# Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells

Authors: Mikhail Z. Iofa
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates. We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications to the Israel junction conditions are discussed.


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## Body

## 1. Introduction

Thin shells provide a useful tool to study dynamics of collapsing body eventually forming black hole and Hawking radiation from the black hole [1, 2].Dynamics of domain walls was studied by Israel [3], Poisson [4], Ipser and Sikivie [5], Berezin et al. [6], Blau et al. [7], Chowdhury [8], Gladush [9], Kraus and Wilczek [10], and many other authors.There are two natural settings to study geometry of the spacetime with a spherical shell: that based on Kodama-Schwarzschild coordinates and that employing the Gaussian normal coordinate system.Kodama found that in any (possibly time-dependent) spherically symmetric spacetime there exists a conserved vector which is timelike in the exterior of the shell [11]. Although the Kodama vector does not reduce to the Killing vector even in the static spacetime, it can be used to define a preferred “time coordinate” and to construct a geometrically preferred coordinate system for a spherically symmetric spacetime [11–15]. Because the Kodama vector is orthogonal to dr, one can construct the time coordinate t so that dr is orthogonal to dt [15]. Using the Schwarzschild radial coordinate r, one arrives at the diagonal, time-dependent spherically symmetric metric which in this parametrization in the (r,t) sector has the metric components grr=1-2mt,r/r-1 and gtt=b2(r,t)(1-2m(t,r)/r).The Gaussian normal coordinate system in the neighborhood of the shell [7, 16] is constructed by using a family of nonintersecting geodesics orthogonal to the surface Σ swept by the shell. Coordinates of a point outside of the shell are introduced as the geodesic distance from the point to the shell along the geodesic orthogonal to the surface Σ and coordinates of the intersection point of the geodesic with Σ.The aim of the present paper is to study connection between two approaches. We find explicit coordinate transformation between the Kodama -Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications of the above results to the Israel junction conditions are discussed.


## 2. Kodama-Schwarzschild Coordinates

The (2+1) dimensional hypersurface Σ swept by a spherically symmetric shell divides 4D spacetime in two regions V±. Any spherically symmetric metric in D=1+3 spacetime has the general form(1)ds2=ga,bxdxadxb=g2i,jxdxidxj+r2xdΩ2.Here xa=(xi,xα), where xi are coordinates in the base space and θ and φ are coordinates on the spherically symmetric fibers. For any spherically symmetric spacetime it is possible to introduce a vector ka=ε⊥ab∇br (Kodama vector) [11–15, 17], which lies in the radial-temporal plane, where ε⊥ab(2)ε⊥ab=ε⊥ij000.By construction Kodama vector k is orthogonal to ∇ar. Choosing the time coordinate t so that ∂t~k [15], one obtains the metric in the diagonal form because k and dt are orthogonal to dr. In the parametrization through the time coordinate t and Schwarzschild radial coordinate r the metric can be expressed as(3)ds±2=-b±2r,t±f±r,t±dt±2+dr2f±r,t±+r2dΩ2,where f±(r,t±)=(1-2m±(r,t±)/r). In this parametrization m(r,t) is interpreted as the quasi-local mass (Misner-Sharp-Hernandez mass) [18, 19]. Note that in this parametrization the metric is diagonal.Position of the surface is defined by parametric equationsr=R(τ),  t±=T±(τ). The metrics induced on the shell from the regions V± are (4)ds±2=-b±2R,T±f±R,T±dT±2+dR2f±R,T±+R2dΩ2.From the requirement that the metrics induced on Σ from both regions V± coincide (first Israel condition) it follows that (5)-b+2R,T+f+R,T+T˙+2+f+-1R,T+R˙2=-b-2R,T-f-R,T-T˙-2+f--1R,T-R˙2.By choosing τ as the proper time on the surface, one obtains(6)-b±2R,T±f±R,T±T˙±2+f±-1R,T±R˙2=-1,and projection of the metric on Σ is(7)ds2=-dτ2+R2τdΩ2.


