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 . However, it does not reduce to the round-sphere energy in spherically symmetric spacetimes , and, in particular, gives zero for the event horizon of a Schwarzschild black hole.
© Max Planck Society and the author(s)