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5.1 The Bartnik mass

5.1.1 The main idea

One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [39Jump To The Next Citation Point38Jump To The Next Citation Point]. His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let S be a compact, connected 3-manifold with connected boundary S, and let hab be a (negative definite) metric and xab a symmetric tensor field on S such that they, as an initial data set, satisfy the dominant energy condition: If 16pGm := R + x2 - xabxab and 8pGja := Db(xab - xhab), then m > (-j ja)1/2 a. For the sake of simplicity we denote the triple (S, h ,x ) ab ab by S. Then let us consider all the possible asymptotically flat initial data sets ^ ^ (S,hab,x^ab) with a single asymptotic end, denoted simply by ^S, which satisfy the dominant energy condition, have finite ADM energy and are extensions of S above through its boundary S. The set of these extensions will be denoted by E(S). By the positive energy theorem ^S has non-negative ADM energy E (^S) ADM, which is zero precisely when ^S is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies, { } inf EADM( S^) | ^S (- E (S), where the infimum is taken on E(S). 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 S by this infimum9. However, it is easy to see that, without further conditions on the extensions of (S, hab,xab), this infimum is zero. In fact, S can be extended to an asymptotically flat initial data set ^S with arbitrarily small ADM energy such that ^ S contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and S. In particular, in the ‘_O_M,m-spacetime’, discussed in Section 4.2.1 on the round spheres, the spherically symmetric domain bounded by the maximal surface (with arbitrarily large round-sphere mass M/G) has an asymptotically flat extension, the _O_M,m-spacetime itself, with arbitrarily small ADM mass m/G. Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of the presence of a horizon hiding S from the outside. This led Bartnik [39Jump To The Next Citation Point38Jump To The Next Citation Point] to formulate his suggestion for the quasi-local mass of S. He concentrated on the time-symmetric data sets (i.e. those for which the extrinsic curvature xab is vanishing), when the horizon appears to be a minimal surface of topology 2 S in ^S (see for example [156Jump To The Next Citation Point]), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature: R > 0. Thus, if E0(S) denotes the set of asymptotically flat Riemannian geometries ^S = (^S,^hab) with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

{ } mB (S) := inf EADM( S^) | ^S (- E0 (S) . (36)
The ‘no-horizon’ condition on ^S implies that topologically S is a 3-ball. Furthermore, the definition of E0(S) in its present form does not allow one to associate the Bartnik mass to those 3-geometries (S, h ) ab that contain minimal surfaces inside S. Although formally the maximal 2-surfaces inside S are not excluded, any asymptotically flat extension of such a S would contain a minimal surface. In particular, the spherically symmetric 3-geometry with line element 2 2 2 2 2 2 dl = - dr - sin r(dh + sin hdf ) with (h,f) (- S2 and r (- [0, r0], p/2 < r0 < p, has a maximal 2-surface at r = p/2, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than 4p sin2 r0. Thus the Bartnik mass (according to the original definition given in [39Jump To The Next Citation Point38]) cannot be associated with every compact time-symmetric data set (S,hab) even if S is topologically trivial. Since for 0 < r0 < p/2 this data set can be extended without any difficulty, this example shows that mB is associated with the 3-dimensional data set S and not only to the 2-dimensional boundary @S.

Of course, to rule out this limitation, one can modify the original definition by considering the set E~0(S) of asymptotically flat Riemannian geometries ^S = (S^, ^hab) (with non-negative scalar curvature, finite ADM energy and with no stable minimal surface) which contain (S, q ) ab as an isometrically embedded Riemannian submanifold, and define ~mB(S) by Equation (36View Equation) with E~0(S) instead of E0(S). Obviously, this ~mB(S) could be associated with a larger class of 2-surfaces than the original mB(S) to compact 3-manifolds, and 0 < ~mB(@S) < mB(S) holds.

In [208Jump To The Next Citation Point41Jump To The Next Citation Point] the set E0(S) was allowed to include extensions ^ S of S having boundaries as compact outermost horizons, whenever the corresponding ADM energies are still non-negative [159Jump To The Next Citation Point], and hence mB(S) is still well-defined and non-negative. (For another definition for E0(S) allowing horizons in the extensions but excluding them between S and the asymptotic end, see [87Jump To The Next Citation Point] and Section 5.2 below.)

Bartnik suggested a definition for the quasi-local mass of a spacelike 2-surface S (together with its induced metric and the two extrinsic curvatures), too [39Jump To The Next Citation Point]. He considered those globally hyperbolic spacetimes M^ := (M^ ,^gab) that satisfy the dominant energy condition, admit an asymptotically flat (metrically complete) Cauchy surface S^ with finite ADM energy, have no event horizon and in which S can be embedded with its first and second fundamental forms. Let E (S) 0 denote the set of these spacetimes. Since the ADM energy ^ EADM( M ) is non-negative for any ^ M (- E0(S) (and is zero precisely for flat ^M), the infimum

{ } mB (S) := inf EADM( M^) |M^ (- E0 (S) (37)
exists and is non-negative. Although it seems plausible that mB(@S) is only the ‘spacetime version’ of mB(S), without the precise form of the no-horizon conditions in E0(S) and that in E0(S) they cannot be compared even if the extrinsic curvature were allowed in the extensions ^S of S.

