where is the jump across the 2-surface of the trace of the extrinsic curvatures of the 2-surface itself
in
. Remarkably enough, the Katz-Lynden-Bell-Israel quasi-local energy
in the form (94
),
associated with the equipotential surface
, is independent of any distributional matter field, and it can
also be interpreted as follows. Let
be the metric on
,
the extrinsic curvature of
in
and
. Then suppose that there is a flat metric
on
such that the induced metric from
on
coincides with that induced from
, and
matches continuously to
on
. (Thus, in particular, the induced area element
determined on
by
, and
coincide.) Let the extrinsic curvature of
in
be
, and
. Then
is the integral on
of
times
the difference
. Apart from the overall factor
, this is essentially the Brown-York
energy.
In asymptotically flat spacetimes tends to the ADM energy [232]. However, it does not
reduce to the round-sphere energy in spherically symmetric spacetimes [277], and, in particular, gives zero
for the event horizon of a Schwarzschild black hole.
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