Junctions and thin shells in general relativity using computer algebra: I. The Darmois Israel formalism

Transcription

1 Class. Quantum Grav. 13 (1996) Printed in the UK Junctions and thin shells in general relativity using computer algebra: I. The Darmois Israel formalism Peter Musgrave and Kayll Lake Department of Physics, Queen s University, Kingston, Ontario, Canada Received 15 December 1995, in final form 4 March 1996 Abstract. We present the GRjunction computer algebra program which allows the study of non-null boundary surfaces and thin shells in general relativity. Implementing the Darmois Israel thin-shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straightforward way. We have used the package to correctly reproduce a wide variety of examples from the literature. In this paper GRjunction is used to perform two new calculations: joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit, and the calculation of the stress energy of a Curzon wormhole. The Curzon wormhole has the interesting property that shells located at radius R < 2m have regions which satisfy the weak energy condition. PACS number: Introduction The Darmois Israel junction/thin-shell formalism has found wide application in general relativity and cosmology [1, 2]. The junction of dust to Schwarzschild by Oppenheimer and Snyder allowed the first insights into the nature of gravitational collapse to a black hole [3]. Since Israel s landmark paper [2] the formalism has been applied in a number of contexts including further studies of gravitational collapse, the evolution of bubbles and domain walls in a cosmological setting and wormholes. In this paper we describe the GRjunction package we have developed to assist relativists in the evaluation of junction conditions and the parameters associated with thin shells (the package is available free of charge [4]). At the present time the package deals only with non-null surfaces although efforts to extend this to null shells are underway. GRjunction runs under the computer algebra system Maple [5] in conjunction with GRTensorII [6]. Our goal in creating the package was to ensure that it could easily recover all the standard shell results in the literature (the bulk of which assume spherical symmetry) without biasing the package towards spherical symmetry in any way allowing users to probe the relatively unstudied area of non-spherical shells and junctions. The package is necessarily interactive, allowing users to manipulate results and determine the conditions for junctions. We begin by outlining the shell formalism to establish notation and motivate choices of algorithms which we describe in the following section. Next we present new results relating musgrave@astro.queensu.ca lake@astro.queensu.ca /96/ $19.50 c 1996 IOP Publishing Ltd 1885

2 1886 P Musgrave and K Lake to the study of shells around slowly spinning black holes. Finally the calculation of the stress energy of a Curzon wormhole is given. We believe this is the first example of an exact general relativistic thin-shell calculation which does not rely on spherical, cylindrical or planar symmetry. The wormhole itself is physically interesting since for some values of shell radius portions of the shell can satisfy the weak energy condition. The intention of this paper is to describe the junction package we have developed and not to review the vast literature on junctions and thin shells. The references we have chosen are not always the first or simplest treatment of a problem and in some cases we have deliberately selected examples which differ from the standard treatments to test the robustness of our package. 2. The formalism In this section we review the junction formalism to establish notation and clarify some issues associated with the computer implementation (see also, e.g., [7 10]). Consider two spacetimes (Lorentzian manifolds with signature (+ + + )) M + and M with metrics g + αβ (xγ +) and g αβ (xγ ) in the coordinate systems x γ + and x. γ Within these spacetimes define two non-null 3-surfaces + and (in M + and M, respectively) with metrics g + ij (ξc + ) and g ij (ξc ) in the coordinates ξc + and ξc which decompose each of the 4-manifolds into two distinct parts. (Greek indices range over the coordinates of the 4- manifold and Latin indices over the coordinates of the 3-surfaces.) We label the distinct parts of M + created by + as M + 1 and M + 2 and likewise for M. The junction/shell formalism constructs a new manifold M by joining one of the distinct parts of M + to one of the distinct parts of M by the identification + =. Clearly there are four possibilities, i.e. M + 1 M 1, M+ 2 M 1, M+ 1 M 2, M+ 2 M 2. The assumed isometry between the points on the surface is often (but need not always be) the identification ξ+ c = ξc. What now follows holds simultaneously for M + and M and so we drop the ± distinction in this paragraph. The parametric equation for is of the form f (x α (ξ a )) = 0. (1) We assume that is non-null. The unit 4-normals to in M are given by ( ) βγ f f 1/2 n α = ± g f x β x γ x. (2) α We assume n α 0 and label as timelike (spacelike) for ǫ n α n α = 1 (1). The three basis vectors tangent to are e α (a) = xα ξ a (3) which give the induced metric on as g ij = xα ξ i x β ξ j g αβ. (4) The extrinsic curvature (second fundamental form) is given by K ij = xα ξ i x β ξ j αn β (5) ( 2 x γ = n γ ξ i ξ + x α x β j Ŵγ αβ ξ i ξ j ). (6)

