Let be a spacelike topological two-sphere in spacetime such that the metric has positive
scalar curvature. Then, by the embedding theorem, there is a unique isometric embedding of
into the flat three-space, and this embedding is unique. Let
be the trace of the
extrinsic curvature of
in this embedding, which is completely determined by
and
is necessarily positive. Let
and
be the trace of the extrinsic curvatures of
in the
physical spacetime corresponding to the outward-pointing unit spacelike and future-pointing
timelike normals, respectively. Then Liu and Yau define their quasi-local energy in [338
] by
Isolating the gauge invariant part of the connection one-form, Liu and Yau defined a
quasi-local angular momentum as follows [338
]. Let
be the solution of the Poisson equation
on
, whose source is just the field strength of
(see Eq. (4.3
)). This
is
globally well defined on
and is unique up to addition of a constant. Then, define
on
the domain of the connection one-form
, which is easily seen to be closed. Assuming the space and time
orientability of the spacetime,
is globally defined on
, and hence, by
the
one-form
is exact:
for some globally defined real function
on
. This function is
unique up to an additive constant. Therefore,
, where the first term is gauge
invariant, while the second represents the gauge content of
. Then for any rotation Killing
vector
of the flat three-space Liu and Yau define the quasi-local angular momentum by
For a definition of the Kijowski–Liu–Yau energy as a quasi-local energy oparator in loop quantum gravity, see [565].
The most important property of the quasi-local energy (10.17) is its positivity. Namely [338
], let
be a
compact spacelike hypersurface with smooth boundary
, consisting of finitely many connected
components
, …,
such that each of them has positive intrinsic curvature. Suppose that the matter
fields satisfy the dominant energy condition on
. Then
is strictly positive
unless the spacetime is flat along
. In this case
is necessarily connected. The proof is based on the
use of Jang’s equation [289], by means of which the general case can be reduced to the results of Shi and
Tam in the time-symmetric case [458], stated in Section 10.1.7 (see also [566]). This positivity result is
generalized in [339], Namely,
is shown to be non-negative for all
, and if
for some
, then the spacetime is flat along
and
is connected. (In fact, since
depends only on
but is independent of the actual
, if the energy condition is satisfied
on the domain of dependence
, then
implies the flatness of the spacetime along
every Cauchy surface for
, i.e., the flatness of the whole domain of dependence as well.) A potential
spinorial proof of the positivity of
is suggested in [12]. This is based on the use of the
Nester–Witten 2–form and a Witten type argumentation. However, the spinor field solving the
Witten equation on the spacelike hypersurface
would have to satisfy a nonlinear boundary
condition.
If is an apparent horizon, i.e.,
, then
is just the integral of
. Then,
by the Minkowski inequality for the convex surfaces in the flat three-space (see, e.g., [519]) one
has
i.e., it is not less than twice the irreducible mass of the horizon. For round spheres coincides
with
, and hence, it does not reduce to the standard round sphere expression (4.8
). In particular,
for the event horizon of the Schwarzschild black hole it is
. (For a more detailed discussion, and, in
particular, the interpretation of
in the spherically-symmetric context, see [400
].)
was
calculated for small spheres both in nonvacuum and vacuum, and for large spheres near the future null
infinity in [575
]. In the leading order in nonvacuum we get the expected result
(see Eq. (4.9
)),
but in vacuum, in addition to the expected Bel–Robinson ‘energy’, there are extra terms in the
leading
order. As could be expected, at null infinity
reproduces the Bondi
energy.
However, can be positive even if
is in the Minkowski spacetime. In fact, for
a given intrinsic metric
on
(with positive scalar curvature)
can be embedded
into the flat
; this embedding is unique, and the trace of the extrinsic curvature
is
determined by
. On the other hand, the isometric embedding of
in the Minkowski
spacetime is not unique. The equations of the embedding (i.e., the Gauss, Codazzi–Mainardi, and
Ricci equations) form a system of six equations for the six components of the two extrinsic
curvatures
and
and the two components of the
gauge potential
.
Thus, even if we impose a gauge condition for the connection one-form
, we have only
six equations for the seven unknown quantities, leaving enough freedom to deform
(with
given, fixed intrinsic metric) in the Minkowski spacetime to get positive Kijowski–Liu–Yau
energy. Indeed, specific two-surfaces in the Minkowski spacetime are given in [401], for which
. Moreover, it is shown in [361] that the Kijowski–Liu–Yau energy for a closed two-surface
in Minkowski spacetime strictly positive unless
lies in a spacelike hyperplane. On the
applicability of
in the formulation and potential proof of Thorne’s hoop conjecture see
Section 13.2.2.
In the definition of one of the assumptions is the positivity of the scalar curvature of the
intrinsic metric on the two-surface
. Thus, it is natural to ask if this condition can be relaxed and
whether or not the quasi-local mass can be associated with a wider class of surfaces. Moreover, though in
certain circumstances
behaves as energy (see [400, 575]), it is the (renormalized) integral of the
length of the mean curvature vector, i.e., it is analogous to mass (compare with Eq. (2.7
)). Hence, it is
natural to ask if a energy-momentum four-vector can be introduced in this way. In addition, in the
calculation of the large sphere limit of
in asymptotically anti-de Sitter spacetimes it seems
natural to choose the reference configuration by embedding
into a hyperbolic rather than Euclidean
three-space. These issues motivate the following generalization [542] of the Kijowski–Liu–Yau
expression.
One of the key ideas is that two-surfaces with spherical topology and scalar curvature that are bounded
from below by a negative constant, i.e., , can be isometrically embedded in a unique way into
the hyperbolic space
with constant sectional curvature
, and hence, this embedding can be
(and in fact is) used to define the reference configuration. Let
denote the mean curvature of
in
this embedding, where the hyperbolic space
is thought of as a spacelike hypersurface
with constant negative curvature in the Minkowski spacetime
. Then the main result is
that, assuming that the mean curvature vector
of
in the spacetime is spacelike,
there exists a function
, depending only on the length
of
the mean curvature vector and the embedding of
into
, such that the four integrals
The proof of the nonspacelike nature of (10.19) is based on a Witten type argumentation, in which ‘the
mass with respect to a Dirac spinor
on
’ takes the form of an integral of
weighted by the norm of
. Thus, the norm of
appears to be a nontrivial lapse function. The
suggestion of [580] for a quasi-local mass-like quantity is based on an analogous expression. Let
be the
boundary of some spacelike hypersurface
on which the intrinsic scalar curvature is positive, let us
isometrically embed
into the Euclidean three-space, and let
be the pull back to
of a
constant spinor field. Suppose that the dominant energy condition is satisfied on
, and consider
the solution
of the Witten equation on
with one of the chiral boundary conditions
, where
are the projections to the space of the right/left handed Dirac
spinors, built from the projections
of Section 4.1.7. Then, by the Sen–Witten identity, a
positive definite boundary expression is introduced, and interpreted as the ‘quasi-local mass’
associated with
. In contrast to Brown–York type expressions, this mass, associated with
the two-spheres of radius
in the
hypersurfaces in Schwarzschild spacetime,
is an increasing function of the radial coordinate, and tends to the ADM mass. In general,
however, this limit is
, rather than the expected ADM mass. This construction is
generalized in [581] by embedding
into some
instead of
. A modified version
of these constructions is given in [582], which tends to the ADM energy and mass at spatial
infinity.
Update
Suggestion (11.12), due to Anco [11
], can also be considered as a generalization of the Kijowski–Liu–Yau
mass.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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