K-Surfaces Simulations in Schwarzschild Geometry M. Ayub Faridi, Fazal-e-Aleem and Haris Rashid
[email protected]
Centre for High Energy Physics, University of the Punjab Lahore, 54590 Pakistan
Abstract Study of the spacetime dynamics in Schwarzschild Geometry (SG) has always been in the lime light. Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K-Surfaces, play an important role in SG. It has been explained in this talk what spacetime foliations are and what is the behavior of K-surfaces for large values of K near essential singularity of Schwarzschild black hole.
Contents of Talk Schwarzschild Geometry Kruskal-Szekeres Diagram Penrose Diagram Curvature Black Blac k Hole Hypersurfaces Spacelike Hypersurfaces Foliation K-Surfaces K-surfaces near Essential Singularity Conclusion References
Schwarzschild Solution Schwarzschild Schwarzsc hild solutio solution n depict depicts s a stat static ic spacetime spacet ime containing a single black hole. The Schwarzschild metric is given by the line element
ds2 = Adt2 A−1 dr 2 r2 dθ2 r2 sin2 θdφ2
−
where
−
2m
A= 1− r
−
Schwarzschild Solution Schwarzschild solution depicts a static spacetime containing a single black hole. The Schwarzschild metric is given by the line element
ds2 = Adt2 A−1 dr 2 r2 dθ2 r2 sin2 θdφ2
−
where
−
−
2m
A= 1− r
m is the mass of the Schwarzschild black hole as measured at spacelike infinity.
Schwarzschild Solution Schwarzschild solution depicts a static spacetime containing a single black hole. The Schwarzschild metric is given by the line element
ds2 = Adt2 A−1 dr 2 r2 dθ2 r2 sin2 θdφ2
−
where
−
−
2m
A= 1− r
m is the mass of the Schwarzschild black hole as measured at spacelike infinity. As r approaches 2m, the coefficient of dt2 approaches zero, and the coefficient of dr 2 approaches infinity.
r = 2m is a coordinate singularity that can be removed by coordinate transformation
Kruskal-Szekeres Coordinates Kruskal-Szekeres(KS ) coordinates (v, u) are defined as ;
u v
= =
r − 1 2m r − 1
1 2
1 2
2m
t r exp cosh 4m 4m t r exp
4m
sinh
2m
The line element of schwarzschild in (KS ) coordinates (v, u) is given by :
ds2 = f 2 (dv 2
2
− du ) − r
2
(dθ2 + sin2 θdφ2 )
where f 2 is given as :
32m3 f = exp r 2
− r 2m
This coordinate system (t, r) changes to (v, u), while θ and φ remain unchanged.
Kruskal-Szekeres Diagram There are four regions of the (v, u) plane, which may be denoted by I , II , III and IV :
I
II
(r ≥ 2m, u ≥ 0) exp( uv == −− 11 exp( (r ≤ 2m, u ≥ 0) exp( uv == 11 −− exp( 1 2
r 2m r 2m
1 2
r 2m r 2m
1 2
1 2
r t ) cosh( ) 4m 4m r t ) sinh( 4m ) 4m
r 4m
r 4m
t ) sinh( 4m )
) cosh( 4tm )
Kruskal-Szekeres Diagram Cont.
III
and
IV
(r ≥ 2m, ≤ 0) − 1 u=− v = − − 1 (r ≤ 2m, u ≤ 0) u=− 1− v = − 1 − r 2m r 2m
r 2m r 2m
1 2 1 2
1 2 1 2
t exp( 4rm ) cosh( 4m )
exp( 4rm ) sinh( 4tm )
t exp( 4rm ) sinh( 4m )
exp( 4rm ) cosh( 4tm )
Kruskal-Szekeres Diagram Cont. For the inverse transformations, t and r are given by :
t=
and
4m tanh 4m tanh u2
−v
−1 v (u) −1 u (v)
2
=(
r 2m
,
(In region I and III )
,
(In region II and IV )
r − 1) exp( 2m )
In KS -coordinate system, the singularity at r = 0 is situated at :
v2
2
−u
=1
and thus for r = 0, there are two spacelike singularities given by :
v = ± 1+ u
2
Kruskal-Szekeres Diagram Cont.
