First law and a variational principle for static asymptotically Randall-Sundrum black holes

Abstract

We give a new, intrinsic, mass definition for spacetimes asymptotic to the Randall-Sundrum braneworld models, RS1 and RS2. For this mass, we prove a first law for static black holes, including variations of the bulk cosmological constant, brane tensions, and RS1 interbrane distance. Our first law defines a thermodynamic volume and a gravitational tension that are braneworld analogs of the corresponding quantities in asymptotically AdS black hole spacetimes and asymptotically flat compactifications, respectively. We also prove the following related variational principle for asymptotically RS black holes: instantaneously static initial data that extremizes the mass yields a static black hole, for variations at fixed apparent horizon area, AdS curvature length, cosmological constant, brane tensions, and RS brane warp factors. This variational principle is valid with either two branes (RS1) or one brane (RS2), and is applicable to variational trial solutions.

Stability and binding energy of small asymptotically Randall-Sundrum black holes

“…In this paper, we applied our variational principle [18] to initial data for small asymptotically RS2 black holes, and found two static black holes. We showed that the well known static black hole on the brane is stable against translations transverse to the brane, and has a large binding energy to the brane due to the brane's orbifold symmetry.…”

“…Here we formulate the initial data for small initially static black holes in RS2. As described in [18], this amounts to solving the constraint R = 2Λ, where R is the Ricci scalar of the spatial metric h ab . The spatial geometry can be visualized as a conventional embedding diagram [20,21] with two asymptotically RS2 regions (instead of asymptotically flat regions), connected by a bridge.…”

“…(73) We defined the mass M for an asymptotically RS geometry in [18], which for the large ρ asymptotics (62) yields…”

Energy extremum principle for charged black holes

“…If all of the black hole charges have the same sign, then our extremum principle is a type of variational principle, and augments a list of existing principles [6][7][8] that interpret static black holes (here, the static Hartle-Hawking solution) as extrema of total energy. Our results are also consistent with the supersymmetric BPS energy minimum (E = |Q|) of [9].…”

“…Our extremum principle (2) adds the case of charged black holes to these earlier variational principles [6][7][8] and identifies the static Hartle-Hawking solution as its energy extremum. In contrast to [6][7][8], in this paper, the number N of black holes that we consider is arbitrary, and our energy extremum varies (rather than holds fixed) the black hole areas.…”