3 Junctions and thin shells in general relativity: I 1887 Define u α as any unit 4-tangent to (u α n α = 0, u α u α = ǫ) and the associated 4-acceleration u α ( u β β u α ). Since u α = u i xα ξ i (7) we can find the associated intrinsic 3-tangent u i. Combining these with the above relations gives [ nα u α] = u i u j [ K ij ] (8) and n α u α = u i u j K ij (9) where [X] X + X and X (X + + X )/2 with X ± denoting the limiting values of X on. The Darmois conditions for the joining of a part of M + to a part of M are [ gij ] = 0 (10) and [ Kij ] = 0. (11) If both (10) and (11) are satisfied we refer to as a boundary surface. If only (10) is satisfied then we refer to as a thin shell. Conditions (10) and (11) require a common coordinate system on and this is easily done if one can set ξ+ a = ξa. Failing this, establishing (10) requires a solution to the threedimensional metric equivalence problem. Condition (11) as it stands is ambiguous since the orientation of the 4-vector field n α n ± α has not been specified. The Israel formalism requires the normals in M to point from M A to M+ B (where A denotes the part of M and B denotes the part of M + we wish to use to form M). Clearly the sign of the normal vectors are crucial since, e.g., n α points away from the portion of M which will be used in forming M. Hence an understanding of which side n α points into is key. In general this can be done by considering a trajectory in M through with tangent n α (and likewise for n + α ). The majority of the existing literature deals with spherical symmetry where the direction of the normal is clear, but in more complicated examples (see below) great care must be taken. Note that while there are two normal vectors in each of M ±, once we have identified and + there is a single unique normal field to in M. There may be circumstances in which one can determine the differential relation between the coordinates of M A and M+ B via n α = xα + x β n + β (12) in an open neigbourhood of in M and this will give the direction of one normal vector relative to the other. (In the case of a boundary surface often only the sign of n α on one side of needs to be determined. Since α n α = Ki i in the case of a boundary surface, (11) gives the useful relation [ α n α ] = 0.) Some studies have left the sign of the normal vectors unspecified to exhaustively study the taxonomies of all possible combinations of M and M + (usually excluding those which require shells which violate energy conditions, e.g. [12, 13]). The junction package allows the user to leave the sign unspecified but we take the view that the typical starting point is to explicitly choose the signs of the normal vectors so that a particular combination of M A and M+ B can be studied.

4 1888 P Musgrave and K Lake Once we have selected signs in (2) there is no ambiguity in (11) and we use (10) and (11) in conjunction with the Einstein tensor G αβ to obtain the identities [ Gαβ n α n β] = 0 (13) and [G αβ x α ξ i nβ ] = 0. (14) (Note that (13) and (14) do not guarantee (10) and (11).) This shows, for example, that for timelike the flux through (as measured comoving with ) is continuous. The Israel formulation of thin shells follows from the Lanczos equation S ij = ǫ 8π ([K ij ] g ij [K i i ]) (15) and we refer to S ij as the surface stress energy tensor of. The ADM constraint j K j i i K = G αβ x α ξ i nβ (16) along with Einstein s equations then gives the conservation identity [ ǫ i Sj i = x α ] T αβ. (17) The Hamiltonian constraint gives the evolution identity ξ i nβ G αβ n α n β = (ǫ( 3 R) + K 2 K ij K ij )/2 (18) ǫs ij K ij = [ T αβ n α n β]. (19) The identities (17) and (19) do not give information about the dynamics of the shell. The evolution of the shell stems from a phenomenological interpretation and the Lanczos equation (15) Phenomenology The GRjunction package does not make assumptions about the phenomenology to be attached to. The user is free to choose the phenomenology. Consequently the Segré type of the intrinsic stress energy tensor is not predetermined by the package. However the package does provide support for what we will term the standard phenomenology. The standard phenomenology associated with is introduced as follows: let ξ a be the coordinates intrinsic to and choose a trajectory ξ a (τ) with associated 3-tangent u a = dξa (20) dτ where u a u a = ǫ. For a timelike surface we view the curve as the worldline of a (possibly hypothetical) particle in the surface with τ as the proper time. (If the surface is spacelike the phenomenology is a formal analogy.) The associated 4-tangent is or equivalently u α ± = xα ± ξ a ua = dxα ± dτ (21) u a = u ± α x α ± ξ a. (22)

