with res#ect to the laoratory frame. An electron mo%es to the ri/ht with a s#eed of .(3 c relati%e to the #roton. :hat is the s#eed of the electron with res#ect to the laoratory frame? "#ace sh!ttle Valiant mo%es to the ri/ht with a s#eed of .00 c with res#ect to the nternational "#ace "tation. "#ace sh!ttle End!rance lea%es the nternational "#ace "tation and heads left at a s#eed of .440 c. ind the s#eed of the Valiant relati%e to the End!rance. or %ery fast s#eeds =!t still m!ch smaller than c> for e+am#le those achie%ed y a s#ace sh!ttle show that the relati%istic %elocity addition is red!ced to the classical %elocity addition form!la. $ocet shi# Gamma lea%es its "tarase Al#ha with a s#eed of .60c when it is #!rs!ed y a hostile $ocet shi# named ;!nter tra%elin/ with a s#eed of .72c relati%e to "tarase Al#ha. The ;!nter la!nches a missile toward Gamma with a relati%e s#eed of .34 c. ;ow fast does the ca#tain of Gamma see the missile a##roachin/ him? Two electrons A and ha%e res#ecti%e s#eeds of .,c and ..( c. ind their relati%e s#eeds =1> if they are mo%in/ in the same direction =2> if they are mo%in/ in o##osite directions.
73 4.2.14 "#aceshi# A is flyin/ #ast the earth with a s#eed of .76 c relati%e to the earth =mo%in/ to the ri/ht>. "#aceshi# is a##roachin/ the earth with a s#eed of L.67 c. relati%e to the earth =mo%in/ left>. ind the s#eed of shi# relati%e to shi# A. 4.2.10 "how that if a #hoton tra%elin/ with %elocityc in frame S’ and frame S’ is tra%elin/ with %elocity c relati%e to the frame S the %elocity of the #hoton with res#ect to S is c. 4.2.16 A s#aceshi# mo%es to the ri/ht at .60 c and a rocet shi# mo%es to the left toward the s#aceshi# at the s#eed of .74 c as meas!red y an Earth oser%er. :hat is the rocet shi#’s s#eed relati%e to the s#aceshi#? 4.2.17 A s#aceshi# mo%es away from #lanet M!#iter at a s#eed of .,70 c and shoots a #roe ac toward M!#iter at the s#eed of .7(, c relati%e to the shi#. ind the s#eed of the #roe relati%e to M!#iter. 4.2.1, oat 1 crosses a stream directly at a distance ? and ret!rns to its destination in t ' time. oat 2 /oes !#stream a distance ? and downstream for the same distance tain/ a time t (. Com#are t ' and t (. 4.2.1( A s#aceshi# while mo%in/ away from )ars with a %elocity of .,c fires a rocet in the direction of the s#aceshi#’s motion which reaches a %elocity of .7 c relati%e to the s#aceshi#. :hat is the rocet’s s#eed relati%e to )ars? 4.2.2 A rocet shi# tra%els to a star 10 + 11, mDs away with a s#eed of 1.270 + 1, mDs as meas!red y an oser%er at the la!nch site. ;ow lon/ will the tri# tae accordin/ to an onoard cloc? 4.2.21 Two /ala+ies are tra%elin/ away from the earth in o##osite directions with the same s#eed of .,0c. :hat wo!ld the s#eed of each /ala+y e as meas!red y an oser%er in the other? 4.2.22 f a 8!asar mo%in/ away from Earth with s#eed of .70c eects matter toward Earth with a s#eed of .40 c what is the s#eed of the eected material as meas!red from Earth? 4.2.23 n a laoratory e+#eriment an electron mo%in/ at .,((0 c collides with a #ositron tra%elin/ at .(,(0 c. At what s#eed do the #articles a##roach each other? 4.2.24 A s#aceshi# is tra%elin/ towards a #oint on Earth when a laser eam from the same #oint fires toward the shi#. At what s#eed does the laser eam tra%el as meas!red y an en/ineer aoard the shi# who recei%es it? $esol%e the #rolem classically. 4.2.20 Two li/ht #!lses are sent o!t in o##osite directions. :hat is the s#eed of either one with res#ect to the other? $esol%e the #rolem classically. 4.2.26 An eartho!nd oser%er finds that two s#aceshi#s tra%elin/ at the s#eed of .,20c are mo%in/ in o##osite directions one toward Earth the other away from Earth. :hat is the s#eed of one s#aceshi# with res#ect to the other? 4.2.27 Two s#aceshi#s are racin/ toward Earth. "hi# A in the lead mo%es at .,(c and se#arates from shi# at .04 c. =a> :ith res#ect to Earth what is shi# ’s s#eed? => Calc!late the relati%e s#eed etween the two shi#s if shi# A increases its s#eed y 0. 4.2.2, Asteroid A and asteroid are recedin/ from Earth with s#eeds relati%e to Earth of 2 + 1, mDs and 2.0 + 1, mDs res#ecti%ely. :hat is their s#eed relati%e to each other? 4.2.2( A #roe is la!nched in the direction of motion of its s#aceshi# with a s#eed of 1 + 1, mDs. f the s#aceshi#’s s#eed relati%e to Earth is 0 + 1, mDs what is the #roe’s s#eed relati%e to Earth? 4.2.3 Two #articles fly in o##osite directions each with the s#eed of .77c. ind the relati%e s#eed of the #articles. $." %orent9 Transformation an Sim!ltaneity 4.3.1 4.3.2
