5.2 Bray’s modifications

UpdateJump To The Next Update Information Another, slightly modified definition for the quasi-local mass is suggested by Bray [110Jump To The Next Citation Point, 113Jump To The Next Citation Point]. Here we summarize his ideas.

Let Σ = (Σ, hab,χab) be any asymptotically flat initial data set with finitely-many asymptotic ends and finite ADM masses, and suppose that the dominant energy condition is satisfied on Σ. Let 𝒮 be any fixed two-surface in Σ, which encloses all the asymptotic ends except one, say the i-th (i.e., let 𝒮 be homologous to a large sphere in the i-th asymptotic end). The outside region with respect to 𝒮, denoted by O (𝒮 ), will be the subset of Σ containing the i-th asymptotic end and bounded by 𝒮, while the inside region, I(𝒮), is the (closure of) Σ − O (𝒮). Next, Bray defines the ‘extension’ ˆ Σe of 𝒮 by replacing O (𝒮 ) by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Similarly, the ‘fill-in’ ˆΣf of 𝒮 is obtained from Σ by replacing I(𝒮) by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Finally, the surface 𝒮 will be called outer-minimizing if, for any closed two-surface &tidle; 𝒮 enclosing 𝒮, one has &tidle; Area (𝒮) ≤ Area (𝒮).

Let 𝒮 be outer-minimizing, and let ℰ(𝒮 ) denote the set of extensions of 𝒮 in which 𝒮 is still outer-minimizing, and ℱ (𝒮 ) denote the set of fill-ins of 𝒮. If ˆΣf ∈ ℱ (𝒮) and A ˆΣ f denotes the infimum of the area of the two-surfaces enclosing all the ends of ˆ Σf except the outer one, then Bray defines the outer and inner mass, mout (𝒮) and min (𝒮), respectively, by

{ } mout (𝒮 ) := inf mADM (ˆΣe) | ˆΣe ∈ ℰ (𝒮 ) , { ∘ ------ } AˆΣf min (𝒮 ) := sup 16πG--| ˆΣf ∈ ℱ (𝒮 ) .
mout (𝒮) deviates slightly from Bartnik’s mass (5.1View Equation) even if the latter would be defined for non–time-symmetric data sets, because Bartnik’s ‘no-horizon condition’ excludes apparent horizons from the extensions, while Bray’s condition is that 𝒮 be outer-minimizing.

A simple consequence of the definitions is the monotonicity of these masses: If 𝒮 2 and 𝒮1 are outer-minimizing two-surfaces such that 𝒮2 encloses 𝒮1, then min (𝒮2 ) ≥ min(𝒮1) and mout(𝒮2) ≥ mout (𝒮1 ). Furthermore, if the Penrose inequality holds (for example, in a time-symmetric data set, for which the inequality has been proven), then for outer-minimizing surfaces mout (𝒮) ≥ min(𝒮 ) [110Jump To The Next Citation Point, 113Jump To The Next Citation Point]. Furthermore, if Σi is a sequence such that the boundaries ∂Σi shrink to a minimal surface 𝒮, then the sequence mout(∂Σi) tends to the irreducible mass ∘ ---------------2-- Area(𝒮 )∕ (16 πG ) [56]. Bray defines the quasi-local mass of a surface not simply to be a number, but the whole closed interval [min(𝒮),mout (𝒮)]. If 𝒮 encloses the horizon in the Schwarzschild data set, then the inner and outer masses coincide, and Bray expects that the converse is also true: If min(𝒮) = mout (𝒮), then 𝒮 can be embedded into the Schwarzschild spacetime with the given two-surface data on 𝒮 [113Jump To The Next Citation Point].

For further modification of Bartnik’s original ideas, see [311].


  Go to previous page Go up Go to next page