Go to previous page Go up Go to next page

5.2 Bray’s modifications

Another, slightly modified definition for the quasi-local mass was suggested by Bray [87Jump To The Next Citation Point90Jump To The Next Citation Point]. Here we summarize his ideas.

Let S = (S, hab,xab) 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 S. Let S be any fixed 2-surface in S 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 S containing the i-th asymptotic end and bounded by S, while the inside region, I(S), is the (closure of) S - O(S). Next Bray defines the ‘extension’ ^Se 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’ ^Sf of S is obtained from S 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 2-surface ~ S enclosing S one has 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 S^ (- F(S) f and A^ Sf denotes the infimum of the area of the 2-surfaces enclosing all the ends of S^f except the outer one, then Bray defines the outer and inner mass, mout(S) and min(S), respectively, by

{ } mout (S) := mADM( S^e) | ^Se (- E (S) , { } min (S) := A^S |^Sf (- F (S) . f
mout(S) deviates slightly from Bartnik’s mass (36View 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 S2 and S 1 are outer-minimizing 2-surfaces such that S 2 encloses S 1, then m (S ) > m (S ) in 2 in 1 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 proved), then for outer-minimizing surfaces mout(S) > min(S) [87Jump To The Next Citation Point90Jump To The Next Citation Point]. Furthermore, if Si is a sequence such that the boundaries @Si shrink to a minimal surface S, then the sequence mout(@Si) tends to the irreducible mass V~ ---------------2-- Area(S)/(16pG ) [41]. 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 2-surface data on S [90Jump To The Next Citation Point].


  Go to previous page Go up Go to next page