One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [54, 53
]. His
idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let
be
a compact, connected three-manifold with connected boundary
, and let
be a (negative definite)
metric and
a symmetric tensor field on
, such that they, as an initial data set, satisfy the
dominant energy condition: if
and
, then
. For the sake of simplicity we denote the triple
by
. Then let
us consider all the possible asymptotically flat initial data sets
with a single
asymptotic end, denoted simply by
, which satisfy the dominant energy condition, have
finite ADM energy and are extensions of
above through its boundary
. The set of these
extensions will be denoted by
. By the positive energy theorem,
has non-negative ADM
energy
, which is zero precisely when
is a data set for the flat spacetime. Then
we can consider the infimum of the ADM energies,
, where the
infimum is taken on
. Obviously, by the non-negativity of the ADM energies, this infimum
exists and is non-negative, and it is tempting to define the quasi-local mass of
by this
infimum.8
However, it is easy to see that, without further conditions on the extensions of
, this infimum
is zero. In fact,
can be extended to an asymptotically flat initial data set
with arbitrarily small
ADM energy such that
contains a horizon (for example in the form of an apparent horizon) between
the asymptotically flat end and
. In particular, in the ‘
-spacetime’ discussed in Section 4.2.1
on round spheres, the spherically symmetric domain bounded by the maximal surface (with
arbitrarily-large round-sphere mass
) has an asymptotically flat extension, the complete spacelike
hypersurface of the data set for the
-spacetime itself, with arbitrarily small ADM mass
.
Obviously, the fact that the ADM energies of the extensions can be arbitrarily small is a consequence of
the presence of a horizon hiding from the outside. This led Bartnik [54
, 53
] to formulate his suggestion
for the quasi-local mass of
. He concentrated on time-symmetric data sets (i.e., those for which the
extrinsic curvature
is vanishing), when the horizon appears to be a minimal surface of topology
in
(see, e.g., [213
]), and the dominant energy condition is just the requirement of the
non-negativity of the scalar curvature of the spatial metric:
. Thus, if
denotes
the set of asymptotically flat Riemannian geometries
with non-negative scalar
curvature and finite ADM energy that contain no stable minimal surface, then Bartnik’s mass is
Of course, to rule out this limitation, one can modify the original definition by considering the set
of asymptotically flat Riemannian geometries
(with non-negative scalar
curvature, finite ADM energy and with no stable minimal surface), which contain
as an
isometrically-embedded Riemannian submanifold, and define
by Eq. (5.1
) with
instead
of
. Obviously, this
could be associated with a larger class of two-surfaces
than the original
can be to compact three-manifolds, and
holds.
In [279, 56
] the set
was allowed to include extensions
of
having boundaries as compact
outermost horizons, when the corresponding ADM energies are still non-negative [217
], and hence
is still well defined and non-negative. (For another description of
allowing horizons in the
extensions but excluding them between
and the asymptotic end, see [110
] and Section 5.2 of this
paper.)
Bartnik suggests a definition for the quasi-local mass of a spacelike two-surface (together with its
induced metric and the two extrinsic curvatures), as well [54
]. He considers those globally-hyperbolic
spacetimes
that satisfy the dominant energy condition, admit an asymptotically flat
(metrically-complete) Cauchy surface
with finite ADM energy, have no event horizon and in which
can be embedded with its first and second fundamental forms. Let
denote the set of these
spacetimes. Since the ADM energy
is non-negative for any
(and is zero precisely
for flat
), the infimum
The first immediate consequence of Eq. (5.1) is the monotonicity of the Bartnik mass. If
, then
, and hence,
. Obviously, by definition (5.1
) one has
for any
. Thus, if
is any quasi-local mass functional that is larger
than
(i.e., that assigns a non-negative real to any
such that
for any allowed
), furthermore if
for any
, then by the definition of the infimum in
Eq. (5.1
) one has
for any
. Therefore,
is the largest mass
functional satisfying
for any
. Another interesting consequence
of the definition of
, due to Simon (see [56
]), is that if
is any asymptotically flat,
time-symmetric extension of
with non-negative scalar curvature satisfying
, then
there is a black hole in
in the form of a minimal surface between
and the infinity
of
. For further discussion of
from the point of view of black holes, as well as
the relationship between the Bartnik mass and other expressions (e.g., the Hawking energy),
see [460
].
