Towards the classification of static vacuum spacetimes with negative cosmological constant

Piotr T. Chruściel ^{∗} Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France

Walter Simon^{†} Institut für theoretische Physik Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria

December 20, 2000

Abstract

We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (“Schwarzschild—anti-de Sitter”) solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such space-times. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.

∗Supported in part by KBN grant # 2 P03B 073 15. E–mail: Chrusciel@Univ-Tours.fr †Supported by Jubiläumsfonds der Österreichischen Nationalbank, project # 6265, and by a grant from Région Centre, France. E–mail: simon@ap.univie.ac.at

Contents

1 Introduction 2

2 The generalized Kottler solutions 8 2.1 k = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Asymptotics 10 3.1 Three dimensional formalism . . . . . . . . . . . . . . . . . . . . 10 3.2 Four dimensional conformal approach . . . . . . . . . . . . . . . 17 3.3 A coordinate approach . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Connectedness of ∂_{∞}Σ 23

5 The mass 25 5.1 A coordinate mass M_{c} . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 The Hamiltonian mass M_{Ham}. . . . . . . . . . . . . . . . . . . . 27 5.3 A generalized Komar mass . . . . . . . . . . . . . . . . . . . . . . 32 5.4 The Hawking mass M_{Haw}(ψ) . . . . . . . . . . . . . . . . . . . . 34

6 The generalized Penrose inequality 35

7 Mass and area inequalities 39 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1 Introduction

Consider the families of metrics

ds^{2} = −(k − \frac{2m}{r} − \frac{Λ}{3}r^{2})dt^{2} + (k − \frac{2m}{r} − \frac{Λ}{3}r^{2})^{−1}dr^{2} + r^{2}dΩ_{k}^{2} , k = 0, ±1 ,

(1.1)

ds^{2} = −(λ − Λr^{2})dt^{2} + (λ − Λr^{2})^{−1}dr^{2} + |Λ|^{−1}dΩ_{k}^{2} , k = ±1 , kΛ > 0 , λ ∈ ℝ

(1.2)

where dΩ_{k}^{2} denotes a metric of constant Gauss curvature k on a two dimensional manifold ^{2}M. (Throughout this work we assume that ^{2}M is compact.) These are well known static solutions of the vacuum Einstein equation with a cosmological constant Λ; some subclasses of (1.1) and (1.2) have been discovered by de Sitter [64] ((1.1) with m = 0 and k = 1), by Kottler [56] (Equation (1.1) with an arbitrary m and k = 1), and by Nariai [60] (Equation (1.2) with k = 1). As discussed in detail in Section 5.4, the parameter m ∈ ℝ is related to the Hawking mass of the foliation t = const, r = const. We will refer to those solutions as the generalized Kottler and the generalized Nariai solutions. The constant Λ is an arbitrary real number, but in this paper we will mostly be interested in Λ < 0, and this assumption will be made unless explicitly stated otherwise. There has been recently renewed interest in the black hole aspects of the generalized Kottler solutions [19, 33, 59, 66]. The object of this paper is to initiate a systematic study of static solutions of the vacuum Einstein equations with a negative cosmological constant.

The first question that arises here is that of asymptotic conditions one wants to impose. In the present paper we consider metrics which tend to the generalized Kottler solutions, leaving the asymptotically Nariai case to future work. We present the following three approaches to asymptotic structure, and study their mutual relationships: three dimensional conformal compactifications, four dimensional conformal completions, and a coordinate approach. We show that under rather natural hypotheses the conformal boundary at infinity is connected.

The next question we address is that of the definition of mass for such solutions, without assuming staticity of the metrics. We review again the possible approaches that occur here: a naive coordinate approach, a Hamiltonian approach, a “Komar–type” approach, and the Hawking approach. We show that the Hawking mass converges to a finite value for the metrics considered here, and we also give conditions on the conformal completions under which the “coordinate mass”, or the Hamiltonian mass, are finite. Each of those masses come with different normalization factor, whenever all are defined, except for the Komar and Hamiltonian masses which coincide. We suggest that the correct normalization is the Hamiltonian one.

Returning to the static case, we recall that appropriately behaved vacuum black holes with Λ = 0 are completely described by the parameter m appearing above [20, 26, 48], and it is natural to enquire whether this remains true for other values of Λ. In fact, for Λ < 0, Boucher, Gibbons, and Horowitz [15] have given arguments suggesting uniqueness of the anti-de Sitter solution within an appropriate class. As a step towards a proof of a uniqueness theorem in the general case we derive, under appropriate hypotheses, 1) lower bounds on (loosely speaking) the area of cross-sections of the horizon, and 2) upper bounds on the mass of static vacuum black holes with negative cosmological constant. When these inequalities are combined the result goes precisely the opposite way as a (conjectured) generalization of the Geroch–Huisken–Ilmanen– Penrose inequality [16, 17, 37, 46, 47, 62] appropriate to space-times with nonvanishing cosmological constant. In fact, such a generalization was obtained by Gibbons [38] along the lines of Geroch [37], and of Jang and Wald [49], i.e. under the very stringent assumption of the global existence and smoothness of the inverse mean curvature flow, see Section 6. We note that it is far from clear that the arguments of Huisken and Ilmanen [46, 47], or those of Bray [16, 17], which establish the original Penrose conjecture can be adapted to the situation at hand. If this were the case, a combination of this with the results of the present work would give a fairly general uniqueness result. In any case this part of our work demonstrates the usefulness of a generalized Penrose inequality, if it can be established at all.

To formulate our results more precisely, consider a static space-time (M,^{4}g) which might — but does not have to — contain a black hole region. In the asymptotically flat case there exists a well established theory (see [22], or [26, Sections 2 and 6] and references therein) which, under appropriate hypotheses, allows one to reduce the study of such space-times to the problem of finding all suitable triples (Σ, g, V), where (Σ, g) is a three dimensional Riemannian manifold and V is a non-negative function on Σ. Further V is required to vanish precisely on the boundary of Σ, when non-empty:

V ≥ 0 ,  V (p) = 0 ⟺ p ∈ ∂Σ .  (1.3)

Finally g and V satisfy the following set of equations on Σ:

ΔV = −ΛV ,  (1.4)
R_{ij} = V^{−1}D_{i}D_{j}V + Λg_{ij}  (1.5)

(Λ = 0 in the asymptotically flat case). Here R_{ij} is the Ricci tensor of the (“three dimensional”) metric g. We shall not attempt to formulate the conditions on (M,^{4}g) which will allow one to perform such a reduction (some of the aspects of the relationship between (Σ, g, V) and the associated space-time are discussed in Section 3.2, see in particular Equation (3.37)), but we shall directly address the question of properties of solutions of (1.4)–(1.5). Our first main result concerns the topology of ∂Σ (cf. Theorem 4.1, Section 4; compare [32, 69]):

Theorem 1.1 Let Λ < 0, consider a set (Σ, g, V) which is C^{3} conformally compactifiable in the sense of Definition 3.1 below, suppose that (1.3)–(1.5) hold. Then the conformal boundary at infinity ∂_{∞}Σ of Σ is connected.

Our second main result concerns the Hawking mass of the level sets of V, cf. Theorem 5.2, Section 5.4:

Theorem 1.2 Under the conditions of Theorem 1.1, the Hawking mass m of the level sets of V is well defined and finite.

It is natural to enquire whether there exist static vacuum space-times with complete spacelike hypersurfaces and no black hole regions; it is expected that no such solutions exist when Λ < 0 and ∂_{∞}Σ ≠ S^{2}. We hope that points 2. and 3. of the following theorem can be used as a tool to prove their non-existence:

Theorem 1.3 Under the conditions of Theorem 1.1, suppose further that

∂Σ = ∅ ,

and that the scalar curvature R′ of the metric g′ = V^{−2}g is constant on ∂_{∞}Σ. Then:

1. If ∂_{∞}Σ is a sphere, then the Hawking mass m of the level sets of V is nonpositive, vanishing if and only if there exists a diffeomorphism ψ : Σ → Σ_{0} and a positive constant λ such that g = ψ^{∗}g_{0} and V = λV_{0} ∘ ψ, with (Σ_{0}, g_{0}, V_{0}) corresponding to the anti-de Sitter space-time.

2. If ∂_{∞}Σ is a torus, then the Hawking mass m is strictly negative.

3. If the genus g_{∞} of ∂_{∞}Σ is higher than or equal to 2 we have

m < −\frac{1}{3\sqrt{−Λ}} ,  (1.6)

with m = m(V) normalized as in Equation (6.7).

A mass inequality similar to that in point 1. above has been established in [15], and in fact we follow their technique of proof. However, our hypotheses are rather different. Further, the mass here is a priori different from the one considered in [15]; in particular it isn’t clear at all whether the mass defined as in [15] is also defined for the metrics we consider, cf. Sections 3.3 and 5.1 below.

We note that metrics satisfying the hypotheses of point 2 above, with arbitrarily large (strictly) negative mass, have been constructed in [45].

As a straightforward corollary of Theorem 1.3 one has:

Corollary 1.4 Suppose that the generalized positive energy inequality

m ≥ m_{crit}(g_{∞})

holds in the class of three dimensional manifolds (Σ, g) which satisfy the requirements of point 1. of Definition 3.1 with a connnected conformal infinity ∂_{∞}Σ of genus g_{∞}, and, moreover, the scalar curvature R of which satisfies R ≥ 2Λ. Then:

1. If m_{crit}(g_{∞} = 0) = 0, then the only solution of Equations (1.4)–(1.5) satisfying the hypotheses of point 1. of Theorem 1.3 is the one obtained from anti-de Sitter space-time.

2. If m_{crit}(g_{∞} > 1) = −1/(3\sqrt{−Λ}), then there exist no solutions of Equations (1.4)–(1.5) satisfying the hypotheses of point 3. of Theorem 1.3.

When ∂_{∞}Σ = S^{2} one expects that the inequality m ≥ 0, with m being the mass defined by spinorial identities can be established using Witten type techniques (cf. [6, 39]), regardless of whether or not ∂Σ = ∅. (On the other hand it follows from [11] that when ∂_{∞}Σ ≠ S^{2} there exist no asymptotically covariantly constant spinors which can be used in the Witten argument.) This might require imposing some further restrictions on e.g. the asymptotic behavior of the metric. To be able to conclude in this case that there are no static solutions without horizons, or that the only solution with a connected non-degenerate horizon is the anti-de Sitter one, requires working out those restrictions, and showing that the Hawking mass of the level sets of V coincides with the mass occuring in the positive energy theorem.

When horizons occur, our comparison results for mass and area read as follows:

Theorem 1.5 Under the conditions of Theorem 1.1, suppose further that the genus g_{∞} of ∂_{∞}Σ satisfies

g_{∞} ≥ 2 , and that the scalar curvature R′ of the metric g′ = V^{−2}g is constant on ∂_{∞}Σ. Let ∂_{1}Σ be any connected component of ∂Σ for which the surface gravity κ defined by Equation (7.1) is largest, and assume that

0 < κ ≤ \sqrt{−\frac{Λ}{3}} .  (1.7)

Let m_{0}, respectively A_{0}, be the Hawking mass, respectively the area of ∂Σ_{0}, for that generalized Kottler solution (Σ_{0}, g_{0}, V_{0}), with the same genus g_{∞}, the surface gravity κ_{0} of which equals κ. Then

m ≤ m_{0} ,  A_{0}(g_{∂_{1}Σ} − 1) ≤ A(g_{∞} − 1) ,  (1.8)

where A is the area of ∂_{1}Σ and m = m(V) is the Hawking mass of the level sets of V. Further m = m_{0} if and only if there exists a diffeomorphism ψ : Σ → Σ_{0} and a positive constant λ such that g = ψ^{∗}g_{0} and V = λV_{0} ∘ ψ.

The asymptotic conditions assumed in Theorems 1.3 and 1.5 are somewhat related to those of [9, 15, 43, 44]. The precise relationships are discussed in Sections 3.2 and 3.3. Let us simply mention here that the condition that R′ is constant on ∂_{∞}Σ is the (local) higher genus analogue of the (global) condition in [9, 43] that the group of conformal isometries of ℐ coincides with that of the standard conformal completion of the anti-de Sitter space-time; the reader is referred to Proposition 3.6 in Section 3.2 for a precise statement.

We note that the hypothesis (1.7) is equivalent to the assumption that the generalized Kottler solution with the same value of κ has non-positive mass; cf. Section 2 for a discussion. We emphasize, however, that we do not make any a priori assumptions concerning the sign of the mass of (Σ, g, V). Our methods do not lead to any conclusions for those values of κ which correspond to generalized Kottler solutions with positive mass.

With m = m(V) normalized as in Equation (6.7), the inequality m ≤ m_{0} takes the following explicit form

m ≤ \frac{(Λ + 2κ^{2})\sqrt{κ^{2} − Λ} + 2κ^{3}}{3Λ^{2}} ,  (1.9)

while A(g_{∞} − 1) ≥ A_{0}(g_{∂_{1}Σ} − 1) can be explicitly written as

A(g_{∞} − 1) ≥ 4π(g_{∂_{1}Σ} − 1) [\frac{κ + \sqrt{κ^{2} − Λ}}{Λ}]^{2} .  (1.10)

(The right-hand sides of Equations (1.9) and (1.10) are obtained by straightforward algebraic manipulations from (2.1) and (2.11).)

It should be pointed out that in [70] a lower bound for the area has also been established. However, while the bound there is sharp only for the generalized Kottler solutions with m = 0, our bound is sharp for all Kottler solutions. On the other hand in [70] it is not assumed that the space-time is static.

If the generalized Penrose inequality (which we discuss in some detail in Section 6) holds,

2M_{Haw}(u) ≥ ∑_{i=1}^{k} ((1 − g_{∂_{i}Σ}) (\frac{A_{∂_{i}Σ}}{4π} )^{1/2} − \frac{Λ}{3} (\frac{A_{∂_{i}Σ}}{4π} )^{3/2})  (1.11)

(with the ∂_{i}Σ’s, i = 1, …, k, being the connected components of ∂Σ, the A_{∂_{i}Σ}’s — their areas, and the g_{∂_{i}Σ}’s — the genera thereof) we obtain uniqueness of solutions:

Corollary 1.6 Suppose that the generalized Penrose inequality (1.11) holds in the class of three dimensional manifolds (Σ, g) with scalar curvature R satisfying R ≥ 2Λ, which satisfy the requirements of point 1. of Definition 3.1 with a connnected conformal infinity ∂_{∞}Σ of genus g_{∞} > 1, and which have a compact connected boundary. Then the only static solutions of Equations (1.4)–(1.5) satisfying the hypotheses of Theorem 1.5 are the corresponding generalized Kottler solutions.

This paper is organized as follows: in Section 2 we discuss those aspects of the generalized Kottler solutions which are relevant to our work. The main object of Section 3 is to set forth the boundary conditions which are appropriate for the problem at hand. In Section 3.1 this is analyzed from a three dimensional point of view. We introduce the class of objects considered in Definition 3.1, and analyze the consequences of this Definition in the remainder of that section. In Section 3.2 four-dimensional conformal completions are considered; in particular we show how the setup of Section 3.1 relates to a four dimensional one, cf. Proposition 3.4 and Theorem 3.5. We also show there how the requirement of local conformal flatness of the geometry of ℐ relates to the restrictions on the geometry of ∂_{∞}Σ considered in Section 3.1. In Section 3.3 a four dimensional coordinate approach is described; in particular, when (M, g) admits suitable conformal completions, we show there how to construct useful coordinate systems in a neighborhood of ℐ — cf. Proposition 3.7. In Section 4 connectedness of the conformal boundary ∂_{∞}Σ is proved under suitable conditions. Section 5 is devoted to the question how to define the total mass for the class of space-times at hand. This is discussed from a coordinate point of view in Section 5.1, from a Hamiltonian point of view in Section 5.2, and using the Hawking approach in Section 5.4; in Section 5.3 we present a generalization of the Komar integral appropriate to our setting. The main results of the analysis in Section 5 are the boundary conditions (5.19) together with Equation (5.22), which gives an ADM-type expression for the Hamiltonian mass for space-times with generalized Kottler asymptotics; we emphasize that this formula holds without any hypotheses of staticity or stationarity of the space-time metric. Theorem 1.2 is proved in Section 5.4. In Section 6 we recall an argument due to Gibbons [38] for the validity of the generalized Penrose inequality. (However, our conclusions are different from those of [38].) In Section 7 we prove Theorems 1.3 and 1.5, as well as Corollary 1.6.

Acknowledgements W.S. is grateful to Tom Ilmanen for helpful discussions on the Penrose inequality. We thank Gary Horowitz for pointing out Reference [45].

2 The generalized Kottler solutions

We recall some properties of the solutions (1.1). Those solutions will be used as reference solutions in our arguments, so it is convenient to use a subscript 0 when referring to them. As already mentioned, we assume Λ < 0 unless indicated otherwise. For m_{0} ∈ ℝ, let r_{0} be the largest positive root of the equation ^{1}

V_{0}^{2} ≡ k − \frac{2m_{0}}{r} − \frac{Λ}{3}r^{2} = 0 .  (2.1)

We set

Σ_{0} = {(r, v)|r > r_{0}, v ∈ ^{2}M} ,  (2.2)
g_{0} = (k − \frac{2m_{0}}{r} − \frac{Λ}{3}r^{2})^{−1}dr^{2} + r^{2}dΩ_{k}^{2} ,  (2.3)

where, as before, dΩ_{k}^{2} denotes a metric of constant Gauss curvature k on a smooth two dimensional compact manifold ^{2}M. We denote the corresponding surface gravity by κ_{0}. (Recall that the surface gravity of a connected component of a horizon 𝒩[X] is usually defined by the equation

(X^{α}X_{α})_{,μ}|_{𝒩[X]} = −2κX_{μ} ,  (2.4)

where X is the Killing vector field which is tangent to the generators of 𝒩[X]. This requires normalizing X; here we impose the normalization^{2} that X = ∂/∂t in the coordinate system of (1.1).) We set

W_{0}(r) ≡ g_{0}^{ij}D_{i}V_{0}D_{j}V_{0} = (\frac{m_{0}}{r^{2}} − \frac{Λr}{3} )^{2} .  (2.5)

When m_{0} = 0 we note the relationship

W_{0} = −\frac{Λ}{3}(V_{0}^{2} − k) ,  (2.6)

which will be useful later on, and which holds regardless of the topology of ^{2}M.

