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
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
[87
, 90
]. 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
].
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