5 Junctions and thin shells in general relativity: I 1889 We view as covered by a 3-vector field u a. Note that in general we cannot ensure that observers on adjacent trajectories have clocks which run at the same rate (as is the usual assumption on spherical surfaces). If this can be done over the entire surface then we label this as a synchronous surface. The surface energy density σ associated with the trajectory u a on is defined by S ab u b = σ(ξ c )u a q a (23) where u a q a = 0. The inclusion of q a represents an intrinsic energy flux orthogonal to the prescribed intrinsic velocity field. The surface energy density σ(ξ c ) is not in general an eigenvalue of S ab, but is given by σ(ξ c ) = ǫs ab u a u b. (24) In analogy to a perfect 4-fluid we can suppose that q a = 0 and that S ab takes the form S ab = ǫ(σ(ξ c ) + p(ξ c ))u a u b + p(ξ c )g ab. (25) This defines the surface pressure ( surface tension) p(ξ c ) = (σ(ξ c ) + Sa a )/2. (26) It now follows from (17) that x σ + (σ + p) = ǫ [T α αβ ξ i nβ where an overdot signifies the intrinsic 3-derivative along u a ( σ = u a a σ ) and is the 3-expansion of u a ( a u a ). From (8), the Lanczos equation (15) and the phenomenological equations (24) and (26), we have [ nα u α] = 8π(p + σ/2). (28) From (9), (15) and the identity (19), the phenomenology gives ] (27) ǫ(σ + p)n α u α + pk i i = ǫ [ T αβ n α n β]. (29) In spherical spacetimes the first integral of the evolution equation (29) is given by the identity [11] ( ) [m] 2 Ṙ 2 = ǫ + 2ǫm ( ) M 2 M R + (30) 2R where R = R(τ), d/dτ and M = 4πσR 2. The effective mass, m ± is given by m ± 1 2 ((4) g ± θθ )3/2 (4) R ± θφ θφ. (31) 3. Computer implementation Specifying the junction formalism for a computer algebra system requires a careful choice of specific object definitions and recognition of several standard calculus manipulations. Joining spacetimes requires the simultaneous consideration of four metrics, so the underlying general relativity software must permit this. In order to make the package reproduce the existing literature on spherical shells, special consideration must be given to the choice of object definitions for one-parameter shells. We first present an overview of the package and then discuss the choices we have made in implementing the junction/thin-shell package for the Maple version of GRTensorII.

6 1890 P Musgrave and K Lake 3.1. Overview The GRjunction package provides the user with a means to specify a surface, calculate intrinsic and extrinsic quantities on the surface, identify two such surfaces and evaluate whether a boundary surface or thin shell results. A manual for the package which includes specific examples of the package in operation (in the form of Maple worksheets) is available [4]. The specification of a surface is done by invoking the command surf which then prompts the user for the necessary information. Once a surface has been specified, objects defined on the surface (e.g. K ij ) can be calculated. The identification of two surfaces is performed via the command join which calculates [ ] [ ] g ij and displays the result. If gij 0 the user can manipulate this expression in an attempt to determine restrictions on metric functions which will ensure [ ] g ij = 0. The jump or mean of any quantity in the joined manifold can be evaluated by means of the operators Jump and Mean. Hence to determine [ ] K ij the user would refer to the object Jump[K(dn,dn)]; to determine K ij, Mean[K(up,up)]. Standard quantities and equations for the joined spacetimes can be calculated in a straightforward manner. For example to determine S j i the user refers to the object S3(dn,up). (By convention we list dn indices ahead of up indices for mixed two-index objects). While we have emphasised the Darmois Israel formalism, GRjunction can trivially evaluate the Lichnerowicz junction conditions [14] which consist of [ ] gαβ = 0 (32) [ gαβ / x γ] = 0 (33) (recall that these conditions require admissible coordinates and consequently are not as general as the Darmois Israel conditions). The object g αβ / x γ is referred to as g(dn,dn,pdn) (pdn denoting the partial derivative index) and so (33) can be evaluated by referring to the object Jump[g(dn,dn,pdn)]. GRTensorII also allows users to work within the Newman Penrose formalism and GRjunction will allow users to evaluate jumps in the spin coefficients across a surface (this technique is employed in, e.g., [15]) The surface and related quantities In this section we describe how to specify a non-null surface to the junction package and the intrinsic (on ) and extrinsic quantities which can be calculated with the package. In this section we restrict attention to quantities which are independent of phenomenology. All the expressions in this section should implicitly carry a ± designation (which we omit) with the exception of the ξ i, the coordinates on. For the package to compare first and second fundamental forms we must have ξ+ i = ξi. The package does not currently consider the 3-metric equivalence problem. In general, to specify a surface in a space M with coordinates x α in sufficient detail to allow calculation of the first and second fundamental forms we require: the coordinates ξ i on ; the coordinate definition of : x α = x α (ξ i ); the parametric definition of : f (x α ) = 0; the choice of normal vector sign in (2). The essential idea of the Darmois Israel formalism is to use intrinsic quantities in the description of all objects of interest on. However many of the definitions of objects on the surface are in terms of quantities in M. For example, (6) uses the normal vector which