2.1 k = −1

Suppose, now, that k = −1, and that m_{0} is in the range

m_{0} ∈ [m_{crit}, 0] ,  (2.7)

1See [66] for an exhaustive analysis, and explicit formulae for the roots of Equation (2.1).

2When ^{2}M = T^{2} a unique normalization of X needs a further normalization of dΩ_{k}^{2}, cf.

Sections 5.1 and 5.2 for a detailed discussion of this point.

where

m_{crit} ≡ −\frac{1}{3\sqrt{−Λ}} .  (2.8)

Here m_{crit} is defined as the smallest value of m_{0} for which the metrics (1.1) can be extended across a Killing horizon [19, 66]. Let us show that Equation (2.7) is equivalent to

r_{0} ∈ [\frac{1}{\sqrt{−Λ}}, \sqrt{−\frac{3}{Λ}}] .  (2.9)

In order to simplify notation it is useful to introduce

\frac{1}{ℓ^{2}} ≡ −\frac{Λ}{3} .  (2.10)

Now, the equation V_{0}(ℓ/\sqrt{3}) = 0 implies m = m_{crit}. Next, an elementary analysis of the function r^{3}/ℓ^{2}−r−2m_{0} (recall that k = −1 in this section) shows that 1) V has no positive roots for m < m_{crit}; 2) for m = m_{crit} the only positive root is ℓ/\sqrt{3}; 3) if r_{0} is the largest positive root of the equation V_{0}(r_{0}) = 0, then for each m_{0} > m_{crit} the radius r_{0}(m_{0}) exists and is a differentiable function of m_{0}. Differentiating the equation r_{0}V_{0}(r_{0}) = 0 with respect to m_{0} gives (\frac{3r_{0}^{2}}{ℓ^{2}} +

k)\frac{∂r_{0}}{∂m_{0}} = (\frac{3r_{0}^{2}}{ℓ^{2}} −1)\frac{∂r_{0}}{∂m_{0}} = 2. It follows that for r ≥ ℓ/\sqrt{3} the function r_{0}(m_{0}) is a monotonically increasing function on its domain of definition [m_{crit}, ∞), which establishes our claim.

We note that the surface gravity κ_{0} is given by the formula

κ_{0} = \sqrt{W_{0}(r_{0})} = \frac{m_{0}}{r_{0}^{2}} + \frac{r_{0}}{ℓ^{2}} ,  (2.11)

which gives

\frac{∂κ_{0}}{∂m_{0}} = \frac{1}{r_{0}^{2}} + (\frac{1}{ℓ^{2}} − \frac{2m_{0}}{r_{0}^{3}} ) \frac{∂r_{0}}{∂m_{0}} .

Equation (2.11) shows that κ_{0} vanishes when m_{0} = m_{crit}. ^{3} Under the hypothesis that m_{0} ≤ 0, it follows from what has been said above a) that \frac{∂κ_{0}}{∂m_{0}} is positive; b) that we have

κ_{0} ∈ [0, \sqrt{−\frac{Λ}{3}}] ,  (2.12)

when (2.7) holds, and c) that, under the current hypotheses on k and Λ, (2.7) is equivalent to (2.12) for the metrics (1.1). While this can probably be established directly, we note that it follows from Theorem 1.5 that (2.12) is equivalent to (2.7) without having to assume that m_{0} ≤ 0.

3The methods of [68] show that in this case the space-times with metrics (1.1) can be extended to black hole space-times with a degenerate event horizon, thus a claim to the contrary in [66] is wrong. It has been claimed without proof in [19] that ℐ^{+}, as constructed by the methods of [68], can be extended to a larger one, say ℐ̂^{+}, which is connected. Recall that that claim would imply that ∂I^{−}(ℐ̂^{+}) = ∅ (see Figure 2 in [19]), thus the space-time would not contain an event horizon with respect to ℐ̂^{+}. Regardless of whether such an extended ℐ̂^{+} exists or not, we wish to point out the following: a) there will still be degenerate event horizons as defined with respect to any connected component of ℐ^{+}; b) regardless of how null infinity is added there will exist degenerate Killing horizons in those space-times; c) there will exist an observer horizon associated to the world-line of any observer which moves along the orbits of the Killing vector field in the asymptotic region. It thus appears reasonable to give those space-times a black hole interpretation in any case.

In what follows we shall need the fact that in the above ranges of parameters the relationship V_{0}(r) can be inverted to define a smooth function r(V_{0}) : [0, ∞) → ℝ. Indeed, the equation \frac{dV_{0}}{dr} (r_{crit}) = 0 yields r_{crit}^{3} = 3m_{0}/Λ; when k = −1, Λ < 0, and when (2.7) holds one finds V_{0}(r_{crit}) ≤ 0, with the inequality being strict unless m = m_{crit}. Therefore, V_{0}(r) is a smooth strictly monotonic function in [r_{0}, ∞), which implies in turn that r(V_{0}) is a smooth strictly monotonic function on (0, ∞); further r(V_{0}) is smooth up to 0 except when m = m_{crit}.

3 Asymptotics

3.1 Three dimensional formalism

As a motivation for the definition below, consider one of the metrics (1.1) and introduce a new coordinate x ∈ (0, x_{0}] by

\frac{r^{2}}{ℓ^{2}} = \frac{1 − kx^{2}}{x^{2}} .  (3.1)

with x_{0} defined by substituting r_{0} in the left-hand-side of (3.1). It then follows that

g = ℓ^{2}x^{−2} [\frac{dx^{2}}{(1 − kx^{2})(1 − \frac{2mx^{3}}{ℓ\sqrt{1−kx^{2}}} )} + (1 − kx^{2})dΩ_{k}^{2}] .

Thus the metric

g′ ≡ (ℓ^{−2}x^{2})g

is a smooth up to boundary metric on the compact manifold with boundary \overline{Σ}_{0} ≡ [0, x_{0}]×^{2}M. Furthermore, xV_{0} can be extended by continuity to a smooth up to boundary function on \overline{Σ}_{0}, with xV_{0} = 1. This justifies the following definition:

Definition 3.1 Let Σ be a smooth manifold^{4}, with perhaps a compact boundary which we denote by ∂Σ when non empty.^{5} Suppose that g is a smooth metric on Σ, and that V is a smooth nonnegative function on Σ, with V (p) = 0 if and only if p ∈ ∂Σ.

1. (Σ, g) will be said to be C^{i}, i ∈ ℕ ∪ {∞}, conformally compactifiable or, shortly, compactifiable, if there exists a C^{i+1} diffeomorphism χ from Σ \ ∂Σ to the interior of a compact Riemannian manifold with boundary (\overline{Σ} ≈ Σ ∪ ∂_{∞}Σ, \overline{g}), with ∂_{∞}Σ ∩ Σ = ∅, and a C^{i} function ω : \overline{Σ} → ℝ^{+} such that

g = χ^{∗}(ω^{−2}\overline{g}) .  (3.2)

We further assume that {ω = 0} = ∂_{∞}Σ, with dω nowhere vanishing on ∂_{∞}Σ, and that \overline{g} is of C^{i} differentiability class on \overline{Σ}.

4All manifolds are assumed to be Hausdorff, paracompact, and orientable throughout.

5We use the convention that a manifold with boundary Σ contains its boundary as a point set.

2. A triple (Σ, g, V) will be said to be C^{i}, i ∈ ℕ∪{∞}, compactifiable if (Σ, g) is C^{i} compactifiable, and if V ω extends by continuity to a C^{i} function on \overline{Σ},

3. with

lim_{ω→0} V ω > 0 .  (3.3)

We emphasize that Σ itself is allowed to have a boundary on which V vanishes,

∂Σ = {p ∈ Σ|V (p) = 0} .

If that is the case we will have

∂\overline{Σ} = ∂Σ ∪ ∂_{∞}Σ .

To avoid ambiguities, we stress that one point compactifications of the kind encountered in the asymptotically flat case (cf., e.g., [13]) are not allowed in our context.

The conditions above are not independent when the “static field equations” (Equations (1.4)–(1.5)) hold:

Proposition 3.2 Consider a triple (Σ, g, V) satisfying Equations (1.3)–(1.5).

1. The condition that |dω|\overline{g} has no zeros on ∂_{∞}Σ follows from the remaining hypotheses of point 1. of Definition 3.1, when those hold with i ≥ 2.

2. Suppose that (Σ, g) is C^{i} compactifiable with i ≥ 2. Then lim_{ω→0} V ω exists. Further, one can choose a (uniquely defined) conformal factor so that ω is the \overline{g}-distance from ∂_{∞}Σ. With this choice of conformal factor, when (3.3) holds a necessary condition that (Σ, g, V) is C^{i} compactifiable is that

(4\overline{R}_{ij} − \overline{R}\overline{g}_{ij})\overline{n}^{i}\overline{n}^{j}|_{∂_{∞}Σ} = 0 ,  (3.4)

where \overline{n} is the field of unit normals to ∂_{∞}Σ.

3. (Σ, g, V) is C^{∞} compactifiable if and only if (Σ, g) is C^{∞} compactifiable and Equations (3.3) and (3.4) hold.

Remarks: 1. When (Σ, g) is C^{∞} compactifiable but Equation (3.4) does not hold, the proof below shows that V ω is of the form α_{0} + α_{1}ω^{2} log ω, for some smooth up-to-boundary functions α_{0} and α_{1}. This isn’t perhaps so surprising because the nature of the equations satisfied by g and V suggests that both \overline{g} and V ω should be polyhomogeneous, rather than smooth. (“Polyhomogeneous” means that \overline{g} and V ω are expected to admit asymptotic expansions in terms of powers of ω and log ω near ∂_{∞}Σ under some fairly weak conditions on their behavior at ∂_{∞}Σ; cf., e.g. [4] for precise definitions and related results.) From this point of view the hypothesis that (Σ, g) is C^{∞} compactifiable is somewhat unnatural and should be replaced by that of polyhomogeneity of \overline{g} at ∂_{∞}Σ.

2. One can prove appropriate versions of point 3. above for (Σ, g)’s which are C^{i} compactifiable for finite i. This seems to lead to lower differentiability of 1/V near ∂_{∞}Σ as compared to \overline{g}, and for this reason we shall not discuss it here.

3. We leave it as an open problem whether or not there exist solutions of (1.3)–(1.5) such that (Σ, g) is smoothly compactifiable, such that V can be extended by continuity to a smooth function on \overline{Σ}, while (3.3) does not hold.

4. We note that (3.4) is a conformally invariant condition because ω and \overline{g} are uniquely determined by g. However, it is not conformally covariant, in the sense that if \overline{g} is conformally rescaled, then (3.4) will not be of the same form in the new rescaled metric. It would be of interest to find a form of (3.4) which does not have this drawback.

5. The result above has counterparts for one-point compactifications in the asymptotically flat case, (cf., e.g., the theorem in the Appendix of [13]).

Proof: Let

α ≡ V ω .

After suitable identifications we can without loss of generality assume that the map χ in (3.2) is the identity. Equations (1.4)–(1.5) together with the definition of \overline{g} = ω^{2}g lead to the following

\overline{Δ}α − 3\frac{\overline{D}^{i}ω\overline{D}_{i}α}{ω} + (\frac{\overline{Δ}ω}{ω} + \frac{\overline{R}}{2}) α = 0 ,  (3.5)
\overline{D}_{i}\overline{D}_{j}α − \frac{\overline{D}^{k}ω\overline{D}_{k}α}{ω} \overline{g}_{ij} = (\overline{R}_{ij} + 2\frac{\overline{D}_{i}\overline{D}_{j}ω}{ω} − (\frac{\overline{Δ}ω}{ω} + \frac{\overline{R}}{2}) \overline{g}_{ij}) α .  (3.6)

We have also used R = 2Λ which, together with the transformation law of the curvature scalar under conformal transformations, implies

ω^{2}\overline{R} = 6|dω|2 ; \overline{g} + 2Λ − 4ω\overline{Δ}ω .  (3.7)

In all the equations here barred quantities refer to the metric \overline{g}. Point 1 of the proposition follows immediately from Equation (3.7).

To avoid factors of −Λ/3 in the remainder of the proof we rescale the metric g so that Λ = −3. Next, to avoid annoying technicalities we shall present the proof only for smoothly compactifiable (Σ, g) — i = ∞; the finite i cases can be handled using the results in [4, Appendix A] and [28, Appendix A]. Suppose, thus, that i = ∞. As shown in [5, Lemma 2.1] we can choose ω and \overline{g} so that ω coincides with the \overline{g}-distance from ∂_{∞}Σ in a neighborhood of ∂_{∞}Σ; we shall use the symbol x to denote this function. In this case we have

\overline{Δ}ω = \overline{p} ,  (3.8)

where \overline{p} is the mean curvature of the level sets of ω = x. Further |dω|\overline{g} = 1 so that (3.8) together with (3.7) give

\overline{R} = −4\frac{\overline{p}}{x} ,  (3.9)

in particular

\overline{p}|_{x=0} = 0 .  (3.10)

We can introduce Gauss coordinates (x^{1}, x^{A}) near ∂_{∞}Σ in which x^{1} = x ∈ [0, x_{0}), while the (x^{A}) = v’s form local coordinates on ∂_{∞}Σ, with the metric taking the form

\overline{g} = dx^{2} + \overline{h} ,  \overline{h}(∂_{x}, ·) = 0 .  (3.11)

To prove point 2, from Equation (3.6) we obtain

ω\overline{D}^{i}ω\overline{D}^{j}ω\overline{D}_{i}(ω^{−1}\overline{D}_{j}α) =
= \overline{D}^{i}ω\overline{D}^{j}ω (\overline{R}_{ij} + 2\frac{\overline{D}_{i}\overline{D}_{j}ω}{ω} − (\frac{\overline{Δ}ω}{ω} + \frac{\overline{R}}{2}) \overline{g}_{ij}) α .  (3.12)

Equations (3.8)–(3.12) lead to

x∂_{x}(x^{−1}∂_{x}α) = (\overline{R}_{xx} − \frac{\overline{R}}{4})α .  (3.13)

At each v ∈ ∂_{∞}Σ this is an ODE of Fuchsian type for α(x, v). Standard results about such equations show that for each v the functions x → α(x, v) and x → ∂_{x}α(x, v) are bounded and continuous on [0, x_{0}). Integrating (3.13) one finds

∂_{x}α = xβ(v) + (\overline{R}_{xx} − \frac{\overline{R}}{4})α(0, v)x ln x + O(x^{2} ln x) ,  (3.14)

where β(v) is a (v-dependent) integration constant. By hypothesis there exist no points at ∂_{∞}Σ such that α(0, v) = 0, Equations (3.13) and (3.14) show that ∂_{x}^{2}α blows up at x = 0 unless (3.4) holds, and point 2. follows.

We shall only sketch the proof of point 3.: Standard results about Fuchsian equations show that solutions of Equation (3.13) will be smooth in x whenever (\overline{R}_{xx} − \frac{\overline{R}}{4})(x = 0, v) vanishes throughout ∂_{∞}Σ. A simple bootstrap argument applied to Equation (3.6) with (ij) = (1A) shows that α is also smooth in v. Commuting Equation (3.6) with (x∂_{x})^{i}∂_{v}^{β}, where β is an arbitrary multi-index, and iteratively repeating the reasoning outlined above establishes smoothness of α jointly in v and x. □

A consequence of condition 3 of Definition 3.1 is that the function

V′ ≡ 1/V ,

when extended to \overline{Σ} by setting V′ = 0 on ∂_{∞}Σ, can be used as a compactifying conformal factor, at least away from ∂Σ: If we set

g′ = V^{−2}g ,

then g′ is a Riemannian metric smooth up to boundary on \overline{Σ} \ ∂Σ. In terms of this metric Equations (1.4)–(1.5) can be rewritten as

Δ′V′ = 3V′W + ΛV ,  (3.15)
R′_{ij} = −2V D′_{i}D′_{j}V′ .  (3.16)

Here R′_{ij} is the Ricci tensor of the metric g′, D′ is the Levi–Civita covariant derivative associated with g′, while Δ′ is the Laplace operator associated with g′. Taking the trace of (3.16) and using (3.15) we obtain

R′ = −6W − 2ΛV^{2} ,  (3.17)

where

W ≡ D_{i}V D^{i}V .  (3.18)

Defining

W′ ≡ g^{′ij}D′_{i}V′D′_{j}V′ = (V′)^{2}W ,  (3.19)

Equation (3.17) can be rewritten as

6W′ = −2Λ − R′(V′)^{2} .  (3.20)

If (Σ, g, V) is C^{2} compactifiable then R′ is bounded in a neighborhood of ∂_{∞}Σ, and since V blows up at ∂_{∞}Σ it follows from Equation (3.17) that so does W, in particular W is strictly positive in a neighborhood of ∂_{∞}Σ. Further Equation (3.20) implies that the level sets of V are smooth manifolds in a neighborhood of ∂_{∞}Σ, diffeomorphic to ∂_{∞}Σ there.

Equations (1.4)–(1.5) are invariant under a rescaling V → λV, λ ∈ ℝ^{∗}. This is related to the possibility of choosing freely the normalization of the Killing vector field in the associated space-time. Similarly the conditions of Definition 3.1 are invariant under such rescalings with λ > 0. For various purposes — e.g.,, for the definition (7.1) of surface gravity — it is convenient to have a unique normalization of V. We note that if (Σ, g, V) corresponds to a generalized Kottler solution (Σ_{0}, g_{0}, V_{0}), then (1.1) and (2.5) together with (3.18) give 6W′_{0} = −2Λ(1 − k(V′_{0})^{2}) + O((V′_{0})^{3}) so that from (3.17) one obtains

R′_{0}|_{∂_{∞}Σ} = −2Λk .  (3.21)

We have the following:

Proposition 3.3 Consider a C^{i}-compactifiable triple (Σ, g, V), i ≥ 3, satisfying equations (1.4)–(1.5).

