One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [54, 53]. His
idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let be
a compact, connected three-manifold with connected boundary , and let be a (negative definite)
metric and a symmetric tensor field on , such that they, as an initial data set, satisfy the
dominant energy condition: if and , then
. For the sake of simplicity we denote the triple by . Then let
us consider all the possible asymptotically flat initial data sets with a single
asymptotic end, denoted simply by , which satisfy the dominant energy condition, have
finite ADM energy and are extensions of above through its boundary . The set of these
extensions will be denoted by . By the positive energy theorem, has non-negative ADM
energy , which is zero precisely when is a data set for the flat spacetime. Then
we can consider the infimum of the ADM energies, , where the
infimum is taken on . Obviously, by the non-negativity of the ADM energies, this infimum
exists and is non-negative, and it is tempting to define the quasi-local mass of by this
infimum.^{8}
However, it is easy to see that, without further conditions on the extensions of , this infimum
is zero. In fact, can be extended to an asymptotically flat initial data set with arbitrarily small
ADM energy such that contains a horizon (for example in the form of an apparent horizon) between
the asymptotically flat end and . In particular, in the ‘-spacetime’ discussed in Section 4.2.1
on round spheres, the spherically symmetric domain bounded by the maximal surface (with
arbitrarily-large round-sphere mass ) has an asymptotically flat extension, the complete spacelike
hypersurface of the data set for the -spacetime itself, with arbitrarily small ADM mass
.

Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding from the outside. This led Bartnik [54, 53] to formulate his suggestion for the quasi-local mass of . He concentrated on time-symmetric data sets (i.e., those for which the extrinsic curvature is vanishing), when the horizon appears to be a minimal surface of topology in (see, e.g., [213]), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature of the spatial metric: . Thus, if denotes the set of asymptotically flat Riemannian geometries with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

The ‘no-horizon’ condition on implies that topologically is a three-ball. Furthermore, the definition of in its present form does not allow one to associate the Bartnik mass to those three-geometries that contain minimal surfaces inside . Although formally the maximal two-surfaces inside are not excluded, any asymptotically flat extension of such a would contain a minimal surface. In particular, the spherically-symmetric three-geometry, with line element with and , , has a maximal two-surface at , and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than . Thus, the Bartnik mass (according to the original definition given in [54, 53]) cannot be associated with every compact time-symmetric data set , even if is topologically trivial. Since for this data set can be extended without any difficulty, this example shows that is associated with the three-dimensional data set , and not only to the two-dimensional boundary .Of course, to rule out this limitation, one can modify the original definition by considering the set of asymptotically flat Riemannian geometries (with non-negative scalar curvature, finite ADM energy and with no stable minimal surface), which contain as an isometrically-embedded Riemannian submanifold, and define by Eq. (5.1) with instead of . Obviously, this could be associated with a larger class of two-surfaces than the original can be to compact three-manifolds, and holds.

In [279, 56] the set was allowed to include extensions of having boundaries as compact outermost horizons, when the corresponding ADM energies are still non-negative [217], and hence is still well defined and non-negative. (For another description of allowing horizons in the extensions but excluding them between and the asymptotic end, see [110] and Section 5.2 of this paper.)

Bartnik suggests a definition for the quasi-local mass of a spacelike two-surface (together with its induced metric and the two extrinsic curvatures), as well [54]. He considers those globally-hyperbolic spacetimes that satisfy the dominant energy condition, admit an asymptotically flat (metrically-complete) Cauchy surface with finite ADM energy, have no event horizon and in which can be embedded with its first and second fundamental forms. Let denote the set of these spacetimes. Since the ADM energy is non-negative for any (and is zero precisely for flat ), the infimum

exists and is non-negative. Although it seems plausible that is only the ‘spacetime version’ of , without the precise form of the no-horizon conditions in and that in they cannot be compared, even if the extrinsic curvature were allowed in the extensions of .

The first immediate consequence of Eq. (5.1) is the monotonicity of the Bartnik mass. If , then , and hence, . Obviously, by definition (5.1) one has for any . Thus, if is any quasi-local mass functional that is larger than (i.e., that assigns a non-negative real to any such that for any allowed ), furthermore if for any , then by the definition of the infimum in Eq. (5.1) one has for any . Therefore, is the largest mass functional satisfying for any . Another interesting consequence of the definition of , due to Simon (see [56]), is that if is any asymptotically flat, time-symmetric extension of with non-negative scalar curvature satisfying , then there is a black hole in in the form of a minimal surface between and the infinity of . For further discussion of from the point of view of black holes, as well as the relationship between the Bartnik mass and other expressions (e.g., the Hawking energy), see [460].

As we saw, the Bartnik mass is non-negative, and, obviously, if is flat (and hence is a data set for flat spacetime), then . The converse of this statement is also true [279]: If , then is locally flat. The Bartnik mass tends to the ADM mass [279]: If is an asymptotically flat Riemannian three-geometry with non-negative scalar curvature and finite ADM mass , and if , , is a sequence of solid balls of coordinate radius in , then . The proof of these two results is based on the use of Hawking energy (see Section 6.1), by means of which a positive lower bound for can be given near the nonflat points of . In the proof of the second statement one must use the fact that Hawking energy tends to the ADM energy, which, in the time-symmetric case, is just the ADM mass.

The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice application of the Riemannian Penrose inequality [279]. Let be a spherically-symmetric Riemannian three-geometry with spherically-symmetric boundary . One can form its ‘standard’ round-sphere energy (see Section 4.2.1), and take its spherically-symmetric asymptotically flat vacuum extension (see [54, 56]). By the Birkhoff theorem the exterior part of is a part of a hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just . Then, any asymptotically flat extension of can also be considered as (a part of) an asymptotically flat time-symmetric hypersurface with minimal surface, whose area is . Thus, by the Riemannian Penrose inequality [279] . Therefore, the Bartnik mass of is just the ‘standard’ round-sphere expression .

Since for any given the set of its extensions is a huge set, it is almost hopeless to parametrize it. Thus, by its very definition, it seems very difficult to compute the Bartnik mass for a given, specific . Without some computational method the potentially useful properties of would be lost from the working relativist’s arsenal.

Such a computational method might be based on a conjecture of Bartnik [54, 56]: The infimum in definition (5.1) of the mass is realized by an extension of such that the exterior region, , is static, the metric is Lipschitz-continuous across the two-surface , and the mean curvatures of of the two sides are equal. Therefore, to compute for a given , one should find an asymptotically flat, static vacuum metric satisfying the matching conditions on , and where the Bartnik mass is the ADM mass of . As Corvino shows [154], if there is an allowed extension of for which , then the extension is static; furthermore, if , and has an allowed extension for which , then is static. Thus, the proof of Bartnik’s conjecture is equivalent to the proof of the existence of such an allowed extension. The existence of such an extension is proven in [360] for geometries close enough to the Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still lacking. (For further partial existence results see [17].) Bartnik’s conjecture is that determines this exterior metric uniquely [56]. He conjectures [54, 56] that a similar computation method can be found for the mass , defined in Eq. (5.2), as well, where the exterior metric should be stationary. This second conjecture is also supported by partial results [155]: If is any compact vacuum data set, then it has an asymptotically flat vacuum extension, which is a spacelike slice of a Kerr spacetime outside a large sphere near spatial infinity.

To estimate one can construct admissible extensions of in the form of the metrics in quasi-spherical form [55]. If the boundary is a metric sphere of radius with non-negative mean curvature , then can be estimated from above in terms of and .

Living Rev. Relativity 12, (2009), 4
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