## 3. Gaussian Normal Coordinates

Gaussian normal coordinate system in 4D spacetime in which a hypersurface swept by the spherical shell divides into two regions is introduced starting from a certain coordinate systemx^μ with a metric g^μν(x^). The surface Σ is parametrized by coordinates xi=(ξ,θ,φ) Consider a neighborhood of Σ with a system of geodesics orthogonal to Σ. The neighborhood is chosen so that the geodesics do not intersect; that is, any point in the neighborhood is located on one and only one geodesic. Let us consider a point in the neighborhood of Σ with the geodesic orthogonal to Σ which goes through this point. The new coordinate system xμ is introduced in the following way. Three coordinates of the point coincide with the coordinates xi of the point of intersection of the geodesic with Σ. The fourth coordinate of a point is equal to the proper geodesic distance along the geodesic from the point to Σ. The proper length along the geodesic is(8)η=∫0ηdη′gμνxμdxμdη′dxνdη′,where η is the affine parameter along the geodesic. Expression (8) is invariant under the coordinate transformations with Jacobian equal to unity, and we can rewrite (8) through the new coordinates xμ and the metric gμν(x). Taking the derivative over η from both sides of (8) over η, one has(9)gμνx∂xμη,xi∂η∂xνη,xi∂η=1,or gηη(η,xi)=1. Orthogonality condition of the tangent vector to geodesic to the tangent surface to Σ is(10)gμνx∂xμη,xi∂η∂xνη,xi∂xiη=0=0,or gηi(0,xi)=0. The tangent vector ∂xμ(η,xi)/∂ηη=0 is orthogonal to Σ and the vector ∂xν(0,xi)/∂xi is in the plane tangent to Σ.The metrics inV± are (below, to simplify formulas, we omit the subscript ± everywhere, where it does not lead to confusion)(11)ds±2=dη2-p2η,ξdξ2+2qη,ξdτdη+ρ2η,ξdΩ2±.Because of condition (10), on the surface Σ the interval reduces to(12)ds2=-p20,ξdξ2+ρ20,τdΩ2.On the surface Σ reparametrization of τ allows setting p2(0,ξ)=1, which is assumed in the following. It is seen that one can identify τ with ξ and R(τ) with ρ(0,ξ). In the following we use the variable τ.


## 4. Transformation between the Coordinate Systems

Coordinate transformationt±=t±η,τ and r±=r(η,τ) from Kodama-Schwarzschild coordinates xμ=(t,r,θ,φ) to Gaussian normal coordinates x^μ=(η,τ,θ,φ) yields the following relations between the components of the metrics (6) and (11):(13)-b2ft˙2+f-1r˙2=-p2,(14)-b2ft′2+f-1r′2=1,(15)-b2ft˙t′+f-1r˙r′=q,where prime and dot denote derivatives over η and τ. On the surface Σ transformations (13)–(15) are of the same form with the substitution t˙→T˙, r˙→R˙ and q=0, p=1.It is straightforward to obtain solution of systems (13)–(15) in the spacetime regions V± as t˙=t˙p,q,r˙, t′=t′p,q,r˙, and r′=r′p,q,r˙. Instead of writing this cumbersome and not instructive general solution, we consider the restriction of the transformation to the surface Σ which we use as follows:(16)t˙2Σ=T˙2=fT,R+R˙2b2T,Rf2T,R,t′2Σ=R˙2b2T,Rf2T,R,r′2Σ=fT,R+R˙2.Because Σ is orientable, on Σ a normal vector can be defined. In Kodama-Schwarzschild coordinates tangent, uμ, and normal, nμ, vectors to the surface Σ at either side of the surface are(17)u±μ=T˙±,R˙,0,0,u2=1,(18)nμ±=±N-R˙,T˙±,0,0,uμnμ=0,(19)n±μ=±NR˙b±2f±,T˙±f±,0,0.Normalizing n2 to unity, we obtain(20)n2=N2-R˙2b2f+T˙2f=N2b2=1.Transformations of the components of the tangent vector from Kodama-Schwarzschild coordinates (t,r) to Gaussian coordinates (τ,η) are(21)u^η±=uμ±∂x±μ∂ηΣ=-b2fT˙t′+R˙r′fΣ±=0,u^τ±Σ=uμ±∂x±μ∂τΣ=-b2fT˙t˙+R˙r˙fΣ±=-1.The corresponding transformations of the components of the normal vector are(22)n^τ±=nμ±∂x±μ∂τΣ=-R˙t˙+T˙r˙Σ±=0,(23)n^η±=nμ±∂x±μ∂ηΣ=N-R˙t′+T˙r′Σ±=±1.In (22) and (23) we used expressions (16), where all the square roots for T˙,t′Σ, and rΣ′ are taken with the same signs. The upper sign in (23) corresponds to the square roots taken with the sign (+).Next, we consider another method to construct the explicit form of the coordinate transformation from Kodama-Schwarzschild coordinates to Gaussian normal coordinates. The problem can be solved in principle by solving the geodesic equations:(24)d2xμdλ2+Γνλμdxνdλdxλdλ=0.In the general case with metric (3) depending on t the system of nonlinear differential equations is not tractable. Explicit relations can be obtained in the case of metric (3) with the components independent of t (in this case it is possible to set b(r)=1: introducing new variable ρ by the relation dρ=dr/b(r) and denoting k(ρ)=b2f(r(ρ)), we obtain the metric (3) with b=1.). In this case the geodesic equations in Kodama -Schwarzschild coordinates are(25)d2tdλ2+f,rfdtdλdrdλ=0,d2rdλ2+f,rf2dtdλ2-f,r2fdrdλ2+frdθdλ2+frsin2⁡θdφdλ2=0,d2θdλ2=0=d2φdλ2,where λ is affine parameter. The first integrals of the system of equations are(26)dtdλ=Ctτfr,drdλ=±fr+Ct2τ-Cθfrr21/2,dθdλ=Cθ,dφdλ=Cφ.Here r=r(λ,τ),  t=t(λ,τ). To maintain spherical symmetry, we take Cθ=Cφ=0. By construction the vector lμ(λ,τ)=(∂t/∂λ,∂r/∂λ,0,0) is tangent to the geodesic. Let us consider the geodesics orthogonal to Σ. In this case the affine parameter λ can be identified with the parameter η. From (26) it follows that the vector lμ is normalized to unity. At the surface Σ the vector lμ up to the sign coincides with the normal vector (19). Thus, at the surface Σ we have Ct2=R˙2. For lμ(λ,τ) we obtain(27)lμ=±R˙fr,fr+R˙2,0,0.On the surface Σ solution (27) coincides with formulas (16).In Kodama-Schwarzschild parametrization the variablesτ and η have a clear geometrical meaning: at the (t,r) plane τ varies along the trajectory of the shell (R(τ),T(τ)), and η varies along the geodesics orthogonal to the surface swept by the shell.