1. We have

2R′|_{x=0} = \frac{1}{3}R′|_{x=0} ,  (3.22)

where ^{2}R′ is the scalar curvature of the metric induced by g′ ≡ V^{−2}g on the level sets of V, and R′ is the Ricci scalar of g′.

2. If R′ is constant on ∂_{∞}Σ, replacing V by a positive multiple thereof if necessary we can achieve

R′|_{∂_{∞}Σ} = −2Λk ,  (3.23)

where k = 0, 1 or −1 according to the sign of the Gauss curvature of the metric induced by g′ on ∂_{∞}Σ.

Remark: When k = 0 Equation (3.23) holds with an arbitrary normalization of V.

Proof: Consider a level set {V = const} of V which is a smooth hypersurface in \overline{Σ}, with unit normal n_{i}, induced metric h_{ij}, scalar curvature ^{2}R, second fundamental form p_{ij} defined with respect to an inner pointing normal, mean curvature p = h^{ij}p_{ij} = h_{i}^{k}h_{j}^{m}D_{(k}n_{m)}; we denote by q_{ij} the trace-free part of p_{ij}: q_{ij} = p_{ij} − 1/2h_{ij}p. Let R_{ijk}, respectively R′_{ijk}, be the Cotton tensor of the metric g_{ij}, respectively g′_{ij}; by definition,

R_{ijk} = 2 (R_{i[j} − \frac{1}{4}Rg_{i[j})_{;k]} ,  (3.24)

where square brackets denote antisymmetrization with an appropriate combinatorial factor (1/2 in the equation above), and a semi-column denotes covariant differentiation. We note the useful identity due to Lindblom [57]

R′_{ijk}R^{′ijk} = V^{6}R_{ijk}R^{ijk}
= 8(V W)^{2}q_{ij}q^{ij} + V^{2}h^{ij}D_{i}W D_{j}W .  (3.25)

When (Σ, g, V) is C^{3} compactifiable the function R′_{ijk}R^{′ijk} is uniformly bounded on a neighborhood of \overline{Σ}, which gives

(V W)^{2}q_{ij}q^{ij} ≤ C  (3.26)

in that same neighborhood, for some constant C. Equations (3.26) and (3.19) give

|q|_{g} = O((V′)^{3}) ,  (3.27)

Let q′_{ij} be the trace-free part of the second fundamental form p′_{ij} of the level sets of V′ with respect to the metric g′_{ij}, defined with respect to an inner pointing normal; we have q′_{ij} = q_{ij}/V, so that

|q′|_{g′} = O((V′)^{2}) .  (3.28)

Throughout we use | · |_{k} to denote the norm of a tensor field with respect to a metric k.

Let us work out some implications of (3.28); Equations (3.15)–(3.17) lead to

(Δ′ + \frac{R′}{2} )V′ = 0 .  (3.29)

Equations (3.19) and (3.20) show that dV′ is nowhere vanishing on a suitable neighborhood of ∂_{∞}Σ. We can thus introduce coordinates there so that

V′ = x .

If the remaining coordinates are Lie dragged along the integral curves of D′x the metric takes the form

g′ = (W′)^{−1} dx^{2} + h′ ,  h′(∂_{x}, ·) = 0 .  (3.30)

Equations (3.29)–(3.30) give then

p′ = −\frac{1}{2\sqrt{W′}} (\frac{∂W′}{∂x} + R′x)
= −\frac{x}{12\sqrt{W′}} (4R′ − x\frac{∂R′}{∂x} ) ,  (3.31)

and in the second step we have used (3.20). Here p′ = \sqrt{W′}∂_{x}(\sqrt{det h′})/\sqrt{det h′} is the mean curvature of the level sets of x measured with respect to the inner pointing normal n′ = \sqrt{W′}∂_{x}. Equation (3.16) implies

R′_{ij}n^{′i}n^{′j} = −2V n^{′i}n^{′j}D′_{i}D′_{j}V′
= −2\frac{D^{′i}V′D^{′j}V′}{V′W′} D′_{i}D′_{j}V′
= −\frac{D^{′i}V′D′_{i}W′}{V′W′} = \frac{−∂_{x}W′}{x}

in the coordinate system of Equation (3.30). From (3.20) we get

R′_{ij}n^{′i}n^{′j} = \frac{R′}{3} + O(x) .  (3.32)

From the Codazzi–Mainardi equation,

(−2R′_{ij} + R′g′_{ij})n^{′i}n^{′j} = ^{2}R′ + q′_{ij}q^{′ij} − \frac{1}{2}p^{′2} ,  (3.33)

where ^{2}R′ is the scalar curvature of the metric induced by g′ on ∂_{∞}Σ, one obtains

(−2R′_{ij} + R′g′_{ij})n^{′i}n^{′j} = ^{2}R′ + O(x) ,  (3.34)

where we have used (3.28) and (3.31). This, together with Equation (3.32), establishes Equation (3.22). In particular R′|_{∂_{∞}Σ} is constant if and only if 2R′ is, and R′ at x = 0 has the same sign as the Gauss curvature of the relevant connected component of ∂_{∞}Σ. Under a rescaling V → λV, λ > 0, we have W → λ^{2}V; Equation (3.17) shows that R′ → λ^{2}R′, and choosing λ appropriately establishes the result. □

We do not know whether or not there exist smoothly compactifiable solutions of Equations (1.4)–(1.5) for which R′ is not locally constant at ∂_{∞}Σ, it would be of interest to settle this question. Let us point out that the remaining Codazzi–Mainardi equations do not lead to such a restriction. For example, consider the following equation:

R′_{1a} = −𝒟′_{a}p′ + 𝒟′_{b}p′_{a}^{b}
= −\frac{1}{2}𝒟′_{a}p′ + 𝒟′_{b}q′_{a}^{b} .  (3.35)

Here we are using the adapted coordinate system of Equation (3.30) with x^{1} = x and with the indices a, b = 2, 3 corresponding to the remaining coordinates; further 𝒟′ denotes the Levi–Civita derivative associated with the metric h′. Since D′_{a}D′_{x}x = 𝒟′_{a}\sqrt{W′}, Equation (3.16) yields

𝒟′_{a}(\sqrt{W′} − \frac{1}{4}xp′) = −\frac{1}{2}x𝒟′_{b}q′_{a}^{b} = O(x^{3}) ;  (3.36)

in the last equality Equation (3.28) has been used. Unfortunately the terms containing R′ exactly cancel out in Equations (3.31) and (3.20) leading to

\sqrt{W′} + \frac{1}{4}xp′ = \sqrt{−\frac{Λ}{3}} + O(x^{3}) ,

which does not provide any new information.

3.2 Four dimensional conformal approach

Consider a space-time (M,^{4}g) of the form M = ℝ × Σ with the metric ^{4}g

4g = −V^{2}dt^{2} + g , g(∂_{t}, ·) = 0 , ∂_{t}V = ∂_{t}g = 0 .  (3.37)

By definition of a space-time ^{4}g has Lorentzian signature, which implies that g has signature +3; it then naturally defines a Riemannian metric on Σ which will still be denoted by g. Equations (1.4)–(1.5) are precisely the vacuum Einstein equations with cosmological constant Λ for the metric ^{4}g. It has been suggested that an appropriate [9, 43] framework for asymptotically anti-de Sitter spacetimes is that of conformal completions introduced by Penrose [61]. The work of Friedrich [31] has confirmed that it is quite reasonable to do that, by showing that a large class of space-times (not necessarily stationary) with the required properties exist; some further related results can be found in [50, 58]. In this approach one requires that there exists a space-time with boundary (\overline{M},^{4}\overline{g}) and a positive function Ω : \overline{M} → ℝ^{+}, with Ω vanishing precisely at ℐ ⊂ ∂\overline{M}, and with dΩ without zeros on ℐ, together with a diffeomorphism Ξ : M → \overline{M} \ ℐ such that

4g = Ξ^{∗}(Ω^{−2} ^{4}\overline{g}) .  (3.38)

The vector field X = ∂_{t} is a Killing vector field for the metric (3.37) on M, and it is well known (cf., e.g., [36, Appendix B]) that X extends as smoothly as the metric allows to ℐ; we shall use the same symbol to denote that extension. We have the following trivial observation:

Proposition 3.4 Assume that (Σ, g, V) is smoothly compactifiable, then M = ℝ × Σ with the metric (3.37) has a smooth conformal completion with ℐ diffeomorphic to ℝ × ∂_{∞}Σ. Further (M,^{4}g) satisfies the vacuum equations with a cosmological constant Λ if and only if Equations (1.4)–(1.5) hold.

The implication the other way round requires some more work:

Theorem 3.5 Consider a space-time (M,^{4}g) of the form M = ℝ × Σ, with a metric ^{4}g of the form (3.37), and suppose that there exists a smooth conformal completion (\overline{M},^{4}\overline{g}) with nonempty ℐ. Then:

1. X is timelike on ℐ; in particular it has no zeros there;

2. The hypersurfaces t =const extend smoothly to ℐ;

3. (Σ, g, V) is smoothly compactifiable;

4. There exists a (perhaps different) conformal completion of (M,^{4}g), still denoted by (\overline{M},^{4}\overline{g}), such that \overline{M} = ℝ × \overline{Σ}, where (\overline{Σ}, \overline{g}) is a conformal completion of (Σ, g), with X = ∂_{t} and with

4\overline{g} = −α^{2}dt^{2} + \overline{g} , \overline{g}(∂_{t}, ·) = 0 , X(α) = L_{X} \overline{g} = 0 .  (3.39)

Remark: The new completion described in point 4. above will coincide with the original one if and only if the orbits of X are complete in the original completion.

Proof: As the isometry group maps M to M, it follows that X has to be tangent to ℐ. On M we have ^{4}\overline{g}(X, X) > 0 hence ^{4}\overline{g}(X, X) ≥ 0 on ℐ, and to establish point 1. we have to exclude the possibility that ^{4}\overline{g}(X, X) vanishes somewhere on ℐ.

Suppose, first, that X(p) = 0 for a point p ∈ ℐ. Clearly X is a conformal Killing vector of ^{4}\overline{g}. We can choose a neighborhood 𝒰 of ℐ so that X is strictly timelike on 𝒰 \ ℐ. There exists ϵ > 0 and a neighborhood 𝒪 ⊂ 𝒰 of p such that the flow ϕ_{t}(q) of X is defined for all q ∈ 𝒪 and t ∈ [−ϵ, ϵ]. The ϕ_{t}’s are local conformal isometries, and therefore map timelike vectors to timelike vectors. Since X vanishes at p the ϕ_{t}’s leave p invariant. It follows that the ϕ_{t}’s map causal curves through p into causal curves through p; therefore they map ∂J^{+}(p) into itself. This implies that X is tangent to ∂J^{+}(p). However this last set is a null hypersurface, so that every vector tangent to it is spacelike or null, which contradicts timelikeness of X on ∂J^{+}(p) ∩ 𝒰 ≠ ∅. It follows that X has no zeros on ℐ.

Suppose, next, that X(p) is lightlike at p. There exists a neighborhood of p and a strictly positive smooth function ψ such that X is a Killing vector field for the metric ^{4}\overline{g}ψ^{2}. Now the staticity condition

X_{[α}∇_{β}X_{γ]} = 0  (3.40)

is conformally invariant, and therefore also holds in the ^{4}\overline{g} metric. We can thus use the Carter–Vishweshvara Lemma [21, 67] to conclude that the set 𝒩 = {q ∈ \overline{M}|X(q) ≠ 0\}∩∂\{^{4}\overline{g}(X, X) < 0\} ≠ ∅ is a null hypersurface. By hypothesis there exists a neighborhood 𝒰 of ℐ in \overline{M} such that 𝒩 ∩M ∩𝒰 = ∅, hence 𝒩 ⊂ ℐ. This contradicts the fact [61] that the conformal boundary of a vacuum spacetime with a strictly negative cosmological constant Λ is timelike. It follows that X cannot be lightlike on ℐ either, and point 1. is established.

To establish point 2., we note that Equation (3.40) together with point 1. show that the one-form

λ ≡ \frac{1}{^{4}\overline{g}_{αβ}X^{α}X^{β}} ^{4}\overline{g}_{μν}X^{μ}dx^{ν}

is a smooth closed one-form on a neighborhood 𝒪 of ℐ, hence on any simply connected open subset of 𝒪 there exists a smooth function \overline{t} such that λ = d\overline{t}.

Now (3.37) shows that the restriction of λ to M is dt, which establishes our claim. From now on we shall drop the bar on \overline{t}, and write t for the corresponding time function on \overline{M}.

Let

\overline{Σ} = \overline{M} ∩ \{t = 0\} , χ = Ξ|_{t=0} , ω = Ω|_{t=0} ,

where Ξ and Ω are as in (3.38); from Equation (3.38) one obtains

g = χ^{∗}(ω^{−2}\overline{g}) ,

which shows that (\overline{Σ}, \overline{g}) is a conformal completion of (Σ, g). We further have V^{2}ω^{2} = ^{4}g(X, X)|_{t=0}ω^{2} = ^{4}\overline{g}(X, X)|_{t=0}, which has already been shown to be smoothly extendible to ℐ^{+} and strictly positive there, which establishes point 3.

There exists a neighborhood 𝒱 of \overline{Σ} in \overline{M} on which a new conformal factor Ω can be defined by requiring Ω|_{t=0} = ω, X(Ω) = 0. Redefining ^{4}\overline{g} appropriately and making suitable identifications so that Ξ is the identity, Equation (3.38) can then be rewritten on 𝒱 as

4\overline{g} = −(V Ω)^{2}dt^{2} + Ω^{2}g .  (3.41)

All the functions appearing in Equation (3.41) are time-independent. The new manifold \overline{M} defined as \overline{Σ}×ℝ with the metric (3.41) satisfies all the requirements of point 4., and the proof is complete. □

In addition to the conditions described above, in [9, 43] it was proposed to further restrict the geometries under consideration by requiring the group of conformal isometries of ℐ to be the same as that of the anti-de Sitter spacetime, namely the universal covering group of O(2, 3); cf. also [58] for further discussion. While there are various ways of adapting this proposal to our setup, we simply note that the requirement on the group of conformal isometries to be O(2, 3) or a covering therof implies that the metric induced on ℐ is locally conformally flat. Let us then see what are the consequences of the requirement of local conformal flatness of ^{ℐ}g in our context; this last property is equivalent to the vanishing of the Cotton tensor of the metric ^{ℐ}g induced by ^{4}\overline{g} on ℐ. As has been discussed in detail in Section 3.1, we can choose the conformal factor Ω to coincide with V^{−1}, in which case Equation (3.41) reads

4g′ ≡ ^{4}g/V^{2}
= −dt^{2} + V^{−2}g
= −dt^{2} + g′ ,  (3.42)

with g′ ≡ V^{−2}g already introduced in Section 3.1. It follows that

ℐg ≡ ^{4}g′|_{ℐ} = −dt^{2} + h′ ,  (3.43)

where h′ is the metric induced on ∂_{∞}Σ ≡ ℐ ∩ \overline{Σ} by g′. Let ^{ℐ} R_{ij} denote the Ricci tensor of ^{ℐ}g; from (3.43) we obtain

ℐ R_{it} = 0 , ^{ℐ} R_{AB} = ^{2}R_{AB} ,  (3.44)

where ^{2}R_{AB} is the Ricci tensor of h′. In particular the xxA component of the Cotton tensor ^{ℐ} R_{ijk} of ^{ℐ}g satisfies

ℐ R_{xxA} = −\frac{^{2}R_{,A}}{4} .

Point 1. of Proposition 3.3, see Equation (3.22), shows that the requirement of conformal flatness of ^{ℐ}g implies that R′ is constant on ∂_{∞}Σ. Conversely, it is easily seen from (3.44) that a locally constant R′ — or equivalently ^{2}R — on ∂_{∞}Σ implies the local conformal flatness of ^{ℐ}g. We have therefore proved:

Proposition 3.6 Let (Σ, g, V) be C^{i} conformally compactifiable, i ≥ 3, and satisfy (1.3)–(1.5). The conformal boundary ℝ × ∂_{∞}Σ of the space-time (M = ℝ×Σ,^{4}g), ^{4}g given by (3.37), is locally conformally flat if and only if the scalar curvature R′ of the metric V^{−2}g is locally constant on ∂_{∞}Σ. This is equivalent to requiring that the metric induced by V^{−2}g on ∂_{∞}Σ has locally constant Gauss curvature.

3.3 A coordinate approach

An alternative approach to the conformal one discussed above is by introducing preferred coordinate systems. As discussed in [44, Appendix D], coordinate approaches are often equivalent to conformal approaches when sufficiently strong hypotheses are made. We stress that this equivalence is a delicate issue when finite degrees of differentiability are assumed, as arguments leading from one approach to the other often involve constructions in which some differentiability is lost.

