Go to previous page Go up Go to next page

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

Let Ka be a hypersurface-orthogonal timelike Killing vector field, S a spacelike hypersurface to which Ka is orthogonal, and K2 := KaKa. Let SK be the set of those points of S where the length of the Killing field is the value K, i.e. S K are the equipotential surfaces in S, and let D < S K be the set of those points where the magnitude of a K is not greater than K. Suppose that DK is compact and connected. Katz, Lynden-Bell, and Israel [232Jump To The Next Citation Point] associate a quasi-local energy to the 2-surfaces SK as follows. Suppose that the matter fields can be removed from intDK and can be concentrated into a thin shell on SK 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 ab T s and assuming that it satisfies the weak energy condition, the total energy of the shell, integral ab DK KaT s tbdS, is positive. Here ta is the future directed unit normal to S. Then, using the Einstein equations, the energy of the shell can be rewritten in terms of geometric objects on the 2-surface as
1 gf EKLI (SK) := -----K [k] dSK, (94) 8pG SK

where [k] is the jump across the 2-surface of the trace of the extrinsic curvatures of the 2-surface itself in S. Remarkably enough, the Katz-Lynden-Bell-Israel quasi-local energy EKLI in the form (94View Equation), associated with the equipotential surface SK, is independent of any distributional matter field, and it can also be interpreted as follows. Let hab be the metric on S, kab the extrinsic curvature of SK in (S,hab) and k := habkab. Then suppose that there is a flat metric h0 ab on S such that the induced metric from 0 hab on SK coincides with that induced from hab, and 0 h ab matches continuously to hab on SK. (Thus, in particular, the induced area element dSK determined on SK by hab, and h0ab coincide.) Let the extrinsic curvature of SK in h0ab be 0kab, and k0 := habk0ab. Then EKLI(SK) is the integral on SK of K times the difference k - k0. Apart from the overall factor K, this is essentially the Brown-York energy.

In asymptotically flat spacetimes EKLI(SK) 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.


  Go to previous page Go up Go to next page