Piotr T. Chrusciel and Gabriel Nagy
Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, F-37200 Tours, France
11.30-j, 04.20.-q, 02.40.Ma
Let be an -dimensional spacelike hypersurface in a
-dimensional Lorentzian space-time
. Suppose that
contains an open set which is covered by a finite number
of
coordinate charts , with
, and with
-- local coordinates on some compact dimensional manifold
, such that
. Assume that the metric
approaches a background metric of the form
We show in [13] that the vacuum Einstein equations together
with the decay conditions (3)-(4) and the asymptotic behavior
of the frame components of the -Killing vector fields,
To define the integrals (6) we have fixed a model background
metric , as well as an orthonormal frame as in (2); this
last equation requires the corresponding coordinate system
as in (1). Hence, the background structure required in our
analysis consists of a background metric and a background
coordinate system. While this is
reasonably satisfactory from a Hamiltonian point of view, in which
each choice of background structure defines an associated phase space,
there is an essential
potential geometric ambiguity in the integrals
(6) that arises as follows: let be any metric such
that its frame components tend to as tends to
infinity, in such a way that the integrals
given by
(6) (labelled by all the background Killing vector fields or
perhaps by a subset thereof) converge. Consider another coordinate
system
with the associated background
metric :
One does not expect all the requirements in (3)-(4) to be
necessary: for metrics which are asymptotically flat at it is
sufficient to impose conditions on the induced metric and the
extrinsic curvature of the hypersurface to obtain a well
defined mass, and one expects the same to be the case here. However,
the decay rates imposed are sharp in the following sense:
consider the metric , with the hatted coordinates
defined as
It should be stressed that we do not know a priori that the
hatted coordinates are related to the unhatted ones by the simple
coordinate transformation (11) with
decaying as , or behaving in some controlled way --
the behavior of
could in principle
be very wild. The main technical result of [13] is the proof
that this is not the case: under the hypothesis that is a
sphere with a round metric, or that has non-positive Ricci
tensor and constant Ricci scalar we show
that all coordinate transformations which leave invariant
(so that ) and that preserve the decay conditions
(3)-(4) are compositions of a map satisfying (12)
with an isometry of the background. In order to obtain a geometric
invariant of it remains thus to study the behavior of the
integrals (6) under isometries of preserving . If
is such an isometry, the fact that
is a
tensor density immediately yields the formula
1. Let be the dimensional sphere with a
round metric ,
normalized so that the
substitution in (1) leads to a metric which is the
dimensional anti-de Sitter metric. The space
of -Killing
vector fields normal to
is spanned by vector fields
which on take the form
,
, where
, and , being the
coordinate which appears in (1), while
. The group
of
isometries of which map into coincides with
the homogeneous Lorentz group ; it acts on
by
push-forward. It can be shown that for each such there exists
a Lorentz transformation
so
that for every
we have
Consider the remaining Killing vector fields
of :
2. Let be a compact dimensional manifold with a metric
of constant scalar curvature and with non-positive Ricci tensor, and
let take the form (1), with
,
and with or according to whether the Ricci scalar of
vanishes or not. We show in [13] that for such metrics the
space of -Killing vector fields normal to consists of vector
fields of the form
,
and that
The number defined in each case above is our proposal for the geometric definition of total mass of in . We note that its multiplicative normalization in dimensions is determined by the requirement of the correct Newtonian limit when and , while the additive one is determined by imposing that the background models have vanishing energy. In higher dimensions it appears appropriate to keep the same multiplicative factor for the Lagrangian , whenever a Kaluza-Klein reduction applies.
The results described above can be reformulated in a purely Riemannian context, this will be discussed elsewhere [12]. Some similar results in that context have been obtained independently by M. Herzlich [20].
It is natural to study the invariance of the mass when is allowed to move in . A complete answer would require establishing an equivalent of our analysis of admissible coordinate transformations in a space-time setting. The difficulties that arise in the corresponding problem for asymptotically Minkowskian metrics [11] suggest that this might be a considerably more delicate problem, which we plan to analyse in the future. It should be stressed that this problem mixes two different issues, one being the potential background dependence of (6), another one being the possibility of energy flowing in or out through the timelike conformal boundary of space-time.
