tceic.com

学霸学习网 这下你爽了

学霸学习网 这下你爽了

- Solutions of the Einstein and Einstein-Maxwell Equations
- A New Class of Vacuum Solutions of Einstein Field Equations
- Time-Periodic Solutions of the Einstein's Field Equations
- The comparsion principle for viscosity solutions of fully nonlinear subelliptic equations i
- Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems
- Family of solutions for axisymmetric electrovacuum Einstein-Maxwell field equations
- Einstein's Field Equations for the Interior of a Uniformly Rotating Stationary Axisymmetric
- On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing
- Boundary Harnack Principle for Positive Solutions of Semilinear Elliptic Equations
- Static Fundamental Solutions of Einstein Equations and Superposition Principle in Relativis

A variational principle for stationary, axisymmetric solutions of Einstein’s equations

arXiv:gr-qc/0508061v1 16 Aug 2005

Sergio Dain Max-Planck-Institut f¨r Gravitationsphysik u Am M¨hlenberg 1 u 14476 Golm Germany February 7, 2008

Abstract Stationary, axisymmetric, vacuum, solutions of Einstein’s equations are obtained as critical points of the total mass among all axisymmetric and (t, φ) symmetric initial data with ?xed angular momentum. In this variational principle the mass is written as a positive de?nite integral over a spacelike hypersurface. It is also proved that if absolute minimum exists then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, initial data with ?xed angular momentum. Arguments are given to support the conjecture that this minimum exists and is the extreme Kerr initial data.

1

Introduction

In an axisymmetric, vacuum, gravitational collapse the total angular momentum is a conserved quantity. Therefore, if we assume, according to the standard picture of the gravitational collapse, that the ?nal state will be a Kerr black hole the following inequality should hold for every axisymmetric, vacuum, asymptotically ?at, complete, initial data set |J| ≤ m, (1)

where m is the mass of the data and J the angular momentum in the asymptotic region. Moreover, the equality in (1) should imply that the data is an 1

slice of the extreme Kerr black hole. A counter example to (1) will provide a regular vacuum data that do not settle down to a Kerr black hole. For a more detailed discussion of the motivations and relevance of (1) and related inequalities see [10], [12] and [8]. Inequality (1) is a property of the spacetime and not only of the data, since both quantities J and m are independent of the slicing. It is in fact a property of axisymmetric, vacuum, black holes spacetimes, because a non zero J (in vacuum) implies a non trivial topology on the data and this is expected to signal the presence of a black hole. Note, however, that the mass in (1) is a global quantity but the angular momentum is a quasilocal quantity because we have assumed axial symmetry. Without axial symmetry we still have J de?ned as a global quantity at spacelike in?nity, but (1) is not longer true in this case. A more subtle question is whether (1) is true where both m and J are quasilocal quantities, that is, whether (1) is in fact a quasilocal property of the black hole. In general there is no unique de?nition of quasilocal mass (see the recent review on the subject [17]). However, a remarkable counter example was found in [1] in which there is a clear quasilocal mass de?nition (the Komar mass) and inequality (1) is violated at the quasilocal level. Finally, let us note that (1) is false for black holes in higher dimensions (see, for example, [13] and reference therein). The inequality (1) suggests the following variational principle: (i) The extreme Kerr initial data is the absolute minimum of the mass among all axisymmetric, vacuum, asymptotically ?at and complete initial data with ?xed angular momentum. So far, there is no proof of (1). A promising strategy to prove it is to use the variational formulation (i). In this article we will prove the following results, which are a step forward in this direction. The ?rst result is the following related variational principle: (ii) The critical points of the mass among all the axisymmetric, (t, φ) symmetric, asymptotically ?at data are the stationary, axisymmetric solutions. A spacetime is de?ned to be (t, φ) symmetric if it is symmetric under a simultaneous change of sign of the time coordinate t and the axial angle φ. A data is called (t, φ) symmetric if its evolution is a (t, φ) symmetric spacetime. These data are also known as “momentarily stationary data” (see [2] for more details). The variational principle (ii) was proved by Bardeen [2], who also included matter in the formulation. It was also studied by Hawking [11] for black holes including boundary terms. However, in all these works 2

the mass is not written as a positive de?nitive integral (see the discussion of section VIII in [2]). Therefore, it is not possible to relate (ii) with (i) in these formulations. In this article we will prove (ii) using the mass formula discovered by Brill [4], which is a positive de?nitive integral over the slice. Using this formulation of (ii) we will be able to prove the following: (i’) If the absolute minimum of the mass among all axisymmetric, (t, φ) symmetric, vacuum, asymptotically ?at and complete initial data with ?xed angular momentum exists, then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, asymptotically ?at and complete initial data with ?xed angular momentum. Moreover, the absolute minimum is stationary. That is, we have essentially reduced the variational problem (i) to the (t, φ) symmetric case. Note that we have included in (i’) the condition that the data are maximal (i.e. the trace of the second fundamental form is zero). This is a technical assumptions which simpli?es considerably the analysis, but the statement is expected to be valid without it. There exist other variational formulations of the stationary, axisymmetric, equations, see [14] [16]. Particularly interesting in the present context is the variational formulation given by Carter [7] which is based in the Ernst formulation [9]. There exist a remarkable connection between (ii) in the present formulation and Carter’s variational principle, we will prove that the Lagrangians di?er only by a (singular) boundary term.