5.1.2 The main properties of mB(S)

The first immediate consequence of Equation (36View Equation) is the monotonicity of the Bartnik mass: If S1 < S2, then E0(S2) < E0(S1), and hence mB(S1) < mB(S2). Obviously, by definition (36View Equation) one has ^ mB(S) < mADM( S) for any ^ S (- E0(S). Thus if m is any quasi-local mass functional which is larger than mB (i.e. which assigns a non-negative real to any S such that m(S) > mB(S) for any allowed S), furthermore if m(S) < m (^S) ADM for any ^S (- E (S) 0, then by the definition of the infimum in Equation (36View Equation) one has mB(S) > m(S) - e > mB(S) - e for any e > 0. Therefore, mB is the largest mass functional satisfying mB(S) < mADM( ^S) for any ^S (- E0(S). Another interesting consequence of the definition of mB, due to W. Simon, is that if ^S is any asymptotically flat, time symmetric extension of S with non-negative scalar curvature satisfying mADM( ^S) < mB(S), then there is a black hole in ^ S in the form of a minimal surface between S and the infinity of ^ S (see for example [41Jump To The Next Citation Point]).

As we saw, the Bartnik mass is non-negative, and, obviously, if S is flat (and hence is a data set for the flat spacetime), then mB(S) = 0. The converse of this statement is also true [208Jump To The Next Citation Point]: If mB(S) = 0, then S is locally flat. The Bartnik mass tends to the ADM mass [208Jump To The Next Citation Point]: If (^S,^h ) ab is an asymptotically flat Riemannian 3-geometry with non-negative scalar curvature and finite ADM mass mADM( ^S), and if {Sn}, n (- N, is a sequence of solid balls of coordinate radius n in ^S, then limn -->o o mB(Sn) = mADM( ^S). The proof of these two results is based on the use of the Hawking energy (see Section 6.1), by means of which a positive lower bound for mB(S) can be given near the non-flat points of S. In the proof of the second statement one must use the fact that the 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 [208Jump To The Next Citation Point]: Let S be a spherically symmetric Riemannian 3-geometry with spherically symmetric boundary S := @S. One can form its ‘standard’ round-sphere energy E(S) (see Section 4.2.1), and take its spherically symmetric asymptotically flat vacuum extension ^SSS (see [39Jump To The Next Citation Point41Jump To The Next Citation Point]). By the Birkhoff theorem the exterior part of ^SSS is a part of a t = const. hypersurface of the vacuum Schwarzschild solution, and its ADM mass is just E(S). Then any asymptotically flat extension ^S of S can also be considered as (a part of) an asymptotically flat time-symmetric hypersurface with minimal surface, whose area is 2 16pG EADM( S^SS). Thus by the Riemannian Penrose inequality [208Jump To The Next Citation Point] EADM( ^S) > EADM( ^SSS) = E(S). Therefore, the Bartnik mass of S is just the ‘standard’ round sphere expression E(S).

5.1.3 The computability of the Bartnik mass

Since for any given S the set E0(S) of its extensions is a huge set, it is almost hopeless to parameterize it. Thus, by the very definition, it seems very difficult to compute the Bartnik mass for a given, specific (S, hab). Without some computational method the potentially useful properties of mB(S) would be lost from the working relativist’s arsenal.

Such a computational method might be based on a conjecture of Bartnik [39Jump To The Next Citation Point41Jump To The Next Citation Point]: The infimum in definition (36View Equation) of the mass mB(S) is realized by an extension (^S, ^hab) of (S, hab) such that the exterior region, ^ ^ (S - S, hab|S^- S), is static, the metric is Lipschitz-continuous across the 2-surface @S < ^S, and the mean curvatures of @S of the two sides are equal. Therefore, to compute mB for a given (S, h ) ab, one should find an asymptotically flat, static vacuum metric ^h ab satisfying the matching conditions on @S, and the Bartnik mass is the ADM mass of ^ hab. As Corvino showed [119], if there is an allowed extension ^S of S for which mADM( ^S) = mB(S), then the extension -- ^S - S is static; furthermore, if S1 < S2, mB(S1) = mB(S2) and S2 has an allowed extension ^S for which mB(S2) = mADM( ^S), then --- S2 - S1 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 [267] for geometries (S, hab) close enough to the Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still lacking. Bartnik’s conjecture is that (S, hab) determines this exterior metric uniquely [41Jump To The Next Citation Point]. He conjectures [3941Jump To The Next Citation Point] that a similar computation method can be found for the mass m (S) B, defined in Equation (37View Equation), too, where the exterior metric should be stationary. This second conjecture is also supported by partial results [120]: If (S, hab,xab) 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 m (S) B one can construct admissible extensions of (S,h ) ab in the form of the metrics in quasi-spherical form [40]. If the boundary @S is a metric sphere of radius r with non-negative mean curvature k, then mB(S) can be estimated from above in terms of r and k.

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