One of the most natural ideas of quasi-localization of the familiar ADM mass is due to Bartnik [39, 38
]. His
idea is based on the positivity of the ADM energy, and, roughly, can be summarized as follows. Let
be
a compact, connected 3-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
infimum9.
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 the round spheres, the spherically symmetric domain
bounded by the maximal surface (with arbitrarily large round-sphere mass
) has an
asymptotically flat extension, 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 [39
, 38
] to formulate his suggestion
for the quasi-local mass of
. He concentrated on the 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 for example [156
]), and the dominant energy condition is just the
requirement of the non-negativity of the scalar curvature:
. 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 Equation (36
) with
instead of
. Obviously, this
could be associated with a larger class of
2-surfaces than the original
to compact 3-manifolds, and
holds.
In [208, 41
] the set
was allowed to include extensions
of
having boundaries as compact
outermost horizons, whenever the corresponding ADM energies are still non-negative [159
], and hence
is still well-defined and non-negative. (For another definition for
allowing horizons in the
extensions but excluding them between
and the asymptotic end, see [87
] and Section 5.2
below.)
Bartnik suggested a definition for the quasi-local mass of a spacelike 2-surface (together with its
induced metric and the two extrinsic curvatures), too [39
]. He considered 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 Equation (36) is the monotonicity of the Bartnik mass: If
,
then
, and hence
. Obviously, by definition (36
) one has
for any
. Thus if
is any quasi-local mass functional which is larger
than
(i.e. which assigns a non-negative real to any
such that
for any allowed
), furthermore if
for any
, then by the definition of the infimum in
Equation (36
) one has
for any
. Therefore,
is the largest
mass functional satisfying
for any
. Another interesting consequence of
the definition of
, due to W. Simon, 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
(see for
example [41
]).
As we saw, the Bartnik mass is non-negative, and, obviously, if is flat (and hence is a data set for
the flat spacetime), then
. The converse of this statement is also true [208
]: If
,
then
is locally flat. The Bartnik mass tends to the ADM mass [208
]: If
is an
asymptotically flat Riemannian 3-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 the Hawking energy
(see Section 6.1), by means of which a positive lower bound for
can be given near the
non-flat points of
. In the proof of the second statement one must use the fact that the
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 [208]: Let
be a spherically symmetric Riemannian
3-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 [39
, 41
]). 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 [208
]
. 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 parameterize
it. Thus, by the 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 [39, 41
]: The infimum in
definition (36
) of the mass
is realized by an extension
of
such that the
exterior region,
, is static, the metric is Lipschitz-continuous across the 2-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 the Bartnik mass is the ADM mass of
. As Corvino showed [119], 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 [267] for geometries
close enough to the Euclidean one and satisfying a certain
reflection symmetry, but the general existence proof is still lacking. Bartnik’s conjecture is that
determines this exterior metric uniquely [41
]. He conjectures [39, 41
] that a similar
computation method can be found for the mass
, defined in Equation (37
), too, where
the exterior metric should be stationary. This second conjecture is also supported by partial
results [120]: 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 [40]. 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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