4.3.3 4.3.4 4.3.0
At time t = t’ = the ori/ins of frame S and S’ o%erla#. S’ mo%es at v 1.2 + 1, mDs with res#ect to S. A #article in S’ comes to a rest at coordinates x’ 30m y’ 4m and z’ . :hat are its coordinates in frame S at =a> t =3.0 s? =a> t =40.0 s? "!##ose two inertial systems S and S’ ha%e the same x*a+is and #arallel y*a+es. rame S’ mo%es with a s#eed of .60 c relati%e to S in the F x*direction. A rocet in S mo%es with a s#eed of .2 c in the F y*direction. =a> :hat does an oser%er in S’ calc!late the s#eed of the rocet to e? => :hat is the direction of the rocet as seen y that oser%er? "how that at e%eryday %elocity v c the &orent' transformation red!ces to the Galilean transformation. A rocet shi# mo%in/ away from Earth with a s#eed of .,24 c fires a #roe in the direction of its motion with the s#eed of .06 c relati%e to the shi#. :hat is the #roe’s s#eed with res#ect to Earth? "how that =a> the &orent' transformation red!ces to the Galilean transformation if v = . => the &orent' transformations #redict len/th contraction of a mo%in/ oect.
74 At the e/innin/ of a s#aceshi#’s interstellar %oya/e to a star 1 ly away x x’ m and t t’ s. ts s#eed as meas!red y oser%ers at its station is .0 c. ind =a> the distance x and the time t as meas!red y home station’s astrona!ts when the s#aceshi# arri%es at the star and => the s#aceshi#’s x’ and t’ at its destination as meas!red y onoard astrona!ts. 4.3.7 Two li/hts are t!rned on sim!ltaneo!sly at the moment fast*mo%in/ 5ser%er 1 is #assin/ stationary 5ser%er 2. =a> -o these li/hts a##ear to flash sim!ltaneo!sly for either oser%er or oth oser%ers? => Add!ce e%idence to s!##ort yo!r answer from a &orent' transformation e+#ression. 4.3., A stationary oser%er sees two firewors e+#lode one 0. seconds after the other. :hat does an oser%er mo%in/ at a s#eed of .00 c #ast the firewors meas!re the time inter%al etween the two e+#losions to e? 4.3.( )o%in/ #ast Earth at %elocity 2.0 + 1, mDs a s#aceshi# eams laser si/nals to Earth e%ery 12 seconds accordin/ to an onoard cloc. :hat does a cloc on Earth record this time inter%al to e? 4.3.1 At the common ori/in of two frames S and S’ a li/ht #!lse is emitted at time t’ . At time t its distance from the ori/in is x’ 2 = c2t’ 2. y !sin/ the &orent' transformation show that the motion is e+actly the same in oth frame S and S’ i.e. that the distance in frame S is x2 = c2t 2. 4.3.6
$.$ (ichelson-(orley EAperiment 4.4.1
4.4.2
4.4.3 4.4.4
"tartin/ from the same air#ort two air#lanes fly for 4 m at the same airs#eed of 3 mDs one headin/ east for itts!r/h the other headin/ north for -etroit. The wind lows at 0 mDh in the easterly direction. ind =a> the time of fli/ht to each city => the time of the ret!rn tri# and =c> the difference of the total fli/ht times etween to two air#lanes. GintM Sse the oat analo/y in i/!re 2* 2. n the )ichelson*)orley e+#eriment eam 1 tra%els across the ether wind and maes the ro!nd tri# in time t ' 2 l & c =1 D =1 L v( D c(> 1D2 > while eam 2 tra%els with and a/ainst the ether wind main/ the ro!nd tri# in time t ( 2 l & c P1 D =1 L v( D c(>Q. "!##ose the )s * ) 2 arm shrins as s!//ested y the &orent'*it'Gerald contraction. "how that Nt = t ( L t ' . n one r!n of the )ichelson*)orley e+#eriment the #er#endic!lar arms of the a##arat!s ha%e len/th & 20 m. Gi%en v = 3 + 14 mDs calc!late =a> the time difference ca!sed y the rotation of the interferometer and => the e+#ected frin/e shift if the li/ht eam !sed has a wa%elen/th of 04 nm. ind the associated with a #article mo%in/ at the s#eed of =a> .0 cR => .10 c =c> .0 c and =d> .6670 c.
