10.3 Epp’s expression

10.3.1 The general form of Epp’s expression

The Brown–York energy expression, based on the original flat space reference, has the highly undesirable property that it gives nonzero energy even in the Minkowski spacetime if the fleet of observers on the spherical is chosen to be radially accelerating (see the second paragraph of Section 10.1.7). Thus, it would be a legitimate aim to reduce this extreme dependence of the quasi-local energy on the choice of the observers. One way of doing this is to formulate the quasi-local quantities in terms of boost-gauge invariant objects. Such a boost-gauge invariant geometric object is the length of the mean extrinsic curvature vector of Section 4.1.2, which, in the notation of this section, is . If is spacelike or null, then this square root is real, and (apart from the reference term in equation (10.9)) in the special case it reduces to times the surface energy density of Brown and York. This observation lead Epp to suggest

as the general definition of the ‘invariant quasi-local energy’ [178]. Here, as in the Brown–York definition, and give the ‘reference term’ that should be fixed in a separate procedure. Note that it is that is referenced and not the mean curvatures and , i.e., is not the integral of . Apart from the fact that of Eq. (2.7) is associated with a three-surface, Epp’s invariant quasi-local energy expression appears to be analogous to rather than to of Eq. (2.6) or to of Eq. (2.5). However, although at first sight appears to be a quasi-local mass, it turns out in special situations that it behaves as an energy expression. In the ‘quasi-local rest frame’, i.e., in which , it reduces to the Brown–York expression, provided is positive. Note that must be spacelike to have a quasi-local rest frame. This condition can be interpreted as a very weak convexity condition on . In particular, is not needed to be positive, only is required. While is sensitive to the sign of , is not. Hence, is not simply the value of the Brown–York expression in the quasi-local rest frame.

10.3.2 The definition of the reference configuration

The subtraction term in Eq. (10.16) is defined through an isometric embedding of into some reference spacetime instead of a three-space. This spacetime is usually Minkowski or anti-de Sitter spacetime. Since the two-surface data consist of the metric, the two extrinsic curvatures and the -gauge potential, for given and ambient spacetime the conditions of the isometric embedding form a system of six equations for eight quantities, namely for the two extrinsic curvatures and the gauge potential (see Section 4.1.2, and especially Eqs. (4.1) and (4.2)). Therefore, even a naïve function counting argument suggests that the embedding exists, but is not unique. To have uniqueness, additional conditions must be imposed. However, since is a gauge field, one condition might be a gauge fixing in the normal bundle, and Epp’s suggestion is to require that the curvature of the connection one-form in the reference spacetime and in the physical spacetime be the same [178]. Or, in other words, not only the intrinsic metric of is required to be preserved in the embedding, but the whole curvature of the connection as well. In fact, in the connection on the spinor bundle both the Levi-Civita and the connection coefficients appear on an equal footing. (Recall that we interpreted the connection to be a part of the universal structure of .) With this choice of reference configuration depends not only on the intrinsic two-metric of , but on the connection on the normal bundle as well.

Suppose that is a two-surface in such that with , and, in addition, can be embedded into the flat three-space with . Then there is a boost gauge (the ‘quasi-local rest frame’) in which coincides with the Brown–York energy in the particular boost-gauge for which . Consequently, every statement stated for the latter is valid for , and every example calculated for is an example for as well [178]. A clear and careful discussion of the potential alternative choices for the reference term, especially their potential connection with the angular momentum, is also given there.

10.3.3 The various limits

First, it should be noted that Epp’s quasi-local energy is vanishing in Minkowski spacetime for any two-surface, independent of any fleet of observers. In fact, if is a two-surface in Minkowski spacetime, then the same physical Minkowski spacetime defines the reference spacetime as well, and hence, . For round spheres in the Schwarzschild spacetime it yields the result that gave. In particular, for the horizon, it is (instead of ), and at infinity it is  [178]. Thus, in particular, is also monotonically decreasing with in Schwarzschild spacetime. The explicit calculation of Epp’s energy in Friedmann–Robertson–Walker spacetimes is given in [6].

Epp calculated the various limits of his expression as well [178]. In the large sphere limit, near spatial infinity, he recovered the Ashtekar–Hansen form of the ADM energy, and at future null infinity, the Bondi–Sachs energy. The technique that is used in the latter calculation is similar to that of [117]. In nonvacuum, in the small sphere limit, reproduces the standard result, but the calculations for the vacuum case are not completed. The leading term is still probably of order , but its coefficient has not been calculated. Although in these calculations plays only the role of fixing the two-surfaces, as a result we get the energy seen by the observer instead of mass. This is why is considered to be energy rather than mass. In the asymptotically anti-de Sitter spacetime (with the anti-de Sitter spacetime as the reference spacetime) gives zero. This motivated Epp to modify his expression to recover the mass parameter of the Schwarzschild–anti-de Sitter spacetime at infinity. The modified expression is, however, not boost-gauge invariant. Here the potential connection with the AdS/CFT correspondence is also discussed (see also [48]).