2

Axially symmetric initial data and Brill proof of the positive mass theorem

In this section we review Brill’s positive mass theorem for axisymmetric data [4]. The original proof was for time-symmetric data in R3 , here we slightly extend it to include maximal data and non-trivial topologies. An initial data set for Einstein’s vacuum equations consists in a 3-manifold ? ? S, a Riemannian metric hab , and a symmetric tensor ?eld K ab such that the vacuum constraint equations ? ? ? ? D b Kab ? Da K = 0, ? ? ? ? R + K 2 ? Kab K ab = 0, (2) (3)

? ? are satis?ed on S; where Da and R are the Levi-Civita connection and the ? ab , K = hab Kab , and the indexes are moved ? ? ? Ricci scalar associated with h ? ? with the metric hab and its inverse hab . 3

We will assume that the initial data are axially symmetric, that is, there exist an axial Killing vector η a such that ? ?η hab = 0, ? ?η Kab = 0, (4)

where ? denotes the Lie derivative. The Cauchy development of such initial data will be an axially symmetric spacetime. ? The Killing vector η a is assumed to be orthogonal with respect to hab ? to a family of 2-surfaces in S. Under these conditions, the metric hab can be characterized by two functions, one is essentially the norm of the Killing vector and the other is a conformal factor on the 2-surfaces. We make explicit the choice of the free functions as follows. Let (ρ, z, φ) be local coordinates in S such that the metric has following form ? hab = ψ 4 hab , where the conformal metric hab is given h = e?2q (dρ2 + dz 2 ) + ρ2 d?2 , (6) (5)

and q, ψ are functions which depend only on z and ρ with ψ > 0. The ? vector η a = (?/??)a is a Killing vector of both metrics hab and hab . The a norm of η with respect to the physical metric will be denoted by X, (i.e. ? X = η a η b hab = ψ 4 ρ2 ), the norm of η a with respect to the conformal metric is given by ρ2 = η a η b hab . We de?ne the following quantity J(Σ) =

Σ

πab η a nb dsh , ? ? ?

(7)

? ? ? where πab = Kab ? hab K, Σ is any closed 2-surface, na is the unit normal ? ? ? ab and ds? is the area element of Σ with respect vector to Σ with respect to h h ? to hab . Equation (2) and the Killing equation imply that the vector πab η a ? is divergence free. If Σ is the boundary of some compact domain ? ? S, by the Gauss theorem, we have J(Σ) = 0. For example, if S = R3 then J(Σ) = 0 for all Σ. In an asymptotically ?at data, J(Σ∞ ) gives the total angular momentum, where Σ∞ is any closed surface in the asymptotic region. Then, the angular momentum will be zero unless Σ∞ is not the boundary of some compact domain contained in S. In order to have non zero angular momentum we will allow S to have many asymptotic ends1 . Let ik a ?nite number of points in R3 . The manifold S is

There is an interesting alternative (not included here) discussed in [10] to allow non zero angular momentum: the interior of the manifold is assumed to be compact and non simply connected with a pseudo axial Killing vector.

1

4

assumed to be R3 \ k ik . The points ik will represent the extra asymptotic ends, at those points we will impose singular boundary conditions for ψ. The be consistent with the axial symmetry assumption the points ik should be located on the axis ρ = 0. In addition to axial symmetry we will assume that the data are maximal ? K = 0. (8) ? By equation (3) this implies that R is positive, this will be essential in order to extend Brill’s proof to non-time symmetric data. ? De?ne the conformal second fundamental form by K ab = ψ 10 K ab . Using (8) and (4) we obtain ?η Kab = 0, K = 0. (9) The constraint equations (2)–(3) can be written as equations for Kab and ψ using the well known conformal method (see, for example, [3] and reference therein) Da K ab = 0, 1 1 D a Da ψ ? Rψ = ? Kab K ab ψ ?7 , 8 8 (10) (11)

where Da and R are the Levi-Civita connection and the Ricci scalar associated with the conformal metric hab . In these equations, the indexes are moved with the conformal metric hab and its inverse hab . The function q is assumed to be smooth with respect to the coordinates (ρ, z). At the axis we impose the regularity condition q(ρ = 0, z) = 0. (12)

Note that condition (12) includes the points ik . These points are assumed to be regular points of the conformal metric hab , that is, hab is well de?ned in R3 . We assume the following fall-o? condition at in?nity q = o(r ?1 ), q,r = o(r ?2 ), (13)

where r = ρ2 + z 2 and a comma denotes partial derivatives. This fall o? conditions imply that the total mass of the conformal metric hab is zero. At in?nity, the conformal factor ψ and the conformal second fundamental form satisfy ψ ? 1 = O(r ?1), ψ,r = O(r ?2), (14) and Kab = O(r ?2). 5 (15)