$.' Time #ilation 4.0.1
4.0.2 4.0.3 4.0.4 4.0.0 4.0.6
Ca#tain A!rore of "#aceshi# &on/tre emars on a ro!nd*tri# %oya/e to a faraway Gala+y Tresloin sit!ated 10.0 ly from Earth tra%elin/ with a %elocity of .70 c. :hen he comes ac to Earth how old is Ca#tain A!rore if he left at a/e 27? GintM Yse v = d & Nt. Commander -esois left his ,*year*old da!/hter on a o!rney to star -iana tra%elin/ with a %elocity of .,0 c. The ro!nd tri# too him 6.,4 years as meas!red from Earth. ;ow old is his da!/hter when he comes home? Ca#tain A/amemnon too a 20*year ro!nd tri# to 9e!la Troy and t!rned ac. :hen he came home his wife Clytemnestra fo!nd him to e 7 years older. :hat was the a%era/e s#eed of his s#aceshi# d!rin/ the tri#? ;!rtlin/ at %elocity .7,,0 c toward star &aas far away s#aceshi# ntre#id reaches its destination and heads ac to Earth. 5n his ret!rn Ca#tain Achilles’ wife fo!nd him , years older than when he had left. ;ow many years did Ca#tain Achilles’ ro!nd tri# last from his wife’s #oint of %iew? E+#edition &eader ;erc!les s#ent 17 years of his starshi# life in search of the ;ea%enly Golden leece. 5n his ret!rn to Earth 2 years had #assed !d/in/ y the historical records e#t at the &irary of :orld Con/ress. :hat was the a%era/e %elocity of his %oya/e? n the #arallel !ni%erse S2 all the #hysical constants are different from the #hysical constants of o!r !ni%erse. "!##ose a #hysicist in S2 standin/ on the #latform clocs a #assin/ train as tra%elin/ with the s#eed of 2 mDs. ;is collea/!e on the train finds the ela#sed time of a flash of li/ht from the train’s floor to its mirrored ceilin/ and ac to e 60 s while he recorded the same two e%ents as lastin/ 0( s. :hat is the s#eed of li/ht in S2? GintM Fhat was an3mal3us ab3ut this scenari3Z
4.0.7
4.0., 4.0.( 4.0.1 4.0.11 4.0.12 4.0.13 4.0.14 4.0.10 4.0.16
4.0.17 4.0.1, 4.0.1( 4.0.2 4.0.21
4.0.22 4.0.23 4.0.24
4.0.20 4.0.26 4.0.27 4.0.2,
70 n the #arallel !ni%erse S3 all the #hysical constants are different from the #hysical constants of o!r !ni%erse. "!##ose a #hysicist in S3 standin/ on the #latform clocs a #assin/ train as tra%elin/ with the s#eed of 2 mDs. ;is collea/!e on the train finds the ela#sed time of a flash of li/ht from the train’s floor to its mirrored ceilin/ and ac to e 70 s while he recorded the same two e%ents as lastin/ ,( s. :hat is the s#eed of li/ht in S3? A cosmona!t’s tri# to a distant star taes 11.7,3 years tra%elin/ with a s#eed of .((2c relati%e to Earth. ;ow many years does the cosmona!t a/e d!rin/ this tri#? An astrona!t s#ends 2 min!tes eatin/ l!nch y his s#aceshi#’s time. Ass!min/ the astrona!t is tra%elin/ at .,,7 c relati%e to Earth =a> how lon/ does the astrona!t’s l!nch last from Earth’s reference frame? => how far does the s#aceshi# tra%el d!rin/ this time from Earth’s #oint of %iew? ;ow m!ch ha%e yo! a/ed flyin/ the distance of ,0 m from ;o!ston to aris in a et airliner tra%elin/ with the s#eed of 1 mDh com#ared to when yo! left ;o!ston? lannin/ a tri# to Andromeda /ala+y 2 million ly away yo! fi/!re it will tae 34 years y a cloc on the s#aceshi#. ;ow fast will the s#aceshi# ha%e to tra%el on a%era/e to com#lete the #lanned tri#? 5n a ro!nd tri# to a /ala+y 3 ly away 42*year*old Astrona!t Mo!%ence tra%els with a s#eed of .(( c. ;is son ;i##olyte was 1( when he left. =a> ;ow old is Mo!