MTPR3

K-Surfaces Simulations inSchwarzschild Geometry

M. Ayub Faridi, Fazal-e-Aleem and Haris Rashid

[email protected]

Centre for High Energy Physics, University of the Punjab Lahore, 54590 Pakistan

MODERN TRENDS IN PHYSICS RESEARCH(MTPR-010) K-Surfaces Simulations in Schwarzschild Geometry Ayub Faridi et al – p. 1/39

Abstract

Study of the spacetime dynamics inSchwarzschild Geometry (SG) has always beenin the lime light. Constant Mean ExtrinsicCurvature (CMEC) hypersurfaces, also known asK-Surfaces, play an important role in SG. It hasbeen explained in this talk what spacetimefoliations are and what is the behavior ofK-surfaces for large values of K near essentialsingularity of Schwarzschild black hole.


Contents of Talk

Schwarzschild Geometry

Kruskal-Szekeres Diagram

Penrose Diagram

Curvature

Black Hole

Hypersurfaces

Spacelike Hypersurfaces

Foliation

K-Surfaces

K-surfaces near Essential Singularity

Conclusion

References


Schwarzschild Solution

Schwarzschild solution depicts a staticspacetime containing a single black hole.

The Schwarzschild metric is given by theline element

ds2 = Adt2−A−1dr2−r2dθ2−r2 sin2 θdφ2

where A =(

1 − 2mr

)

m is the mass of the Schwarzschild blackhole as measured at spacelike infinity.

As r approaches 2m, the coefficient of dt2

approaches zero, and the coefficient of dr2

approaches infinity.

r = 2m is a coordinate singularity that can beremoved by coordinate transformation


Kruskal-Szekeres Coordinates

Kruskal-Szekeres(KS) coordinates (v, u) are defined as ;

u =( r

2m− 1

) 1

2exp

( r

4m

)

cosh

(

t

4m

)

v =( r

2m− 1

) 1

2exp

(

t

4m

)

sinh( r

2m

)

The line element of schwarzschild in (KS) coordinates (v, u) is given by :

ds2 = f2(dv2 − du2) − r2(dθ2 + sin2 θdφ2)

where f2 is given as :

f2 =32m3

rexp

(

− 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

(r ≥ 2m,u ≥ 0)

u =(

r2m

− 1) 1

2 exp( r4m

) cosh( t4m

)

v =(

r2m

− 1) 1

2 exp( r4m

) sinh( t4m

)

II

(r ≤ 2m,u ≥ 0)

u =(

1 − r2m

) 1

2 exp( r4m

) sinh( t4m

)

v =(

1 − r2m

) 1

2 exp( r4m

) cosh( t4m

)


Kruskal-Szekeres Diagram Cont.

III

(r ≥ 2m,≤ 0)

u = −(

r2m

− 1) 1

2 exp( r4m

) cosh( t4m

)

v = −(

r2m

− 1) 1

2 exp( r4m

) sinh( t4m

)

and

IV

(r ≤ 2m,u ≤ 0)

u = −(

1 − r2m

) 1

2 exp( r4m

) sinh( t4m

)

v = −(

1 − r2m

) 1

2 exp( r4m

) cosh( t4m

)



For the inverse transformations, t and r are given by :

t =

4m tanh−1( vu) , (In region I and III)

4m tanh−1( uv) , (In region II and IV )

and

u2 − v2 = (r

2m− 1) exp(

r

2m)

