The following is a confusing problem because simultaneity is a frame-dependent concept, and because Jackson is a little disingenuous about not needing to consider explicit Lorentz Lo rentz transforms. However, the basic idea is very important, and you must master it: if you work out a tensor-quantity in any frame and express your result as a tensor-function of well-defined tensors, this same expression must be correc t in any frame. Woodard:
The electric and magnetic fields of a charge in uniform motion are obtained fr om coulomb’s law in the charge’s rest frame, and the fact that the field -strength -strength F is an antisymmetric-tensor of rank 2,
Jackson:
;
;
[I.1]
This can be done without w ithout considering the explicit Lorentz transform! The idea is the following: for a charge in uniform motion, the only relevant variables are the charge’s 4-velocity 4-velocity U and the relative coordinate X x p xq , where { x p , xq } are the 4-vector coordinates of the observation point and the charge, respectively. The only antisymmetric tensor that can be formed is X U X U . Thus, the electromagnetic field (tensor) F must be this tensor multiplied by some scalar function of the possible scalar products X X , X U , or U U , all of which are invariants. a) consider the following geometry,
[I.2] The coordinates of P and q at a common time in K can be written, x p ct, b ; xq ct, vt ;
bv
0
[I.3]
By considering the general form of F in the rest-frame of the charge, show that, th at,
F
X U X U
q
2
[I.4]
c (c (U X ) X X ) 2
3/ 2
Verify that [I.4] that [I.4] yields yields [I.1] in frame K. Woodard:
note that boosting to the rest frame of the charge and using v b 0 yields, 1
x p ct , b vt ;
xq 1 ct , 0
First use undergraduate-physics to work out E and B in the primed frame at position Lorentz-boost to q-rest frame, 1
I’m not sure what b is — it it
seems to be some sort of length- scale…
[I.5] b vt .
Considering
vb
m
n ˆ
m
ˆ
n
mn
m n m ˆ
ˆ
m n ( 1) mn n
ˆ
[I.6]
ˆ
0 b 0 , we get, ˆ
x p ct , b vt
xq
1
ct , 0
get the fields : at x p ; they are
E B
q( b vt ) 3 0 b vt 1
[I.7]
Next: assemble them into the primed field strength tensor, and then express F as X U U X times a function of the invariants U X and X X ; note that U ( c, 0) and X x p xq , even though the two
events are not simultaneous in the primed frame.
X U
[I.8]
Finally: show that the unprimed result agrees with.
Woodard: you
are to carry out the same exercise, but now express your answer for the primed field strength tensor in terms of the 4-vector Y , which his deinfed as the separation between observation and charge events which are simultaneous in the rest frame of the charge. The point is, of course, that you still get a completelycorrect expression.
are to express the result in terms of the 4-vector Z , which is defined as the separation between observation and charge when the observation events occurs at the time needed for light to propagate from the charge event. The point is: you get a correct expression for the field-strength tensor. It doesn’t matter how you express it in terms of tensors. Any va lid expression will be right. You can exploit this method by choosing whatever tensors are most convenient for whatever problem you are solving. Woodard: you