%ence on his ret!rn to Earth? => ;ow old is his son ;i##olyte when he sees his father a/ain? )ission Control on Earth meas!res Astrona!t rina’s hearteat to e , eats #er min!te. "he is tra%elin/ with a %elocity of .60 c relati%e to Earth. :hat does she meas!re her heart rate to e? A et #lane flyin/ with a constant %elocity of 33 mDs re8!ires 3 h y the onoard cloc to reach its destination. :hat is the d!ration of the tri# y the clocs on the /ro!nd? The lifetime of a ne!tron as meas!red on Earth is 6.42 + 12 s. :hat is the lifetime of the ne!tron as meas!red y a cloc on a s#aceshi# flyin/ #ast at a %elocity of 2 + 1, mDs? As s#aceshi# ntre#id cr!ises at a constant s#eed of .,(4 c #ast #lanet -iana its ca#tain Goddard oser%es that the -iana cloc and his own cloc oth read 2 #.m. At 33 #.m. y his cloc Ca#tain Goddard’s shi# #asses y #lanet Athena. f the -iana and Athena clocs are synchroni'ed with each other what time do Athena’s clocs show when s#aceshi# ntre#id #asses y? Ca#tain ;!le wants to mae a tri# to a star 30 ly away. ;ow fast does he ha%e to tra%el to mae the tri# in only , years? :hat is the mean life of a m!on fallin/ down with a s#eed of .((((c relati%e to Earth if its mean life at rest is 2.2 s? The !nstale m!on decays in 2.3 s at rest. =a> ind the m!on’s mean life in a laoratory if it tra%els with a s#eed of .(( c with res#ect to the la. => ;ow far has the m!on #article tra%eled d!rin/ its lifetime as meas!red in the la? A certain #article which decays after 16 s has tra%eled at distance of 00 m. :hat is its mean %elocity d!rin/ is lifetime? After a 4*year inter/alactic %oya/e as meas!red y his yo!n/er rother on Earth Ca#tain ;awin/ came home to find he is 0 years older than when he left and his yo!n/er rother twice as old as he is. ind =a> the a%era/e s#eed he achie%ed d!rin/ his o!rney. => his rother’s a/e on his de#art!re and his a/e after the %oya/e if their total a/e at re!nion is 70 years. Gi%en a qF meson’s mean life of 2.0 + 1 L, s if it tra%els with a %elocity of .((3 c =a> what is the distance it co%ers efore decay? => "!##ose there were no time dilation effects what wo!ld this distance e? The a%era/e life of a ne!tron as meas!red on Earth is 6.40 + 12 s. :hat is the mean life of this ne!tron if meas!red y an oser%er on a s#aceshi# tra%elin/ #ast with a s#eed of 2.0 + 1, mDs? q*mesons s!atomic #articles #rod!ced in #article #hysics e+#eriments and in cosmic radiation ha%e a lifetime of .,4 + 1 L*16 s in low*ener/y deris of a #article #hysics e+#eriment and a mean lifetime of 1.4 + 1 L*16 s as cosmic rays hittin/ o!r atmos#here. =a> Gi%e a reason for the discre#ancy in these lifetimes. => Calc!late the and v of each ty#e of meson. f an oect mo%es at a s#eedv 1 D 4 c as meas!red y a stationary oser%er find its . :hat is the s#eed of a m!on which has a mean life at rest of 2.2 s if its mean life in motion is 0.( s? An onoard atomic cloc records 42 ho!rs on the aro!nd*the*world tri# made y a et #lane flyin/ at the s#eed of 1 mDh. :hat is the difference etween this time and the time meas!red y an identical and synchroni'ed atomic cloc in the laoratory on Earth? f a s#ace*farin/ astrona!t wants to come ac to Earth only 0 years older y her cloc than when she left at what s#eed m!st she tra%el on a 3*ly ro!nd tri#?