In any case, the coordinate approach has been used by Boucher, Gibbons and Horowitz [15] in their argument for uniqueness of the anti-de Sitter metric within a certain class of static space-times. More precisely, in [15] one considers metrics which are asymptotic to generalized Kottler metrics with k = 1 in the following strong sense: if g_{0} denotes one of the metrics (1.1) with k = 1, then one assumes that there exists a coordinate system (t, r, x^{A}) such that

g = g_{0} + O(r^{−2})dt^{2} + O(r^{−6})dr^{2}

+O(r) (remaining differentials not involving dr)

+O(r^{−1}) (remaining differentials involving dr) .  (3.45)

We note that in the uniqueness assertions of [15] one makes appeal to the positive energy theorem to conclude. Now we are not aware of a version of such a theorem which would hold without some further hypotheses on the behavior of the metric. For example, in such a theorem one is likely to require that the derivatives of the metric also fall off at some sufficiently high rates. In any case the argument presented in [15] seems to implicitly assume that the asymptotic behavior of g^{tt} described above is preserved under differentiation, so that the corrections terms in (3.45) give a vanishing contribution when calculating |dV |_{g}^{2} − |dV_{0}|_{g_{0}}^{2} and passing to the limit r → ∞, with g_{0} — the anti-de Sitter metric. While it might well be possible that Equations (1.4)–(1.5) force the metrics satisfying (3.45) to have sufficiently good asymptotic properties to be able to justify this, or to apply a positive energy theorem^{6}, this remains to be established.^{7}

It is far from being clear whether or not a general metric of the form (3.45) has any well behaved conformal completions. For example, the coordinate transformation (3.1) together with a multiplication by the square of the conformal factor ω = x brings the metric (3.45) to one which can be continuously extended to the boundary, but if only (3.45) is assumed then the resulting metric will not be differentiable up to boundary on the compactified manifold in general. There could, however, exist coordinate systems which lead to better conformal behavior when Equations (1.4)–(1.5) are imposed.

In any case, it is natural to ask whether or not a metric satisfying the requirements of Section 3.1 will have a coordinate representation similar to (3.45). A partial answer to this question is given by the following^{8}:

Proposition 3.7 Let (Σ, g, V) be a C^{i} compactifiable solution of Equations (1.4)–(1.5), i ≥ 3. Define a C^{i−2} function k̃ = k̃(x^{A}) on ∂_{∞}Σ by the formula

R′|_{∂_{∞}Σ} = −2Λk̃ .  (3.46)

1. Rescaling V by a positive constant if necessary, there exists a coordinate system (r, x^{A}) near ∂_{∞}Σ in which we have

V^{2} = \frac{r^{2}}{ℓ^{2}} + k̃ ,  (3.47)

g = \frac{dr^{2}}{(\frac{r^{2}}{ℓ^{2}} + k̃ − \frac{2μ}{r})} + O(r^{−3})dr dx^{A} + (r^{2}ȟ_{AB} + O(r^{−1})) dx^{A}dx^{B} (3.48)

(recall that ℓ^{2} = −3Λ^{−1}), for some r-independent smooth two-dimensional metric ȟ_{AB} with Gauss curvature equal to k̃ and for some function μ = μ(r, x^{A}). Further

ȟ^{AB}g_{AB} = 2 (r^{2} − \frac{μ_{∞}}{r} + O(r^{−2})) ,  (3.49)

where ȟ^{AB} denotes the matrix inverse to ȟ_{AB} while

μ_{∞} ≡ lim_{r→∞} μ = \frac{ℓ^{3}}{12} \frac{∂R′}{∂x} |_{x=0} .  (3.50)

6Recall that in the asymptotically flat case one can derive an asymptotic expansion for stationary metrics from rather weak hypotheses on the leading order behavior of the metric [25, 52, 63]. See especially [2, 3], where the Lichnerowicz theorem is proved without any hypotheses on the asymptotic behavior of the metric, under the condition of geodesic completeness of space-time.

7The key point of the argument in [15] is to prove that the coordinate mass is negative.

When ∂_{∞}Σ = S^{2}, and the asymptotic conditions are such that the positive energy theorem applies, one can conclude that the initial data set under consideration must be coming from one in anti-de Sitter space-times provided one shows that the coordinate mass coincides with the mass which occurs in the positive energy theorem. To our knowledge such an equality has not been proved so far for metrics with the asymptotics (3.45), or else.

8See [44, Appendix] for a related discussion. While the conclusions in [44] appear to be weaker than ours, it should be stressed that in [44] staticity of the space-times under consideration is not assumed.

2. If one moreover assumes that R′ is locally constant on ∂_{∞}Σ, then Equation (3.48) can be improved to

g = \frac{dr^{2}}{(\frac{r^{2}}{ℓ^{2}} + k − \frac{2μ}{r})} + (r^{2}ȟ_{AB} + O(r^{−1})) dx^{A}dx^{B} ,  (3.51)

with ȟ_{AB} having constant Gauss curvature k = 0, ±1 according to the genus of the connected component of ∂_{∞}Σ under consideration.

Remarks: 1. The function (x, x^{A}) → μ(r = 1/x, x^{A}) is of differentiability class C^{i−3} on \overline{Σ}, with the function (x, x^{A}) → (μ/r)(r = 1/x, x^{A}) being of differentiability class C^{i−2} on \overline{Σ}.

2. In Equations (3.48) and (3.51) the error terms O(r^{−j}) satisfy

∂_{r}^{s}∂_{A_{1}} … ∂_{A_{t}}O(r^{−j}) = O(r^{−j−s})

for 0 ≤ s + t ≤ i − 3.

3. We emphasize that the function k̃ defined in Equation (3.46) could a priori be x^{A}-dependent. In such a case neither the definition of coordinate mass of Section 5.1 nor the definition of Hamiltonian mass of Section 5.2 apply.

4. It seems that to be able to obtain (3.45), in addition to the hypothesis that R′ is locally constant on ∂_{∞}Σ one would at least need the quantity appearing at the right hand side of Equation (3.50) to be locally constant on ∂_{∞}Σ as well. We do not know whether this is true in general; we have not investigated this question as this is irrelevant for our purposes.

Proof: Consider, near ∂_{∞}Σ, the coordinate system of Equation (3.30); from Equations (3.31) and (3.20) we obtain

∂_{x} (ln \sqrt{det h′_{AB}}) = −2k̃x − \frac{3μ_{∞}}{ℓ} x^{2} + O(x^{3}) ,  (3.52)

ℓ as in (2.10), k̃ as in (3.46), μ_{∞} as in (3.50). This, together with Equation (3.28), leads to

\frac{∂h′_{AB}}{∂x} = −2xk̃h′_{AB} + O(x^{2}) ⟹

h′_{AB} = (1 − k̃x^{2})ℓ^{2}ȟ_{AB} + O(x^{3}) ,

where

ȟ_{AB} ≡ \frac{1}{ℓ^{2}} h′_{AB}|_{x=0} .

Proposition 3.3 shows that k̃ is proportional to the Gauss curvature of ȟ_{AB}. It follows now from (3.20) that

g = x^{−2}g′ = \frac{ℓ^{2}}{x^{2} (1 − \frac{R′ℓ^{2}x^{2}}{6} )}dx^{2} + \{\frac{(1 − k̃x^{2})}{x^{2}}h′_{AB}|_{x=0} + O(x^{3})\} dx^{A}dx^{B} .

The above suggests to introduce a coordinate r via the formula^{9}

\frac{r^{2}}{ℓ^{2}} = \frac{1 − k̃x^{2}}{x^{2}} .  (3.53)

Suppose, first, that k̃ is locally constant on ∂_{∞}Σ, then k̃ equals k = 0, ±1 according to the genus of the connected component of ∂_{∞}Σ under consideration, and one finds

g = \frac{dr^{2}}{(\frac{r^{2}}{ℓ^{2}} + k) \{1 + \frac{ℓ^{2}}{r^{2}} (k − \frac{R′ℓ^{2}x^{2}}{6} )\}} + (\frac{r^{2}}{ℓ^{2}} h′_{AB}|_{x=0} + O(r^{−1})) dx^{A}dx^{B}

= \frac{dr^{2}}{(\frac{r^{2}}{ℓ^{2}} + k − \frac{2μ}{r})} + (\frac{r^{2}}{ℓ^{2}} h′_{AB}|_{x=0} + O(r^{−1})) dx^{A}dx^{B} ,

where the “mass aspect” function μ = μ(r, x^{A}) is defined as

μ ≡ −\frac{r}{2} (1 + k \frac{ℓ^{2}}{r^{2}} ) (k − \frac{R′ℓ^{2}x^{2}}{6} )
= −\frac{r}{2} (k − \frac{R′ℓ^{2}}{6} + \frac{k^{2}ℓ^{2}}{r^{2}} )
= \frac{rℓ^{2}}{2} (\frac{1}{6}(R′ − R′|_{x=0}) − \frac{k^{2}}{r^{2}} ) .  (3.54)

This establishes Equations (3.47) and (3.51). When k̃ is not locally constant an identical calculation using the coordinate r defined in Equation (3.53) establishes Equation (3.48) — the only difference is the occurrence of non-vanishing error terms in the drdx^{A} part of the metric, introduced by the angle dependence of k̃. It follows from Equation (3.54) — or from the k̃ version thereof when k̃ is not locally constant — that

μ = \frac{ℓ^{3}}{12} \frac{∂R′}{∂x} |_{x=0} + O(r^{−1}) ,

which establishes Equation (3.50). Equation (3.49) is obtained by integration of Equation (3.52).

4 Connectedness of ∂_{∞}Σ

The class of manifolds considered so far could in principle contain Σ’s for which neither ∂_{∞}Σ nor ∂Σ are connected. Under the hypothesis of staticity the question of connectedness of ∂Σ is open; we simply note here the existence of dynamical (non-stationary) solutions of Einstein–Maxwell equations with a nonconnected black hole region with positive cosmological constant Λ [18, 51]. As far as ∂_{∞}Σ is concerned, we have the following:

9We note that k̃ is of differentiability class lower by two orders as compared to the metric itself, which leads to a loss of three derivatives when passing to a new coordinate system in which r is defined by Equation (3.53). One can actually introduce a coordinate system closely related to (3.53) with a loss of only one degree of differentiability of the metric by using the techniques of [4, Appendix A], but we shall not discuss this here.

Theorem 4.1 Let (Σ, g, V) be a C^{i} compactifiable solution of Equations (1.4)– (1.5), i ≥ 3. Then ∂_{∞}Σ is connected.

Proof: Consider the manifold M = ℝ×Σ with the metric (3.37); its conformal completion \overline{M} = ℝ × \overline{Σ} with the metric ^{4}g/V^{2} is a stably causal manifold with boundary. We wish to show that it is also globally hyperbolic in the sense of [33], namely that 1) it is strongly causal and 2) for each p, q ∈ M the set J^{+}(p)∩ J^{−}(q) is compact. The existence of the global time function t clearly implies strong causality, so it remains to verify the compactness condition. Now a path Γ(s) = (t(s), γ(s)) ∈ ℝ × Σ is an achronal null geodesic from p = (t(0), γ(0)) to q = (t(1), γ(1)) if and only if γ(s) is a minimizing geodesic between γ(0) and γ(1) for the “optical metric” V^{−2}g. Compactness of J^{+}(p) ∩ J^{−}(q) is then equivalent to compactness of the V^{−2}g-distance balls; this latter property will hold when (Σ∪∂_{∞}Σ, V^{−2}g) is a geodesically complete manifold (with boundary) by (an appropriate version of) the Hopf–Rinow theorem.

Let us thus show that (Σ, V^{−2}g) is geodesically complete. Suppose, first, that ∂Σ = ∅; the hypothesis that Σ has compact interior together with the fact that V tends to infinity in the asymptotic regions implies that V ≥ V_{0} > 0 for some constant V_{0}. This shows that (Σ, V^{−2}g) is a compact manifold with boundary ∂_{∞}Σ, and the result follows. (When the metric induced by V^{−2}g on ∂_{∞}Σ has positive scalar curvature connectedness of ∂_{∞}Σ can also be inferred from [69].)

Consider, next, the case ∂Σ ≠ ∅. It is well known that |dV |_{g} is a nonzero constant on every connected component of ∂Σ (cf. the discussion around Equation (7.2)); therefore we can introduce coordinates near ∂Σ so that V = x, with the metric taking the form

V^{−2}g = x^{−2} ((dx)^{2} + h_{AB}(x, x^{A})dx^{A}dx^{B}) ,  (4.1)

where the x^{A}’s are local coordinates on ∂Σ. It is elementary to show now from (4.1) that (Σ ∪ ∂_{∞}Σ, V^{−2}g) is a complete manifold with boundary, as claimed.

When (Σ, g) is smoothly compactifiable we can now use [33, Theorem 2.1] to infer connectedness of ∂_{∞}Σ, compare [32, Corollary, Section III]. For compactifications with finite differentiability we argue as follows: For small s let λ be the mean curvature of the sets ≡ {x = s}, where x is the coordinate of Equation (3.11). In the coordinate system used there the unit normal to those sets pointing away from ∂_{∞}Σ equals x∂_{x}; if (Σ, g, V) is C^{3} compactifiable the tensor field \overline{h} appearing in Equation (3.11) will be C^{1} so that^{10}

λ = \frac{1}{\sqrt{det g}} ∂_{i} (\sqrt{det g} n^{i})
= \frac{x^{3}}{\sqrt{det \overline{h}}}∂_{x} (x^{−2}\sqrt{det \overline{h}})
= −2 + O(x) .

10The differentiability threshold k = 3 can be lowered using the “almost Gaussian coordinate systems” of [4, Appendix A], we shall however not be concerned with this here.

It follows that for s small enough the sets {x = s, t = τ} are trapped, with respect to the inward pointing normal, in the space-time ℝ × Σ with the metric (3.37). Suppose that ∂_{∞}Σ were not connected, then those (compact) sets would be outer trapped with respect to every other connected component of ∂_{∞}Σ. This is, however, not possible by the usual global arguments, cf., e.g., [34, 35] or [28, Section 4] for details. □

5 The mass

5.1 A coordinate mass M_{c}

There exist several proposals how to assign a mass M to a space-time which is asymptotic to an anti-de Sitter space-time [1, 8, 9, 38, 44]; it seems that the simplest way to do that (as well as to extend the definition to the generalized Kottler context considered here) proceeds as follows: consider a metric defined on a coordinate patch covering the set

Σ_{ext} ≡ {t = t_{0}, r ≥ R, (x^{A}) ∈ ^{2}M}  (5.1)

(which we will call an “end”), and suppose that in this coordinate system the functions g_{μν} are of the form (1.1) plus lower order terms^{11}

g_{tt} = −(k − \frac{2m}{r} − \frac{Λ}{3}r^{2}) + \frac{o(1)}{r} , g_{rr} = 1/(k − \frac{2m}{r} − \frac{Λ}{3}r^{2} + \frac{o(1)}{r}) ,
g_{tμ} = o(1) , μ ≠ t , g_{rμ} = o(1) , μ ≠ r, t ,
g_{AB} − r^{2}h_{AB} = o(r^{2}) ,  (5.2)

for some constant m, and for some constant curvature (t and r independent) metric h_{AB}dx^{A}dx^{B} on ^{2}M. Then we define the coordinate mass M_{c} of the end Σ_{ext} to be the parameter m appearing in (1.1). This procedure gives a unique prescription how to assign a mass to a metric and a coordinate system on Σ_{ext}.

There is an obvious coordinate-dependence in this definition when k = 0: Indeed, in that case a coordinate transformation r → λr, t → t/λ, dΩ_{k}^{2} → λ^{−2}dΩ_{k}^{2}, where λ is a positive constant, does not change the asymptotic form of the metric, while m gets replaced by λ^{−3}m. When ^{2}M is compact this freedom can be removed e.g. by requiring that the area of ^{2}M with respect to the metric dΩ_{k}^{2} be equal to 4π, or to 1, or to some other chosen constant. For k = ±1 this ambiguity does not arise because in this case such rescalings will change the asymptotic form of the metric, and are therefore not allowed.

It is far from being clear that the above definition will assign the same parameter M_{c} to every coordinate system satisfying our requirements: if that is the case, to prove such a statement one might perhaps need to further require that the coordinate derivatives of the coordinate components of g in the above described coordinate system have some appropriate decay properties; further one might perhaps have to replace the o(1)’s by o(r^{−σ})’s or O(r^{−σ})’s, for some appropriate σ’s, perhaps as in (3.45); this is however irrelevant for our discussion at this stage.

11Because the natural length of the vectors ∂_{A} is O(r) it would actually be natural to require g_{rμ} = o(r), μ ≠ r, t instead of g_{rμ} = o(1), μ ≠ r, t.

A possible justification of this definition proceeds as follows: when ^{2}M = S^{2} and Λ = 0 it is widely accepted that the mass of Σ_{ext} equals m, because m corresponds to the active gravitational mass of the gravitational field in a quasi-Newtonian limit. (It is also known in this case that the definition is coordinate-independent [10, 24].) For Λ ≠ 0 and/or ^{2}M ≠ S^{2} one calls m the mass by extrapolation.

Consider, then, the metric (3.37), with V and g as in (3.47)–(3.48); suppose further that the limit

μ_{∞} ≡ lim_{r→∞} μ

exists and is a constant. To achieve the form of the metric ^{4}g just described one needs to replace the coordinate r of (3.47)–(3.48) with a new coordinate ρ defined as

r^{2} + k = ρ^{2} + k + \frac{μ_{∞}}{ρ} .

This leads to

4g = − (\frac{ρ^{2}}{ℓ^{2}} + k + \frac{μ_{∞}}{ρ} ) dt^{2} + \frac{dρ^{2}}{(\frac{ρ^{2}}{ℓ^{2}} + k + \frac{μ_{∞}}{ρ} + O(\frac{1}{ρ^{2}} ))}
+O(ρ^{−3})dρ dx^{A} + (ρ^{2}ȟ_{AB} + O(ρ^{−1})) dx^{A}dx^{B} ,  (5.3)

and therefore

M_{c} ≡ −\frac{μ_{∞}}{2} = −\frac{ℓ^{3}}{24} \frac{∂R′}{∂x} |_{x=0} ,  (5.4)

where the second equality above follows from (3.50). We note that the approach described does not give a definition of mass when lim_{r→∞} μ does not exist, or is not a constant function on ∂_{∞}Σ.

