### 12.3 The Katz-Lynden-Bell-Israel energy for static spacetimes

Let be a hypersurface-orthogonal timelike Killing vector field, a spacelike hypersurface to
which is orthogonal, and . Let be the set of those points of where the length
of the Killing field is the value , i.e. are the equipotential surfaces in , and let be
the set of those points where the magnitude of is not greater than . Suppose that is
compact and connected. Katz, Lynden-Bell, and Israel [232] associate a quasi-local energy to the 2-surfaces
as follows. Suppose that the matter fields can be removed from and can be concentrated into
a thin shell on in such a way that the space inside be flat but the geometry outside remain the
same. Then, denoting the (necessarily distributional) energy-momentum tensor of the shell by
and assuming that it satisfies the weak energy condition, the total energy of the shell,
, is positive. Here is the future directed unit normal to . Then, using the Einstein
equations, the energy of the shell can be rewritten in terms of geometric objects on the 2-surface as
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.