Under these assumptions the total mass of the physical metric is given by m= ?1 lim 2π r→∞ na Da ψ dsh ,

Σr

(16)

where Σr are the 2-spheres r = constant, na is the unit normal, with respect to hab , pointed outwards and dsh is the area element of Σ with respect to hab . The Ricci scalar R of the conformal metric (6) is given by R = 2e2q (q,ρρ + q,zz ). We have the important equation R d?h = 0,

R3

(17)

(18)

where d?h is the volume element of the metric hab To prove this, note that d?h = ρe?2q dρdzdφ, then

∞ ∞

R d?h = 4π

R3 0 ∞

dρ

?∞ ∞

dz (q,ρρ + q,zz )ρ dz ((ρq,ρ ? q),ρ + (ρq,z ),z ) ,

(19) (20)

= 4π

0

dρ

?∞

we use the divergence theorem in two dimension to transform this volume integral in a boundary integral over the axis ρ = 0 and in?nity. The boundary integral at the axis vanishes since q satis?es (12) and at in?nity it also vanishes because of (13). Since limr→∞ ψ = 1, we have an equivalent expression for the mass m= We use the identity Da Da ψ ψ = D a Da ψ Da ψDa ψ ? , ψ ψ2 (22) ?1 lim 2π r→∞ na Da ψ dsh . ψ (21)

Σr

the constraint equation (11), equation (18) and the mass formula (21) to obtain the ?nal expression m= 1 2π K ab Kab Da ψDa ψ + 8ψ 8 ψ2 6 d?h , (23)

R3

which is de?nite positive. To obtain (23) from (22) we have assumed that the boundary integral around the singular points ik vanishes, that is na Da ψ lim dsh = 0, (24) rk →0 Σ ψ rk where rk is the distance to the point ik . This condition (which is, of course, trivially satis?ed when the topology of the physical data is R3 ) allows for a singular behavior of ψ at ik which in particular include the case where ik are asymptotically ?at ends. Near an asymptotically ?at end ik the conformal ?1 ?2 factor satis?es ψ = O(rk ), ψ,r = O(rk ) which imply (24). To illustrate this, consider the following two examples. The Schwarzschild initial data in isotropic coordinates is time-symmetric (Kab = 0) and conformally ?at (q = 0). In this case we have one point i0 located at the origin and the conformal factor is given by m0 , (25) ψ =1+ 2r where m0 is the Schwarzschild mass. We have ∞ (ψ,r )2 2 m=2 r dr, (26) ψ2 0 = m0 . (27) Note that the integral is taken over the two asymptotic regions. The second example is the Brill-Lindquist[5] initial data. In this case the data is also time-symmetric and conformally ?at, but here we have n ends ik and the conformal factor is given by

n

ψ =1+

k

mk , 2rk

(28)

where mk are arbitrary positive constants. The conformal factor (28) satis?es (24) and we have that 1 m= 2π

ab

R3

Da ψDa ψ ψ2

n

d?h =

k

mk .

(29)

In the non time-symmetric case, we have assumed that the integral of K Kab ψ ?8 over R3 is bounded. At in?nity, the integral converges because the assumptions (15) and (14). At the points ik the conformal second fundamental form will, in general, be singular. However the integral will be bounded because the singular behavior of Kab will be canceled out by the singular behavior of ψ. For example, in the asymptotically ?at case, ?4 Kab = O(rk ) near ik and then we have that K ab Kab ψ ?8 is bounded. In appendix A we prove that Kerr initial data satisfy these conditions. 7

3

The variational principle

In the integral (23) the mass depends on the metric variables ψ, q (the function q appears in the volume element and in the indexes contractions) and on the conformal second fundamental form K ab . These functions are not independent, they have to satisfy the constraint equations (10) and (11). In order to formulate the variational principle we want to express the mass in terms of functions that can be freely varied. We analyze ?rst the conformal second fundamental form K ab and the constraint (10). Consider the following vector ?eld S a Sa = Kab η b ? ρ?2 ηa Kbc η b η c . (30)

Using equations (9), (10) and the Killing equation for η a it follows that S a satis?es ?η S a = 0, S a ηa = 0, Da S a = 0. (31) From (7) we deduce an equivalent expression for the total angular momentum J=? 1 8π Sa na dsh ,

Σ∞

(32)

where we have used that the second term in the right-hand side of (30) does not contribute to the angular momentum because we can always chose a closed surface at in?nity such that na ηa = 0. The conformal metric hab can be decomposed into hab = qab + ρ?2 ηa ηb , where is the intrinsic metric of the planes orthogonal to η a . Using this decomposition and the de?nition of S a we obtain the following expression for the square of the conformal second fundamental form K ab Kab = Kab Kcd q ac q bd + ρ?4 (Kab η a η b )2 + 2ρ?2 S a Sa . (35) qab ≡ e?2q (dρ2 + dz 2 ), (34) (33)