## 5. Extrinsic Curvature

The extrinsic curvatures at either side ofΣ are (28)Kij±=hμλnν;λdxμdxidxνdxjΣ±,where xi=(τ,θ,φ) are coordinates on Σ, hμν=gμν-nμnν, and (;) denote covariant derivative with respect to gμν±. In Kodama-Schwarzschild parametrization xμΣ=(T,R,θ,φ). In Kodama-Schwarzschild parametrization the nonzero components of the extrinsic curvature are(29)Kττ±=hμλnν;λuμuν±=nν;λuλuν±,Kθθ±=hμλnν;λdxμdθdxνdθ±=hθλnθ;λ±=nθ;θ±±=-ΓθθRnR±,Kφφ±=nφ;φ±=-ΓφφRnR±.Using the identity nμ;νuμuν=-nμu;νμuν, we have Kττ=-uνu;νμnμ. From the identity uνuν=1 it follows that uμuμ;νuν=0 or uTu;νTuν+uRu;νRuν=0. Direct calculation yields (30)u;νRuν=R¨+T˙2ΓTTR+R˙2ΓRRR+2T˙R˙ΓRTR.Kττ is expressed as(31)Kττ=-uνu;νμnμ=u;νRuνnTuRuT-nR=-u;νRuνT˙bR,TfR,T.Using the expressions of Section 4 for uμ and nμ with the upper signs, we obtain the extrinsic curvature in Kodama-Schwarzschild coordinates(32)Kττ±=-1R˙2+fR,TR¨+f,rR,T2+fR,T+R˙2b,rR,TbR,T-R˙T˙f,tR,TfR,T±,(33)Kθθ±=RR˙2+fR,T±,(34)Kφφ±=RR˙2+fR,Tsin2θ±.Here transforming from (31) to (32) we substituted (35)T˙±bR,T±f±R,T±=f±R,T±+R˙21/2which follows from (6).In Gaussian normal coordinates the components of the extrinsic curvature are(36)K^ττ±=n^τ;τ±=-Γττηn^ηΣ±=-12p2,ητ,ηn^ηη=0±,K^θθ±=n^θ;θ±=-Γθθηn^ηη=0±=12r,η2n^ηη=0±,K^φφ±=n^φ;φ±=-Γφφηn^ηη=0±=12r,η2n^ηsin2θη=0±.In the general case of the functions f(t,r) depending on t calculation is straightforward but cumbersome. Below we perform calculation for the case of f±(r) independent of t. Using solutions (27), (37)t′=R˙fr,r′=fr+R˙2,we have(38)t˙′Σ=R¨fR-f,rRR˙2f2R,r˙′Σ=f,rRR˙+2R˙R¨2fR+R˙2.Using (13), we obtain(39)p,η2Σ=ft˙2-f-1r˙2;ηΣ=f,rRr′T˙2+2fRT˙t˙′+f,rRr′R˙2f2R-2R˙r˙′fRΣ.Substituting expressions (38), we obtain(40)Kττ±=-2R¨+f,r±R2f±R+R˙2,K^θθ±=Rf±R+R˙2,K^φφ±=Rf±R+R˙2sin2θ.It is seen that extrinsic curvatures in both parametrizations coincide.


