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 š’® 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 a Q of Section 4.1.2, which, in the notation of this section, is √ -2---2- k − l. If a Q is spacelike or null, then this square root is real, and (apart from the reference term k0 in equation (10.9View Equation)) in the special case l = 0 it reduces to − 8πG times the surface energy density of Brown and York. This observation lead Epp to suggest

1 āˆ® ( āˆ˜ ------------ √ ------) EE (š’® ) := ----- (k0)2 − (l0)2 − k2 − l2 dš’® (10.16 ) 8πG š’®
as the general definition of the ‘invariant quasi-local energy’ [178Jump To The Next Citation Point]. Here, as in the Brown–York definition, k0 and l0 give the ‘reference term’ that should be fixed in a separate procedure. Note that it is EE (š’® ) that is referenced and not the mean curvatures k and l, i.e., EE (š’®) is not the integral of āˆ˜ -2----2 šœ€ − jāŠ¢. Apart from the fact that M Σ of Eq. (2.7View Equation) is associated with a three-surface, Epp’s invariant quasi-local energy expression appears to be analogous to M Σ rather than to EΣ [ξa] of Eq. (2.6View Equation) or to Q š’®[K ] of Eq. (2.5View Equation). However, although at first sight EE (š’® ) 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 l = 0, it reduces to the Brown–York expression, provided k is positive. Note that Qa must be spacelike to have a quasi-local rest frame. This condition can be interpreted as a very weak convexity condition on š’®. In particular, k is not needed to be positive, only 2 2 k > l is required. While EBY is sensitive to the sign of k, EE is not. Hence, EE(š’® ) 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. (10.16View Equation) is defined through an isometric embedding of (š’®, qab) 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 SO (1,1)-gauge potential, for given (š’®,q ) ab and ambient spacetime (M 0,g0 ) ab 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 Ae (see Section 4.1.2, and especially Eqs. (4.1View Equation) and (4.2View Equation)). 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 A e 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 Ae in the reference spacetime and in the physical spacetime be the same [178Jump To The Next Citation Point]. Or, in other words, not only the intrinsic metric qab of š’® is required to be preserved in the embedding, but the whole curvature f abcd of the connection δe as well. In fact, in the connection δe on the spinor bundle A S (š’® ) both the Levi-Civita and the SO (1,1) connection coefficients appear on an equal footing. (Recall that we interpreted the connection δe to be a part of the universal structure of š’®.) With this choice of reference configuration EE (š’® ) depends not only on the intrinsic two-metric qab of š’®, but on the connection δe on the normal bundle as well.

Suppose that š’® is a two-surface in M such that 2 2 k > l with k > 0, and, in addition, (š’®,qab) can be embedded into the flat three-space with 0 k ≥ 0. Then there is a boost gauge (the ‘quasi-local rest frame’) in which EE (š’®) coincides with the Brown–York energy EBY (š’®,ta) in the particular boost-gauge ta for which taQa = 0. Consequently, every statement stated for the latter is valid for E (š’® ) E, and every example calculated for E (š’®,ta) BY is an example for EE (š’® ) as well [178Jump To The Next Citation Point]. 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 š’® is a two-surface in Minkowski spacetime, then the same physical Minkowski spacetime defines the reference spacetime as well, and hence, EE (š’® ) = 0. For round spheres in the Schwarzschild spacetime it yields the result that EBY gave. In particular, for the horizon, it is 2m āˆ•G (instead of m āˆ•G), and at infinity it is m āˆ•G [178Jump To The Next Citation Point]. Thus, in particular, EE is also monotonically decreasing with r 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, EE (š’®) reproduces the standard 4π 3 a b 3 r Tabtt result, but the calculations for the vacuum case are not completed. The leading term is still probably of order 5 r, but its coefficient has not been calculated. Although in these calculations ta plays only the role of fixing the two-surfaces, as a result we get the energy seen by the observer ta instead of mass. This is why E (š’® ) E 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) EE 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]).


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