The two ?rst terms in the right hand side of this equation are positive, then we have K ab Kab ≥ 2ρ?2 S a Sa . (36) Equations (32) and (36) are important because they show that S a contains the angular momentum of K ab and its square is a lower bound for the square of K ab . 8

We de?ne the tensor

2 ? K ab = S (a η b) , η 2S a Sa ? ? . K ab Kab = ρ2

(37)

we have

(38)

It is interesting to note (but we will not make use of it) that this tensor is trace free and divergence free. To prove this we use the Killing equation D(a ηb) = 0, the fact that η a is hypersurface orthogonal, (i.e.; it satis?es Da ηb = ?η[a Db] ln η) and equations (31). A data will be (t, φ) symmetric if and only if the following conditions hold (see [2]) Kab q ac q bd = 0, Kab η a η b = 0. (39) ? This is equivalent to K ab = K ab . a The vector S can be expressed in terms of a free potential. De?ne the rescaled vector sa by sa = e?2q S a , (40) then ?η sa = 0, sa ηa = 0, ?a sa = 0, (41) where ?a is the connexion with respect to the ?at metric δ = dρ2 + dz 2 + ρ2 d?2 , (42)

and in equation (41) the indexes are moved with this metric and its inverse. The same will apply to all the equations from now on: all of them will be given in term of the ?at metric δab and its connexion ?a . An arbitrary vector sa , which satis?es equations (41), can be written in term of a potential Y in the following form sa = 1 abc ? ηb ?c Y, 2ρ2 (43)

where ?abc is the volume element of the ?at metric (42) and ?η Y = 0. The motivation of the normalization factor 1/2 in (43) will be clear in the next section. We have the relation 2sa sa ? a Y ?a Y ? ? K ab Kab = = . ρ2 2ρ4 The angular momentum (31) is given in terms of the potential Y by J= 1 (Y (ρ = 0, ?z) ? Y (ρ = 0, z)) , 8 9 (45) (44)

where z is taken to be larger than the location of any point ik . Motivated by Brill’s formula (23), we de?ne the mass functional as follows M(v, Y ) = 1 32π 16?a v? a v + ρ?4 e?8v ? a Y ?a Y

R3

d?0 ,

(46)

where v = ln ψ and d?0 is the ?at volume element. Note that in the integral (46) the metric function q does not appear. From equation (23) and (44) we see that for every axisymmetric and (t, φ) symmetric data we have m = M(v, Y ). From (36) we see that for every axisymmetric, maximal data, we have m ≥ M(v, Y ). (47)

We emphasize that the functions (v, Y ) can be computed for an arbitrary axisymmetric data (in the construction of the potential Y we have not used the maximal condition) and then the functional M(v, Y ) can be also calculated for arbitrary data (provided, of course, the integral is well de?ned). However, only for maximal data we can use the Brill formula (23) to conclude (47) and only for (t, φ) symmetric data we have that M(v, Y ) is in fact the mass. For the present calculations is more convenient to write the functional M in the form (46), where the axial symmetry is not explicit. For completeness, we also write it in a manifest axisymmetric form M(v, Y ) = 1 16

∞ ∞

dρ

0 ?∞

2 2 2 2 dz (16ρ( v,z + v,ρ +ρ?3 e?8v Y,z + Y,ρ )) .

(48)

Let us de?ne A as the set of all functions (v, Y ) such that the integral (46) is bounded. Although M(v, Y ) is well de?ned in A, not for every function in A we will have that M(v, Y ) is equal to the mass of some (t, φ) symmetric initial data. This is a subtle and important point, let us discuss it in detail. We have seen that all axisymmetric and (t, φ) symmetric data can be generated by three functions (v, q, Y ). They are coupled by the Hamiltonian constraint (3). In coordinates, this equation is given by 4 ?ψ ? a Y ?a Y ? (q,ρρ + q,zz ) = , ψ ρ4 ψ 8 (49)

where ? is the ?at Laplacian with respect to (42). For given (v, Y ) (remember that v = ln ψ) this is a linear, two dimensional, Poisson equation for q. The delicate point are the boundary conditions. In order to obtain Brill’s formula we have required that q satis?es (12) and (13). But we cannot impose this two equations as boundary conditions for a two dimensional Poisson equation. 10