It should be pointed out that there exist several alternative methods of defining mass in asymptotically anti-de Sitter space-time -- using coordinate systems [6,18], preferred foliations [17], generalized Komar integrals [26], conformal techniques [2,3,4], or ad-hoc methods [1]; an extended discussion can be found in [14, Section 5]. Each of those approaches suffers from some potential ambiguities, so that the question of the geometric character of the definition of mass given there arises as well. Let us briefly describe the relationship of the results presented here to some of those works. Consider, first, the Abbott-Deser approach [1], which seems to have been most often used. A precise comparison is difficult to carry out because in Ref. [1] the boundary conditions which should be used are not spelled out in detail. As already pointed out, the coordinate transformation (13) gives a non-zero Abbott-Deser mass to the anti-de Sitter metric, so the boundary conditions do matter. Assuming that the authors of [1] had in mind boundary conditions which are at least as restrictive as our conditions (3)-(4), a straightforward but tedious calculation shows that the Abbott-Deser mass coincides with the Hamiltonian mass advocated here. One needs then to face the same ambiguities as we do, and our results in [13] can be interpreted as proving the existence of a geometric invariant which can be calculated using Abbott-Deser type integrals. It should be stressed, however, that while the Hamiltonian approach is universal and leads to unique -- up to a constant -- functionals on each phase space, no results about either uniqueness or universality of the approach of [1] are known to us.
Consider, next, the Hamiltonian approach of [19]; here the question of equivalence of the different Hamiltonian formalisms used in [9] and in [19] arises. Those formalisms are identical in spirit -- both are Hamiltonian -- but differ in various details. We note that the boundary conditions (3)-(4) are weaker than the conditions imposed in [19]. It can be checked that under (3)-(4) the formalism of [19] is a special case of the geometric Hamiltonian formalism of [9], when applied to asymptotically anti-de Sitter space-times, so the Hamiltonian part of our analysis in [13] can be thought of as an extension of the results of [19] to larger phase spaces arising naturally in this context. An advantage of the geometric Hamiltonian formalism of Kijowski and Tulczyjew is that it allows the use of both a manifestly four-dimensional covariant formalism, and of a ADM one. The potential ambiguities in a geometric definition of mass that occur in [19] are identical to the ones described above.
As another example, we note the potential ambiguity in the mass defined by the conformal methods in [2,3], related to the possibility of existence of smooth conformal completions which are not smoothly conformally equivalent. In those works one assumes existence of smooth completions, while our conditions would -- roughly speaking -- correspond to conformal completions, , hence our conditions (3)-(4) are satisfied in the setup of [2,3] (compare [14] for a detailed discussion in the static case). It can be shown (A. Ashtekar, private communication) that the mass of [2,3] coincides with the Abbott-Deser one; what has been said above implies that -- under the asymptotic conditions of [2,3] -- it also coincides with the Hamiltonian mass described here. The results proved in [13] can be used to show that no inequivalent conformal completions of the kind considered in [2,3] exist, establishing the invariant character of the framework of [2,3].
Let us, finally, turn our attention to the AdS/CFT motivated definitions of mass (cf., e.g., [2,4,15,27] and references therein). It seems that there might be a belief in the string community that, at least in odd space-time dimensions, there is no systematic way to start from the Einstein-Hilbert action and arrive unambiguously at conserved quantities. Our results in [13] show that such a belief is incorrect, in the space of metrics asymptotic to the backgrounds considered here, as well as for those which asymptote to the more general backgrounds considered in [13]. It should be pointed out, however, that the Hamiltonian approach gives Hamiltonians which are only defined up to a constant, which leaves ample room for non-zero ``Casimir energies'', the occurrence of which might well be justified by other physical considerations. On the other hand, in a Hamiltonian approach the only natural choice of the additive constant seems to be zero. In any case, it is not clear whether or not the AdS/CFT based approaches lead to geometric invariants of hypersurfaces in space-times, in the sense presented here.
We note that a similar problem for the ADM mass of asymptotically flat initial data sets has been settled in [5,10] (see also [11]). Our treatment in [13] is a non-trivial extension to the current setup of the methods of [10].
Acknowledgements: PTC wishes to thank M. Herzlich, S. Ilias, A. Polombo and A.El Soufi for useful discussions. Suggestions from an anonymous referee for improvements of the manuscript are acknowledged.