7 Junctions and thin shells in general relativity: I 1891 involves partial derivatives with respect to the coordinates of M and Christoffel symbols of M. This can frequently be resolved simply by substituting the coordinate definition of the surface (i.e. x α = x α (ξ i )) but this is not always desirable. Consider a metric which has an arbitrary function u(r, θ) and a definition of with coordinates ξ a = ( θ, φ, τ) and which has a coordinate definition which includes r = f (ξ i ) and θ = θ. The Christoffel symbols used in defining K ij may contain partial derivatives of u with respect to both r and θ. Substitution of θ = θ will merely change the variable, but the r substitution will result in u(f (ξ i ), θ)/ f (ξ i ). This is more than notationally ugly it constitutes an error in Maple; you cannot take a partial derivative with respect to a function. In these cases we make use of the Maple operator D (instead of the Maple procedure diff). For example, the representations of the r derivatives of f (r) and ν(r, θ) on the surface are represented in Maple as d dr f (r) r=r(t) D(f )(R(t)) (34) d dr ν(r, θ) r=r(t) D [1] (ν)(r(t), θ) (35) (the [1] indicates that the derivative acts on the first argument of the functional ν). It is essential that we make use of this form of representing derivatives to ensure that their time dependence is handled properly in subsequent calculations. While we could use the D derivative throughout we prefer to minimize its use since in the Maple output we find it preferable to have expressions with f (r)/ r instead of D(f)(r). Consequently object definitions which take derivatives and then evaluate values on the surface make use of code which substitutes in the coordinate relations and ensures that the substitution does not cause errors in derivatives. This code then minimizes the number of D derivatives which appear by converting extraneous occurrances back into Maple s usual diff derivative Synchronous surfaces A large part of the existing shell literature has dealt with timelike spherical 3-surfaces within spherical 4-manifolds. The majority of these analyses define one of the ξ a to be the proper time on the surface (what we referred to above as synchronous shells). We must take some care in choosing algorithms for objects in these cases, since some of the routine steps in a hand calculation are best avoided in a computer algebra approach. All of the algorithm issues arise in the specification of a surface in one of M ± so we limit our discussion to the specification of one surface. First we describe two standard timelike surfaces we will make reference to in our discussion: a static spherically symmetric surface and the dynamic counterpart within a spherically symmetric 4-manifold. In the 4-manifold we take coordinates (r, θ, φ, t) and on the 3-surface coordinates ( θ, φ, τ). For the static shell we use the coordinate definitions r = R, θ = θ, φ = φ, t = T s (τ) (36) and a surface equation r R = 0. For the dynamic shell we use r = R(τ), θ = θ, φ = φ, t = T d (τ) (37) and a surface equation r R(τ) = 0.

8 1892 P Musgrave and K Lake In both cases it is conventional to ensure that the coefficient of dτ 2 on is 1 (hence τ is the proper time on the shell). For this reason we express t as a function of τ and then make use of the constraint u α u α = 1 to eliminate, e.g., T d (τ)/ τ from the quantities we calculate on. This facility is built into the surf command. The user is asked if the surface has a parameter which governs its evolution. If the user so indicates then the package evaluates u α u α and asks if there is a quantity to be eliminated. An attempt is then made to solve the tangent constraint so that this quantity can be eliminated. The tangent constraint is then automatically applied during the calculation of n α and K ij. We must specify an algorithmic approach to calculating quantities such as K ij and u α which recover the results for one-parameter shells in a straightforward way. Not all object definitions are equivalent because some require steps which are obvious in a hand calculation but are difficult for a computer algebra system. One example of this is the calculation of u α. For a synchronous shell with x α = x α (τ) we would proceed with the definition: u β β u α = u β ( u α (τ) x β + Ŵ α βγ uγ ) = dxβ (τ) u α (τ) + Ŵ α dτ x β βγ uγ u β = duα dτ + Ŵα βγ uγ u β. (39) While in a hand calculation using the definition of a total derivative to get the last line is an obvious step, this is not so in a computer algebra system. Once specific components are used in (38) and partial derivatives are taken, recognizing terms which can be collected into total derivatives with respect to τ becomes a problem in pattern matching. To avoid such problems we require that the user indicates the variable to be used for total derivatives and we use relation (39) instead of (38). The same issue arises in the calculation of K ij for the dynamic shell discussed above if we use (5), so we use the definition (6). This definition has the advantage that it is expressed in terms of derivatives with respect to intrinsic quantities (although we still need to evaluate the Christoffel symbols of the 4-manifold on ). Another computer algebra issue arises in the calculation of the normal vector. For oneparameter surfaces it is customary to write the equation of the surface in terms of a function of a parameter, (e.g. r R(τ) = 0). Yet the covariant derivative in M will require partial derivatives with respect to the x α. The junction package requires that the surface equation be in terms of the x α (so for the dynamic shell example above we specify r R(t) instead of r R(τ)). In these cases the package then applies the chain rule dr dt dr dt dr(t) dτ dτ dt (38) dr(τ)/dτ dt (τ)/dτ. (40) where we have made use of the relations r = R(τ) and t = T (τ) and that Ṫ 0. If the surface equation is governed by a parameter other than τ then the normal vector and K ij can be expressed in terms of intrinsic quantities only if the function dependence on the other ξ i can be given explicitly in the surface equation (e.g. r r(t) cos θ = 0). If all that is known is, e.g., r R(t, θ) then we cannot make use of (40) and n α and K ij will be left in terms of the partial derivative with respect to the coordinates of M. With these definitions and constraints we now have available the quantities: n α, u α, u α, g ij and K ij in terms of intrinsic quantities. The remaining objects and equations in section 2 are calculated in a straightforward way. All of these objects and equations can be evaluated in the junction package.