The above described dogmatic approach suffers from various shortcomings. First, when ^{2}M ≠ S^{2}, the arguments given are compatible with M_{c} being any function M_{c}(m, Λ) with the property that M_{c}(m, 0) = m. Next, the transition from Λ ≠ 0 to Λ = 0 has dramatic consequences as far as global properties of the corresponding space-times are concerned, and one can argue that there is no reason why the function M_{c}(m, Λ) should be continuous at zero. For example, according to [44, Equation (III.8c)], the mass of the metric (1.1) with ^{2}M = S^{2} should be 16πmℓ, with ℓ as in (2.10), which blows up when Λ tends to zero with m being held fixed. Finally, when ^{2}M ≠ S^{2} the Newtonian limit argument does not apply because the metrics (1.1) with Λ = 0 and ^{2}M ≠ S^{2} do not seem to have a Newtonian equivalent. In particular there is no reason why M_{c} should not depend upon the genus g_{∞} of ^{2}M as well.

All the above arguments make it clear that a more fundamental approach to the definition of mass would be useful. It is common in field theory to define energy by Hamiltonian methods, and this is the approach we shall pursue in the next section.

5.2 The Hamiltonian mass M_{Ham}.

The Hamiltonian approach allows one to define the energy from first principles. For a solution of the field equations, we can simply take as the energy the numerical value of the Hamiltonian. It must be recognized, however, that the Hamiltonians might depend on the choice of the phase space, if several such choices are possible, and they are defined only up to an additive constant on each connected component of the phase space. They also depend on the choice of the Hamiltonian structure, if more than one such structure exists.

Let us start by briefly recalling the results of the analysis of [23], based on the Hamiltonian approach of Kijowski and Tulczyjew [54, 55], see also [53]. One assumes that a manifold M on which an (unphysical) background metric b is given, and one considers metrics ^{4}g which asymptote to b in the relevant asymptotic regions of M. We stress that the background metric is only a tool to prescribe the asymptotic boundary conditions, and does not have any physical significance. Let X be any vector field on M and let Σ be any hypersurface in M. By a well known procedure the motion of Σ along the flow of X can be used to construct a Hamiltonian dynamical system in an appropriate phase space of fields over Σ; the reader is referred to [29, 53–55] for a geometric approach to this question. In [23] it is also assumed that X is a Killing vector field of the background metric; this is certainly not necessary (cf., e.g., [29] for general formulae), but is sufficient for our purposes, as we are going to take X to be equal to ∂/∂t in the coordinate system of Equation (3.37). In the context of metrics which asymptote to the generalized Kottler metrics at large r, a rigorous functional description of the spaces involved has not been carried out so far, and lies outside the scope of this paper. Let us simply note that one expects, based on the results in [29, 31, 50], to obtain a well defined Hamiltonian system in this context. Therefore the formal calculations of [23] lead one to expect that on an appropriate space of fields, such that the associated physical spacetime metrics ^{4}g asymptote to the background metric b at a suitable rate, the Hamiltonian H(X, Σ) will coincide with (or, at worse, will be closely related to) the one given by the formula derived in [23]:

H(X, Σ) = \frac{1}{2} ∫_{∂Σ} 𝕌^{αβ}dS_{αβ} ,  (5.5)

where the integral over ∂Σ should be understood by a limiting process, as the limit as R tends to infinity of integrals of coordinate spheres t = 0, r = R on Σ_{ext}. Here dS_{αβ} is defined as \frac{∂}{∂x^{α}} ⌟\frac{∂}{∂x^{β}} ⌟dx^{0} ∧ ⋯ ∧ dx^{n}, with ⌟ denoting contraction, and 𝕌^{αβ} is given by

𝕌^{νλ} = 𝕌^{νλ}_{β}X^{β} + \frac{1}{8π} (\sqrt{| det g_{ρσ}|} g^{α[ν} − \sqrt{| det b_{ρσ}|} b^{α[ν}) δ_{β}^{λ]}X^{β}_{;α} ,(5.6)
𝕌^{νλ}_{β} = \frac{2| det b_{μν}|}{16π\sqrt{| det g_{ρσ}|}}g_{βγ}(e^{2}g^{γ[ν}g^{λ]κ})_{;κ} .  (5.7)

Here, and throughout this section, g stands for the space-time metric ^{4}g unless explicitly indicated otherwise. Further, a semicolon denotes covariant differentiation with respect to the background metric b, while

e ≡ \begin{matrix}√ & \overline{| det g_{ρσ}|} \\ √ & \overline{| det b_{μν}|} \end{matrix} .  (5.8)

Some comments concerning Equation (5.6) are in order: in [23] the starting point of the analysis is the Hilbert Lagrangian for vacuum Einstein gravity,

L = \sqrt{− det g_{μν}} \frac{g^{αβ}R_{αβ}}{16π} .

As the normalization factors play an important role in giving a correct definition of mass, we recall that the factor 1/16π is determined by the requirement that the theory has the correct Newtonian limit (units G = c = 1 are used throughout). With our signature (− + ++) the Einstein equations with a cosmological constant read

R_{μν} − \frac{g^{αβ}R_{αβ}}{2} g_{μν} = −Λg_{μν} ,

so that one needs to repeat the analysis in [23] with L replaced by

\frac{\sqrt{− det g_{μν}}}{16π} (g^{αβ}R_{αβ} − 2Λ) .

The general expression for the Hamiltonian (5.5) in terms of X^{μ}, g_{μν} and b_{μν} ends up to coincide with that obtained with Λ = 0, which can be seen either by direct calculations, or by the Legendre transformation arguments of [23, end of Section 3] together with the results in [53]. Note that Equation (5.6) does not exactly coincide with that derived in [23], as the formulae there do not contain the term −\sqrt{| det b_{ρσ}|} b^{α[ν}δ_{β}^{λ]}X^{β}_{;α}. However, this term does not depend on the metric g, and such terms can be freely added to the Hamiltonian because they do not affect the variational formula that defines a Hamiltonian. From an energy point of view such an addition corresponds to a choice of the zero point of the energy. We note that in our context H(X, Σ) would not converge if the term −\sqrt{| det b_{ρσ}|} b^{α[ν}δ_{β}^{λ]}X^{β}_{;α} were not present in (5.6).

In order to apply this formalism in our context, we let b be any t-independent metric on M = ℝ × Σ such that (with 0 ≠ Λ = −3/ℓ^{2})

b = −(k + \frac{r^{2}}{ℓ^{2}} )dt^{2} + (k + \frac{r^{2}}{ℓ^{2}} )^{−1}dr^{2} + r^{2}ȟ  (5.9)

on ℝ × Σ_{ext} ≈ ℝ × [R, ∞) × ^{2}M, for some R ≥ 0, where ȟ = ȟ_{AB}dx^{A}dx^{B} denotes a metric of constant Gauss curvature k = 0, ±1 on the two dimensional connected compact manifold ^{2}M.

Let us return to the discussion in Section 5.1 concerning the freedom of rescaling the coordinate r by a positive constant λ. First, if k in Equation (5.9) is any constant different from zero, then there exists a (unique) rescaling of r and t which brings k to one, or to minus one. Next, if k = 0 we can — without changing the asymptotic form of the metric — rescale the coordinate r by a positive constant λ, the coordinate t by 1/λ, and the metric ȟ by λ^{−2}, so that there is still some freedom left in the coordinate system above; a unique normalization can then be achieved by asking e.g. that the area

A_{∞} ≡ ∫_{^{2}M} d^{2}μȟ  (5.10)

equals 4π — this will be the most convenient normalization for our purposes. Here d^{2}μȟ is the Riemannian measure associated with the metric ȟ. We wish to point out that that regardless of the value of k, the Hamiltonian H(X, Σ) is independent of this scaling: this follows immediately from the fact that 𝕌^{αβ} behaves as a density under linear coordinate transformations. An alternative way of seeing this is that in the new coordinate system X equals λ∂/∂t, which accounts for a factor 1/λ in the transformation law of the coordinate mass, as discussed at the beginning of Section 5.1. The remaining factor 1/λ^{2} needed there is accounted for by a change of the area of ∂_{∞}Σ under the rescaling of the metric ȟ which accompanies that of r.

In order to evaluate H it is useful to introduce the following b-orthonormal frame:

eˆ0 = \frac{1}{\sqrt{k + \frac{r^{2}}{ℓ^{2}}}} ∂_{t} , eˆ1 = \sqrt{k + \frac{r^{2}}{ℓ^{2}}} ∂_{r} , eÂ = \frac{1}{r}ěÂ ,  (5.11)

where ěÂ is an ON frame for the metric ȟ. The connection coefficients, defined by the formula ∇_{eâ}eb̂ = ωĉ_{b̂â}eĉ, read

ω_{ˆ0ˆ1ˆ0} = −_{\frac{r}{ℓ^{2}\sqrt{k+ \frac{r^{2}}{ℓ^{2}}}}} = −\frac{1}{ℓ} + O(r^{−2}) ,

ω_{ˆ1ˆ2ˆ2} = ω_{ˆ1ˆ3ˆ3} = −^{\frac{\sqrt{k+ \frac{r^{2}}{ℓ^{2}}}}{r}} = −\frac{1}{ℓ} + O(r^{−2}) ,
ω_{ˆ2ˆ3ˆ3} = \{ \begin{matrix}−\frac{coth θ}{r} , & k = −1 , \\ 0 , & k = 0 , \\ −\frac{cot θ}{r} , & k = 1 . \end{matrix}  (5.12)

Those connection coefficients which are not obtained from the above ones by permutations of indices are zero; we have used a coordinate system θ, φ on ^{2}M in which ȟ takes, locally, the form dθ^{2} + sinh^{2} θ dφ^{2} for k = −1, dθ^{2} + dφ^{2} for k = 0, and dθ^{2} + sin^{2} θ dφ^{2} for k = 1. We also have

Xˆ0 = \sqrt{k + \frac{r^{2}}{ℓ^{2}}} = \frac{r}{ℓ} + O(r^{−1}) ,  (5.13)
eˆ1(Xˆ0) = Xˆ0_{;ˆ1} = −X_{ˆ0;ˆ1} = X_{ˆ1;ˆ0} = \frac{r}{ℓ^{2}} ,  (5.14)

with the third equality in (5.14) following from the Killing equations X_{μ;ν} + X_{ν;μ} = 0; all the remaining Xμ̂’s and X_{μ̂;ν̂}’s are zero. Let the tensor field e^{μν} be defined by the formula

e^{μν} ≡ g^{μν} − b^{μν} .  (5.15)

We shall use hatted indices to denote the components of a tensor field in the frame eâ defined in (5.11), e.g. e^{âĉ} denotes the coefficients of e^{μν} with respect to that frame:

e^{μν}∂_{μ} ⊗ ∂_{ν} = e^{âĉ}eâ ⊗ eĉ .

Suppose that the metric ^{4}g is such that the e^{âĉ}’s tend to zero as r tends to infinity. By a Gram–Schmidt procedure we can find a frame fã, ã = 0, …, 3, orthonormal with respect to the metric g, such that f_{0} is proportional to e_{0}, and such that the eâ components of f_{0} − e_{0}, …, f_{3} − e_{3} tend to zero as r tends to infinity:

fã = fãâeâ →_{r→∞} δâ ; ãeâ .  (5.16)

From (5.5) and (5.16) we expect that^{12}

H(X, Σ) = lim_{R→∞} ∫_{Σ∩{r=R}} r^{2}𝕌^{ˆ1ˆ0}d^{2}μ_{r} ,  (5.17)

where d^{2}μ_{r} is the Riemannian measure induced on Σ ∩ {r = R} by ^{4}g. We wish to analyze when the above limit exists; we have

r^{2}𝕌^{ˆ1ˆ0}_{β}X^{β} = r^{2}𝕌^{ˆ1ˆ0}ˆ0Xˆ0 ≈ \frac{r^{3}}{ℓ} 𝕌^{ˆ1ˆ0}ˆ0 ,

hence we need to keep track of all the terms in 𝕌^{ˆ1ˆ0}ˆ0 which decay as r^{−3} or slower. Similarly one sees from Equations (5.13)–(5.14) that only those terms in

Δ^{α̂ν̂} ≡ \sqrt{| det g_{ρ̂σ̂}|} g^{α̂ν̂} − \sqrt{| det b_{ρ̂σ̂}|} b^{α̂ν̂}

which are O(r^{−3}), or which are decaying slower, will give a non-vanishing contribution to the term involving the derivatives of X in the integral (5.17). This suggests to consider metrics ^{4}g such that

e^{μ̂ν̂} = o(r^{−3/2}) , eρ̂(e^{μ̂ν̂}) = o(r^{−3/2}) .  (5.18)

The boundary conditions (5.18) ensure that one needs to keep track only of those terms in 𝕌^{ˆ1ˆ0} which are linear in e^{μ̂ν̂} and eρ̂(e^{μ̂ν̂}), when 𝕌^{ˆ1ˆ0} is Taylor expanded around b. For a generalized Kottler metric (1.1) we have

e^{ˆ0ˆ0} ≈ e^{ˆ1ˆ1} ≈ −\frac{2mℓ^{2}}{r^{3}} ,  eˆ1(e^{ˆ0ˆ0}) ≈ eˆ1(e^{ˆ1ˆ1}) ≈ \frac{6mℓ}{r^{3}} ,  (5.19)

with the remaining e^{μ̂ν̂}’s and eσ̂(e^{μ̂ν̂})’s vanishing, so that Equations (5.18) are satisfied. Under (5.18) one obtains

g_{âĉ} = η_{âĉ} − η_{âr̂}η_{ĉŝ}e^{r̂ŝ} + o(r^{−3}) ,  (5.20)
\sqrt{| det g_{μν}|} = \sqrt{| det b_{μν}|} (1 + \frac{1}{2}(e^{ˆ0ˆ0} − e^{ˆ1ˆ1} − e^{ÂÂ}) + o(r^{−3})) ,
𝕌^{ˆ1ˆ0}ˆ0 = −\frac{1}{16π} (2e_{;ˆ1} + e^{ˆ1î}_{;î} − e^{ˆ0ˆ0}_{;ˆ1}) + o(r^{−3})

12Equation (5.17) will indeed turn out to be correct under the conditions (5.18) imposed below.

= \frac{1}{16π} (eˆ1(e^{ÂÂ}) + \frac{1}{ℓ}(e^{ÂÂ} − 2e^{ˆ1ˆ1}) − \frac{1}{r}ĎÂe^{ˆ1Â}) + o(r^{−3}) ,
\frac{1}{8π} Δ^{α[ˆ1}X^{ˆ0]}_{;α} = \frac{1}{16π} (Δ^{ˆ1ˆ1} − Δ^{ˆ0ˆ0}) Xˆ0_{;ˆ1}
= \frac{r}{16πℓ^{2}} (Δ^{ˆ1ˆ1} − Δ^{ˆ0ˆ0}) + o(r^{−3})
= −\frac{r}{16πℓ^{2}} e^{ÂÂ} + o(r^{−3}) .  (5.21)

The indices î run from 1 to 3 while the indices Â run from 2 to 3; ĎÂ denotes the covariant derivative on ^{2}M, and ĎÂe^{ˆ1Â} is understood to be the covariant derivative associated with the metric ȟ of a vector field on ^{2}M, with repeated Â indices being summed over. In Equation (5.20) η_{μ̂ν̂} =diag(−1, +1, +1, +1), while the g_{μ̂ν̂}’s are the components of the tensor g_{μ̂ν̂} in a co-frame dual to (5.11). Inserting all this into (5.17) one is finally led to the simple expression

M_{Ham} ≡ H(\frac{∂}{∂t}, \{t = 0\})
= lim_{R→∞} \frac{r^{3}}{16πℓ^{2}} ∫_{Σ∩\{r=R\}} (r \frac{∂e^{ÂÂ}}{∂r} − 2e^{ˆ1ˆ1}) d^{2}μȟ .  (5.22)

In particular if ^{4}g is the generalized Kottler metric (1.1) one obtains (cf. Equation (5.19))

M_{Ham} = \frac{A_{∞}m}{4π} ,  (5.23)

A_{∞} defined in (5.10). If ^{2}M = T^{2} with area normalized to 4π we obtain M_{Ham} = m. For k = ±1 it follows from the Gauss–Bonnet theorem that A_{∞} = 4π|1 − g_{∞}|, where g_{∞} is the genus of ^{2}M, hence

M_{Ham} = |1 − g_{∞}|m .  (5.24)

This gives again M_{Ham} = m for ^{2}M = S^{2}, but this will not be true anymore for ^{2}M’s of higher genus. We believe that the Hamiltonian approach is the one which provides the correct definition of mass in field theories, and therefore Equations (5.23)–(5.24) are the ones which provide the correct normalization of mass.

Let us finally consider static metrics ^{4}g of the form (3.37), and suppose that the hypotheses of point 2. of Proposition 3.7 hold. We can then use the coordinates of that proposition to calculate M_{Ham}, and obtain

M_{Ham} = −\frac{1}{8π} ∫_{∂_{∞}Σ} μ_{∞} d^{2}μȟ .  (5.25)

If we further assume that μ_{∞} is constant on ∂_{∞}Σ, Equation (5.25) gives

M_{Ham} = −\frac{μ_{∞}}{2} = M_{c}

for ^{2}M = S^{2} and for an appropriately normalized T^{2}, while

M_{Ham} = −|1 − g_{∞}|\frac{μ_{∞}}{2} = |1 − g_{∞}|M_{c}

for higher genus ∂_{∞}Σ’s. Here M_{c} is the coordinate mass as defined in Section 5.1.

