The Brown–York energy expression, based on the original flat space reference, has the highly undesirable
property that it gives nonzero energy even in the Minkowski spacetime if the fleet of observers on the
spherical is chosen to be radially accelerating (see the second paragraph of 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 notation of this
section, is
. If
is spacelike or null, then this square root is real, and (apart from
the reference term
in equation (10.9
)) 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 Eq. (10.16) is defined through an isometric embedding of
into some
reference spacetime instead of a three-space. This spacetime is usually Minkowski or anti-de
Sitter spacetime. Since the two-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 Eqs. (4.1
) and (4.2
)). Therefore, even a naïve 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 one-form
in
the reference spacetime and in the physical spacetime be the same [178
]. 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 reference configuration
depends not only
on the intrinsic two-metric
of
, but on the connection
on the normal bundle as
well.
Suppose that is a two-surface in
such that
with
, and, in addition,
can be embedded into the flat three-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 [178
]. 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
two-surface, independent of any fleet of observers. In fact, if is a two-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
[178
].
Thus, in particular,
is also monotonically decreasing with
in Schwarzschild spacetime.
The explicit calculation of Epp’s energy in Friedmann–Robertson–Walker spacetimes is given
in [6].
Epp calculated the various limits of his expression as well [178]. In the large sphere limit, near spatial
infinity, he recovered the Ashtekar–Hansen form of the ADM energy, and at future null infinity, the
Bondi–Sachs energy. The technique that is used in the latter calculation is similar to that of [117]. In
nonvacuum, 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 only the role
of fixing the two-surfaces, as a result we get the energy seen by the observer
instead of
mass. This is 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 infinity. The modified expression is, however, not boost-gauge
invariant. Here the potential connection with the AdS/CFT correspondence is also discussed (see
also [48]).
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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