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 two-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 two-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 two-surfaces enclosing all the ends of except the outer one, then Bray defines the outer and inner mass, and , respectively, by
A simple consequence of the definitions is the monotonicity of these masses: If and are outer-minimizing two-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 proven), then for outer-minimizing surfaces [110, 113]. Furthermore, if is a sequence such that the boundaries shrink to a minimal surface , then the sequence tends to the irreducible mass . 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 two-surface data on .
For further modification of Bartnik’s original ideas, see .
Living Rev. Relativity 12, (2009), 4
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