### 5.2 Bray’s modifications

Another, slightly modified definition for the quasi-local mass was suggested by Bray [8790]. Here we summarize his ideas.

Let 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 . Let be any fixed 2-surface in which encloses all the asymptotic ends except one, say the -th (i.e. let be homologous to a large sphere in the -th asymptotic end). The outside region with respect to , denoted by , will be the subset of containing the -th asymptotic end and bounded by , while the inside region, , is the (closure of) . Next Bray defines the ‘extension’ of by replacing by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Similarly, the ‘fill-in’ of is obtained from by replacing by a smooth asymptotically flat end of any data set satisfying the dominant energy condition. Finally, the surface will be called outer-minimizing if for any closed 2-surface enclosing one has .

Let be outer-minimizing, and let denote the set of extensions of in which is still outer-minimizing, and denote the set of fill-ins of . If and denotes the infimum of the area of the 2-surfaces enclosing all the ends of except the outer one, then Bray defines the outer and inner mass, and , respectively, by

deviates slightly from Bartnik’s mass (36) 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 be outer-minimizing.

A simple consequence of the definitions is the monotonicity of these masses: If and are outer-minimizing 2-surfaces such that encloses , then and . 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  [8790]. Furthermore, if is a sequence such that the boundaries shrink to a minimal surface , then the sequence tends to the irreducible mass  [41]. Bray defines the quasi-local mass of a surface not simply to be a number, but the whole closed interval . If 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 then can be embedded into the Schwarzschild spacetime with the given 2-surface data on  [90].