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 [54Jump To The Next Citation Point, 53Jump To The Next Citation Point]. His idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let Σ be a compact, connected three-manifold with connected boundary 𝒮, and let hab be a (negative definite) metric and χab a symmetric tensor field on Σ, such that they, as an initial data set, satisfy the dominant energy condition: if 16πG μ := R + χ2 − χabχab and 8πGja := Db (χab − χhab), then μ ≥ (− j ja)1∕2 a. For the sake of simplicity we denote the triple (Σ, h ,χ ) ab ab by Σ. Then let us consider all the possible asymptotically flat initial data sets ˆ ˆ (Σ,hab,χˆab) with a single asymptotic end, denoted simply by ˆΣ, which satisfy the dominant energy condition, have finite ADM energy and are extensions of Σ above through its boundary 𝒮. The set of these extensions will be denoted by ℰ(Σ ). By the positive energy theorem, Σˆ has non-negative ADM energy ˆ EADM (Σ ), which is zero precisely when ˆ Σ is a data set for the flat spacetime. Then we can consider the infimum of the ADM energies, { } inf EADM (Σˆ) | ˆΣ ∈ ℰ (Σ ), where the infimum is taken on ℰ(Σ). 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 Σ by this infimum.8 However, it is easy to see that, without further conditions on the extensions of (Σ, hab,χab), this infimum is zero. In fact, Σ can be extended to an asymptotically flat initial data set ˆΣ with arbitrarily small ADM energy such that ˆ Σ contains a horizon (for example in the form of an apparent horizon) between the asymptotically flat end and Σ. In particular, in the ‘ΩM,m-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 M ∕G) has an asymptotically flat extension, the complete spacelike hypersurface of the data set for the ΩM,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 Σ from the outside. This led Bartnik [54Jump To The Next Citation Point, 53Jump To The Next Citation Point] to formulate his suggestion for the quasi-local mass of Σ. He concentrated on time-symmetric data sets (i.e., those for which the extrinsic curvature χab is vanishing), when the horizon appears to be a minimal surface of topology 2 S in Σˆ (see, e.g., [213Jump To The Next Citation Point]), and the dominant energy condition is just the requirement of the non-negativity of the scalar curvature of the spatial metric: R ≥ 0. Thus, if ℰ0(Σ ) denotes the set of asymptotically flat Riemannian geometries ˆΣ = (ˆΣ,ˆhab) with non-negative scalar curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is

{ } mB (Σ ) := inf EADM (Σˆ) | ˆΣ ∈ ℰ0 (Σ ) . (5.1 )
The ‘no-horizon’ condition on ˆΣ implies that topologically Σ is a three-ball. Furthermore, the definition of ℰ0(Σ) in its present form does not allow one to associate the Bartnik mass to those three-geometries (Σ, h ) ab that contain minimal surfaces inside Σ. Although formally the maximal two-surfaces inside Σ are not excluded, any asymptotically flat extension of such a Σ would contain a minimal surface. In particular, the spherically-symmetric three-geometry, with line element dl2 = − dr2 − sin2 r(d 𝜃2 + sin2𝜃d ϕ2) with (𝜃,ϕ) ∈ S2 and r ∈ [0,r0], π ∕2 < r0 < π, has a maximal two-surface at r = π∕2, and any of its asymptotically flat extensions necessarily contains a minimal surface of area not greater than 2 4 πsin r0. Thus, the Bartnik mass (according to the original definition given in [54Jump To The Next Citation Point, 53]) cannot be associated with every compact time-symmetric data set (Σ,hab), even if Σ is topologically trivial. Since for 0 < r0 < π ∕2 this data set can be extended without any difficulty, this example shows that mB is associated with the three-dimensional data set Σ, and not only to the two-dimensional boundary ∂ Σ.