In KS-coordinate system, the singularity at r = 0 is situated at :

v2 − u2 = 1

and thus for r = 0, there are two spacelike singularities given by :

v = ±√

1 + u2



We notice that r ≥ 2m is given by u2 ≥ v2. This shows that correspond to r ≥ 2m thereare two exterior regions u ≥ +|v| and u ≤ −|v|. On the same way, r ≤ 2m is given byv2 ≥ u2 representing two interior regions, v ≥ +|u| and v ≤ −|v|, both corresponding tor ≤ 2m.The sphere r = 2m is physically a null surface in which all the vectors on the surface arenull vectors and a geodesic with a null tangent vector lying on the surface will continue tolie on it. It is a trapped surface in which all regions interior to this surface have geodesicswhich cannot emerge out. This is also a red shift horizon (on account of the infinite redshift 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 firstdeveloped by R. Penrose. This technique is based on the conformal transformation ofspacetime which brings infinity to a finite value. Consequently, we can convertasymptotic calculations to finite calculations. This technique also provides precisedefinitions for several types of infinity (spacelike, timelike and null) when one takes upasymptotically flat spacetime. In order to observe the asymptotic structure of spacetime,KS coordinates are used to feature finite coordinate values to infinity. The KScoordinates (v, u, θ, φ) for Schwarzschild background are transformed to CompectfiedKruskal and Szekeres coordinates (ψ, ξ, θ, φ)

v + u = tan

(

ψ + ξ

2

)

v − u = tan

(

ψ − ξ

2

)

and

v2 − u2 =(

1 − r

2m

)

exp( r

2m

)

= tan

(

ψ + ξ

2

)

tan

(

ψ − ξ

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 thenormal curvature of the surface. Maximum and minimum values of normal curvature(independent of the choice of the curvature) are known as principal curvatures. Theaverage of principal curvatures is called Mean Extrinsic Curvature (MEC) denoted by Kand their product is called Intrinsic Curvature. MEC can be computed by the trace ofextrinsic curvature tensor as

K = −nµ; µ, (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 ahomomorphism from each element of its open cover into Rn. A space is said to beseparable if there exist a countable infinite subspace of it whose closure is the entirespace. A space is said to be connected if there does not exist A,B ⊂ X such thatA ∪B = X,A ∩B = Φ = A ∩B where Φ is an empty set. A space is said to beHousdroff if ∀ x, y ∈ X,x 6= y there exist neighborhoods η1(x) and η2(x) such thatη1(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 thecoordinates f(xα) = 0 or by giving parametric representation of the equationsxα = xα(ya) where a = 1, 2, 3..... and ya are coordinates intrinsic to a particularΣahypersurface.


Foliation

"A foliation is a process of decomposition of the spacetime manifold in to a sequences ofone parameter spacelike hypersurfaces of smaller dimension" . The word foliationoriginated from the Greek word "folia", that means leaves. Therefore, foliation is aprocess of splitting the spacetime into space and time. The key idea dates back to thebeginning of the theory of differential equations where trajectories of the solution spacecan be thought of as the leaves. Foliation reintroduces the idea of dynamical systemsdeveloping with time. Spacelike foliation are used to study the evolution of cosmologicaldensity perturbations, where different choices of the parameter are usually referred to asgauges. There is no unique way of performing the foliation process.General Relativitytells us that we must get the same physical outcome regardless of which method ofslicing we use. However calculations may be much simpler with a cleverly chosen slicingmethodology that respects some symmetry of the problem.


Foliation by K-Surfaces

Foliation of geometric manifolds in curved spacetime became an excellent tool forunderstanding of the spacelike hypersurfaces

Constant Mean Extrinsic Curvature (CMEC) hypersurfaces, also known asK-Surfaces, play an important role in the dynamics of curved spacetime

Eardley and Smarr initiated foundational work in solving EFE’s using numericmethods. Brill et al numerically demonstrated the foliation of K-Surfaces

Due to the limitation of computational tools, their calculation was restricted toselective values of K(= 1.2, 2). They conjectured that "if the full set of values of Kis used, then a (complete) foliation can be achieved".