5.3 A generalized Komar mass

Recall that the Komar mass is a number which can be assigned to every stationary, asymptotically flat metric the energy-momentum tensor of which decays sufficiently rapidly:

M_{K} = lim_{R→∞} \frac{1}{8π} ∫_{S_{R,T}} \sqrt{| det g_{αβ}|} ∇^{μ}X^{ν} dS_{μν} ,  (5.26)

where X^{μ}∂_{μ} is the Killing vector field which asymptotes to ∂/∂t in the asymptotically flat region, and the S_{R,T} ≡ {t = T, r = R}’s are coordinate spheres in that region. The normalization factor 1/(8π) has been chosen so that M_{K} reproduces the familiar mass parameter m when Schwarzschild metrics are considered. For metrics considered here with Λ ≠ 0 the integral (5.26) diverges when X^{μ}∂_{μ} = ∂/∂t and when the S_{R,T} ’s are taken to be coordinate spheres in the region Σ_{ext} where the metric exhibits the generalized Kottler asymptotics. An obvious way to generalize M_{K} to the situation considered in this paper is to remove the divergent part of the integral using a background metric b:

M_{K} = lim_{R→∞} \frac{1}{8π} ∫_{S_{R,T}} (\sqrt{| det g_{αβ}|} ∇^{μ}X^{ν} − \sqrt{| det b_{αβ}|} \overline{∇}^{μ}X^{ν}) dS_{μν} . (5.27)

Here \overline{∇} denotes a covariant derivative with respect to the background metric. More precisely, let Σ_{ext}, b, ȟ, etc., be as in Equation (5.9), and consider timeindependent metrics g which in the coordinate system of Equation (5.9) are of the form (3.37) with

V^{2} = \frac{r^{2}}{ℓ^{2}} + k̃ − \frac{2β}{r} + o(\frac{1}{r}) ,
∂_{r}(V^{2} − \frac{r^{2}}{ℓ^{2}} − k̃ + \frac{2β}{r}) = o(\frac{1}{r^{2}} ) ,
g^{rr} = \frac{r^{2}}{ℓ^{2}} + k̃ − \frac{2γ}{r} + o(\frac{1}{r}) ,
\sqrt{| det g_{αβ}|} = (r^{2} + \frac{2δℓ^{2}}{r} + o(\frac{1}{r})) \sqrt{| det ȟ_{AB}|} ,  (5.28)

for some r-independent differentiable functions k̃ = k̃(x^{A}), β = β(x^{A}), γ = γ(x^{A}) and δ = δ(x^{A}) defined on a coordinate neigbhourhood of ∂_{∞}Σ. (The conditions (5.28) roughly reflect the behavior of the metric in the coordinate system of Proposition 3.7). Under (5.28) the limit as R tends to infinity in the definition (5.27) of M_{K} exists, and one finds

M_{K} = lim_{R→∞} \frac{1}{4π} ∫_{S_{R,T}} (\sqrt{| det g_{αβ}|} g^{rμ}g^{νt}∂_{[μ}g_{ν]t} − \sqrt{| det b_{αβ}|} b^{rμ}b^{νt}∂_{[μ}b_{ν]t}) dx^{2}dx^{3}
= lim_{R→∞} \frac{1}{8π} ∫_{S_{R,T}} (\sqrt{| det g_{αβ}|} g^{rr}g^{tt}∂_{r}g_{tt} − \sqrt{| det b_{αβ}|} b^{rr}b^{tt}∂_{r}b_{tt}) dx^{2}dx^{3}
= \frac{1}{4π} ∫_{∂_{∞}Σ}(3β − 2γ + 2δ) d^{2}μȟ .  (5.29)

It turns out that the value of M_{K} so obtained depends on the background metric chosen. (Our definition of background, Equation (5.9), is tied to the choice of a particular coordinate system, so another way of stating this is that the number M_{K} as defined so far is assigned to a metric and to a coordinate system, in a manner somewhat similar to the coordinate mass of Section 5.1). Indeed, given any differentiable function α(x^{A}) there exists a neighborhood of ∂_{∞}Σ on which a new coordinate r̂ can be introduced by the formula

\frac{r̂^{2}}{ℓ^{2}} − 2\frac{α}{r̂} = \frac{r^{2}}{ℓ^{2}} .  (5.30)

We can then chose the new background to be b = −(k+ \frac{r̂^{2}}{ℓ^{2}} )dt^{2} +(k+ \frac{r̂^{2}}{ℓ^{2}} )^{−1}dr̂^{2} + r̂^{2}ȟ, and obtain a new M_{K} which will in general not coincide with the old one. (It is noteworthy that the coordinate transformation (5.30) with the associated change of background do not change the value of the Hamiltonian mass M_{Ham}.) For example, if α is constant, using hats to denote the corresponding functions appearing in the metric expressed in the new coordinate system we obtain

β̂ = β + α ,  γ̂ = γ + 3α ,  δ̂ = δ − 2α
⟹  M̂_{K} = M_{K} − \frac{7αA_{∞}}{4π} ,

where A_{∞} is the area of ∂_{∞}Σ with respect to the metric ȟ. It turns out that one can remove this coordinate dependence in an appropriate class of metrics, tailoring the prescription in such a way that Equation (5.29) reproduces, up to a genus dependent factor, the coordinate mass M_{c}. In order to do that we shall suppose that the metric ^{4}g satisfies the hypotheses of point 2. of Proposition 3.7 (in particular k̃ = k = 0, ±1 according to the genus of the connected component of ∂_{∞}Σ under consideration), and we let the background be associated with a coordinate system (ρ, x^{A}) with ρ given by (3.47). It follows from Equations (5.3) and (3.49) that in this coordinate system it holds

\sqrt{| det g_{αβ}|} = r^{2} + o(\frac{1}{r}) ,  (5.31)

where we have used the generic symbol r to denote the coordinate ρ. We then impose (5.31) as a restriction on the coordinate system in which the generalized Komar mass M_{K} has to be calculated. When this condition is imposed we obtain from (5.3) and (5.25)

M_{K} = −\frac{1}{8π} ∫_{∂_{∞}Σ} μ_{∞} d^{2}μȟ = M_{Ham} .

We have thus proved

Proposition 5.1 Consider a metric ^{4}g satisfying the hypotheses of point 2. of Proposition 3.7, then the generalized Komar mass (5.27) associated to a background metric (5.9) such that (5.31) holds equals the Hamiltonian mass (5.22).

Proposition 5.1 is the Λ < 0 analogue of the theorem of Beig [12], that for static Λ = 0 vacuum metrics which are asymptotically flat in spacelike directions the ADM mass and the Komar masses coincide. Our treatment here is inspired by, and somewhat related to, the analysis of [58].

5.4 The Hawking mass M_{Haw}(ψ)

Let ψ be a function defined on the asymptotic region Σ_{ext}, with Σ_{ext} defined as in (5.1), such that the level sets of ψ are smooth compact surfaces diffeomorphic to each other (at least for ψ large enough), with ψ →_{r→∞} ∞. Following Hawking [42], Gibbons [38, Equation (17)] assigns a mass M_{Haw}(ψ) to such a foliation via the formula

M_{Haw}(ψ) ≡ lim_{ϵ→0} \frac{\sqrt{A_{1/ϵ}}}{32π^{3/2}} ∫_{\{ψ=1/ϵ\}}(^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ)dA ,  (5.32)

where A_{α} is the area of the connected component under consideration of the level set {ψ = α}

By considering simple examples in Minkowski space-times it can be seen that this definition is ψ dependent. However, when ^{2}M = S^{2}, Λ = 0, and the coordinate system on Σ_{ext} is such that the ADM mass m_{ADM} (which equals m_{H} as defined in Section 5.2) of Σ_{ext} is well defined (see [10, 24]), then M_{Haw}(ψ) will be independent of ψ, in the class of ψ’s singled out by the condition that the level sets of ψ approach round spheres at a suitable rate. No results of this kind are known when Λ ≠ 0.

The definition (5.32) applied to the function ψ = r and the metric (1.1) with k ≠ 0 gives

M_{Haw} = m|1 − g_{∞}|^{3/2} .

We have also used the Gauss–Bonnet theorem to calculate \sqrt{A_{1/ϵ}}. Thus the definition (5.32) differs from the coordinate one by the somewhat unnatural factor |1 − g_{∞}|^{3/2}. It is not clear why such a factor should be included in the definition of mass.

Consider, next, the metrics (3.37) with V and g given by (3.47)–(3.48). Let ψ = V; from the Codazzi–Mainardi Equation (3.33), the Equation (1.5), and the definition (3.18) of W we obtain, for V large enough so that |dV | > 0,

2R − \frac{1}{2}p^{2} − \frac{2}{3}Λ = (−2R_{ij} + Rg_{ij})n^{i}n^{j} − |q_{ij}|_{g}^{2} − \frac{2}{3}Λ
= −2\frac{D^{i}V D^{j}V}{V W} D_{i}D_{j}V − |q_{ij}|_{g}^{2} − \frac{2}{3}Λ
= −\frac{D^{i}V D_{i}W}{V W} − |q_{ij}|_{g}^{2} − \frac{2}{3}Λ .

In the coordinate system of Equation (3.30), where V = 1/x, one is led to

2R − \frac{1}{2}p^{2} − \frac{2}{3}Λ = x^{3} \frac{∂W}{∂x} − \frac{2}{3}Λ + O(x^{6})
= −\frac{x^{3}}{6} \frac{∂R′}{∂x} + O(x^{6}) ,

and we have used (3.27) and (3.17). From A_{1/ϵ} ≈ x^{−2}A′_{∂_{∞}Σ} we finally obtain

M_{Haw}(V) = −\frac{\sqrt{A′_{∂_{∞}Σ}}}{32π^{3/2}} ∫_{∂_{∞}Σ} \frac{1}{6} \frac{∂R′}{∂x} d^{2}μ_{h′}

= −\frac{\sqrt{A′_{∂_{∞}Σ}}}{32π^{3/2}} ∫_{∂_{∞}Σ} ℓ\frac{n′(R′)}{6}d^{2}μ_{h′} ,  (5.33)

where d^{2}μ_{h′} is the Riemannian area element induced by g′ on ∂_{∞}Σ, and n′ denotes the inward-pointing g′-unit normal to ∂_{∞}Σ. We have thus proved the following result:

Theorem 5.2 Let a triple (Σ, g, V) satisfying Equations (1.3)–(1.5) be C^{i} compactifiable, i ≥ 3. Then the Hawking mass M_{Haw}(V) of the V -foliation is finite and well defined; it is given by the formula (5.33), with R′ — the curvature scalar of the metric g′ = V^{−2}g.

We can relate M_{Haw}(V) to the coordinate mass M_{c} if we assume in addition that the latter is well defined; recall that this required R′ and ∂_{x}R′ to be constant on ∂_{∞}Σ. In this case Equation (5.4) gives

M_{Haw}(V) = (\frac{A′_{∂_{∞}Σ}}{4πℓ^{2}} )^{3/2} M_{c} .  (5.34)

From Equation (3.22) we have^{2}R′|_{x=0} = 2k/ℓ^{2}, and the Gauss–Bonnet theorem implies

∫_{∂_{∞}Σ} ^{2}R′d^{2}μ_{h′} = \frac{2k}{ℓ^{2}} A′_{∂_{∞}Σ} = 8π(1 − g_{∞}) ,

so that when g_{∞} ≠ 1 we obtain

M_{Haw}(V) = |1 − g_{∞}|^{3/2}M_{c} .  (5.35)

We emphasize that M_{Haw}(V) is finite and well defined even when the conditions of Section (5.1), which we have set forth to define M_{c}, are not met.

Similarly, the Hamiltonian mass M_{Ham}, associated to the background singled out by the coordinate system of Proposition 3.7, can be defined when R′ is constant on ∂_{∞}Σ. (This holds regardless of whether or not ∂_{x}R′ is constant on ∂_{∞}Σ.) Proceeding as above, making use of Equations (3.46)–(3.51), one is led to

g_{∞} ≠ 1 ⟹ M_{Haw}(V) = |1 − g_{∞}|^{1/2}M_{Ham} ,
g_{∞} = 1 , A′_{∞} = 4πℓ^{2} ⟹ M_{Haw}(V) = M_{Ham} .  (5.36)

6 The generalized Penrose inequality

We recall here an argument of Geroch [37], as extended by Jang and Wald [49] and Gibbons [38], for the validity of the Penrose inequality^{13}:

Proposition 6.1 Assume we are given a three dimensional manifold (Σ, g) with connected boundary ∂Σ such that:

13The argument we review has been used by Gibbons in [38] to obtain a somewhat different inequality, in which the genus factors are not present. The inequality in [38] is violated for generalized Kottler metrics with g_{∞} ≥ 3.

1. R ≥ 2Θ for some strictly negative constant Θ.

2. There exists a smooth, global solution of the inverse mean curvature flow without critical points, i.e., there exists a surjective function u : Σ → [0, ∞) such that du has no zeros and

\{ \begin{matrix}u|_{∂Σ} = 0 , \\ ∇_{i} (\frac{∇^{i}u}{|du|}) = |du| . \end{matrix}  (6.1)

3. The level sets of u

N_{s} = {u(x) = s}

are compact.

4. The boundary ∂Σ = u^{−1}(0) of Σ is minimal.

5. The Hawking mass of the level sets of u as defined in (5.32) exists.

Then

2M_{Haw}(u) ≥ (1 − g_{∂Σ}) (\frac{A_{∂Σ}}{4π} )^{1/2} − \frac{Θ}{3} (\frac{A_{∂Σ}}{4π} )^{3/2} .  (6.2)

Here A_{∂Σ} is the area of ∂Σ and g_{∂Σ} is the genus thereof.

Remarks: 1. The Proposition above can be applied to solutions of (1.4) and (1.5) with Θ = Λ: in this case we have R = 2Λ; further Equation (1.5) multiplied by V and contracted with two vectors tangent to ∂Σ shows that the boundary {V = 0} is totally geodesic and hence minimal.

2. Equation (6.2) is sharp — the inequality there becomes an equality for the generalized Kottler metrics.

Proof: Let A_{s} denote the area of N_{s}, and define

σ(s) = \sqrt{A_{s}} ∫_{N_{s}}(^{2}R_{s} − \frac{1}{2}p_{s}^{2} − \frac{2}{3}Θ)d^{2}μ_{s} ,  (6.3)

where ^{2}R_{s} is the scalar curvature of the metric induced on N_{s}, d^{2}μ_{s} is the Riemannian volume element associated to that same metric, and p_{s} is the mean curvature of N_{s}. The hypothesis that du is nowhere vanishing implies that all the objects involved are smooth in s. At s = 0 we have

σ(0) = \sqrt{A_{∂Σ}} ∫_{∂Σ}(^{2}R_{0} − \frac{2}{3}Θ)d^{2}μ_{0}
= \sqrt{A_{∂Σ}} (8π(1 − g_{∂Σ}) − \frac{2}{3}ΘA_{∂Σ}) .  (6.4)

On the other hand,

lim_{s→∞} σ(s) = 32π^{3/2}M_{Haw}(u) .

The generalization in [38] of the classical calculation of [37] gives

\frac{∂σ}{∂s} ≥ 0 .  (6.5)

This implies lim_{s→∞} σ(s) ≥ σ(0), which gives (6.2). □

To be able to carry out the above argument one had to assume that du had no zeros, which implies in particular that ∂_{∞}Σ is connected with g_{∂Σ} = g_{∞}. It is not known whether or not the other hypotheses of Proposition 6.1, or the conditions of Definition 3.1 together with Equations (1.3)–(1.5), force ∂Σ to be connected. If they do not, one is tempted to conjecture that the right inequality should be

2M_{Haw}(u) ≥ ∑_{i=1}^{k} ((1 − g_{∂_{i}Σ}) (\frac{A_{∂_{i}Σ}}{4π} )^{1/2} − \frac{Θ}{3} (\frac{A_{∂_{i}Σ}}{4π} )^{3/2}) .  (6.6)

Here the ∂_{i}Σ’s, i = 1, …, k, are the connected components of ∂Σ, A_{∂_{i}Σ} is the area of ∂_{i}Σ, and g_{∂_{i}Σ} is the genus thereof. This would be the inequality one would obtain from the Geroch–Gibbons argument if it could be carried through for u’s which are allowed to have critical points, on manifolds with ∂_{∞}Σ connected but ∂Σ — not connected.

We note that when Λ = 0 there is a version of the proof of Proposition 6.1 due to Huisken and Ilmanen in which du is allowed to have zeros (with ∂Σ — connected)^{14}. Note that at points where du vanishes Equation (6.1) does not make sense classically, and has to be understood in a proper way. Further the monotonicity calculation of [37] breaks down at critical level sets of u, as those do not have to be smooth submanifolds. Nevertheless (when Λ = 0) existence of appropriate functions u (perhaps with critical points) together with the monotonicity of σ can be established [46, 47] when ∂Σ is an outermost (necessarily connected) minimal sphere. It is conceivable that the argument of Huisken and Ilmanen can be modified to include the case Λ ≠ 0. One of the difficulties here is to handle the possibly changing genus of the level sets of u.

Let us discuss some of the consequences of the (hypothetical) Equation (6.6). To proceed further it is convenient to introduce a mass parameter m defined as follows:

m = \{ \begin{matrix}M_{Haw} , & ∂_{∞}Σ = S^{2} , \\ M_{Haw} , & ∂_{∞}Σ = T^{2}, with the normalization A′_{∞} = 4πℓ^{2} , \\ \frac{M_{Haw}}{|g_{∂_{∞}Σ} − 1|^{3/2}} , & g_{∂_{∞}Σ} > 1 . \end{matrix}

(6.7)

Strictly speaking, we should write m(u) if M_{Haw}(u) is used above, m(V) if M_{Haw}(V) is used, etc.; we shall do this when confusions are likely to occur. For generalized Kottler metrics the mass m = m(u) so defined coincides with the mass parameter appearing in (1.1) when u is the radial solution u = u(r) of the problem (6.1); m(V) coincides with the coordinate mass M_{c} for the metrics considered here when M_{c} is defined, cf. Equation (5.34).

Note, first, that if all connected components of the horizon have spherical or toroidal topology, then the lower bound (6.6) is strictly positive. For example, if ∂Σ = T^{2}, and ∂_{∞}Σ = T^{2} as well we obtain

14Bray’s proof [17] of the inequality (6.6) with Θ = 0 but ∂Σ — not necessarily connected, uses a completely different technique; in particular it makes appeal to the positive energy theorem which does not hold in the class of manifolds considered here.

2m ≥ \frac{1}{ℓ^{2}} (\frac{A_{∂Σ}}{4π} )^{3/2} .

