The first law of black brane mechanics

TL;DR

This paper derives a first law of black brane mechanics by establishing surface integrals for energy, tension, and angular momentum, and introduces an effective tension concept for charged branes.

Contribution

It provides the first derivation of a Smarr formula and first law for black branes using ADM and Komar integrals, including the concept of effective tension.

Findings

Derived ADM and Komar surface integrals for black branes.

Established a Smarr-type formula for energy density.

Formulated a first law of black brane mechanics.

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.