The Hamiltonian mass of asymptotically Schwarzschild-de Sitter space-times

We derive the Hamiltonian mass for general relativistic initial data sets with asymptotically Schwarzschild-de Sitter ends.


I. INTRODUCTION
There is growing astrophysical evidence that spacetimes with positive cosmological constant should be given serious consideration. Large families of such noncompact, vacuum, general-relativistic models can be constructed using singular solutions of the Yamabe problem (see [1,2] and references therein). In particular one thus obtains initial data sets with one or more ends of cylindrical type, in which the metric becomes periodic when one recedes to infinity along half-cylinders, approaching the Schwarzschild-de Sitter metric in the limit, with the extrinsic curvature tensor approaching zero. This construction can be carried out in any number of space dimensions n ! 3. (See [3][4][5][6][7] for further families of vacuum initial data sets with various ends of cylindrical type.) This raises the question of existence of a natural notion of mass in this context. The object of this work is to show that the numerical value of a natural Hamiltonian H for a class of such metrics is proportional to the parameter m appearing in the asymptotic metric. We further prove that the contribution to the Hamiltonian from each asymptotically Schwarzschild-de Sitter end can be calculated as (1.1) Here we assume that the space metric is asymptotic to the space part of a Birmingham metric on ½0; 1Þ Â M as in (A1) and (A2), for a compact Riemannian (n À 1)dimensional Einstein manifold ðM ; h Þ; is the lapse function as in (2.3); k is the mean curvature of fx ¼ x 0 g as defined in (2.11); and is the (n À 1)-volume element on fx ¼ x 0 g. The fields 0 and k 0 are the corresponding quantities for the Birmingham metric with vanishing mass (the ''de Sitter solution''); see (3.23) and (3.24). Finally is a dimension-dependent coupling constant; see (D2) in Appendix D below, related to the ''(n þ 1)-dimensional Newton constant'' as in (D6).
We note that a Hamiltonian is always defined up to a constant. Our choice in (1.1) is precisely what is needed for positivity of H ; compare Theorem III.1 below.
See [8][9][10] and references therein for alternative approaches to a definition of mass in the presence of a positive cosmological constant.

II. THE BASIC VARIATIONAL FORMULA
In order to present our results some notation is needed. Let S be a smooth spacelike hypersurface in an (n þ 1)dimensional space-time ðM; gÞ, n ! 2. Consider a spacetime domain with smooth timelike boundary such that V : ¼ \ S is compact. Let x n be a coordinate such that x n is constant on @V, and let ðx a Þ ¼ ðx 0 ; x A Þ be local coordinates on @ such that x 0 is constant on S. Let L ab denote the extrinsic curvature tensor of @, and let Q ab be its ''Arnowitt-Deser-Misner (ADM) counterpart,'' whereĝ ab is the n-dimensional inverse with respect to the induced metric g ab on the world tube @. Let and A denote the ''lapse'' and the ''shift'' in the n-dimensional geometry g ab of the boundary of the world tube @, whereg AB is the (n À 1)-dimensional metric on @V, inverse with respect to the induced metric g AB . We have the identity One can define the following (n À 1)-dimensional objects on @V: a scalar density and a covector density It is further useful to introduce the field The n-dimensional Lorentzian metric g ab on @ can be parametrized as : The corresponding inverse metric readŝ We also have with the trace L of L ab being equal to where k : ¼ ÀL ABg AB (for the Birmingham metrics of Appendix A 3, k is the signed length of the extrinsic curvature vector), and where we use the symbol to denote the curvature (''acceleration'') of the world lines which are geodesic within @ and orthogonal to @ \ S. It holds that Let P ij be the usual ADM momentum on V. Denote by ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det g AB p the (n À 1)-volume element on @V. Let be the hyperbolic angle between @ and V: in the adapted coordinates above, In [11] the following variational formula has been proved for Ricci-flat Lorentzian metrics in dimension 3 þ 1: (2.13) with ¼ 8. It can be checked that the formula remains true for vacuum metrics, possibly with a cosmological constant, in any space dimension n ! 2, with a constant which depends upon dimension; see Appendix D for a discussion. In fact, several terms proportional to (n À 3) appear when generalizing the calculations in [11], but they end up giving no contribution to (2.13).
We will not dwell upon the Hamiltonian interpretation of this identity; the reader is referred to [11][12][13] for details.
In the nonvacuum case (2.13) has to be supplemented by terms involving variations of the matter fields and their momenta. Nevertheless, the formula (1.1) for the Hamiltonian remains valid for a large class of matter models [11] without any further explicit contributions from the matter sources. (Obviously, there is an implicit contribution of the sources via the constraint equations.)