Perusing this work further, Qadir, Pervez and Azad extended this work toK = −0.2 to +0.2 in step size of 0.02. It was claimed that foliation of K surfaces,for |K| ≤ 1 is possible by using comactified KS coordinates. However, inadequatecomputational techniques did not permit to further extend this work for large valuesof |K| > 1.

we developed a precise and user-friendly procedure that numerically allowsfoliation 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, theline element is

ds2 = gµνdxµdxν

For the spherical symmetric spacetime the metric has 10 independent components

gµν = gνµ =

g00 g01 g02 g03

g01 g11 g12 g13

g02 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 S1 at t = 0, an arbitrary hypersurface S at any time tenclose a 4-volume of the spacetime V (S,S1) defined as

V (S, S1) =

∫ t

0

√−gdtdrdθdφ

Any spacelike surface S, described by an implicit function f(t, r, θ, φ) = 0 impliesthat

dt = trdr + tθdθ + tφdφ



The line element with induced metric γij takes the form

dσ2 = γijxixj

The components of γij can be written as

γrr = grr − gtt(tr)2 − 2gtrtr,

γθθ = gθθ − gtt(tθ)2 − 2gtθtθ,

γφφ = gφφ − gtt(tφ)2 − 2gtφtφ,

γrθ = grθ − gtt(tr)(tθ) − gtrtθ − gtθtr,

γrφ = grφ − gtt(tr)(tφ) − gtrtφ − gtφtr,

γθφ = gθφ − gtt(tθ)(tφ) − gtθtφ − gtφtθ



The 3 -Dimensional area, A(S), of the hypersurface is

A(s) =

∫

√

|γij |drdθdφ

Using variational principle general differential equation satisfied by K-surfaces can bewritten as

∂

∂r

( ∂Ω

∂tr

)

+∂

∂θ

( ∂Ω

∂tθ

)

+∂

∂φ

( ∂Ω

∂tφ

)

= λ√−g (1)

where λ is a constant known as Lagrange multiplier and

Ω2 = |detγij |


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 − 1

3Kr3)2 (2)

Where H is constant depending upon fixed parametric values of r and K is the extrinsiccurvature defined using Lie derivative as K = − 1

2Ln|γij |. In Kruskal-Szekeres (KS)

coordinates (v, u) and Compactified Kruskal-Szekeres coordinates(ξ,ψ), Brill et alnumerically demonstrated regular K-Surfaces in Kruskal and Penrose diagrams asshown in Figures on next slide.The slope of regular surfaces in Penrose diagram near I+ is determined by the choice ofK values. These surfaces dip in the region r < 2m, thus avoiding the singularity. Theyhave further pointed out that irregular surfaces for different values of K cannot be usedfor 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) forK-Surfaces with compactified KS coordinates. They demonstrated regular K-Surfacesin Penrose diagrams as shown in Figure below:


Our Work

Extending the earlier work , a different methodology for the foliation of K-Surfaceshas been adopted by us.

This work reiterates that foliation of K-Surfaces is possible including large valuesof K, which has not been done earlier.

we compute K-Surfaces in Schwarzschild geometry using different initialconditions.

Eq. (1) establishes differential equations for K-Surfaces both inside and outsidethe horizon of Schwarzschild spacetime.In order to solve Eq. (1); we use the Kand H values suggested by Eardley and Smarr.

K = ∓ 2r − 1.5

r√r − r2

,

H = ±√

r

1 − r

(2r2 − 3r

6

)


Our Work Cont.

These results have been plotted in Figures below, which depicts the behavior of K andH with variation in r graphically.


Our Work Cont.

Curvature invariants R = R1 and R2, defined as

R1 = Rµνµν

R2 = RρπµνR

µνρπ

etc.,where Rµνρπ the Riemannian tensor, it can be verified that at r = 2m, R1, 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 γij given by Eq. (1), for the Schwarzschild geometry, takes the form

γij =

r4

r4−2mr3+E2 0 0

0 r2 00

0 0 r2sin2θ


Our Work Cont.