9 Junctions and thin shells in general relativity: I 1893 Once the user has loaded two 4-manifolds and specified two 3-surfaces the command join is used to identify the two surfaces. A variety of objects relating to the boundary surface or shell can now be calculated (e.g. S ij, equations (15) (30)). The user now has four active metrics in the GRTensorII session ( ±, M ± ). After using join the default metric is +. The operators Jump and Mean will by default make reference to objects in ±. In general these operators can be used to take the jump or mean of objects from the default metric to any user specified metric. For example, if the user had changed the default metric to M + (say Schw) and wished to evaluate (32) with M as, e.g., SchwInterior this would be done by using the metric name of M as a second parameter to Jump, i.e. Jump[g(dn,dn),SchwInterior]. In practice users may wish to structure calculations and determine results in a variety of ways and this can require minor differences in the definitions of objects. Consider a user who wishes to study the dynamics of a Minkowski shell Schwarzschild scenario. After specifying the two surfaces and joining them they might elect to examine the density and pressures of the shell as given by (24) and (26), which is done by referring to the objects sigma and P. When considering the dynamics they may wish to have σ and P in explicit form or they might opt to have them appear simply as σ(ξ a ) and P(ξ a ), since this allows them to set P(ξ a ) = 0 to study the dust case. Consequently there are two versions of the history and conservation equations in the GRjunction package. Further variations are provided to allow users to specify a stress energy of the 4- manifold for use in the conservation and history equations. A user might opt to specify, e.g., Tθ θ = p(r) instead of using Gθ θ /8π which might be a less transparent combination of functions of the 4-metric. This requires the definition of separate GRjunction objects for each case. The implementation of (30) is complicated by the fact that the definition requires coordinates θ and φ in M ± and makes reference to the shell function R(τ). Users must adhere to these conventions (by using coordinate names theta, phi and r and specifying r = R(τ)) if they wish to make use of this relation for the first integral of the evolution equation. The junction package checks for compliance with these conventions before it will evaluate (30) to ensure that spurious results will not be generated. 4. Verification In developing the junction package we have re-executed a number of calculations performed in the literature to verify that the package can reproduce these results. We list some of the verifications we have performed in table 1 (Maple worksheets for these verifications are available by ftp [4], the dust shell and Oppenheimer Snyder collapse are used as tutorial examples in the GRjunction manual). In addition we have verified the identities (17) and (19) for a variety of spacetimes including a number of non-spherical wormhole spacetimes. 5. Non-spherical junctions and shells In this section we demonstrate the application of GRjunction to some nonspherical/planar/cylindrical thin-shell problems. These examples make clear the utility of having a computer algebra tool to perform the often quite laborious algebraic manipulations. Explicit examples of the use of the package, including ready to run Maple worksheets, are available electronically [4].