III. THE MASS OF ASYMPTOTICALLY BIRMINGHAM METRICS
We consider (2.13) for metrics which, as x tends to infinity, asymptote to Similarly we will assume that the derivatives of the metric g asymptote to those of the metric g . The coordinate x n of the calculations above will be taken to be equal to x, and the boundary @V % M in (2.13) will be assumed to be given by the equation x ¼ x 0 for a constant x 0 . We will let x 0 tend to infinity; this implies Above, and in what follows, we assume that @ x is pointing outwards from the region V of the previous section; some sign adjustments are needed otherwise. Using these formulas, the last line in (2.13) approaches (3.10) Let us assume that f and @ x take the Birmingham form (A1) [14], where 2 f0; AE1g is related to the scalar curvature, assumed to be constant, of the metric h [see (A5)]. Finally, ' À2 is related to the cosmological constant as in (C17).
When h is the unit round metric on the sphere, then ¼ 1 and one recovers the familiar Schwarzschild-de Sitter metrics. Equation (3.11) allows us to express @ x ¼ @ x in terms of and m. Perhaps surprisingly, all the terms cancel out and (3.10) becomes we conclude that for any family of metrics which asymptote to Birmingham metrics as the variable x recedes to infinity it holds that This is the first main result of this work. We wish, next, to provide a geometric formula for the Hamiltonian H . The integrand of the boundary term in (2.13), can be rearranged using the identity As before, we assume that the metric asymptotes to a Birmingham metric as x tends to infinity, similarly for first derivatives. We then have Inserting those relations into the underbraced terms in (3.17) one finds We thus obtain the following formula for the Hamiltonian: Here should be viewed as a function of , and hence of the metric: Choose a constant m 0 2 R and set This leads to the following rewriting of (3.22): h . This is the second main result of this work.
Note that the parameter m 0 has been introduced only to define the reference fields k 0 and 0 , and that the left-hand side is independent of m 0 .
See Appendix B for an alternative derivation of (3.25). One can simply disregard the last term in (3.25), or use reference fields associated with the solution equal to m 0 ¼ 0 there. Here one should keep in mind that a Hamiltonian analysis always defines a Hamiltonian up to a constant, and the choice of this constant is equivalent to the decision of which field configuration (if any) has zero energy. As such, the subtraction of the term k 0 0 can be viewed as a comparison term, where one compares the given field configuration with that time-independent solution which is determined by the parameter m 0 .
One could argue that reference fields corresponding to the solution with m 0 ¼ 0 make no sense because, in the M ¼ S nÀ1 case, the initial data surface is compact, so comparing with a solution with asymptotically periodic ends is unnatural from a Hamiltonian perspective. However, one can adopt the point of view that energy in general relativity is not assigned to a volume V but rather to a surface @V. Given a level set of r in a Schwarzschildde Sitter solution, we can find a surface with identical induced metric in the de Sitter solution [15], m 0 ¼ 0, and use the corresponding values 0 and k 0 in (3.25).
In any case, somewhat surprisingly, the choice of the value of m 0 is irrelevant, in that the numerical value of H as given by (3.25) does not depend upon that choice. This is related to the fact that the mass parameter m is the (unique) ''constant of motion'' for the spherically symmetric Yamabe equation; cf. (C16).
We note that the time-symmetric Birmingham metrics lead to the periodic metrics (3.1) with a strictly positive parameter m; see the discussion in Appendix A. This leads to the following trivial observation: Theorem III.1 (''positive energy theorem'') For all asymptotically periodic metrics as above, the numerical value of the Hamiltonian H given by (3.25) is positive. Now, Theorem III.1 requires neither positivity of matterenergy nor regularity of initial data (in particular, interior boundaries are allowed without any geometric restrictions), and is based purely on asymptotic properties of the solutions. As such it does not carry much nontrivial information: the positivity of the mass has been built into the hypotheses on the asymptotic behavior of the metric.