Kruskal-Szekeres Diagram Cont. 2m is given by u2 v2 . This shows that correspond to r 2m there We notice that r v . On the same way, r 2m is given by are two exterior regions u + v and u v2 u2 representing two interior regions, v + u and v v , both corresponding to r 2m. The sphere r = 2m is physically a null surface in which all the vectors on the surface are null vectors and a geodesic with a null tangent vector lying on the surface will continue to lie on it. It is a trapped surface in which all regions interior to this surface have geodesics which cannot emerge out. This is also a red shift horizon (on account of the infinite red shift at r = 2m) and, hence, it is an event horizon.
≥
≥ ≤
≥ ||
≥
≤ −| | ≥ ||
≤ −| |
≤
≥
Penrose Diagram A mathematical frame work for asymptotic forms of the fields at infinity was first developed by R. Penrose. This technique is based on the conformal transformation of spacetime which brings infinity to a finite value. Consequently, we can convert asymptotic calculations to finite calculations. This technique also provides precise definitions for several types of infinity (spacelike, timelike and null) when one takes up asymptotically flat spacetime. In order to observe the asymptotic structure of spacetime, KS coordinates are used to feature finite coordinate values to infinity. The KS coordinates (v, u, θ, φ) for Schwarzschild background are transformed to Compectfied Kruskal and Szekeres coordinates (ψ, ξ, θ, φ)
ψ +ξ v + u = tan 2 ψ −ξ v
− u = tan
2
and
v2
−u
2
ψ +ξ ψ −ξ r r = 1− exp = tan tan 2m
2m
2
2
Penrose Diagram
Curvature "The rate of change of the tangent vector with arc length"[MTW] is known as curvature. The ratio of the second fundamental form to the first fundamental form is called the normal curvature of the surface. Maximum and minimum values of normal curvature (independent of the choice of the curvature) are known as principal curvatures. The average of principal curvatures is called Mean Extrinsic Curvature (MEC) denoted by K and their product is called Intrinsic Curvature. MEC can be computed by the trace of extrinsic curvature tensor as
K =
− nµµ , ;
(µ = 0, 1, 2, 3)
where nµ is the unit normal surface to the hypersurface.
Hypersurfaces A manifold, M of dimension n is a separable, connected, Housdroff space with a homomorphism from each element of its open cover into R n . A space is said to be separable if there exist a countable infinite subspace of it whose closure is the entire X such that space. A space is said to be connected if there does not exist A, B A B = X, A B = Φ = A B where Φ is an empty set. A space is said to be Housdroff if x, y X, x = y there exist neighborhoods η 1 (x) and η2 (x) such that η1 (x) η2 (x) = Φ. In a 4D spacetime manifold M , a hypersurface Σ is 3D submanifold . Hypersurfaces are of three types:
∪
∩
∀
∩
∈
∩
⊂
Timelike Hypersurfaces Spacelike Hypersurfaces Null Hypersurfaces A particular hypersurface Σa can be obtained by imposing certain constraints on the coordinates f (xα ) = 0 or by giving parametric representation of the equations xα = xα (y a ) where a = 1, 2, 3..... and y a are coordinates intrinsic to a particular Σa hypersurface.
Foliation "A foliation is a process of decomposition of the spacetime manifold in to a sequences of one parameter spacelike hypersurfaces of smaller dimension" . The word foliation originated from the Greek word "folia", that means leaves. Therefore, foliation is a process of splitting the spacetime into space and time. The key idea dates back to the beginning of the theory of differential equations where trajectories of the solution space can be thought of as the leaves. Foliation reintroduces the idea of dynamical systems developing with time. Spacelike foliation are used to study the evolution of cosmological density perturbations, where different choices of the parameter are usually referred to as gauges. There is no unique way of performing the foliation process.General Relativity tells us that we must get the same physical outcome regardless of which method of slicing we use. However calculations may be much simpler with a cleverly chosen slicing methodology that respects some symmetry of the problem.