Using the reduced metric the curvature invariants R and R2 become

R =6H2

r6− 2K2

3

R2 = 2P 2(r)S(r) − 8mK

r3P (r) +

12m2

r6

where

P (r) =H

r3− K

3

and

S(r) =9H2

r2+

2KH

r3+K2

3− 12m

r3

For finite values of r(6= 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 KScoordinates representing K- Surfaces inside and outside event horizon at r = 2m in theSchwarzschild Spacetime as

For the region r > 2m

dψ

dξ=

(A + E )Sin(ψ + ξ) + (A − E )Sin(ψ − ξ)

(A + E )Sin(ψ + ξ) − (A − E )Sin(ψ − ξ)(3)

For the region r < 2m

dξ

dψ=

(A + E )Sin(ψ + ξ) − (A − E )Sin(ψ − ξ)

(A + E )Sin(ψ + ξ) + (A − E )Sin(ψ − ξ)(4)

where

E =3H −Kr3

3,A 2 = E

2 + r3(r − 2m).


Our Work Cont.

Numerical solutions of Eq.(3) and Eq.(4) provide complete foliation ofSchwarzschild black hole spacetime by K-Surfaces.

For a particular value of K = K1, one K-Surface can be obtained by using initialcondition ψ(π − ε) = 0 + ω and setting ε and ω very very small approaching zero.

Using Eq.(3), each K-Surface begins from A and reaches B and D in region I ofPenrose diagram. From B to C, K-Surfaces satisfy Eq.(3) in region II of Penrosediagram. In a similar manner, K -Sufaces are simulated in the lower half of thePenrose diagram.

Each K-Surface is generated by three parameters H,K, r and the sign of A . Itmay 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 SeeTable.

The simulated graphs for different values of H,K and r are presented. Eachhorizontal side represents ξ-axes and vertical side represents ψ-axes.


Table for H, K, r

No. K r H ψ = ψatξ=0 ψmax/min δψ

±1 ±0.44137 0.68000 ∓0.270945 ±0.0984 ±0.14 0.0416

±2 ±0.575728 0.66000 ∓0.257475 ±0.207 ±0.279 0.072

±3 ±0.716146 0.64000 ∓0.244622 ±0.315 ±0.41 0.095

±4 ±0.863961 0.62000 ∓0.232304 ±0.45 ±0.52 0.07

±5 ±1.02062 0.60000 ∓0.220454 ±0.562 ±0.632 0.07

±6 ±1.18772 0.58000 ∓0.209018 ±0.673 ±0.731 0.058

±7 ±1.36702 0.56000 ∓0.197953 ±0.772 ±0.819 0.047

±8 ±1.56056 0.54000 ∓0.187224 ±0.873 ±0.884 0.011

±9 ±1.77065 0.52000 ∓0.176803 ±0.954 ±0.954 0

±10 ±2.00000 0.50000 ∓0.166667 ±1.02 ±1.02 0

±11 ±3.57217 0.45000 ∓0.119753 ±1.14 ±1.14 0

±12 ±43.333 0.10000 ∓0.01555 ±1.25 ±1.25 0

±13 ±540 0.0135 ∓0.001077 ±1.45 ±1.45 0MODERN TRENDS IN PHYSICS RESEARCH(MTPR-010) K-Surfaces Simulations in Schwarzschild Geometry Ayub Faridi et al – p. 30/39

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 dependenceon the parametric values of H,K and r.

K-Surfaces completely foliate the spacetime and these are very clearly shownnear past and future essential singularities.

The difference of the local maximum and minimum values of ψ (say δψ) vary fromsurface 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 thepresent work provides a complete foliation.

As the K-Surfaces pass smoothly through r = 2m, the foliation of K-Surfaces arevery 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 studyquantization 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 sons1973)

Eric Poisson, A Relativist’s Toolkit, (Cambridge University Press, 2007)

James B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, AddisonWesley(2003)

C.Bona and C. Palenuela-Luque, Elements of Numerical Relativity, (Springer 2005)

Anosov, D.V. (2001), "Foliation", in Hazewinkel, Michiel, Encyclopaedia ofMathematics, 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., 119(1960)1743; G. Szekeres, Publ. Mat. Debrecen,7(1960)285

R. Penrose, Phys. Rev. Lett. 14, 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 Punjaband 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.


Thanks

MODERN TRENDS in PHYSICSRESEARCH (MTPR- 2010) OrganizingCommittee

Audience for Listening and Bearing


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