A. Several ends, black hole boundaries
So far we have assumed that the initial data manifold is the union of a compact manifold without boundary and an asymptotically cylindrical end. The generalization of our analysis to a finite number of asymptotically flat, asymptotically cylindrical and asymptotically hyperboloidal ends is straightforward: in such a case each end contributes its respective Hamiltonian mass (as defined here for asymptotically Birmingham ends, and as defined in e.g. [13,[16][17][18] and references therein for the remaining ones) to the total Hamiltonian of the system.
Yet another generalization is of interest, that to manifolds with horizon boundaries. For this purpose, suppose that the boundary of the domain of Sec. II consists of a timelike ''world tube'' S þ and of a null hypersurface S À . Accordingly, the boundary @V of the intersection V of the Cauchy surface S with is composed of two disjoint manifolds, @V þ ¼ V \ S þ , and @V À ¼ V \ S À , assumed to be compact, each of them contributing to the boundary terms in variational formula (2.13). Assume that the spacetime metric asymptotes to a Birmingham metric as the ''external'' boundary @V þ recedes to infinity. The corresponding contribution to (2.13) is handled as in the previous section. The contribution to (2.13) from the null component S À was calculated in [19] in considerable generality. However, for the sake of simplicity, we restrict attention to stationary solutions with Killing horizons, as arising in a thermodynamical analysis of stationary black holes. Then the volume term in (2.13) vanishes identically (since the time derivatives vanish) and the entire formula reduces to [see Eq. (4.2) in [19]] where the right-hand side is the (only remaining) boundary term [20] corresponding to the cross section @V À of the horizon S À . Here H is our Hamiltonian (3.25), s ¼ AE1 is a constant which depends upon the time orientation of the Killing vector so that Às is the surface gravity in usual circumstances [one should also keep in mind further negative signs in (3.26) which might arise from the orientation of the boundary; see Fig. 1]. The field W A is defined on the horizon by the formula where K is a Killing vector field which is null on a horizon, assuming that the horizon is located at x n ¼ const, and that x 0 is a coordinate on the horizon satisfying dx 0 ðKÞ ¼ 1: It is conceivable that the only such vacuum black hole space-times which are asymptotic to the Birmingham metrics are the Birmingham metrics themselves, in which case the ''thermodynamical identity'' (3.26) can be derived by the trivial calculation of Appendix A 4. However, this is not clear, and unlikely in higher dimensions in any case.
As already emphasized, the positive energy theorem III.1 remains valid in the black hole setting.

ACKNOWLEDGMENTS
We are grateful to Christa Raphaela Ö lz for providing the figures, and to Bobby Beig for useful discussions and bibliographical advice. This work was supported in part by Narodowe Centrum Nauki (Poland) under Grant No. DEC-2011/03/B/ST1/02625 and the Austrian Science Fund (FWF) under Project No. P 23719-N16. J. J. and J. K. wish to thank the Erwin Schrödinger Institute, Vienna, for hospitality and support during part of the work on this paper.

APPENDIX A: BIRMINGHAM METRICS
Consider an (n þ 1)-dimensional metric, n ! 3, of the form where h is a Riemannian metric on a compact manifold M with constant scalar curvature R ; we denote by x A local coordinates on M . As discussed in [21], for any m 2 R and ' 2 R Ã the function leads to a vacuum metric, thus ' is a constant related to the cosmological constant as in (C17) below. (Clearly, the case n ¼ 2 would require separate considerations, and we will therefore ignore this dimension in our work.) The multiplicative factor 2 in front of m is convenient in dimension 3 when h is a unit round metric on S 2 , and we will keep this factor regardless of topology and dimension of M .
There is a rescaling of the coordinate r ¼ b" r, with b 2 R Ã , which leaves (A1) and (A2) unchanged (up to ''adding bars'') if moreover We can use this to achieve which will be assumed from now on. The set fr ¼ 0g corresponds to a singularity when m Þ 0. Except in the case m ¼ 0 and ¼ À1, by an appropriate choice of the sign of b we can always achieve r > 0 in the regions of interest. This will also be assumed from now on. For reasons which should be clear from the main text, we will now be seeking functions f which, after a suitable extension of the space-time manifold and metric, lead to spatially periodic solutions.

