Extended black hole thermodynamics in a DGP braneworld

TL;DR

This paper extends black hole thermodynamics to a DGP braneworld by treating brane tension as a thermodynamic variable, introducing a new pressure-volume term, and analyzing static and stationary black holes in this framework.

Contribution

It develops a consistent thermodynamic framework for black holes on a DGP brane, incorporating brane tension as a pressure and deriving associated volume and first law modifications.

Findings

Brane tension acts as a pressure variable in black hole thermodynamics.

The conjugate volume matches the geometric volume for static black holes.

The framework applies to asymptotically flat, ghost-free DGP braneworlds.

Abstract

We develop extended black-hole thermodynamics on a Dvali--Gabadadze--Porrati (DGP) brane by promoting the brane tension $\sigma$ to a thermodynamic variable within the extended Iyer--Wald framework. The brane tension acts as a localized vacuum energy with pressure $P_\sigma \equiv -\sigma$, yielding a new work term $V_\sigma\,\mathrm{d}P_\sigma$ in the first law and the corresponding Smarr relation. For static, spherically symmetric black holes we show that the conjugate volume equals the geometric volume $V_\sigma=\tfrac{4\pi}{3}r_h^3$; for stationary, axisymmetric solutions it admits a covariant, slice-independent definition and evaluates to $V_\sigma=\tfrac{4\pi}{3}\!\left(r_+^3+a^2 r_+\right)$. Working on the ghost-free normal branch, the brane is asymptotically flat with a single horizon, so the construction avoids de Sitter obstructions. Along a flat-brane path, asymptotic flatness is preserved by co-varying the bulk cosmological constant, and induced-gravity effects are suppressed by $r_h/r_c$. These results establish a consistent flat-braneworld realization of black-hole chemistry in which brane tension provides the physically motivated pressure variable.