Brown-York quasilocal energy in Lanczos-Lovelock gravity and black hole horizons

Abstract

A standard candidate for quasilocal energy in general relativity is the BrownYork energy, which is essentially a two dimensional surface integral of the extrinsic curvature on the two-boundary of a spacelike hypersurface referenced to flat spacetime. Several years back one of us had conjectured that the black hole horizon is defined by equipartition of gravitational and non-gravitational energy. By employing the above definition of quasilocal Brown-York energy, we have verified the equipartition conjecture for static charged and charged axi-symmetric black holes in general relativity. We have further generalized the Brown-York formalism to all orders in Lanczos-Lovelock theories of gravity and have verified the conjecture for pure Lovelock charged black hole in all even d = 2m + 2 dimensions, where m is the degree of Lovelock action. It turns out that the equipartition conjecture works only for pure Lovelock, and not for Einstein-Lovelock black holes.

1/r potential in higher dimensions

“…Finally pure Lovelock theories, i.e., a single term in the Lovelock polynomial, exhibit very interesting features. These include -(a) there is a close connection between pure Lovelock and dimensionally continued black holes [15,16], (b) gravity is kinematic in all critical d = 2m +1 dimensions, i.e., vacuum is pure Lovelock flat [17], (c) bound orbits exist for a given m in all 2m +1 < d < 4m +1 dimensions, in contrast, for Einstein gravity they do so only in four dimensions [18] and finally (d) equipartition of gravitational and non-gravitational energy defines location of black hole horizon [19].…”

Quasi-local energy and ADM mass in pure Lovelock gravity

“…In [28], a definition of quasi-local energy in pure Lovelock gravity was proposed. For m = 1 it correctly reduces to the Einstein version [8], however for m ≥ 2, the regularization does not preserve the form of the Hamilton's equations.…”

“…For generic metrics, the limit differs from the ADM mass (2.16) as we will now show. 14 The regularization presented in [28] involves replacing all the extrinsic curvatures in the boundary term B (m) (3.11) with the vacuum subtracted ones ∆ K ij = K ij − K ij | (0) . Clearly it is a non-linear procedure, which is the reason why the form of the Hamilton's equations is not preserved.…”

“…Instead of using background subtraction, there are also other ways to renormalize the quasi-local energy: one example is the use of counterterms [26,27]. Another method was presented in [28] where it was proven that the large surface limit of the resulting renormalized energy is proportional to the ADM mass in spherically symmetric geometries. However, the exact prefactor of proportionality was not specified.…”

“…Note that the B → ∞ of (3.23) does not vanish in contrast to(3.22). The reason is thatP ab cd(m) in (3.23) is not evaluated on the flat background, but instead interpreted as a limit, like in the ADM mass (2.16).14 A schematic proof of the limit for spherically symmetric metrics was presented in[28]. We will prove this explicitly and find the exact prefactor in section 4.…”