Cylindrical solutions
Consider, first, the case where f has no zeros. Since f is negative for large jrj, f is negative everywhere. It therefore makes sense to rename r to > 0, t to x, and Àf to F > 0, leading to the metric The level sets of the time coordinate are infinite cylinders with topology R Â M , with a product metric. Note that the extrinsic curvature of those level sets is never zero because of the 2 term in front of h , except possibly for the f ¼ 0g slice in the case ¼ À1 and m ¼ 0.
Assuming that m Þ 0, the region r 2 ð0; 1Þ is a ''big bang-big freeze'' space-time with cylindrical spatial sections. A ð; xÞ-projection diagram (in the sense of [22]) is an infinite horizontal strip with a singular spacelike boundary at ¼ 0, and a smooth conformal spacelike boundary at ¼ 1; see Fig. 2.
In the case m ¼ 0 and ¼ 0 the spatial sections are again cylindrical, with the boundary f ¼ 0g being now at infinite temporal distance: indeed, setting T ¼ ln , when m ¼ 0 and ¼ 0 we can write When h is a flat torus, this is one of the forms of the de Sitter metric ( [23], p. 125).
The next case which we consider is f 0, with f vanishing precisely at one positive value r ¼ r 0 . This occurs if and only if ¼ 1 and A ðr ¼ ; t ¼ xÞ-projection diagram can be found in Fig. 3. No nontrivial, periodic, time-symmetric (K ij ¼ 0) spacelike hypersurfaces occur in all space-times above. Periodic spacelike hypersurfaces with K ij 60 arise, but a Hamiltonian analysis of initial data asymptotic to such hypersurfaces goes beyond the scope of this work.
From now on we assume that f has positive zeros.

Spheres and naked singularities
Assuming that m ¼ 0 but Þ 0, we must have ¼ 1 in view of our hypothesis that f has positive zeros. For r ! 0 the function f has exactly one zero, r ¼ '. The boundaries fr ¼ 0g correspond either to regular centers of symmetry, in which case the level sets of t are S n 's or their quotients, or to conical singularities. See Fig. 4.
If m < 0 the function f: ð0; 1Þ ! R is monotonously decreasing, tending to minus infinity as r tends to zero, where a naked singularity occurs, and to minus infinity when r tends to 1; hence f has then precisely one zero. The ðt; rÞ-projection diagram can be seen again in Fig. 4.
No spatially periodic time-symmetric spacelike hypersurfaces occur in the space-times above.

Spatially periodic time-symmetric initial data
We continue with the remaining cases, that is, f having zeros and m > 0. [When ¼ 1 this implies 0 < m 1 n ð1 À 2 n Þ n 2 À1 ' nÀ2 .] The function f: ð0; 1Þ ! R is then concave and thus has precisely two first-order zeros, except when m attains its maximal allowed value, a case already discussed [see (A7)]. A projection diagram for a maximal extension of the space-time, for the cases with two first-order zeros, is provided by Fig. 5. The level sets of t within each of the diamonds in that figure can be smoothly continued across the bifurcation surfaces of the Killing horizons to smooth spatially periodic Cauchy surfaces.

Killing horizons
The locations of Killing horizons of the Birmingham metrics are defined, in space dimension n, by the condition Thus, variations of the metric on the horizons satisfy where r nÀ1 nÀ1 is the h -volume of the cross section of the horizon. Let us check that : ¼ ð@ r fÞ 2 j r¼r 0 coincides with the surface gravity of the horizon, defined through the usual formula where K is the Killing vector field which is null on the horizon. For this, we rewrite the space-time metric (A1) in the familiar form where du ¼ dt À f À1 dr. The Killing field K ¼ @ u ¼ @ t is indeed tangent to the horizon and null on it. Formula (A11) implies that The inverse metric equals whence g u ¼ À r , and