## 6. Israel Junction Conditions

Next, we consider the Einstein equations and the Israel junction conditions. The energy-momentum tensor is taken in the form(41)Tμν=θηSμν++θ-ηSμν-+δηδμiδνjSij.Because the values of the extrinsic curvatures at the opposite sides of Σ are different, the derivative of the extrinsic curvature through the surface Σ contains δ-singularity. From the singular part of the ij component of the Einstein equations projected on Σ, (42)Rij3-δji3R-Kji-δjiK,η-KKij+K2+KjiKij/2=8πGTij,follow the relations,(43)Kij-δijK=-8πGSij,where Kij=Ki+j-Ki-j and  K=Kii. The (ηi) component of the Einstein equations, (44)-Kji∣j+K,i=8πGTηi,(vertical bar stands for covariant derivative with respect to metric (7)) yields(45)Kj±i∣j-Ki±=8πGθ±ηSiη±.From the (ηη) component of the Einstein equations, (46)-R32-KjiKij2-K22=8πGTηη,it follows that(47)-12KjiKij+K2±=8πGθ±ηSη±η.Further restrictions on Sμν± follow from the conservation equations of the energy-momentum tensor [7].Projections of the components of the bulk metricgμν defined as g~ij=gμν∂xμ/∂xi∂xν/∂xjΣ are(48)g~ττ=-1,g~θθ=R2τ,g~φφ=R2τsin2⁡θ.Projections of the tangent vectoru~i=uμ∂xμ/∂xiΣ are(49)u~τ=-1,u~θ=u~φ=0.Assuming that the energy-momentum tensorSij has the form Sij=σ~u~iu~j+ζ~g~ij-u~iu~j, from (43), we obtain(50)Sττ=-σ~+2ζ~,Sθθ=Sφφ=ζ~.From Israel conditions (45) written as (51)Kij=-8πGSij-12δjiSwe have(52)Kττ=4πGσ~,Kθθ=Kφφ=-8πGσ~2-ζ~.In the case of metric (3) with the function f independent of t from (40) one obtains(53)Kτ±τ=1R˙ddτf±R+R˙2,Kθ±θ=Kφ±φ=1Rf±R+R˙2.Israel conditions take a simple form in the case ζ~=0. From the Israel conditions it follows that(54)1R˙ddτR˙2+f+R-R˙2+f-R=4πσ~,1RR˙2+f+R-R˙2+f-R=-4πσ~.Solving this system of equations, we find that Rσ~=const and(55)RR˙2+f+R-R˙2+f-R=M=const.Relation (55) can be rewritten as (cf.[8])(56)R˙2+f±R-1M2m+-m-∓M22R2=0.


## 7. Conclusions

Geometry and connection between the two coordinate systems used to study dynamics of thin shells, Kodama-Schwarzschild coordinates and Gaussian normal coordinate system, were studied. Transformation between the coordinate systems is studied and explicitly constructed for the case of Kodama-Schwarzschild metric independent of time. Extrinsic curvatures of the surface swept by the shell in the ambient space are calculated for a general time-dependent metric in both Kodama-Schwarzschild and normal Gaussian parametrizations and are shown to give the same result. Application to the Israel junction conditions is discussed.



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