Let be a spacelike topological 2-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 3-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 [253
] by
Isolating the gauge invariant part of the connection 1-form Liu and Yau defined a quasi-local
angular momentum as follows [253
]. Let
be the solution of the Poisson equation
on
, whose source is just the field strength of
(see Equation (22
)). This
is globally well-defined
on
and is unique up to addition of a constant. Then define
on the domain of the
connection 1-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 1-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 3-space Liu and Yau define the quasi-local angular momentum by
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 3-space (see for example [380]) 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 (27
). In particular, for the
event horizon of the Schwarzschild black hole it is
. Although the strict mathematical analysis is
still lacking,
probably reproduces the correct large sphere limits in asymptotically flat spacetime
(ADM and Bondi-Sachs energies), because the difference between the Brown-York, Epp, and
Kijowski-Liu-Yau definitions disappear asymptotically.
However, can be positive even if
is in the Minkowski spacetime. In fact,
for 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, the Codazzi-Mainardi,
and the 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 1-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 2-surfaces in the Minkowski spacetime are given in [292] for which
.
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