The Brown-York energy expression, based on the original flat space reference, has the highly undesirable
property that it gives non-zero energy even in the Minkowski spacetime if the fleet of observers on the
spherical is chosen to be radially accelerating (see the second paragraph in Section 10.1.7). Thus it
would be a legitimate aim to reduce this extreme dependence of the quasi-local energy on the choice
of the observers. One way of doing this is to formulate the quasi-local quantities in terms of
boost-gauge invariant objects. Such a boost-gauge invariant geometric object is the length of the
mean extrinsic curvature vector
of Section 4.1.2, which, in the notations of the present
section, is
. If
is spacelike or null, then this square root is real, and (apart from
the reference term
in Equation (73
)) in the special case
it reduces to
times the surface energy density of Brown and York. This observation lead Epp to suggest
The subtraction term in Equation (80) is defined through an isometric embedding of
into some reference spacetime instead of a 3-space. This spacetime is usually Minkowski or
anti-de-Sitter spacetime. Since the 2-surface data consist of the metric, the two extrinsic curvatures and
the
-gauge potential, for given
and ambient spacetime
the
conditions of the isometric embedding form a system of six equations for eight quantities, namely for
the two extrinsic curvatures and the gauge potential
(see Section 4.1.2, and especially
Equations (20
, 21
)). Therefore, even a naive function counting argument suggests that the
embedding exists, but is not unique. To have uniqueness, additional conditions must be imposed.
However, since
is a gauge field, one condition might be a gauge fixing in the normal bundle,
and Epp’s suggestion is to require that the curvature of the connection 1-form
in the
reference spacetime and in the physical spacetime be the same [133
]. Or, in other words, not only
the intrinsic metric
of
is required to be preserved in the embedding, but the whole
curvature
of the connection
as well. In fact, in the connection
on the spinor
bundle
both the Levi-Civita and the
connection coefficients appear on an
equal footing. (Recall that we interpreted the connection
to be a part of the universal
structure of
.) With this choice of the reference configuration
depends not only
on the intrinsic 2-metric
of
, but on the connection
on the normal bundle as
well.
Suppose that is a 2-surface in
such that
with
, and, in addition,
can be embedded into the flat 3-space with
. Then there is a boost gauge (the
‘quasi-local rest frame’) in which
coincides with the Brown-York energy
in the
particular boost-gauge
for which
. Consequently, every statement stated for the
latter is valid for
, and every example calculated for
is an example for
as well [133
]. A clear and careful discussion of the potential alternative choices for the
reference term, especially their potential connection with the angular momentum, is also given
there.
First, it should be noted that Epp’s quasi-local energy is vanishing in Minkowski spacetime for any
2-surface, independently of any fleet of observers. In fact, if is a 2-surface in Minkowski
spacetime, then the same physical Minkowski spacetime defines the reference spacetime as well, and
hence
. For round spheres in the Schwarzschild spacetime it yields the result that
gave. In particular, for the horizon it is
(instead of
), and at infinity it is
[133
]. Thus, in particular,
is also monotonically decreasing with
in Schwarzschild
spacetime.
Epp calculated the various limits of his expression too [133]. In the large sphere limit near spatial
infinity he recovered the Ashtekar-Hansen form of the ADM energy, at future null infinity the Bondi-Sachs
energy. The technique that is used in the latter calculations is similar to that of [93]. In non-vacuum in the
small sphere limit reproduces the standard
result, but the calculations for
the vacuum case are not completed. The leading term is still probably of order
, but its
coefficient has not been calculated. Although in these calculations
plays the role only of
fixing the 2-surfaces, as a result we got energy seen by the observer
instead of mass. It is
this reason why
is considered to be energy rather than mass. In the asymptotically
anti-de-Sitter spacetime (with the anti-de-Sitter spacetime as the reference spacetime)
gives zero. This motivated Epp to modify his expression to recover the mass parameter of the
Schwarzschild-anti-de-Sitter spacetime at the infinity. The modified expression is, however, not boost-gauge
invariant. Here the potential connection with the AdS/CFT correspondence is also discussed (see
also [33]).
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