The first law of black brane mechanics

TL;DR

This work extends the ADM and Komar notions of energy to p-brane spacetimes by defining energy density, tension, and angular momentum density per unit p-volume under transverse asymptotic flatness. It derives covariant surface integrals and a Smarr-type formula for neutral and charged branes, introducing an effective horizon area ${\cal A}_{eff}$ and a corresponding first law with a new conjugate pair ${\cal T}_{eff}$ and worldvolume volume $V_p$. The charged case yields an energy bound ${\cal E} \ge \Phi_H {\cal Q}$, with supersymmetric (boost-invariant) branes saturating the bounds, and the first law takes the form dE = κ dA_eff + Ω_H·dJ + Φ_H dQ + T_eff dV_p. The results illuminate the thermodynamics of black branes, linking higher-dimensional p-brane physics to lower-dimensional black holes via dimensional reduction and providing a framework for future extensions to more general brane configurations and field content.

Abstract

We obtain ADM and Komar surface integrals for the energy density, tension and angular momentum density of stationary $p$-brane solutions of Einstein's equations. We use them to derive a Smarr-type formula for the energy density and thence a first law of black brane mechanics. The intensive variable conjugate to the worldspace p-volume is an `effective' tension that equals the ADM tension for uncharged branes, but vanishes for isotropic boost-invariant charged branes.