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K -Surfaces, play an important role in the dynamics of curved spacetime
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K -Surfaces, play an important role in the dynamics of curved spacetime Eardley and Smarr initiated foundational work in solving EFE’s using numeric methods. Brill et al numerically demonstrated the foliation of K -Surfaces
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K -Surfaces, play an important role in the dynamics of curved spacetime Eardley and Smarr initiated foundational work in solving EFE’s using numeric methods. Brill et al numerically demonstrated the foliation of K -Surfaces Due to the limitation of computational tools, their calculation was restricted to selective values of K (= 1.2, 2). They conjectured that "if the full set of values of K is used, then a (complete) foliation can be achieved".
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K -Surfaces, play an important role in the dynamics of curved spacetime Eardley and Smarr initiated foundational work in solving EFE’s using numeric methods. Brill et al numerically demonstrated the foliation of K -Surfaces Due to the limitation of computational tools, their calculation was restricted to selective values of K (= 1.2, 2). They conjectured that "if the full set of values of K is used, then a (complete) foliation can be achieved". Perusing this work further, Qadir, Pervez and Azad extended this work to K = 0.2 to +0.2 in step size of 0.02. It was claimed that foliation of K surfaces, 1 is possible by using comactified KS coordinates. However, inadequate for K computational techniques did not permit to further extend this work for large values of K > 1.
− | |≤
| |
Foliation by K -Surfaces Foliation of geometric manifolds in curved spacetime became an excellent tool for understanding of the spacelike hypersurfaces Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known as K -Surfaces, play an important role in the dynamics of curved spacetime Eardley and Smarr initiated foundational work in solving EFE’s using numeric methods. Brill et al numerically demonstrated the foliation of K -Surfaces Due to the limitation of computational tools, their calculation was restricted to selective values of K (= 1.2, 2). They conjectured that "if the full set of values of K is used, then a (complete) foliation can be achieved". Perusing this work further, Qadir, Pervez and Azad extended this work to K = 0.2 to +0.2 in step size of 0.02. It was claimed that foliation of K surfaces, 1 is possible by using comactified KS coordinates. However, inadequate for K computational techniques did not permit to further extend this work for large values of K > 1.
− | |≤
| |
we developed a precise and user-friendly procedure that numerically allows foliation of Penrose diagram for small as well as large values of K .
Differential Equation for K -Surfaces To develop a general differential equation for K -Surfaces in any spacetime, the line element is
ds2 = gµν dxµ dxν
Differential Equation for K -Surfaces To develop a general differential equation for K -Surfaces in any spacetime, the line element is
ds2 = gµν dxµ dxν For the spherical symmetric spacetime the metric has 10 independent components
gµν = gνµ
g g = g
00
g01
g02
g03
01
g11
g12
g13
02
g12
g22
g23
g03
g13
g23
g33
and each component depends upon coordinates r,θ,φ.