Of course, to rule out this limitation, one can modify the original definition by considering the set ℰ&tidle;(𝒮 ) 0 of asymptotically flat Riemannian geometries ˆΣ = (Σˆ, ˆh ) ab (with non-negative scalar curvature, finite ADM energy and with no stable minimal surface), which contain (𝒮, qab) as an isometrically-embedded Riemannian submanifold, and define m&tidle;B (𝒮 ) by Eq. (5.1View Equation) with &tidle;ℰ0(𝒮) instead of ℰ0(Σ). Obviously, this m&tidle;B (𝒮 ) could be associated with a larger class of two-surfaces than the original mB (Σ) can be to compact three-manifolds, and 0 ≤ m&tidle;B (∂ Σ) ≤ mB (Σ ) holds.

In [279Jump To The Next Citation Point, 56Jump To The Next Citation Point] the set ℰ0 (Σ ) was allowed to include extensions ˆ Σ of Σ having boundaries as compact outermost horizons, when the corresponding ADM energies are still non-negative [217Jump To The Next Citation Point], and hence mB (Σ) is still well defined and non-negative. (For another description of ℰ0 (Σ ) allowing horizons in the extensions but excluding them between Σ and the asymptotic end, see [110Jump To The Next Citation Point] and Section 5.2 of this paper.)

Bartnik suggests a definition for the quasi-local mass of a spacelike two-surface 𝒮 (together with its induced metric and the two extrinsic curvatures), as well [54Jump To The Next Citation Point]. He considers those globally-hyperbolic spacetimes Mˆ := (Mˆ ,ˆgab) that satisfy the dominant energy condition, admit an asymptotically flat (metrically-complete) Cauchy surface Σˆ with finite ADM energy, have no event horizon and in which 𝒮 can be embedded with its first and second fundamental forms. Let ℰ0(𝒮 ) denote the set of these spacetimes. Since the ADM energy EADM (M ˆ ) is non-negative for any ˆM ∈ ℰ0(𝒮) (and is zero precisely for flat ˆM), the infimum

{ } mB (𝒮) := inf EADM (Mˆ) |Mˆ ∈ ℰ0 (𝒮) (5.2 )
exists and is non-negative. Although it seems plausible that mB (∂ Σ) is only the ‘spacetime version’ of m (Σ ) B, without the precise form of the no-horizon conditions in ℰ (Σ ) 0 and that in ℰ (𝒮) 0 they cannot be compared, even if the extrinsic curvature were allowed in the extensions ˆ Σ of Σ.

5.1.2 The main properties of mB (Σ )

The first immediate consequence of Eq. (5.1View Equation) is the monotonicity of the Bartnik mass. If Σ1 ⊂ Σ2, then ℰ0(Σ2 ) ⊂ ℰ0(Σ1), and hence, mB (Σ1 ) ≤ mB (Σ2). Obviously, by definition (5.1View Equation) one has mB (Σ ) ≤ mADM (ˆΣ) for any ˆΣ ∈ ℰ0(Σ ). Thus, if m is any quasi-local mass functional that is larger than m B (i.e., that assigns a non-negative real to any Σ such that m (Σ ) ≥ m (Σ ) B for any allowed Σ), furthermore if ˆ m (Σ) ≤ mADM (Σ ) for any ˆ Σ ∈ ℰ0 (Σ ), then by the definition of the infimum in Eq. (5.1View Equation) one has mB (Σ ) ≥ m (Σ) − 𝜀 ≥ mB (Σ ) − 𝜀 for any 𝜀 > 0. Therefore, mB is the largest mass functional satisfying mB (Σ) ≤ mADM (ˆΣ ) for any Σˆ ∈ ℰ0(Σ). Another interesting consequence of the definition of mB, due to Simon (see [56Jump To The Next Citation Point]), is that if ˆΣ is any asymptotically flat, time-symmetric extension of Σ with non-negative scalar curvature satisfying ˆ mADM (Σ ) < mB (Σ ), then there is a black hole in ˆ Σ in the form of a minimal surface between Σ and the infinity of ˆΣ. For further discussion of mB (Σ ) 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 [460Jump To The Next Citation Point].