Singularities
Consider a metric of the form with h as before. For A ¼ 1; . . . ; n let A be an orthonormal and let ! AB and AB be the associated connection and curvature forms, as in the Cartan structure equations: : Let be the following g-ON coframe: The condition of vanishing of torsion is solved by setting This gives the following curvature two-forms: Suppose that g is a Birmingham metric with m ¼ 0; thus for a constant , and then If h is a space-form, with consistently with (A5), we obtain If, however, h is not a space-form, we have for some nonidentically vanishing tensor r A BCD , with all traces zero. Hence where the functions r are independent in the current frame, and vanish whenever one of the indices is 0 or n. This gives which is singular at ¼ 0.

APPENDIX B: A CONTROL-RESPONSE CALCULATION
To give our considerations a precise Hamiltonian meaning we need to define explicitly the family of metrics considered, as well as the time parameter with respect to which the Hamiltonian will be determined. The latter is closely related to a choice of the lapse function.
Here we will consider two distinct settings: (a) a boundary @V at finite distance with prescribed induced metric there, and (b) a family of metrics which asymptote, along the asymptotically periodic ends, to Birmingham metrics.
At the boundary, or asymptotically, we make the following choice of the lapse function: as already mentioned, this corresponds to a choice of the boundary time, or asymptotic time. The choice is motivated by the fact that (B1) holds for all metrics in the Birmingham family; see (B4)-(B7) below.
In the case of a boundary at finite distance, we choose an (n À 1)-dimensional metric r 2 h on @V, as in (A1), and consider the collection of all initial metrics which induce It should be mentioned that the definition of a phase space requires describing also the space of canonical momenta. In the finite-boundary case this issue will be ignored in this work. Concerning asymptotically cylindrical metrics, we will only consider asymptotically vanishing extrinsic curvature tensors K ij . We plan to return to asymptotically periodic tensors K ij in future work.
In view of (B1), when Q A ¼ 0 and Q ? AB is pure trace at @V, it is useful [using (2.12) and (3.16)] to rewrite the boundary form (3.15) as whereas k 0 and 0 are the corresponding quantities calculated on a ''reference configuration'' corresponding to m ¼ m 0 . For configurations without boundaries with the above equalities should be understood in the limit x ! 1.
For the Birmingham metrics we have This implies that c vanishes identically on @V with the above boundary conditions, so that the entire boundary form reduces to The last term vanishes because of the time gauge (B1), whereas the first term represents the variation of mass: indeed, for all Birmingham metrics we have when the boundary data on @V are as above and where, as before, j@Vj h denotes the volume of @V in the metric h . In particular the integrand is independent of x 0 . Similarly, (B11) along each asymptotically periodic end.

APPENDIX C: THE YAMABE EQUATION ON CYLINDERS
In this section we relate the parameter m appearing in the Schwarzschild-de Sitter metrics to a Hamiltonian for the spherically symmetric Yamabe equation. The reader should note that the Hamiltonian here is a Hamiltonian for the dynamics in x, not to be confused with that for the dynamics in time, as used elsewhere in this work. Let Recall the vacuum Lichnerowicz equation with cosmological constant Ã, in space dimension n, Ág' À n À 2 4ðn À 1ÞR ' ¼ À 2 ' ð2À3nÞ=ðnÀ2Þ þ' nþ2 nÀ2 ; (C2) wherẽ 2 : ¼ n À 2 4ðn À 1Þ jLj 2 g ; : ¼ n À 2 4n 2 À n À 2 2ðn À 1Þ Ã: HereL ij isg-transverse traceless, and is the trace of the extrinsic curvature tensor ¼ g ij K ij , assumed to be constant, with K ij obtained fromL ij by the usual formula. Suppose thatg where h is as in (A1). We then haveR ¼ R , and when is a constant we can seek an x A -independent solution of (C2) withL ij ¼ 0: We apply the above to the Birmingham metrics with f ! 0; as discussed in Appendix A, the metrics with f 0 do not occur as asymptotic models in our context. We only consider regions where f > 0; the final formulas remain valid at f ¼ 0 by continuity.
The field of unit normals N to the static slices t ¼ const, which we denote by S t , is given by For those slices we have ¼ 0.
The volume form d