76 4.0.2( ;ow far does a #article tra%el after reachin/ its destination in ,7.2 s from its rest #osition where it decays in 4 s? :hat is its s#eed of tra%el? 4.0.3 A et air#lane flyin/ with a s#eed of 0 mDs carries an atomic cloc which meas!res a time inter%al of 30 s. :hat does an identical cloc in a laoratory on the /ro!nd meas!re the same inter%al to e? $.) %en&th Contraction 4.6.1 4.6.2 4.6.3
4.6.4 4.6.0 4.6.6
4.6.7 4.6., 4.6.(
4.6.1 4.6.11 4.6.12 4.6.13 4.6.14 4.6.10 4.6.16
4.6.17
4.6.1,
The distance etween two #lanets is meas!red y an Earth oser%er to e (,7 m. f a s#acecraft taes 2 seconds y the clocs on the shi# to fly with a !niform %elocity from one #lanet to the other what is its a%era/e s#eed? :hat is the &orent' contraction of a car tra%elin/ at 10 mDh? As a 10*meter*lon/ rocet mo%es away from Earth at a constant %elocity a li/ht #!lse is eamed to its nose and tail and is reflected ac to Earth. The reflected si/nal from the tail reaches a detector on Earth 20 s after emission and the si/nal reflected from the nose follows 1.0 + 1 L6 s later. ind the distance etween the rocet and the earth and the rocet’s %elocity relati%e to Earth. )o%in/ #ast a meter stic at a s#eed of .00 c an oser%er meas!res its len/th. :hat len/th does the oser%er meas!re the meter stic to e? Calc!late the len/th of a meter stic that mo%es alon/ its len/th with a s#eed of 2.,( + 1, mDs? A rocet shi# mo%es at the s#eed of .40 c to an earth oser%er. =a> ind its . => ind its contracted len/th ? as oser%ed y the earth oser%er. =c> ind the N ? the chan/e in the shi#’s len/th in the direction of motion as oser%ed y the same oser%er. =d> ind its N G, the chan/e in its hei/ht in the direction #er#endic!lar to its motion as oser%ed y the earth oser%er. =e> ind its contracted len/th ? as oser%ed y an onoard astrona!t. The s#ace sh!ttle is 37.23 m on the r!nway. As it h!rtles at 2730, mDh into s#ace how lon/ does it a##ear to an oser%er on Earth? A meter stic on a s#aceshi# meas!res .70 m. ;ow fast does the s#aceshi# tra%el relati%e to an Earth oser%er? Ca#tain ;i/hs#eed tra%els y yo! in a 20*m*lon/ s#aceshi# at a %ery hi/h s#eed. @o! are tra%elin/ in a s#aceshi# that meas!res 2 meters on the /ro!nd. rom yo!r #oint of %iew =a> how lon/ is Ca#tain ;i/hs#eed’s s#aceshi# => how lon/ is yo!r shi# in fli/ht and =c> how fast is Ca#tain ;i/hs#eed’s s#aceshi#? A 3*m*lon/ car tra%els at a s#eed of 12 miDh. :hat is the decrease in the car’s len/th as meas!red y a stationary oser%er? ;ow fast m!st a rocet shi# tra%el so that its len/th shrins y two*thirds to an oser%er at rest on the earth? ;ow lon/ is a s#aceshi# at rest if an oser%er on the earth meas!res its len/th to e (0 m as it flies y the oser%er at the s#eed of .70 c? To mae a tri# to a star 40 ly away how fast wo!ld an astrona!t ha%e to tra%el to mae the tri# in 4 years? To mae a tri# to a star 0 ly away how fast wo!ld an astrona!t ha%e to tra%el so that the distance wo!ld e only 10 ly? "ittin/ in yo!r atmoile yo! see yo!r friend #ass yo! in her ,*m*lon/ atmoile at a s#eed of .12 c. @o!r friend says she meas!res yo!r car to e ,.0 m lon/. :hat do yo! meas!re for the len/ths of the two cars? At what s#eed wo!ld an astrona!t ha%e to tra%el to reach the closest star Al#ha Centa!ri 4. ly from Earth in only 2.0 years as meas!red y a cloc in the s#aceshi#? At that s#eed how far did the astrona!t tra%el in his reference frame? :hat wo!ld an eartho!nd oser%er meas!re for the time the astrona!t s#ent to reach the star? A s#aceshi# 3 m in hei/ht and 2 m in len/th tra%els #ast an oser%er on Earth at a s#eed of .,70 c. =a> :hat does the oser%er meas!re for the hei/ht and len/th of the s#aceshi#? => f the oser%er meas!res an e+#eriment on the earth to last for 20 seconds on her watch how m!ch time did she find the same e+#eriment to last as meas!red y the s#aceshi#’s cloc? =c> ;ow fast did the oser%er a##ear to e mo%in/ accordin/ to the astrona!t? =d> ;ow many seconds did the astrona!t see ela#se on the oser%er’s cloc when 20 seconds #assed on hers? n a laoratory e+#eriment an electron tra%els 4 cm in .20 ns. =a> :hat is the electron’s s#eed? => n the reference frame of the electron what was the distance the la tra%eled?