10 1894 P Musgrave and K Lake Table 1. Results verified with GRjunction. Problem Verified Comments Dust shell [2] K ij Minkowski shell Schw. evolution equation dynamics FRW dust Schw. Schwarzschild [3] Derive the required junction conditions Spherical K ij Checked form of K ij for general symmetry [9] spherical symmetry Inhomogeneous slab cosmology [16] [ ] [ gij ] = 0 Planar symmetry Kij = 0 Schwarzschild S ij, σ, p Thin-shell wormhole wormhole [17] formed by joining Schw. exterior to Schw. exterior Schw. desitter S ij Spherical symmetry but uses shell [18] coordinates (r, v, θ, φ) Rotating dust [ ] gij = 0, K + ij to order a 3 Match to Kerr exterior shell [19] S ij to order a in small a limit (non-spherical surface) Collapsing, rotating S ij Match to Kerr to order a dust shell [20] (dynamic shell) 5.1. A complicated Minkowski junction In this section we describe the junction of the Minkowski metric in the form ds 2 = r2 + u 2 r 2 + a 2 dr2 + r2 + u 2 a 2 u 2 du2 + (a2 u 2 )(r 2 + a 2 ) dφ 2 dt 2 (41) a 2 to a metric ds+ 2 of the same form, but with coordinates (R, U,, T ) and parameter A. These metrics arise from setting m = 0 in a Kerr metric and choosing u = a cos θ. The coordinate names r and R, while standard, are quite misleading since surfaces of constant radius describe spheroids with oblateness governed by a or A. We seek to join a surface of constant radius in M to M +. The surface in M + will be some function of R and U. This demonstrates GRJunction s ability to handle non-spherical surfaces in a case where we know a boundary surface must result. The transformations from, e.g., (41) to the Minkowski metric in spherical form (with coordinates r and θ) are given by u = a cos θ (42) r 2 = r 2 + a 2 sin θ 2 r cos θ = r cos θ. (43) If we take the definition of as r = X (X a constant) then we can use (42) to determine that the the definition of the corresponding surface in M + has the parametric equation: 0 = X (R 2 + A 2 U 2 a ( 2R2 A 2 + 2a 2 A 2 2A 4 + 2A 2 U 2 + 2(R 4 A 4 2R 2 A 4 a 2 + 2R 2 A 6 2R 2 A 4 U 2 + a 4 A 4 2a 2 A 6 + 2a 2 A 4 U 2 + A 8 2A 6 U 2 + A 4 U 4 + 4R 2 U 2 a 2 A 2 ) (1/2) )/A 2 ) (1/2) (44)

11 Junctions and thin shells in general relativity: I 1895 and the relations x α + = xα + (ξi ) are R = ( A ( X2 + A 2 + u 2 a 2 + (X 4 2X 2 A 2 2X 2 u 2 + 2X 2 a 2 + A 4 + 2A 2 u 2 2a 2 A 2 + u 4 2a 2 u 2 + a 4 + 4X 2 u 2 A 2 a 2 ) 1/2 ) + X 2 + a 2 u 2 ) 1/2 (45) U = 1 2 ( X 2 + A 2 + u 2 a 2 + (X 4 2X 2 A 2 2X 2 u 2 + 2X 2 a 2 + A 4 + 2A 2 u 2 2a 2 A 2 + u 4 2a 2 u 2 + a 4 + 4X 2 u 2 A 2 a 2 ) 1/2 ) 1/2 (46) = φ (47) t = T + (τ). In this case it not clear which sign we should choose for the normal vector in M +. However since we know a priori that a boundary surface must result if the package does not reach this result then we can consider the other choice of normal sign for n + α. (In this case we choose the plus sign in (2).) The package determines [ ] g ij = 0 and that K ij is: Kuu = X X 2 + a 2 (a 2 u 2 ) K X 2 + u 2 φφ = X X2 + a 2 (a 2 u 2 ) a 2. (49) X 2 + u 2 With careful direction of the computer simplification the junction package determines that K + ij is also given by (49) Joining Kerr to Kerr In this section we consider joining the interior region of one Kerr spacetime M in Boyer Lindquist form ( dr ds 2 2 ) = ρ + dθ 2 + (r 2 δ + a2 ) sin θ 2 dφ 2 dt 2 + 2mr ρ (a sin θ 2 dφ dt ) 2 (50) ρ = r 2 + a2 cos θ 2, δ = r 2 2mr + a 2 with mass m and angular momentum a to the exterior region of another Kerr spacetime M + with mass M and angular momentum A (we use coordinates (r ±, θ ±, φ ±, t ± ) for M ± ). Such a problem would arise if a thin shell of matter was constructed around a Kerr black hole (i.e. a Dyson sphere). The general problem is an extremely difficult one as the toy problem m = M = 0 indicates (see section 5.1). In this section we limit the discussion to an expansion to first order in a and A, since to this order the surface in M ± is spherical. In the treatment of timelike spherical surfaces it is customary to use transformations such as r = R(τ) and t = T (τ) and then use u α u α = 1 to eliminate T (τ)/ τ. Note that in these spherical cases this produces g ττ = 1 on as desired. In all spherical cases T will be strictly a function of τ but this fails to be true in Kerr. For Kerr spacetimes to In an earlier attempt to determine K ij + (using Boyer Lindquist coordinates) we encountered the Maple error Object too Large which occurs on 32-bit machines when an expression contains more than terms. On present day workstations this limit can be reached in a matter of minutes and often little can be done to circumvent this Maple limitation. (48)

