### 5.2 Bray’s modifications

Update
Another, slightly modified definition for the quasi-local mass is suggested by Bray [110, 113]. 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 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

deviates slightly from Bartnik’s mass (5.1) 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 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
[56]. 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
[113].

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