Let say that we impose (12) and we ask for solutions which fall o? at in?nity. This problem can be solved with an explicit Green function. However, in general, the fall o? of the solution will be q = O(r ?1) which is weaker than (13). Only for some particular source functions (v, Y ) the solution q will satisfy (13). Let us denote by A1 the subset of A of those functions (v, Y ) such that the solution q of equation (49) satis?es (12) and (13). Only for functions in A1 the functional M(v, Y ) can be written as a the boundary integral (16) and hence gives the mass of some initial data. A function v of compact support (such that ψ = 1 near in?nity) is an example of a function which is in A but not in A1 (we can take Y = 0), since in this case clearly M(v, Y ) is strictly positive and the boundary integral (16) is zero. We want to make variations of M(v, Y ). At ?rst sight, it appears that the appropriate set for admissible functions is A1 and not A. However, it seems to be di?cult to characterize A1 . It is known how to characterize the set of those q such that (49) has a solution ψ (for an, essentially, arbitrary Y ) which satis?es (14), in this case a non-linear equation must be solved (see [6] and [15]). However, this set is not very useful in the present context since for the Brill formula is natural to use (v, Y ) as independent functions and not (q, Y ). Instead, what we will do is to take A as the set of admissible functions. Remarkably, it will turn out that the critical equations in this bigger set are only the stationary, axially symmetric equations. Let α and y be compact supported functions in R3 with support in S and such that the support of y does not contain the axis. By equation (45) we see that this condition implies that the perturbation Y + y does not change the angular momentum of Y . De?ne i(?) = M(v + ?α, Y + ?y). The ?rst variation of M(v, Y ) is given by i′ (0) = 1 16π

R3

(50)

16?a v? a α ? 4αρ?4 e?8v ? a Y ?a Y + ρ?4 e?8v ? a Y ?a y d?0 ,

(51) where a prime denotes derivative with respect to ?. Integrating by parts, we obtain that the condition i′ (0) = 0, (52) for all α and y is equivalent to the following Euler-Lagrange equations 4?v + ρ?4 e?8v ? a Y ?a Y = 0, ? a (ρ?4 e?8v ?a Y ) = 0. (53) (54)

11

The second variation is given by i′′ (0) = 1 16π {16?a α? a α+ 32α2 ? a Y ?a Y ? 16α? a Y ?a y + ? a y?a y ρ?4 e?8v d?0 . (55) There is an equivalent way of deducing equations (53)–(54). Instead of taking Y as variable we take the vector sa , which should satisfy the constraints (41). The mass functional is given by M(v, s) = 1 8π 4?a v? a v + ρ?2 e?8v sa sa d?0 .

R3

R3

(56)

Let γ a be a compact supported vector in S such that the support of γ a does not contain the axis. We assume that γ a satis?es the constraint ?a γ a = 0. We de?ne i in analogous way as in (50). The ?rst variation is given by i′ (0) = 1 4π 1 4π

R3

(57)

4?a v? a α ? 4αρ?2 e?8v sa sa + ρ?2 e?8v sa γa d?0,

(58)

integrating by parts we get i′ (0) =

R3

?4α(?v + ρ?2 e?8v sa sa ) + ρ?2 e?8v sa γa d?0. ?v + ρ?2 e?8v sa sa = 0, 1 ρ?2 e?8v sa = ?a ?, 2

(59)

From this we deduce the Euler-Lagrange equations (60) (61)

for some function ?. Equation (60) follows because we can make arbitrary variations in α. On the other hand, variations in γ a should satisfy the constraint (57). Writing γ a as the curl of an arbitrary vector and integrating by parts we get ?[a Hb] = 0, (62) where Ha = ρ?2 sa e?8v . (63) Equation (62) is equivalent to (61). Using the constraint ?a sa = 0, we deduce the following equations which does not involve sa 4?v + ρ2 e8v ? a ??a ? = 0, ? a (ρ2 e8v ?a ?) = 0. 12 (64) (65)

Equations (64)–(65) are equivalent to equations (53)–(54), the relation between ? and Y is given by ?a ? = ρ?4 e?8v ?abc η b ? c Y. (66)

In the next section we will prove that these equations are precisely the stationary, axisymmetric, vacuum equations. This will provide also an interpretation for the potential Y and the velocity ? in the stationary case. Note that Y is de?ned for arbitrary data, in contrast ? is only de?ned for solutions of the critical equations, that is, for stationary axisymmetric data. If we take Y = 0, then these equations reduce to ?v = 0, (67)

which is Weyl equation for axisymmetric, static, spacetimes. This is of course consistent with the result that we are going to prove in next section. However, it is important to note that the Schwarzschild data in the form (25) does not satisfy (67). Schwarzschild satis?es (67) in Weyl coordinates where v and ? the metric function q are given by ? 1 v = ? ln ? 4 r+ + r? ? 2m ? ? r+ + r? + 2m ? ? , q= ? 1 ln 2 (?+ + r? )2 ? 4m2 r ? 4?+ r? r ? , (68)

with r± = ρ2 + (? ± m)2 . The relation with the isotropic coordinates (r, θ) ?2 ? z used in (25) is ρ=ρ 1? ? m2 4r 2 , z =z 1+ ? m2 4r 2 , (69)

where z = r cos θ and ρ = r sin θ. Since X is an scalar independent of coordinates we have X = ρ2 ψ 4 = ρ2 ψ 4 . The function q satis?es our assumptions ? ? ? ? ? = ev does not satis?es (24). (12) and (13), however the conformal factor ψ The conformal factor is singular on the rod ρ = 0, ?m ≤ z ≤ m (which ? ? represent the horizon of Schwarzschild data) and not just on singular points ik . The integral M(?, 0) diverges. Note that R3 in Weyl coordinates (?, z ) v ρ ? represent the exterior of the black holes, in contrast to coordinates (ρ, z) where R3 represent both asymptotic regions.

