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
[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].
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |