Let be a hypersurface-orthogonal timelike Killing vector field, a spacelike hypersurface to which is orthogonal, and . Then , a field equation for , follows from Einstein equations (see, e.g., pp. 71 – 74 of [240] or [199]). Here and , the energy density and the average spatial pressure of the matter fields, respectively, seen by the observer at rest with respect to (or ).

In the study of (‘quasi-static’) equilibrium configurations of self-gravitating systems Tolman [520, 521] found the integral

to be the energy of the system. Here is a compact domain with smooth boundary , is the outward pointing unit normal of in ; and the second expression follows from the field equation for above. can in fact be interpreted as some form of a quasi-local energy [2, 3], called the Tolman energy. Clearly, for matter fields with non-negative energy density and average pressure and non-positive cosmological constant this is non-negative on the domain where is timelike. Using the defining equation of in terms of the two-surface integral, one can show that in asymptotically flat spacetimes it tends to the ADM energy as a non-decreasing set function. The second expression in Eq. (12.1) implies that, similarly to Komar’s expression, in vacuum depends only on the homology class of the 2-surface . Thus, in particular, it associates zero energy with vacuum domains. For spherically symmetric configurations on round spheres with the area radius (see Section 4.2.1) it is , which in vacuum reduces to the Misner–Sharp energy.The Tolman energy appeared to be a useful tool in practice: By means of Abreu and Visser gave remarkable entropy bounds for localized, but uncollapsed bodies [2, 3]. (We discuss this bound in Section 13.4.3.)

Let , the set of those points of where the length of the Killing field is the value , i.e., are the equipotential surfaces in . 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 [309] associate a quasi-local energy to the two-surfaces as follows. Suppose that the matter fields can be removed from and concentrated into a thin shell on in such a way that the space inside is flat but the geometry outside remains 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 two-surface as

where is the jump across the two-surface of the trace of the extrinsic curvatures of the two-surface itself in . Remarkably enough, the Katz–Lynden-Bell–Israel quasi-local energy in the form (12.2), associated with the equipotential surface , is independent of any distributional matter field, and 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 [309]. However, it does not reduce to the round-sphere energy in spherically-symmetric spacetimes [374], and, in particular, gives zero for the event horizon of a Schwarzschild black hole.

The Newtonian limit of general relativity is defined in [240], pp. 71 – 74, via static and (at spatial infinity) asymptotically flat spacetimes. The Newtonian scalar potential is identified with the logarithm of the length of the Killing vector, i.e., (in traditional units) it is . Decomposing the energy density as the sum of the rest-mass energy and the internal energy, , Einstein’s equations yield the field equation

Identifying the last term as , the sum of the energy density and three times of the average spatial stress of the field , (see the definitions (3.2) in the Newtonian case), the exact field equation (12.3) can be compared with the naïve, relativistically corrected Newtonian field equation (3.3); and one can read off the various relativistic corrections [199]. Then (3.4) motivates the definition of an ‘effective’ quasi-local energy as the integral of all the effective source terms on the right hand side of the exact field equation (12.3): If the spacetime is asymptotically flat at spatial infinity (in which case ) such that the hypersurface extends to spatial infinity, then, using e.g. the result of [60], one can show that tends to the ADM energy as is enlarged to exhaust the whole [199]. Since in the vacuum region the integrand of the 3-dimensional integral is negative definite, near infinity tends to the ADM energy as a monotonically decreasing set function. However, because of the extra relativistic correction term in the source, the rate of change of this set function deviates from the one in the naïve relativistically corrected Newtonian theory of Section 3.1.1. In fact, for a two-sphere of radius in the Schwarzschild spacetime with mass parameter the quasi-local energy, for large , is , rather than (see Section 3.1.1).Though is negative in the vacuum regime, for spherically symmetric configurations, when the material source of the gravitational ‘field’ is contained in , it is positive if an energy condition is satisfied; and it is zero if and only if the domain of dependence of in the spacetime is flat. (For the details see [199].)

Living Rev. Relativity 12, (2009), 4
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