12 1896 P Musgrave and K Lake achieve g ττ = 1 we can use the same idea but if we blindly use t = T (τ) we discover that in actuality t = T (τ, θ). We can try again using θ as an argument and we do get τ as the proper time on the shell but now a T (τ, θ)/ θ appears in g θ θ, g θτ and g θ φ. Since we know T/ τ is merely a function of θ we can integrate trivially but this makes those components with a T/ θ depend linearly on τ and the metric components are explicitly dependent on proper time. Hence forcing τ to be the proper time on the surface comes at considerable expense. Fortunately these problems do not arise in the order (a, A) expansions. To facilitate the matching of the g ij on we eliminate the g φτ terms on by using a transformation to the zero angular momentum (ZAMO) frame in the definition of the surface. The transformations are r ± = R, θ ± = θ, φ ± = φ ± T ± (τ, θ), t ± = T ± (τ, θ) (51) where g φt /g φφ. Using GRjunction we calculate the metric and second fundamental form on the surface and only then expand to order a, identify ± and calculate S ij. The package first determines ds 2 ± = R 2 d θ 2 + R 2 sin θ 2 d φ 2 dτ 2 (52) and consequently [ ] g ij = 0. The resulting stress energy of the shell is (from this point on we drop the tildes on θ and φ): Sθ θ f (R M) F(R m) = 8πR 2 f F S φ φ = Sθ θ S τ φ = R2 sin 2 θ S φ τ S φ τ S τ τ = 3(am AM) 8πR 2 = f (R 2M) F(R 2m) 4πR 2 f F where f = 1 2m/R, F = 1 2M/R. Note the appearance of the mixed term Sφ τ which precludes a standard phenomenological interpretation (i.e. σ as given by (24) is not an eigenvector of S ij ). To first order we have = am R, 3 + = AM (53) R 3 and if we require = + then the mixed terms vanish. This now permits a standard phenomenological interpretation of the shell and we can now interpret the density and pressures in the usual way. Note that in this case the pressure is isotropic and hoop stresses do not arise. We can easily extend this calculation to the dynamic case with R R(τ) in the above. The resulting stress energy tensor is: S θ θ = R2 R(f F) + f (R M) F(R m) 8πR 2 f F S φ φ = Sθ θ S τ φ = R2 sin 2 θ S φ τ

13 Junctions and thin shells in general relativity: I 1897 Sτ φ = 3 AMf (R 5 (Ṙ 2 R + R 2M)) 1/2 amf(r 5 (Ṙ 2 R + R 2m)) 1/2 8π R 7 f F S τ τ = f (R 2M) F(R 2m) 4πR 2 f F where = d/dτ. In this case the condition = + does not result in a diagonal stress energy tensor. Hence to find the density of the shell we must find a timelike eigenvector of S j i and the associated eigenvalue σ. To proceed we consider a vector (0, u φ, u t ), u i u u = 1 in the equation S j i ui = σu j. (54) Calculation of the somewhat lengthy components of u i is straightforward. The resulting shell density is X 2 σ = σ Sτ φ X + 2 X = (Sφ τ ) 2 R 2 sin 2 θ 2 (Sθ θ + (55) σ)2 σ = S τ τ. (56) Here we can see that the resulting density of the shell deviates from the static value by a correction factor given by the second term in (55). This factor has an angular dependence and the energy density is not constant over the shell Curzon wormhole In this section we construct a non-spherical thin-shell wormhole by a cut and paste of two copies of the Curzon metric [21] in the form ds 2 = e 2(k U)( dr 2 + r 2 dθ 2) + e 2U r 2 sin 2 θ dφ 2 e 2U dt 2 (57) with u = m/r, k = m 2 sin 2 θ/(2r 2 ). In these coordinates surfaces of constant r are not spherical nor is θ the usual angular coordinate. We construct our thin-shell wormholes in the standard way (see e.g. [22]) by joining the r > R region of one copy of the Curzon spacetime to the same region of a second copy of the spacetime (we use the sub/superscripts (1) and (2) to distinguish these copies). We do this by taking the parametric definitions of the surfaces as f (1) = r (1) R = 0 and f (2) = r (2) R = 0. The choice of signs in (2) determines which portions of the two spaces are pasted together. We choose the plus sign for n α (1) and the minus sign for n α (2) to construct a wormhole comprised of the two asymptotically flat regions of the initial Curzon spacetime. We take (θ, φ, τ) as the coordinates on the surface and then use GRjunction to evaluate g ij, K ij and S i j and confirm [ g ij ] = 0. The line element of the surface is The stress energy is ds 2 = e 2(k U) R 2 dθ 2 + e 2U R 2 sin 2 θ dφ 2 dτ 2. (58) S θ θ = em2 sin 2 θ/(2r 2 ) m/r 4πR (59)