As we saw, the Bartnik mass is non-negative, and, obviously, if is flat (and hence is a data set for
flat spacetime), then
. The converse of this statement is also true [279
]: If
,
then
is locally flat. The Bartnik mass tends to the ADM mass [279
]: If
is an
asymptotically flat Riemannian three-geometry with non-negative scalar curvature and finite ADM mass
, and if
,
, is a sequence of solid balls of coordinate radius
in
, then
. The proof of these two results is based on the use of Hawking energy
(see Section 6.1), by means of which a positive lower bound for
can be given near
the nonflat points of
. In the proof of the second statement one must use the fact that
Hawking energy tends to the ADM energy, which, in the time-symmetric case, is just the ADM
mass.
The proof that the Bartnik mass reduces to the ‘standard expression’ for round spheres is a nice
application of the Riemannian Penrose inequality [279]. Let
be a spherically-symmetric Riemannian
three-geometry with spherically-symmetric boundary
. One can form its ‘standard’ round-sphere
energy
(see Section 4.2.1), and take its spherically-symmetric asymptotically flat vacuum
extension
(see [54
, 56
]). By the Birkhoff theorem the exterior part of
is a part
of a
hypersurface of the vacuum Schwarzschild solution, and its ADM mass is
just
. Then, any asymptotically flat extension
of
can also be considered as
(a part of) an asymptotically flat time-symmetric hypersurface with minimal surface, whose
area is
. Thus, by the Riemannian Penrose inequality [279
]
. Therefore, the Bartnik mass of
is just the ‘standard’ round-sphere expression
.
Since for any given the set
of its extensions is a huge set, it is almost hopeless to parametrize
it. Thus, by its very definition, it seems very difficult to compute the Bartnik mass for a given, specific
. Without some computational method the potentially useful properties of
would be lost
from the working relativist’s arsenal.
Such a computational method might be based on a conjecture of Bartnik [54, 56
]: The infimum in
definition (5.1
) of the mass
is realized by an extension
of
such that the
exterior region,
, is static, the metric is Lipschitz-continuous across the two-surface
, and the mean curvatures of
of the two sides are equal. Therefore, to compute
for a
given
, one should find an asymptotically flat, static vacuum metric
satisfying the matching
conditions on
, and where the Bartnik mass is the ADM mass of
. As Corvino shows [154], if
there is an allowed extension
of
for which
, then the extension
is static; furthermore, if
,
and
has an allowed
extension
for which
, then
is static. Thus, the proof of
Bartnik’s conjecture is equivalent to the proof of the existence of such an allowed extension. The
existence of such an extension is proven in [360] for geometries
close enough to the
Euclidean one and satisfying a certain reflection symmetry, but the general existence proof is still
lacking. (For further partial existence results see [17].) Bartnik’s conjecture is that
determines this exterior metric uniquely [56
]. He conjectures [54, 56
] that a similar computation
method can be found for the mass
, defined in Eq. (5.2
), as well, where the exterior
metric should be stationary. This second conjecture is also supported by partial results [155]:
If
is any compact vacuum data set, then it has an asymptotically flat vacuum
extension, which is a spacelike slice of a Kerr spacetime outside a large sphere near spatial
infinity.
To estimate one can construct admissible extensions of
in the form of the
metrics in quasi-spherical form [55]. If the boundary
is a metric sphere of radius
with
non-negative mean curvature
, then
can be estimated from above in terms of
and
.
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Living Rev. Relativity 12, (2009), 4
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