4

Stationary axisymmetric ?elds

The spacetime metric of a vacuum, stationary and axially symmetric spacetime can be written, in Weyl coordinates, in the following form (see, for example, [18]) g = ?V (dt ? σdφ)2 + V ?1 ρ2 dφ2 + e2γ (dρ2 + dz 2 ) , 13 (70)

where the functions V , σ and γ depend only on (ρ, z). The two Killing vectors are ? ? ? ? ξ? = , η? = , ?t ?φ they de?ne the scalars V = ?ξ ? ξ ν g?ν , X = η ? η ν g?ν , W = η ν ξ ? gν? , (71)

where ?, ν are spacetime indexes. We have the following relations W = V σ, ρ2 = V X + W 2 . (72)

The vacuum ?eld equations are given by ? a V ?1 ?a V + ρ?2 V 2 σ?a σ = 0, ? a ρ?2 V 2 ?a σ = 0. (73) (74)

We want to prove that these equations are equivalent to equations (64)–(65). We ?rst compute the relation between (V, σ) and (v, ?). Take an slice t = constant of the metric (70). The intrinsic metric of this surface is given by ? h = V ?1 (ρ2 ? σ 2 V 2 )dφ2 + V ?1 e2γ (dρ2 + dz 2 ). To write this metric in to the form (5)–(6) set ψ4 = and e2q = From (76) we deduce X X (ρ2 ? V 2 σ 2 ) = 2 = , 2 Vρ ρ (V X + W 2 ) e2γ ρ2 e2γ ρ2 = . (ρ2 ? V 2 σ 2 ) VX 1 (ρ2 ? V 2 σ 2 ) ln . 4 V ρ2 (76) (75)

(77)

v(V, σ) =

(78)

In order to compute ?(v, σ) we need to calculate the second fundamental form of this foliation. The lapse and the shift of the foliation t = constant are given by ρ (79) N = √ = ψ ?2 , Na = W (dφ)a . X 14

and the second fundamental form is 1 ? ? Kab = ? D(a Nb) . 2N (80)

We write Na in terms of the Killing vector η a , as in the previous section we ? de?ne ηa = hab η b where ? ab is given by (75), then we have ? h ηa = ? (ρ2 ? V 2 σ 2 ) (dφ)a . V (81)

Using this expression we write Na as Na = ??a , η where ? is given by ?(V, σ) = W V 2σ = . 2 ? V 2 σ2) 2(ρ X (83) (82)

The scalar ? can be interpreted as the angular velocity of the locally non rotating observers (see [2] and also [18] p. 187). We want to prove that this function ? is precisely the potential ? of the previous section. In order to see this let us compute the vector sa ? sa = η b Kab = ψ 2 η b Kab = 1 2 1 ψ X?a ? = ψ 8 ρ2 ?a ?. 2N 2 (84)

Where we have used sa = sb δab = S b hab . Equation (84) is identical to equation (61). Using the relations (78) and (83), after a long but straightforward computation, we conclude that equations (73) and (74) for the functions (V, σ) are equivalent to equations (60)–(61) for (v, ?). There is another way to prove the equivalence with the stationary equations, using the potential Y . We replace v by X, that is we consider X, Y as variables. From equation (76) we get v= 1 1 ln X ? ln ρ. 4 2 (85)

Take the functional M de?ned in (46) but let us perform the integral on a bounded domain B, in terms of the variables X, Y we get M(X, Y ) = M′ (X, Y )? 1 8π ln

B

ρ ? ln ρ d?0 + X

ρ?1 ln

?B

ρ na ?a ρ ds, X (86)

15

where we have de?ned M′ (X, Y ) = But we have ? ln ρ = 0, (88) for ρ = 0. Then M and M′ di?er only by a boundary term. Hence they give the same Euler-Lagrange equations. Note, however, that the boundary term is singular at the axis ρ = 0: if we take a cylinder ρ = constant near the axis we have, X = O(ρ2 ), ds = ρ dzdφ, na ?a ρ = 1, then the boundary term diverges like O(ln ρ) as ρ → 0. In [7] Carter formulates a variational principle for the axisymmetric, stationary equations. This formulation is, essentially, a modi?cation of the [9] formulation in which the norm of the axial Killing vector (and not of the stationary one) is taken to be the principal variable. Carter’s Lagrangian is precisely M′ (we use the same notation for X and Y , this is the reason for the normalization factor 1/2 in (43)). In [7] it is proved that the critical equations of M′ are the stationary, axisymmetric equations. Therefore, the same is valid for M. There are, however, some important points that we want to stress. If we ignore boundary terms, then equation (86) provides an interpretation for Carter Lagrangian. Also, it gives an interpretation of the space of admissible functions in which the variations are made for the following reason. In Carter’s formulation Y is de?ned in terms of W and X by ?abc η b ? c Y = X?a W ? W ?a X. (89) 1 32π ?a X? a X + ? a Y ?a Y X2 d?0. (87)