On the other hand if ∂Σ = T^{2} but g_{∂_{∞}Σ} > 1 from Equation (6.6) one obtains

2m ≥ \frac{1}{ℓ^{2}|g_{∞} − 1|} (\frac{A_{∂Σ}}{4π} )^{3/2} .

Let us return to the case^{15} where Equations (1.3)–(1.5) hold; we can then use the Galloway–Schleich–Witt–Woolgar inequality [33]

∑_{i=1}^{k} g_{∂_{i}Σ} ≤ g_{∞} .  (6.8)

It implies that if ∂_{∞}Σ has spherical topology, then all connected components of the horizon must be spheres. Similarly, if ∂_{∞}Σ is a torus, then all components of the horizon are spheres, except perhaps for at most one which could be a torus. It follows that to have a component of the horizon which has genus higher than one we need g_{∞} > 1 as well.

When some — or all — connected components of the horizon have genus higher than one the right hand side of Equation (6.6) might become negative. Minimizing the generalized Penrose inequality (6.6) with respect to the areas of the horizons gives the following interesting inequality

M_{Haw}(u) ≥ −\frac{1}{3\sqrt{−Λ}} ∑_{i} |g_{∂_{i}Σ} − 1|^{3/2} ,  (6.9)

where the sum is over those connected components ∂_{i}Σ of ∂Σ for which g_{∂_{i}Σ} ≥ 1. Equation (6.9), together with the elementary inequality ∑_{i=1}^{N} |λ_{i}|^{3/2} ≤

(∑_{i=1}^{N} |λ_{i}|)^{3/2}, lead to

m ≥ −\frac{1}{3\sqrt{−Λ}} .  (6.10)

The Geroch–Gibbons argument establishing the inequality (6.4) when a suitable u exists can also be formally carried through when ∂Σ = ∅. In this case one still considers solutions u of the differential equation that appears in Equation (6.1), however the Dirichlet condition on u at ∂Σ is replaced by a condition on the behavior of u near some chosen point p_{0} ∈ Σ. If the level set of u around p_{0} approach distance spheres centered at p_{0} at a suitable rate, then σ(s) tends to zero when the N_{s}’s shrink to p_{0}, which together with the monotonicity of σ leads to the positive energy inequality:

M_{Haw}(u) ≥ 0 .  (6.11)

15The discussion that follows actually applies to all (Σ, g)’s that can be isometrically embedded into a globally hyperbolic space-time M in which the null convergence condition holds; further the image of Σ should be a partial Cauchy surface in M. Finally the intersection of Σ with ℐ should be compact. The global hyperbolicity here, and the notion of Cauchy surfaces, is understood in the sense of manifolds with boundary, see [33] for details.

It should be emphasized that the Horowitz-Myers solutions [45] with negative mass show that this argument breaks down when g_{∞} = 1.

When ∂_{∞}Σ = S^{2} one expects that (6.11), with M_{Haw}(u) replaced by the spinorially defined mass (which might perhaps coincide with M_{Haw}(u), but this remains to be established), can be proved by Witten type techniques, compare [6, 39]. On the other hand it follows from [11] that when ∂_{∞}Σ ≠ S^{2} there exist no asymptotically covariantly constant spinors which can be used in the Witten argument. The Geroch–Gibbons argument has a lot of “ifs” attached in this context, in particular if ∂_{∞}Σ ≠ S^{2} then some level sets of u are necessarily critical and it is not clear what happens with σ when a jump of topology from a sphere to a higher genus surface occurs. We note that the area of the horizons does not occur in (6.10) which, when g_{∂_{∞}Σ} > 1, suggests that the correct inequality is actually (6.10) rather than (6.11).

7 Mass and area inequalities

7.1 Preliminaries

We first give here a sketch of the content of this section. We define W_{0} via a suitably chosen generalized Kottler solution, (called the ”reference solution”, RS) and W̃ = Ψ^{−4}W and W̃_{0} = Ψ^{−4}W_{0} for a certain function Ψ(V). We then establish three lemmas. The first one (Lemma 7.1) expresses the surface integral at infinity of the normal derivative n^{i}D_{i}(W̃ − W̃_{0}) in terms of the mass difference between the given solution and a RS, while Lemma 7.2 expresses this same normal derivative taken on the horizon in terms of the difference of the areas of the given and the RS, with appropriate genus factors. We next recall from [14], an elliptic equation of the form (△ − a)(W̃ − W̃_{0}) ≥ 0, for some function a. This equation is first employed in Lemma 7.3 where we show that the generalized Kottler solutions can be characterized either by the condition W̃ = W̃_{0} or by conformal flatness of (Σ, g) (this Lemma is actually formulated more genrerally such as to include the Nariai case as well). The crucial step in the proofs then consists of applying the maximum principle to the elliptic equation for W̃ − W̃_{0}. This is possible if the function a is non-negative, which is the case in the present situation (Λ < 0) iff the mass of the reference Kottler solution is non-positive. By the asymptotic conditions (and by a suitable choice of the RS) we can achieve that W̃ − W̃_{0} takes its maximum value (namely zero) both at the horizon (if there is one) and at infinity. The maximum principle then yields that the derivatives n^{i}D_{i}(W̃ − W̃_{0}) with respect to the outward normals at the horizon and at infinity are positive, and zero precisely if W̃ = W̃_{0}. Theorems 1.3 and 1.5 then readily follow from the lemmata. As a final step we combine the mass and area inequalities to derive the inverse Penrose inequality.

Turning, then, to the details, let (Σ, g, V) satisfy (1.3)–(1.5) together with the topological, the differential, and the asymptotic requirements spelled out in the statements of Theorems 1.3 or 1.5. (As mentioned above, Lemma 7.3 holds under more general conditions). We first introduce the surface gravity κ of ∂Σ to be the corresponding restriction of the function \sqrt{W} defined by (3.18) to ∂Σ:

κ ≡ |dV |_{g}|_{∂Σ} .  (7.1)

where we have normalized V so that Equation (3.23) holds, cf. Proposition 3.3. By the strong maximum principle [40, Lemma 3.4] W is nowhere vanishing on ∂Σ. Moreover, it is well known (and easily seen using Equation (1.5)) that κ is locally constant on ∂Σ:

0 = n^{j}D_{i}D_{j}V |_{V =0} = \frac{D^{j}V}{\sqrt{W}} D_{i}D_{j}V |_{V =0}
= \frac{1}{2\sqrt{W}} D_{i}W |_{V =0} .  (7.2)

Here n^{i} is the unit normal to ∂Σ, where V vanishes.

It is convenient to introduce the notion of a reference solution (RS) as follows: This is a generalized Kottler solution with the same genus g_{∞} as (Σ, g, V). Moreover, if ∂Σ ≠ ∅, the surface gravity κ of the RS is chosen to be equal to the maximum of the surface gravities of (Σ, g, V). On the other hand, if ∂Σ = ∅, the mass of the RS will be defined suitably below, in the proof of theorem 1.3. We only consider RS with mass m_{0} in the range (2.7) (if ∂Σ ≠ ∅, this property follows from the restriction (1.7) on κ). It should be stressed that we are not comparing manifolds and/or metrics, but we are only using the resulting scalar functions V and W.

Let r(·) be the function V_{0} → r(V_{0}) constructed at the end of Section 2. Composing r with V we obtain functions r(V (·)) and W_{0}(r(V (·)) defined on Σ. By an abuse of notation we shall still denote those functions by r and W_{0}.

In the same manner, we can define a RS from other solutions with the property that W is a function of V only. (In Lemma 7.3 below we will also include the Nariai case).

Following [14] we define ψ(V) to be that unique solution of the equation

ψ^{−1} \frac{dψ}{dV} = −V W_{0}^{−1} \frac{m_{0}}{r^{3}}  (7.3)

which goes^{16} to 1 as V goes to ∞. (In particular ψ ≡ 1 when m_{0} = 0.) Here r = r(V) is again the function defined at the end of Section 2. Standard results on ODE’s show that solutions of (7.3) have no zeros unless identically vanishing, and that

Ψ ≡ ψ ∘ V

can be extended by continuity to a smooth function on \overline{Σ}, still denoted by Ψ, which satisfies

Ψ > 0 , Ψ|_{∂_{∞}Σ} = 1 .

16Using the asymptotic behavior of V (r) and r(V) it is not too difficult to show that solutions of (7.3) are uniformly bounded on [0, ∞), and approach a non-zero constant at infinity unless identically vanishing. Since solutions of (7.3) are defined up to a multiplicative constant, we can choose this constant so that our normalization holds.

We also define

g̃_{ij} = V^{−2}Ψ^{4}g_{ij} ,
W̃ = Ψ^{−4}W ,
W̃_{0} = Ψ^{−4}W_{0} .  (7.4)

We proceed with a computation which is required in Lemma 7.1 as well as in Lemma 7.2. Consider a level set {V = const} of V which is a smooth hypersurface in \overline{Σ}, with unit normal n_{i}, induced metric h_{ij}, scalar curvature 2R, second fundamental form p_{ij} defined with respect to an inner pointing normal, mean curvature p = h^{ij}p_{ij}; we denote by q_{ij} the trace-free part of p_{ij}: q_{ij} = p_{ij} − \frac{1}{2}h_{ij}p. Using Equation (2.5), the Equation (1.4) with g = g_{0} and V = V_{0}, together with the relation

\frac{dV_{0}}{dr} = \frac{\sqrt{W_{0}}}{V_{0}}  (7.5)

we obtain

V^{−1} \frac{dW_{0}}{dV} = −\frac{2}{3}Λ − \frac{4m_{0}}{r^{3}} .  (7.6)

To obtain (7.7) we use, in this order, the definitions (7.4), the Equations (1.4)– (1.5), Equations (7.6) and (7.3), and the Codazzi-Mainardi equation:

V^{−1}W̃^{−1}D^{i}V D_{i}(W̃ − W̃_{0})

= V^{−1}W^{−1}D^{i}V (D_{i}W) − V^{−1} \frac{dW_{0}}{dV} − 4V^{−1}Ψ^{−1} \frac{dΨ}{dV}(W − W_{0})

= (2R_{ij} − Rg_{ij})n^{i}n^{j} + \frac{2}{3}Λ + \frac{4m_{0}}{r^{3}} − \frac{4m_{0}}{r^{3}} (1 − W_{0}^{−1}W)

= −^{2}R − q_{ij}q^{ij} + \frac{1}{2}p^{2} + \frac{2}{3}Λ + \frac{4m_{0}}{r^{3}} − \frac{4m_{0}}{r^{3}} (1 − W_{0}^{−1}W) .  (7.7)

Lemma 7.1 Under the conditions of Theorem 1.1, suppose further that the scalar curvature R′ of the metric g′ = V^{−2}g is constant on ∂_{∞}Σ. Let V be normalized so that (3.23) holds, with A′_{∞} = 4πℓ^{2} when ∂_{∞}Σ = T^{2}. If m is the Hawking mass parameter defined as in (6.7), then

∫_{∂_{∞}Σ} D′_{i}(W̃ − W̃_{0})dS^{′i} = − (\frac{2Λ}{3})^{2} A′_{∂_{∞}Σ}(m − m_{0}) ,  (7.8)

where dS^{′i} denotes the outer-oriented area element of the metric g′ = V^{−2}g, and A′_{∂_{∞}Σ} is the area of ∂_{∞}Σ with respect to that metric.

Proof: Using

D^{′i}(W̃ − W̃_{0})n′_{i} = \frac{1}{\sqrt{W′}} D_{i}(W̃ − W̃_{0})D^{i}V  (7.9)

and (7.7), the left hand side of (7.8) reads

∫_{∂_{∞}Σ} \frac{V W̃}{\sqrt{W′}} [−^{2}R − q_{ij}q^{ij} + \frac{1}{2}p^{2} + \frac{2}{3}Λ + \frac{4m_{0}}{r^{3}} − \frac{4m_{0}}{r^{3}} (1 − W_{0}^{−1}W)] d^{2}μ_{g′}

(7.10) where d^{2}μ_{g′} is the two-dimensional surface measure associated with the metric g′. Chasing through the definitions one finds that

\frac{V W̃}{\sqrt{W′}} ≈ \sqrt{−\frac{Λ}{3}}V^{3}  (7.11)

near ∂_{∞}Σ. From the definition of V_{0} we further have

r ≈ \sqrt{−\frac{3}{Λ}}V ,

again near ∂_{∞}Σ, so that lim_{V →∞} V W̃/(\sqrt{W′}r^{3}) = (−Λ/3)^{2}. It follows that the second to last term in (7.10) gives a contribution

(\frac{2Λ}{3})^{2} A′_{∂_{∞}Σ}m_{0}  (7.12)

where A′_{∂_{∞}Σ} denotes the g′-area of the connected component of ∂_{∞}Σ under consideration. Equation (3.17) and its equivalent with W replaced by W_{0} show that

(1 − W_{0}^{−1}W) →_{V →∞} 0

so that the last term drops out from (7.10). Furthermore, by Equation (3.27) we have

\frac{V W̃}{\sqrt{W′}} q_{ij}q^{ij} = O(V^{−3}) →_{V →∞} 0 ,

and it remains to analyze the contribution of −V W̃ (^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ) /\sqrt{W′} to the integral (7.8). To do this, note that

A_{1/ϵ} ≡ A({V = 1/ϵ}) = ∫_{V′=ϵ} d^{2}μ_{g}
= ∫_{V′=ϵ} V^{2}d^{2}μ_{g′} ≈ ϵ^{−2}A′_{∂_{∞}Σ} ,

where d^{2}μ_{g} is the induced measure on ∂_{∞}Σ associated with the metric g. It follows that

− ∫_{V′=ϵ} \frac{V W̃}{\sqrt{W′}} (^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ) d^{2}μ_{g′}

≈  −\sqrt{−\frac{Λ}{3}} \frac{1}{ϵ} ∫_{V′=ϵ} (^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ) d^{2}μ_{g}

≈  −\sqrt{−\frac{Λ}{3}}\sqrt{\frac{A_{1/ϵ}}{A′_{∂_{∞}Σ}}} ∫_{V′=ϵ} (^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ) d^{2}μ_{g}

→_{ϵ→0} − (\frac{2Λ}{3})^{2} A′_{∂_{∞}Σ}m ,  (7.13)

where

m ≡ lim_{ϵ→0} \frac{1}{4} (−\frac{ΛA′_{∂_{∞}Σ}}{3} )^{−3/2} \sqrt{A_{1/ϵ}} ∫_{\{V =1/ϵ\}}(^{2}R − \frac{1}{2}p^{2} − \frac{2}{3}Λ)dA . (7.14)

To finish the proof we need to show that m in (7.14) is indeed the Hawking mass as defined in Equation (6.7). In the torus case this follows immediately from the normalization condition A′_{∞} = 4πℓ^{2}; for the remaining topologies this can be seen as follows: if V is normalized so that (3.23) holds, then (3.22) implies

2R′|_{x=0} = −\frac{2}{3}Λk .

When g_{∞} ≠ 1 the Gauss–Bonnet theorem gives

8π|1 − g_{∞}| = |∫ ^{2}R′d^{2}μ_{g′} |
= −\frac{2}{3}ΛA′_{∂_{∞}Σ} ,

which shows that the mass defined by Equation (7.14) coincides with that of (6.7). □

As to the subsequent Lemma, recall from Theorem 1.5 that ∂_{1}Σ refers to the the component of ∂Σ of the given solution with the largest surface gravity.

Lemma 7.2 Under the conditions of Theorem 1.1, we have

∫_{∂_{1}Σ} W̃^{−1/2}D̃_{i}(W̃ − W̃_{0})dS̃^{i} = 8π [(g_{∂_{1}Σ} − 1) − \frac{A_{∂_{1}Σ}}{A_{0}} (g_{∞} − 1)]  (7.15)

Proof: We integrate (7.7) over ∂_{1}Σ. We note that Equation (1.5) multiplied by V and contracted with two vectors tangent to ∂Σ shows that ∂Σ is totally geodesic; equivalently, q_{ij} = 0. We introduce

2R_{0} = \frac{2}{3}Λ + \frac{4m_{0}}{r_{0}^{3}} ,

the scalar curvature of the metric dΩ_{k}^{2}. Using (7.7) and the Gauss–Bonnet theorem, the left hand side of (7.15) can be written as

∫_{∂_{1}Σ}(−^{2}R + \frac{2}{3}Λ + \frac{4m_{0}}{r_{0}^{3}} )dA = ∫_{∂_{1}Σ}(−^{2}R +^{2}R_{0})dA = 8π (g_{∂_{1}Σ} − 1) +^{2}R_{0}A_{∂_{1}Σ}

(7.16)

Equation (7.15) is then obtained by eliminating ^{2}R_{0} from (7.16), using the Gauss–Bonnet theorem for the generalized Kottler metrics:

8π(1 − g_{∞}) = ^{2}R_{0}A_{0} .
□

The following elliptic equation for W̃ − W̃_{0} will be the crucial ingredient in the proof of the theorems. It is also useful for Lemma 7.3.

(Δ̃ − a)(W̃ − W̃_{0}) =
= \frac{1}{4}W̃^{−1}R̃_{ijk}R̃^{ijk} + \frac{3}{4}W̃^{−1}D̃_{i}(W̃ − W̃_{0})D̃^{i}(W̃ − W̃_{0}) , (7.17)

with

a = \frac{5}{3r^{3}} m_{0}ΛV^{4}W_{0}^{−2} W̃ ,  (7.18)

Δ̃ being the Laplace operator of the metric g̃_{ij}, and R̃_{ijk} — the Cotton tensor of g̃_{ij}. This equation is obtained by specializing^{17} Equation (5.4) of [14] (which has been used in that paper in the context of a uniqueness proof for static perfect fluid solutions) to the present case with 8πρ = −8πp = Λ.