Diff. Eq. for K -Surfaces Cont. For a fixed hypersurface S 1 at t = 0, an arbitrary hypersurface S at any time t enclose a 4-volume of the spacetime V (S, S 1 ) defined as t
V (S, S 1 ) =
√ 0
−gdtdrdθdφ
Diff. Eq. for K -Surfaces Cont. For a fixed hypersurface S 1 at t = 0, an arbitrary hypersurface S at any time t enclose a 4-volume of the spacetime V (S, S 1 ) defined as t
V (S, S 1 ) =
√ 0
−gdtdrdθdφ
Any spacelike surface S , described by an implicit function f (t,r,θ,φ) = 0 implies that
dt = tr dr + tθ dθ + tφ dφ
Diff. Eq. for K -Surfaces Cont. The line element with induced metric γ ij takes the form
dσ 2 = γ ij xi xj The components of γ ij can be written as
γ rr
=
γ θθ
=
γ φφ
=
γ rθ
=
γ rφ
=
γ θφ
=
grr
2
− gtt (tr ) − 2gtr tr , gθθ − gtt (tθ ) − 2gtθ tθ , gφφ − gtt (tφ ) − 2gtφ tφ , grθ − gtt (tr )(tθ ) − gtr tθ − gtθ tr , grφ − gtt (tr )(tφ ) − gtr tφ − gtφ tr , gθφ − gtt (tθ )(tφ ) − gtθ tφ − gtφ tθ 2
2
Diff. Eq. for K -Surfaces Cont. The 3 -Dimensional area, A(S ), of the hypersurface is
A(s) =
|γ ij |drdθdφ
Using variational principle general differential equation satisfied by K -surfaces can be written as ∂ ∂ Ω ∂ ∂ Ω ∂ ∂ Ω + + =λ g (1) ∂r ∂tr ∂θ ∂tθ ∂φ ∂tφ
where λ is a constant known as Lagrange multiplier and
Ω2 = detγ ij
|
|
√−
K -Surfaces Foliation we first briefly review the work of Brill et al., Qadir et al. Brill et al provided the Schwarzschild version of Eq. (1)
dr = dt
±(1 − 1/r)
1 + r3 (r
− 1)/(H − 13 Kr
3 )2
(2)
Where H is constant depending upon fixed parametric values of r and K is the extrinsic curvature defined using Lie derivative as K = 12 L n γ ij . In Kruskal-Szekeres (KS) coordinates (v, u) and Compactified Kruskal-Szekeres coordinates (ξ, ψ), Brill et al numerically demonstrated regular K -Surfaces in Kruskal and Penrose diagrams as shown in Figures on next slide. The slope of regular surfaces in Penrose diagram near I + is determined by the choice of K values. These surfaces dip in the region r < 2m, thus avoiding the singularity. They have further pointed out that irregular surfaces for different values of K cannot be used for complete foliation.
−
| |
Work of Brill et al.
Work of Pervez et al. Following Brill et al , Pervez et al adopted the procedure to plot differential Eq. (2) for K -Surfaces with compactified KS coordinates. They demonstrated regular K -Surfaces in Penrose diagrams as shown in Figure below:
Our Work Extending the earlier work , a different methodology for the foliation of K -Surfaces has been adopted by us. This work reiterates that foliation of K -Surfaces is possible including large values of K , which has not been done earlier. we compute K -Surfaces in Schwarzschild geometry using different initial conditions. Eq. (1) establishes differential equations for K -Surfaces both inside and outside the horizon of Schwarzschild spacetime.In order to solve Eq. (1); we use the K and H values suggested by Eardley and Smarr.
K =
r2r√r−−1.5r
2
,
r 2r H = ± 1
−r
2
− 3r
6
Our Work Cont. These results have been plotted in Figures below, which depicts the behavior of K and H with variation in r graphically.
Our Work Cont. Curvature invariants R = R1 and R2 , defined as µν R1 = Rµν ρπ µν R2 = Rµν Rρπ µ
etc.,where Rνρπ the Riemannian tensor, it can be verified that at r = 2m, R 1 , R2 , R3 , remain finite. At r = 0, the Schwarzschild metric displays an essential singularity, where second and third curvature invariants become infinite. For the intrinsic curvature, the reduced metric γ ij given by Eq. (1), for the Schwarzschild geometry, takes the form
···
γ ij
=
r4 r 4 −2mr3 +E 2
0
0
0
r2
00
0
0
r2 sin2 θ
Our Work Cont. Using the reduced metric the curvature invariants R and R 2 become
6H 2 R= 6 r 2
R2 = 2P (r)S (r)
−
−
2K 2 3
8mK 12m2 P (r) + r3 r6
where
P (r) =
H r3
− K 3
and
9H 2 2KH K 2 S (r) = 2 + + r r3 3
− 12m r 3
For finite values of r(= 0), K and H the curvature invariants are finite even at r = 2m and go to infinity at the past and future essential singularities at r = 0.