77 4.6.1( @o!r s#aceshi# and yo!r friend’s s#aceshi# are tra%elin/ toward a /ala+y. @o!r s#eed is .70c relati%e to yo!r friend’s. @o!r friend meas!res the len/ths of oth s#aceshi#s and finds them to e the same. =a> f yo! meas!re the len/th of yo!r friend’s shi# do yo! find yo!r shi# lon/er or shorter than her shi#’s len/th? E+#lain. => Calc!late the ratio of the len/th of yo!r shi# to the len/th of yo!r friend’s. 4.6.2 Ca#tain Kir and Ca#tain Co!stea! tra%el thro!/h s#ace in identical s#aceshi#s. After main/ meas!rements Ca#tain Kir finds his shi# ,D7th times as lon/ as of Ca#tain Co!stea!’s shi#. =a> rom Ca#tain Co!stea!’s #oint of %iew how lon/ is Ca#tain Kir’s shi# if the shi#s at rest is 12m? => :hat is Ca#tain Kir’s s#eed relati%e to Ca#tain Co!stea!’s? 4.6.21 5n a tri# to a star an astrona!t mo%in/ with the s#eed of ., c relati%e to Earth finds the distance co%ered from her reference frame to e 7 ly from Earth. 5n the ret!rn tri# the astrona!t tra%els at a s#eed of .(7 c relati%e to Earth. :hat is the distance in li/ht*years the astrona!t tra%eled on the ret!rn tri# as meas!red y the astrona!t? 4.6.22 A laoratory t!e in which #rotons mo%e at a mean s#eed of .0c meas!res 7 m lon/. n the #rotons’ reference frame how lon/ does the t!e meas!re? 4.6.23 A rectan/!lar s#aceshi# 20 m hi/h and 14 m lon/ is tra%elin/ #arallel to its len/th. :hat is the s#eed of the shi# when an earth oser%er finds the shi# to e a s8!are? 4.6.24 The hei/ht and len/th of a trian/!lar s#aceshi# meas!re 3 m and 1 m res#ecti%ely. At what s#eed m!st it tra%el in a direction #arallel to its len/th for its len/th*to*hei/ht ratio to e 0D3? 4.6.20 An astrona!t tra%elin/ to a star , ly away with a s#eed s!ch that the /amma factor is 7D6. :hat is the distance from Earth to the star as meas!red y an eartho!nd instr!ment? 4.6.26 f a 0*m*lon/ rocet flyin/ #ast a stationary oser%er a##ears to meas!re only half its ori/inal len/th what is its s#eed? 4.6.27 ;ow fast does a s#orts car ha%e to tra%el in order for a stationary oser%er to find its len/th to e shortened y 1 com#ared to its len/th at rest? 4.6.2, An electron tra%els a strai/ht distance of 10 m as meas!red in its own frame of reference. An e+#erimenter finds the electron’s s#eed in the #article accelerator to e .((((((c. :hat is the distance tra%eled y the electron as meas!red y the e+#erimenter? 4.6.2( A s!#ersonic et #lane meas!rin/ 1 m in len/th on the r!nway flies at the s#eed of 3 mDh. =a> :hat is the difference in the len/ths of the et #lane when it is at rest and when it is flyin/ as oser%ed y an air traffic controller? => ;ow fast does the #lane ha%e to tra%el for this difference to 8!adr!#le? 4.6.3 E%ent A and e%ent occ!r in frame S at the same #oint. E%ent occ!rs 3 s after e%ent A. n frame S’ mo%in/ relati%e to S e%ent occ!rs 0 s after e%ent A. :hat is the distance etween the #ositions of the two e%ents as meas!red in frame S’ ? $.* Relativistic (oment!m 4.7.1
Calc!late the moment!m of a #roton mo%in/ with the s#eed of =a>..1c => .67 c =c> .70 c and =d> .(0 c. Calc!late the moment!m of an electron mo%in/ with the same s#eeds. 4.7.2 f an electron’s relati%istic moment!m is (0 lar/er than its classical moment!m =a> what is its s#eed? => f the #article were a #roton what wo!ld e the #roton’s s#eed? 4.7.3 To achie%e the same moment!m as an electron mo%in/ at a s#eed of .(0c at what s#eed m!st the #roton mo%e? 4.7.4 At what s#eed m!st an oect tra%el so that its relati%istic moment!m is fi%e times /reater than its classical moment!m? 4.7.0 A rocet shi# with a mass of 0 + 1,*/ mo%es away from )ars with a s#eed of .70 c. :hat are its classical moment!m and its relati%istic moment!m? 4.7.6 ind the relati%istic moment!m of a #roton with a mass of 1.673 + 1*27 / mo%in/ at s#eed of 2D3 c. 4.7.7 A ne!tron mo%es with s#eed v .70 c and has a mass of 1.670 + 1*27 /. -etermine its relati%istic moment!m. 4.7., f an electron has a moment!m ,0 lar/er than its classical moment!m how fast is it mo%in/? 4.7.( f a #roton has 1,36 times the mass of an electron calc!late the s#eed at which the electron will ha%e the same relati%istic moment!m as the #roton mo%in/ at .2 c. 4.7.1 Calc!late the s#eed of an oect whose relati%istic moment!m is fi%e times /reater than its classical moment!m.