As we saw, the Bartnik mass is non-negative, and, obviously, if Σ is flat (and hence is a data set for flat spacetime), then mB (Σ ) = 0. The converse of this statement is also true [279Jump To The Next Citation Point]: If mB (Σ ) = 0, then Σ is locally flat. The Bartnik mass tends to the ADM mass [279Jump To The Next Citation Point]: If (ˆΣ,ˆhab) is an asymptotically flat Riemannian three-geometry with non-negative scalar curvature and finite ADM mass m (ˆΣ ) ADM, and if {Σ } n, n ∈ ℕ, is a sequence of solid balls of coordinate radius n in ˆΣ, then ˆ limn → ∞ mB (Σn ) = mADM (Σ). 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 mB (Σ ) can be given near the nonflat points of Σ. 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 [279Jump To The Next Citation Point]. Let Σ be a spherically-symmetric Riemannian three-geometry with spherically-symmetric boundary 𝒮 := ∂Σ. One can form its ‘standard’ round-sphere energy E (𝒮 ) (see Section 4.2.1), and take its spherically-symmetric asymptotically flat vacuum extension ˆ ΣSS (see [54Jump To The Next Citation Point, 56Jump To The Next Citation Point]). By the Birkhoff theorem the exterior part of ˆ ΣSS 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 Σˆ of Σ can also be considered as (a part of) an asymptotically flat time-symmetric hypersurface with minimal surface, whose area is 16 πG2 E (Σˆ ) ADM SS. Thus, by the Riemannian Penrose inequality [279Jump To The Next Citation Point] E (Σˆ) ADM ˆ ≥ EADM (ΣSS ) = E(𝒮 ). Therefore, the Bartnik mass of Σ is just the ‘standard’ round-sphere expression E (𝒮).

5.1.3 The computability of the Bartnik mass

Since for any given Σ the set ℰ0(Σ ) 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 (Σ, hab). Without some computational method the potentially useful properties of mB (Σ ) would be lost from the working relativist’s arsenal.

Such a computational method might be based on a conjecture of Bartnik [54Jump To The Next Citation Point, 56Jump To The Next Citation Point]: The infimum in definition (5.1View Equation) of the mass m (Σ ) B is realized by an extension (ˆΣ,ˆh ) ab of (Σ,h ) ab such that the exterior region, ˆ ˆ (Σ − Σ, hab|ˆΣ− Σ), is static, the metric is Lipschitz-continuous across the two-surface ∂ Σ ⊂ ˆΣ, and the mean curvatures of ∂Σ of the two sides are equal. Therefore, to compute mB for a given (Σ, hab), one should find an asymptotically flat, static vacuum metric ˆhab satisfying the matching conditions on ∂Σ, and where the Bartnik mass is the ADM mass of ˆh ab. As Corvino shows [154], if there is an allowed extension ˆ Σ of Σ for which ˆ mADM (Σ ) = mB (Σ), then the extension -- ˆΣ − Σ is static; furthermore, if Σ1 ⊂ Σ2, mB (Σ1 ) = mB (Σ2) and Σ2 has an allowed extension ˆΣ for which mB (Σ2) = mADM (ˆΣ ), then --- Σ2 − Σ1 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 (Σ, hab) 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 (Σ, hab) determines this exterior metric uniquely [56Jump To The Next Citation Point]. He conjectures [54, 56Jump To The Next Citation Point] that a similar computation method can be found for the mass m (𝒮 ) B, defined in Eq. (5.2View Equation), as well, where the exterior metric should be stationary. This second conjecture is also supported by partial results [155]: If (Σ, hab,χab) 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 (Σ ) B one can construct admissible extensions of (Σ,h ) ab in the form of the metrics in quasi-spherical form [55]. If the boundary ∂Σ is a metric sphere of radius r with non-negative mean curvature k, then mB (Σ ) can be estimated from above in terms of r and k.

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