Our Work Cont. Our task is to convert Eq. (1) into two initial value problems in compactified KS coordinates representing K - Surfaces inside and outside event horizon at r = 2m in the Schwarzschild Spacetime as For the region r > 2m
dψ (A + E )Sin(ψ + ξ) + (A = dξ (A + E )Sin(ψ + ξ) (A
−
− E )Sin(ψ − ξ) − E )Sin(ψ − ξ)
(3)
− E )Sin(ψ − ξ) − E )Sin(ψ − ξ)
(4)
For the region r < 2m
dξ (A + E )Sin(ψ + ξ) (A = dψ (A + E )Sin(ψ + ξ) + (A
−
where E
=
3H
− Kr 3
3
, A 2 = E 2 + r3 (r
− 2m).
Our Work Cont. Numerical solutions of Eq. (3) and Eq.(4) provide complete foliation of Schwarzschild black hole spacetime by K -Surfaces. For a particular value of K = K 1 , one K -Surface can be obtained by using initial condition ψ(π ) = 0 + ω and setting and ω very very small approaching zero.
−
Using Eq.(3), each K-Surface begins from A and reaches B and D in region I of Penrose diagram. From B to C, K-Surfaces satisfy Eq. (3) in region II of Penrose diagram. In a similar manner, K -Sufaces are simulated in the lower half of the Penrose diagram. Each K-Surface is generated by three parameters H,K,r and the sign of A . It may be noted that as K-Surfaces rise from ψ = 0, r decreases with an increase of K .
| |
The behavior of K -Surfaces are plotted for different values of H, K and r See Table. The simulated graphs for different values of H, K and r are presented. Each horizontal side represents ξ -axes and vertical side represents ψ -axes.
K -Surface
in Upper Half
K -Surface in Lower Half
K -Surface in Penrose Diagram
Sequence of K -Surface
Comparison of Our Work
Conclusion and Discussion The global structure of K -Surfaces in Schwarzschild geometry has dependence on the parametric values of H, K and r.
K -Surfaces completely foliate the spacetime and these are very clearly shown near past and future essential singularities. The difference of the local maximum and minimum values of ψ (say δψ ) vary from surface to surface. δψ has certain finite value as r varies between 0.52 to 0.7. But, as we approach r = 0, δψ becomes zero. Comparing the results of BCI and Pervez et al. with our work, it is clear that the present work provides a complete foliation. As the K -Surfaces pass smoothly through r = 2m, the foliation of K -Surfaces are very helpful for the procedure of canonical quantization of massive scalar fields. Very useful for the attempts to quantize gravity. They have also been used to study quantization of other fields in curved spacetime backgrounds.
References S. Weinberg, Gravitation and Cosmology (Wiley, 1972). C. W. Misner, K. S. Thorne ans J.A. Wheeler, Gravitation (W. H. Freeman and sons 1973) Eric Poisson, A Relativist’s Toolkit, (Cambridge University Press, 2007) James B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, Addison Wesley(2003) C.Bona and C. Palenuela-Luque, Elements of Numerical Relativity, (Springer 2005) Anosov, D.V. (2001), "Foliation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 L.D. Landau and E. M. Lifshitz, The Classical Theory of Fields, (Pergamon Press, Oxford, 1975) M. D. Kruskal, Phys. Rev., 7(1960)285
119(1960)1743;
R. Penrose, Phys. Rev. Lett.
14,
G. Szekeres, Publ. Mat. Debrecen,
57 (1965).
References Cont. G. Reeb, Actualites Sci. Indust.(1952) L. Smarr and J W. Jork Jr, Phys. Rev., D17(1978)1945. Brill et al, J.Math. Phys.
21(1980) 2789.
A. Pervez, Ph.D. thesis, Quaid-I-Azam University, (1994). A. Pervez, A. Qadir and A. Siddiqui, Phys. Rev.D51 (1995)4598. Ayub Faridi, M.Phil thesis "Foliation of Spacetime" 2002 University of the Punjab and references there in. A. Faridi et al, Preprints PU-CHEP-2010/12 A. Faridi et al, Chin. Phys. Lett. Vol.
23(2006) 3161;
A. Faridi et al. AIP Conf. Proc. 888 (2007) pp.274-277.