14 1898 P Musgrave and K Lake S φ φ = Sθ θ (R 2 + m 2 sin 2 θ) R 2 (60) Sτ τ = (2R 2 + m 2 sin 2 θ 2mR) Sθ θ (61) R 2 which can be interpreted as an anisotropic fluid. For R > 2m the weak energy condition is violated on the entire shell. However for R < 2m the weak energy condition is not violated for some angular positions on the shell. For example, if we take m = 1 and R = 0.5 then the energy density on the shell is positive for θ up to about 0.8 rad at which point it becomes negative (recall θ = 0 is the equator). Hence null geodesics approaching in this region will be focused by the shell (and if the shell were not present must have been defocusing). Radial null geodesics in the equatorial plane approaching the origin of the Curzon spacetime will be defocused near the origin. To see this consider the null vector field v r = λ, v θ = v φ = 0 v t = λe m/r /e m2 sin 2 θ/(2r 2 ) m/r (with affine parameter λ) which is geodesic in the equatorial (θ = 0) plane. Evaluating the expansion u r ;r gives = λ(m 2 sin 2 θ mr + r 2 )/r 3 (62) which changes sign at r = m in the θ = 0 plane. Thus the density of the shell is consistent with an analysis of null geodesics. Constructions with R < m are possible in principle since in the Curzon metric surfaces of constant r remain timelike in this region. Note that while moving R close to zero maximizes the portion of the shell which satisfies the weak energy condition the values of the energy density and pressures grow rapidly. Thin-shell wormholes with regions of non-negative stress energy have been constructed by Visser [23]. However the construction of these wormholes relies on using polyhedral surfaces. The Curzon wormhole presented above is to the best of our knowledge the first example of an exact non-spherical (non-planar/non-cylindrical) thin shell in general relativity and the first example of a smooth static thin-shell wormhole which has regions which do not violate the weak energy condition. Note added in proof. After this paper was accepted, we became aware of the exact non-spherical shells in Kerr Newman due to Israel [24] and Lopez [25]. Acknowledgments It is a pleasure to thank Malcolm MacCallum and Werner Israel for a number of useful suggestions and comments. This work was supported in part by grants (to KL) from the Natural Sciences and Engineering Research Council of Canada and the Advisory Research Committe of Queen s University. PM would like to thank the Ministry of Colleges and Universities of Ontario for support in the form of an Ontario Graduate Scholarship. References [1] Darmois G 1927 Mémorial de Sciences Mathématiques, Fascicule XXV Les equations de la gravitation einsteinienne ch V [2] Israel W 1966 Thin shells in general relativity Nuovo Cimento 66 1 (erratum B )

15 Junctions and thin shells in general relativity: I 1899 [3] Oppenheimer J R and Snyder H 1939 Phys. Rev [4] The package can be obtained by anonymous ftp from astro.queensu.ca ( ) in the directory /pub/grtensor or from the web site [5] Waterloo Maple Software MapleV Release 3 (Waterloo: Canada) [6] Musgrave P, Pollney D and Lake K 1995 GRTensorII software available as indicated in [4] [7] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (New York: Freeman) [8] Lake K th Brazilian School of Cosmology and Gravitation ed M Novello (Singapore: World Scientific) pp 1 82 [9] Berezin V A, Kuzmin V A and Tkachev I I 1987 Phys. Rev. D [10] Barrabès C and Israel W 1991 Phys. Rev. D [11] Lake K 1979 Phys. Rev. D [12] Sato H 1986 Prog. Theor. Phys [13] Sakai N and Maeda K 1994 Phys. Rev. D [14] Lichnerowicz A 1955 Théories Relativistes de la Gravitation et de l Electromagnétisme (Paris: Masson) ch III [15] Redmount I H 1985 Prog. Theor. Phys [16] Lake K 1992 Astophys. J. 401 L1 [17] Poisson E and Visser M 1995 Phys. Rev. D [18] Frolov V P, Markov M A and Mukhanov V F 1990 Phys. Rev. D [19] De La Cruz V and Israel W 1967 Phys. Rev [20] Lindblom L and Brill D R 1974 Phys. Rev. D [21] Kramer D et al 1980 Exact Solutions of Einstein s Field Equations (Cambridge: Cambridge University Press) section 18.1 [22] Visser M 1989 Nucl. Phys. B [23] Visser M 1989 Phys. Rev. D [24] Israel W 1970 Phys. Rev. D [25] Lopez C W 1984 Phys. Rev. D