It is important to stress that Equation (7.17), as it stands, makes only sense on the set {dV ≠ 0}, because of the factors W̃^{−1} appearing there. However, it follows from Equation (1.4) that the set {dV = 0} has no interior: indeed, if dV vanishes on a connected open set then V is constant there, which is compatible with Equation (1.5) only if V vanishes there. This contradicts our hypothesis that V vanishes only^{18} on ∂Σ . Hence Equation (7.17) holds on an open dense set of Σ. Since the left hand side of Equation (7.17) is a smooth function on Σ \ ∂Σ, the right hand side thereof is smoothly extendible by continuity to a smooth function on Σ \ ∂Σ, and Equation (7.17) holds everywhere on this set with the right hand side being understood in the sense explained here.

Lemma 7.3 Let Λ ∈ ℝ, and let (Σ, g, V) be a solution of (1.3)–(1.5) such that

a. either W ≡ W_{0} for W_{0} defined from the generalized Kottler or from the Nariai solution (1.2), or

b. (Σ, g) is locally conformally flat.

Suppose further that Σ is a union of compact boundary-less level sets of V. Then:

1. Every connected component 𝒱 of the set {p ∈ Σ | dV (p) ≠ 0} “corresponds to” one of the generalized Kottler solutions (1.1), or to one of the generalized Nariai solutions (1.2), or is flat. More precisely, there exists an interval J ⊂ ℝ, a two-dimensional compact Riemannian manifold (^{2}M, dΩ_{k}^{2}), with dΩ_{k}^{2} an (r-independent) metric of constant Gauss curvature k = 0, ±1, and a diffeomorphism ψ : 𝒱 → J ×^{2}M such that, transporting g and V to J ×^{2}M using ψ, we have:

(i) Either there exists a constant λ > 0 such that V = λV_{0} and

g = V_{0}^{−2}dr^{2} + r^{2}dΩ_{k}^{2} , r ∈ J ,  (7.19)
V_{0}^{2} = k − \frac{2m}{r} − \frac{Λ}{3}r^{2} ,  (7.20)

(ii) or, when kΛ > 0, there exists a constant λ ∈ ℝ (λ > 0 if Λ > 0) such that

g = V^{−2}dz^{2} + |Λ|^{−1}dΩ_{k}^{2} , z ∈ J ,  (7.21)
V^{2} = λ − Λz^{2} ,  (7.22)

17The assumption of spherical symmetry of the level sets of the reference solution made in [14] is not needed to obtain Equation (7.18).

18Let us mention that if V is zero on an open set, then the Aronszajn unique continuation theorem [7] shows in any case that V must be identically zero on Σ.

(iii) or, when k = Λ = 0, there exists a constant λ > 0 such that V = λz and

g = dz^{2} + dΩ_{k}^{2} , z ∈ J .  (7.23)

(In each case the interval J is constrained by the condition that V and V^{2} be non-negative).

2. Under condition a. above, if Σ is connected and if W_{0} (considered as a function of V) has no zeros in the interval where V takes its values,

∀ p ∈ Σ  W_{0}(V (p)) ≠ 0 ,  (7.24)

then 𝒱 = Σ, thus Equations (7.21) or (7.19) hold globally on Σ.

Remarks: 1. Here we do not make any hypotheses on the sign of Λ.

2. The result is local, in particular it is sufficient to be able to invert r_{0}(V_{0}) locally on the range of the values of V under consideration to obtain W_{0}(V).

3. The set (Σ, g, V) corresponding to the metric (7.23) arises from a boost Killing vector in suitably identified Minkowski space-time.

4. We note that the set 𝒱 could be empty, as is the case for ℝ × T^{3} with the obvious flat metric. Our analysis does not say anything about the metric on regions where dV vanishes.

5. We note that the generalized Kottler and the generalized Nariai metrics also arise naturally in the generalized Birkhoff theorem, see [30, 41], and also [65] for a very clear treatment in the Λ > 0 case.

6. The Lemma can easily be reformulated by taking any conformally flat solution of (1.4) and (1.5) as a reference solution. The condition of conformal flatness is required to ensure that (7.17) holds and excludes, in particular, the Horowitz-Myers solutions with toroidal ℐ^{+} [45] as RS.

Proof: The proof is an adaptation of an argument of [27] to the current setting. Suppose that W = W_{0} for some W_{0}; Equation (7.17) shows then that R̃_{ijk}R̃^{ijk} vanishes, so that (Σ, g) is locally conformally flat. It then follows that condition b. holds in both cases.

We start by removing from Σ some undesirable points: set

Σ_{sing} ≡ {p ∈ Σ| the connected component of the set {q|V (q) = V (p)}

containing p contains a point r such that dV (r) = 0. } ,

Σ′ ≡ Σ \ Σ_{sing} .

Σ_{sing} is a closed subset of Σ, so that Σ′ is still a manifold. It follows from Sard’s theorem that Σ′ ≠ ∅. We note that Σ′ still satisfies all the hypotheses of the Lemma, except perhaps for being connected. By construction all the level sets of V are non-critical in Σ′. (Recall that a level set {V = c} of V is non-critical if dV is nowhere vanishing on {V = c}.)

Let 𝒰 to be any connected component of Σ′. Compactness of the level sets of V implies^{19} that 𝒰 is diffeomorphic to I × ^{2}M, for some two-dimensional compact connected manifold ^{2}M and some interval I ⊂ ℝ, with V equal to c on {c} ×^{2}M, c ∈ I, and that on 𝒰 the function V can be used as a coordinate. Further we can introduce on ^{2}M a finite number of coordinate patches with coordinates x^{A}, A = 1, 2, so that on 𝒰 the metric takes the form

19The possibility that 𝒰 is diffeomorphic to S^{1} ×^{2}M (or some twisted version thereof) is excluded by the fact that dV does not vanish on 𝒰.

g = W^{−1}dV^{2} + h_{AB}dx^{A}dx^{B} .  (7.25)

Let, as before, q_{AB}dx^{A}dx^{B} be the trace free part of the extrinsic curvature tensor of the level sets of V — in the coordinate system of (7.25)

q_{AB} = \sqrt{W} (\frac{∂h_{AB}}{∂V} − \frac{1}{2}h^{CD} \frac{∂h_{CD}}{∂V} h_{AB}) .  (7.26)

Equations (7.26) and (3.25) imply that q_{AB} vanishes hence \frac{∂h_{AB}}{∂V} is pure trace, that W = W (V), and that det γ_{AB} is a product of a function of V with a function of the remaining coordinates. We thus have

h = W (V)^{−1}dV^{2} + r(V)^{2}dΩ^{2} .  (7.27)

for some positive function r(V), where dΩ^{2} is a V -independent metric on 2M. Next, from (1.5) and from the Codazzi-Mainardi equations (3.35), respectively (3.33), applied to ^{2}M ⊂ 𝒰 , we find that the mean curvature p of all level surfaces, respectively their Ricci scalars, are constant. Hence (^{2}M, dΩ^{2}) is a space of constant curvature, and scaling r appropriately we can without loss of generality assume that the Gauss curvature k of the metric dΩ^{2} equals 0, ±1, as appropriate to the genus of ^{2}M. We define

L = \frac{dW}{dV} + 2ΛV .  (7.28)

Evaluating (1.4) for the metric (7.27), we find

\frac{dr}{dV} = −\frac{rL}{4W} .  (7.29)

Equations (1.4)–(1.5) for the metric (7.27) are equivalent to (7.28)–(7.29) together with

2W (Λ − \frac{k}{r^{2}} ) = L (V^{−1}W − \frac{L}{8}) ,  (7.30)
W \frac{dL}{dV} = \frac{3}{4}L^{2} + (V^{−1}W − ΛV)L .  (7.31)

These equations arise e.g. by adapting Equations (3.16) and (3.17) of [14] to the present case (namely by setting 8πρ = −8πp = Λ, L_{1} = L and C^{2} = k, and allowing the constant k to take negative values). Suppose, first, that there exists V_{∗} such that L(V_{∗}) = 0. Equation (7.31) shows then that L ≡ 0, and from (7.30) one obtains

Λ = \frac{k}{r^{2}} .  (7.32)

If k = 0 then Λ vanishes as well; further r is constant by Equation (7.29) and can therefore be absorbed into dΩ^{2}. Integrating Equation (7.28) one finds that there exists a strictly positive constant λ such that W = λ^{2}, defining a coordinate z by the equation z = V/λ proves point 1iii. on 𝒰 . Next, if k ≠ 0 Equation (7.32) gives kΛ > 0 as desired, together with r^{2} = −1/|Λ|. Integrating Equation (7.28) one obtains

W = Λ(λ − V^{2}) ,

for some constant λ ∈ ℝ. Introducing the coordinate z via the equation V^{2} = λ − Λz^{2} establishes point 1ii. on 𝒰 .

In the case of L without zeros we obtain, from (7.28), (7.29) and (7.31), that

\frac{d}{dV} (\frac{V \sqrt{W}}{rL} ) = 0 ,

which implies that there exists a non-vanishing constant α such that

L = αV \frac{\sqrt{W}}{r} .  (7.33)

Using (7.29) one is led to

\frac{dV}{dr} = −\frac{4\sqrt{W}}{αV} .  (7.34)

Next we define

m(V) = −\frac{α}{4}r^{2}\sqrt{W} + \frac{Λr^{3}}{3} ;  (7.35)

from (7.28), (7.33) and (7.34) we obtain dm/dV = 0, i.e. m is a constant. Equation (7.30) gives

V^{2} = \frac{16}{α^{2}} (k − \frac{2m}{r} − \frac{Λ}{3}r^{2}) .

Equation (7.29) shows that we can use r as a coordinate, and Equation (7.34) implies that the metric is of the desired form (7.19). This establishes point 1i. on 𝒰 .

Let 𝒱 be the connected component of {dV ≠ 0} ⊂ Σ that contains 𝒰 . To establish point 1. of the Lemma we need to show that 𝒱 = 𝒰 . We claim that 𝒰 is open in 𝒱 — and hence in Σ — which can be seen as follows: Let p ∈ 𝒰 , we thus have dV (q) ≠ 0 for all q such that V (p) = V (q). By Equation (7.27) |dV |_{g} = \sqrt{W} is constant on the intersection with 𝒰 of the level set V^{−1}(V (p)) of V through p, so that

inf_{V^{−1}(V (p))∩𝒰} |dV |_{g} > 0 ,

which easily implies that all nearby level sets in 𝒰 ⊂ Σ′ are non-critical.

Let us show now that 𝒰 is closed in 𝒱 . To see that, consider a sequence p_{i} ∈ 𝒰 such that p_{i} → p ∈ 𝒱 . By definition of 𝒱 the function |dV |_{g} has no zeros on 𝒱 , hence dV (p) ≠ 0. Now it follows from (3.25) that |dV |_{g} is locally constant on smooth subsets of level sets of V, which easily implies a) that the connected component of V^{−1}(V (p)) containing p is smooth and b) that |dV |_{g} is nowhere vanishing there. Compactness of the level sets of V implies that all the connected components of level sets intersecting a neighborhood of p are non-critical, and hence are in Σ′. It then follows that p ∈ 𝒰 .

We have thus shown that 𝒰 is both open and closed in 𝒱 ; connectedness of 𝒱 shows that 𝒰 = 𝒱 , and point 1. is established.

To prove part 2., we note that the equality W (p) = W_{0}(V (p)) together with Equation (7.24) shows that V has no critical points on Σ; as Σ is connected the set 𝒱 of point 1. coincides with Σ, and point 2. follows from point 1. □

7.2 Proofs

Proof of Theorem 1.3: Suppose that ∂Σ = ∅. We first consider as RS a generalized Kottler solution with m_{0} = 0 (see Equation (2.6)). This leads to

Ψ ≡ 1 ,  W̃_{0}(V_{0}) = −\frac{Λ}{3}(V_{0}^{2} − k) .  (7.36)

We further normalize V as in Proposition 3.3, so that by (3.17), (3.21) and (3.23) we have

W̃ − W̃_{0} →_{r→∞} 0 .

(Actually when ∂_{∞}Σ = T^{2}, the normalization of V does not play any role, as we make claims only about the sign of m in this case.) Equation (7.17) together with the maximum principle shows that

W̃ − W̃_{0} ≤ 0 on Σ ,  (7.37)
n^{′ i}D′_{i}(W̃ − W̃_{0})|_{∂_{∞}Σ} ≥ 0 ,  (7.38)

where n′ is the outer pointing g′-unit normal to ∂_{∞}Σ. Further, equality is attained in (7.37) or in (7.38) if and only if W ≡ W_{0} [40, Theorems 3.5 and 3.6]. Thus Lemma 7.1 together with Equation (7.38) shows that

m ≤ 0 .

Assume now that m = 0 in the case ∂_{∞}Σ = S^{2}; as an indirect argument, we also assume that m = 0 in the T^{2} case, or that m ≥ m_{crit} in the remaining cases. In the sphere or torus case from the strong maximum principle we obtain

W ≡ W_{0} .  (7.39)

In the higher genus cases we consider (7.17) again but take here as RS a generalized Kottler solution with the same mass as the given one, m_{0} = m. Equations (7.37)–(7.38) hold again; then Lemma 7.1 shows that equality must hold in (7.38). Applying the maximum principle again yields Equation (7.39). We note that both point a. as well as the structural hypotheses of Lemma 7.3 hold under the hypotheses of Theorem 1.3. Equation (7.39) and the discussion of Section 2 show that point 2. of that Lemma applies, so that the given solution must be a member of the generalized Kottler family with m in the range (2.7) (the generalized Nariai metrics are excluded as they do not satisfy the asymptotic hypotheses of Theorem 1.3). In the case ∂_{∞}Σ = S^{2} point 1. readily follows. In the remaining cases none of these solutions has the topology required in Theorem 1.3, which gives a contradiction and establishes Theorem 1.3. □

Proof of Theorem 1.5: By choice of the RS we have (W̃ − W̃_{0})|_{∂Σ} = 0. We normalize V again so that lim_{→∞}(W̃ − W̃_{0}) = 0 holds, cf. Proposition 3.3 and equation (3.17). Negativity of m_{0} implies that a in (7.17) is nonnegative. The maximum principle applied to Equation (7.17) gives W̃ − W̃_{0} ≤ 0 on Σ, with equality being achieved somewhere if and only if W ≡ W_{0}. Moreover, as in the proof of part 2, the boundary version of the strong maximum principle [40, Theorem 3.6] implies that n^{i ′}D′_{i}(W̃ −W̃_{0}) > 0 on ∂_{∞}Σ unless W = W_{0}. Lemma 7.1 allows us to conclude that either m < m_{0} or W ≡ W_{0}. In that last case point 2. of Lemma 7.3 implies that (Σ, g, V) corresponds to a generalized Kottler solution. In any case there holds m ≤ m_{0}.

To prove the area inequality in (1.8) requires some care as the metric g̃ defined in Equation (7.4) is singular at Σ, so that standard maximum principle arguments such as [40, Theorem 3.6] do not apply. We proceed as follows. By choice of W_{0} we have W̃ = W̃_{0} on ∂_{1}Σ. Further, Equation (7.2) shows that n^{i}D_{i}(W̃ − W̃_{0}) vanishes there. De l’Hospital’s rule, the non-vanishing of dV at ∂Σ, and the requirement W̃ − W̃_{0} ≤ 0 lead to

n^{j}n^{i}D_{i}D_{j}(W̃ − W̃_{0})|_{∂Σ} = lim_{V →0} \frac{D^{i}V D_{i}(W̃ − W̃_{0})}{V} ≤ 0 .

It follows that the left-hand-side of Equation (7.15) is non-positive, which establishes the second part of (1.8). □

Proof of Corollary 1.6: Assume that ∂Σ is connected and that (6.2) holds; we want to show that (1.8) implies an inequality inverse to (6.2). In order to do this, note first that by (1.8) the mass m is non-positive, and Equation (6.2) implies that g_{∂Σ} > 1. It is useful to introduce a genus-rescaled area radius r_{∂Σ} by the formula

r_{∂Σ} = \sqrt{\frac{A_{∂Σ}}{4π(g_{∂Σ} − 1)}} .

In terms of this object, the inequality (6.2) reads

2m|g_{∞} − 1|^{3/2} + (r_{∂Σ} + \frac{Λ}{3}r_{∂Σ}^{3}) |g_{∂Σ} − 1|^{3/2} ≥ 0 ,  (7.40)

It follows that r_{∂Σ} + \frac{Λ}{3}r_{∂Σ}^{3} ≥ 0, and the Galloway–Schleich–Witt–Woolgar inequality g_{∂Σ} ≤ g_{∞} implies

2m + r_{∂Σ} + \frac{Λ}{3}r_{∂Σ}^{3} ≥ 0 ,  (7.41)

Let us denote by r_{0} the r_{∂Σ} corresponding to the relevant generalized Kottler solution:

r_{0} = \sqrt{\frac{A_{0}}{4π(g_{∂_{∞}Σ} − 1)}} .

The inequality (7.41) is actually an equality for the generalized Kottler solutions, therefore it holds that

2m_{0} + r_{0} + \frac{Λ}{3}r_{0}^{3} = 0 .

We have r_{0} ≥ 1/\sqrt{−Λ} from (2.9), and m ≤ m_{0} , r_{∂Σ} ≥ r_{0} from (1.8), so that

2m + r_{∂Σ} + \frac{Λ}{3}r_{∂Σ}^{3} = 2m + r_{∂Σ} + \frac{Λ}{3}r_{∂Σ}^{3} − 2m_{0} − r_{0} − \frac{Λ}{3}r_{0}^{3} =

= 2(m − m_{0}) + (r_{∂Σ} − r_{0})[1 + \frac{Λ}{3}(r_{∂Σ}^{2} + r_{∂Σ}r_{0} + r_{0}^{2})] ≤

≤ (r_{∂Σ} − r_{0})(1 + Λr_{0}^{2}) ≤ 0 .  (7.42)

It follows from Equations (7.41)–(7.42) that r_{∂Σ} = r_{0}, m = m_{0}, and the rigidity part of Theorem 1.5 establishes Corollary 1.6. □

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