B

This equation can easily be obtained from (66), (76) and (83). That is, Y is de?ned only for stationary axisymmetric spacetimes. From the discussion of section 3 we have seen that Y can be de?ned for arbitrary, axisymmetric data, and the variation of Y and X are in fact variation among axisymmetric and (t, φ) symmetric data. Let us consider boundary terms. The behavior of X near the axis implies that M′ is singular if the domain of integration includes the axis. On the other hand we have seen that M is ?nite. In particular, in appendix A we have explicitly checked that Kerr initial data in quasi-isotropic coordinates satisfy all our assumptions and then M is ?nite and equal to the mass for Kerr. However, is important to note that the relevant domains of integration are di?erent in Carter’s formulation and in the present one. In [7], the domain is the black hole exterior region, in which the inner boundary is the horizon. In section 3 we have not included any inner boundary conditions, 16

the domain of integration is the whole manifold which can include many asymptotic ends. This di?erence is re?ected in the choice of the coordinate system. We have discussed this with Schwarzschild data in section 3. The same apply to non-extreme Kerr initial data in Weyl coordinates: M is singular in this coordinates. However, for extreme Kerr, the Weyl coordinates and the quasi-isotropic coordinates coincides. In this case both domains of integration coincides and M is ?nite whether M′ is not.

5

Final comments

We have analyzed the ?rst variation of the, positive de?nite, mass functional M (de?ned by (46)) over axisymmetric and (t, φ) symmetric initial data with ?xed angular momentum. We have shown that the critical points are the stationary, axial symmetric equations. This proves the variational principle (ii). The functional is a lower bound for the mass (inequality (47)) for all maximal, axisymmetric data. This proves (i’). In order to prove (i), and hence inequality (1), we should prove that extreme Kerr is the unique absolute minimum of M over axisymmetric and (t, φ) symmetric initial data with ?xed angular momentum. This will require the study of the second variation of M, given in equation (55).

Acknowledgments

It is a pleasure to thank Abhay Ashtekar, Marc Mars and Walter Simon for valuable discussions. This work has been supported by the Sonderforschungsbereich SFB/TR7 of the Deutsche Forschungsgemeinschaft.

A

Kerr initial data

Consider the Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ). The ? scalars (71) are given by ? ? a2 sin2 θ 2ma? sin2 θ r , W =? , Σ Σ (?2 + a2 )2 ? ?a2 sin2 θ r X= sin2 θ, Σ V = where ? = r 2 + a2 ? 2m?, ? r 17 Σ = r 2 + a2 cos2 θ, ? (92) (90) (91)

and m is the total mass and a is the angular momentum per unit mass (i.e. J = ma). ? The intrinsic metric hab of a hypersurface t = constant in these coordinates is given by Σ ? h = d?2 + Σdθ2 + ηdφ2 . r (93) ? The metric (93) has a coordinate singularity when ? = 0. The solutions of the equation ? = 0 are given by √ √ r+ = m + m2 ? a2 , r? = m ? m2 ? a2 . ? ? (94) By the following coordinate transformation we extend the metric to a complete manifold with two asymptotic ends. Let us de?ne the quasi-isotropic radius r as the positive root of the following equation r =r+m+ ? m2 ? a2 . 4r (95)

Note that when a = 0 this reduce to the isotropic radius for the Schwarzschild metric. The manifold (like in the Schwarzschild case) has to isometric asymptotically ?at components (the region r ≥ r+ of the metric (93)) joined at the ? ? ? minimal surface (the horizon) r = r+ . The components of hab in the coordi? ? nates (r, θ, φ) are given by Σ ? h = 2 dr 2 + Σdθ2 + ηdφ2. r The metric (96) has the form (5)–(6) with ψ4 = X , ρ2 e?2q = sin2 θΣ , X (97) (96)

where ρ = r sin θ and z = r cos θ. Assume m > |a|. Then, from (97) we see that in the limit r → 0 we have √ m2 ? a2 m ψ= + √ + O(r), ψ,r = O(r ?2), (98) 2 ? a2 r 2 m and at in?nity ψ =1+ From (97) we also have that q(ρ = 0) = 0. 18 (100) m + O(r ?2), 2r q = O(r ?2). (99)

Hence, q satis?es (12), (13) and ψ satis?es (14) and (24). The velocity ? can be calculated from equation (83) using (90) and (91) ?=? (?2 r + a2 )2 2ma? r . ? ?a2 sin2 θ 2ma3 cos θ sin4 θ . Σ (101)