7, 4.7.11 Calc!late the s#eed at which a #roton has a moment!m e8!al to that of an electron mo%in/ at s#eed of .70 c. 4.7.12 y how m!ch is classical moment!m in error %is**%is relati%istic moment!m if an electron with mass (.11 + 1*31 mo%es at s#eed of =a> .1 c => .40 c =c> .(1 c? 4.7.13 :hat is the ratio of relati%istic moment!m to classical moment!m for a m!on tra%elin/ at .(0c? 4.7.14 :hat is the s#eed in terms of c of a m!on s!ch that its relati%istic moment!m is 20 /reater than its nonrelati%istic moment!m? 4.7.10 or an al#ha #article with mass of 6.644 + 1*27 / to ha%e the same moment!m as a ne!tron with mass of 1.674 + 1*27 / tra%elin/ at s#eed of v .(2 c how fast m!st the al#ha #article tra%el? 4.7.16 ;ow fast m!st a #article tra%el for its mass to e fi%e times its rest mass? 4.7.17 A ne!tron with mass 1.674 + 1*27 / has a moment!m 2., + 1*1( /.mDs. :hat is its s#eed? 4.7.1, At what s#eed does the moment!m of an al#ha #article ecomes three times as /reat as its classical moment!m? 4.7.1( Calc!late the classical and relati%istic momenta of a 3.4*/ mass mo%in/ at .,(c. 4.7.2 An electron with a mass of (.11 + 1*31 / mo%es with s#eed v .77 c. ind its classical and relati%istic momenta. $.+ Relativistic (ass 4.,.1 4.,.2 4.,.3 4.,.4 4.,.0 4.,.6 4.,.7 4.,., 4.,.( 4.,.1 4.,.11 4.,.12 4.,.13 4.,.14 4.,.10 4.,.16 4.,.17 4.,.1, 4.,.1( 4.,.2
;ow m!ch does a et airliner increase in mass if it flies at 1 mDh? f an electron mo%es with a s#eed of .0 c what is its relati%istic mass? A mo%in/ #article do!les its mass with res#ect to its rest mass. ;ow fast does the #article mo%e? To ha%e a relati%istic mass of .0 / how fast m!st a .2*/ aseall e thrown? After a collision with an asteroid at rest a rocet with a mass of 6.7 + 17 / tra%elin/ at a s#eed of .6 c remains emedded in the asteroid. Calc!late the asteroid’s mass if the rocet*asteroid system has the s#eed of .40 c. :hat is the mass of a #roton mo%in/ at .60c? :hat wo!ld e the #ercent increase in mass of a 4.0 + 16 / s#ace root tra%elin/ at 4 mDh? ;ow fast m!st an oect mo%e in order for its mass to e 2 #ercent /reater than its rest mass? :hat is the mass of a #roton that is tra%elin/ atv .(2 c? At what s#eed m!st a .2*/ aseall e thrown to ha%e a mass of .0 /? ;ow m!ch mass is con%erted to ener/y #er ho!r in a n!clear reactor that #rod!ces 10 + 11 : of #ower? ;ow m!ch mass does the s!n !rn e%ery second to radiate ener/y at the rate of 3.(2 + 126 :? Ass!min/ a mass of 1.(( + 13 how lon/ will the s!n contin!e to radiate ener/y efore all its mass is totally cons!med? :hat is the mass while in fli/ht of a .0*/ all thrown atv .60 c? Accelerated to .(3 c alon/ a strai/ht #ath in an accelerator an electron has an increased mass 1 times its rest mass when it reaches its final s#eed. :hat is the electron’s final s#eed? At what s#eed will the mass of an electron e .,20 )eV? At what s#eed will the mass of an oect e si+ times its rest mass? f all the mass of an oect is con%erted into ener/y and the ener/y oser%ed is (.123 + 116 M what is the oect’s mass? An ener/y of 6 )eV is /enerated with all the mass of a certain #article. :hat was its mass? ;ow m!ch ener/y in M and )eV can a 0*/ coin #rod!ce ass!min/ all its mass is !sed !# in the #rocess? A f!t!re rocet shi# is ca#ale of con%ertin/ chicens into ener/y. Ass!min/ it carries 1 3*/ chicens on oard how m!ch ener/y in M and )eV can the chicens /enerate?
$.6 Relativistic Ener&y 4.(.1 4.(.2
"!##ose an electron’s inetic ener/y is 2 lar/er than the %al!e that wo!ld e meas!red had the relati%istic effects not e+isted. :hat is the electron’s s#eed? An electron of mass (.00 + 1 L*31 / has a s#eed of .,(0 c. Calc!late =a> the electron’s total relati%istic ener/y in Mo!les and )eV. => its total relati%istic inetic ener/y in Mo!les and )eV. =c> its relati%istic moment!m. =d> its classical inetic ener/y and moment!m.
4.(.3 4.(.4 4.(.0 4.(.6 4.(.7 4.(., 4.(.( 4.(.1 4.(.11 4.(.12 4.(.13 4.(.14 4.(.10 4.(.16 4.(.17 4.(.1, 4.(.1( 4.(.2 $.1D
7( Calc!late the ratio of relati%istic inetic ener/y to classical inetic ener/y of a #article tra%elin/ at %elocity .,0 c. At what s#eed does a tra%elin/ #article’s rest ener/y e8!al its relati%istic inetic ener/y? "!##ose yo! co!ld con%ert all the mass of 3*/ #enny into ener/y. :hat is the ener/y #rod!ced in M and in )eV? Ass!min/ yo! want to #rod!ce a total ener/y of 2 )eV from a #roton. ind its s#eed moment!m and inetic ener/y. A ne!tral #ion at rest decays into two o##ositely directed /amma*ray #hotons. ind the ener/y and moment!m of the #hotons. Calc!late the rest ener/y of an electron whose mass is 7 times its rest mass. ind the rest ener/y of a ne!tron in M and in )eV. :hat is the s#eed and moment!m of a #roton whose inetic ener/y is 2D3 its rest ener/y? A n!clear reactor #rod!ces an o!t#!t of 1.20 + 13 ): of #ower a year. :hat is the chan/e in mass in its reactor f!el ass!min/ that all the ener/y released y fission is con%erted to electricity? =n reality only a small fraction of the ener/y is con%erted > An oect is tra%elin/ at a s#eed s!ch that C = .6 and = 1.2. ind its total ener/y if its mass is 1.0 + 1*2 /. ind the total ener/y the moment!m and the s#eed of an electron with a rest ener/y of .011 )eV and a = 1.70. Calc!late the total ener/y the rest ener/y the inetic ener/y and the moment!m of a #article with a mass of 1.0 + 1*6 / tra%elin/ with s#eed of .30 c. "!##ose the total ener/y needed is 12 + 12 M. ;ow m!ch mass annihilation wo!ld occ!r in n!clear reaction to s!##ly that ener/y? f an electron’s inetic ener/y is fo!r times its rest ener/y find =a> its total ener/y in )eV and in M => its s#eed in terms of c. ind an electron’s total ener/y )eV and in M its inetic ener/y )eV and in M and its moment!m if it mo%es with s#eed .70 c. A #roton has a inetic ener/y e8!al to 2D3 its rest ener/y. ind the #roton’s moment!m and s#eed. Calc!late the inetic ener/y and moment!m of a ne!tron with a mass of 1.670 + 1*27 / tra%elin/ 1 + 17 mDs. f a s#aceshi# with a mass of 0.2 + 17 / has a relati%istic inetic ener/y of 4.3 + 17 M find its s#eed. Relativistic #oppler Effect
4.1.1 A /reen li/ht from a star has a fre8!ency 0.0 + 114 ;' meas!red in its rest frame. :hat fre8!ency does Ca#tain Eteocles oser%e for the star if his s#aceshi# tra%els toward the star with the s#eed of .2 c? 4.1.2 A certain star has a meas!red wa%elen/th of 4,6.112 nm. Accordin/ to the records in the laoratory this star has a wa%elen/th of 4,6.133 nm. s the star mo%in/ toward or away from !s? :hat is its s#eed? 4.1.3 f a 8!asar has a redshift of .16 what is its radial %elocity? :hat is its distance? 4.1.4 An astronomer st!dyin/ the red li/ht of hydro/en emitted from a star meas!res its wa%elen/th to e lon/er than the wa%elen/th of 60, nm meas!red in the laoratory. s the star mo%in/ toward or away from Earth? "!##ose the star’s s#eed is 3 mDs what does the astronomer meas!re for the wa%elen/th of the li/ht? 4.1.0 5n oard a s#aceshi# mo%in/ at .3 c toward olynices "tar radio si/nals from the star are recei%ed at a fre8!ency of 16 );'. 5n what fre8!ency are the radio si/nals recei%ed on the star? 4.1.6 f a certain /ala+y which emits an oran/e li/ht with a fre8!ency of 0.1 + 114 ;' is recedin/ from Earth with s#eed 31 mDh what is the li/ht’s fre8!ency as oser%ed y an eartho!nd oser%er? 4.1.7 Two s#aceshi#s are a##roachin/ each other head*on. The distance etween them is diminishin/ at the rate of (0 mDs. "hi# 1 sends a laser eam toward "hi# 2. :hat was the wa%elen/th of the laser at transmission if it has a wa%elen/th of 630 nm when recei%ed y "hi# 2? 4.1., :hat is the fre8!ency of a li/ht si/nal sent y "#ace "tation Creon to the a##roachin/ s!##ly shi# Anti/one tra%elin/ at .3 c when the si/nal recei%ed has a wa%elen/th of 640 nm? 4.1.( ind the wa%elen/th of a li/ht si/nal with an oser%ed fre8!ency of 0.(0 + 114 ;' if the so!rce is mo%in/ away at the s#eed of .20 c.
, 4.1.1 A star is oser%ed to ha%e a redshift of .30. ;ow fast is it mo%in/? :hat is its distance at the time of oser%ation?
$eferences
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