The potential Y is given by Y = 2ma(cos3 θ ? 3 cos θ) ? (102)

Note that equation (45) is satis?ed for z = 0. To see that the integral of ? a Y ?a Y ρ?4 ψ ?8 over R3 is bounded we need to check the behavior of this function at in?nity and at the axis ρ = 0. At in?nity we have ? a Y ?a Y ρ?4 = O(r ?6), and at the axis ? a Y ?a Y ρ?4 ψ ?8 = O(r 2), (104) where we have used the (98). Then, the integral is bounded and therefore we have proved that the Kerr initial data satis?es our assumptions which implies that M(v, Y ) = m. Weyl coordinates (?, z ) are related to the coordinates (r, θ) by ρ ? √ ? r (105) ρ = ? sin θ, z = (? ? m) cos θ. ? Consider now the extreme case m = |a|. In this case we have r = r ? m, ? ? = r2 , (106) (103)

and the coordinates (r, θ) are equal to the Weyl coordinates. Equations (100) and (99) are still valid in this case. The fall o? of the conformal factor near r = 0 is however di?erent √ 2m √ + O(r 1/2), ψ,r = O(r ?3/2 ), ψ= (107) 2 θ)1/4 r (1 + cos this is because r = 0 is not an asymptotically ?at in this case. Nevertheless ψ satis?es (24). The fall of behavior of Y at in?nity is the same as in the non-extreme case. Near the axis, because of (107), we have ? a Y ?a Y ρ?4 ψ ?8 = O(r ?2 ), and hence we conclude that ? a Y ?a Y ρ?4 ψ ?8 is integrable over R3 . 19 (108)

References

[1] M. Ansorg and D. Petro?. Black holes surrounded by uniformly rotating rings. Phys. Rev. D, 72:024019, 2005, gr-qc/0505060. [2] J. M. Bardeen. A variational principle for rotating stars in general relativity. Astrophys. J., 162:71–95, 1970. [3] R. Bartnik and J. Isenberg. The constraint equations. In P. T. Chru?ciel s and H. Friedrich, editors, The Einstein equations and large scale behavior of gravitational ?elds, pages 1–38. Birkh¨user Verlag, Basel Boston a Berlin, 2004, gr-qc/0405092. [4] D. Brill. On the positive de?nite mass of the Bondi-Weber-Wheeler time-symmetric gravitational waves. Ann. Phys., 7:466–483, 1959. [5] D. Brill and R. W. Lindquist. Interaction energy in geometrostatics. Phys. Rev., 131:471–476, 1963. [6] M. Cantor and D. Brill. The Laplacian on asymptotically ?at manifolds and the speci?cation of scalar curvature. Compositio Mathematica, 43(3):317–330, 1981. [7] B. Carter. Black hole equilibrium states. In Black holes/Les astres occlus ? ?e (Ecole d’Et? Phys. Th?or., Les Houches, 1972), pages 57–214. Gordon e and Breach, New York, 1973. [8] S. Dain, C. O. Lousto, and R. Takahashi. New conformally ?at initial data for spinning black holes. Phys. Rev. D, 65(10):104038, 2002, gr-qc/0201062. [9] F. J. Ernst. New formulation of the axially symmetric gravitational ?eld problem. Phys. Rev., 167:1175–1179, 1968. [10] J. L. Friedman and S. Mayer. Vacuum handles carrying angular momentum; electrovac handles carrying net charge. J. Math. Phys., 23(1):109– 115, 1982. [11] S. W. Hawking and G. F. R. Ellis. The large scale structure of spacetime. Cambridge University Press, Cambridge, 1973. [12] G. T. Horowitz. The positive energy theorem and its extensions. In F. J. Flaherty, editor, Asymptotic behavior of mass and spacetime geometry (Corvallis, Ore., 1983), volume 202 of Lecture Notes in Phys., pages 1–21. Springer, Berlin, 1984. 20

[13] G. T. Horowitz. Higher dimensional generalizations of the Kerr black hole. 2005, gr-qc/0507080. [14] R. A. Matzner and C. W. Misner. Gravitational ?eld equations for sources with axial symmetry and angular momentum. Phys. Rev., 154(5):1229–1232, 1967. [15] D. Maxwell. Solutions of the Einstein constraint equations with apparent horizon boundaries. Commun. Math. Phys., 253(3):561–583, 2005, gr-qc/0307117. [16] G. Neugebauer and H. Herold. Gravitational ?elds of rapidly rotating neutron stars: theoretical foundation. In J. Ehlers and G. Schfer, editors, Relatividtic Gravity Research, number 410 in Lectures notes in physics, pages 303–318. Springer-Verlag, 1992. [17] L. B. Szabados. Quasi-local energy-momentum and angular momentum in GR: A review article. Living Rev. Relativity, 7(4), 2004. cited on 8 August 2005. [18] R. M. Wald. General Relativity. The University of Chicago Press, Chicago, 1984.

21