Extended black hole thermodynamics in a DGP braneworld

TL;DR

This work shows that extended black hole thermodynamics can be naturally realized on a DGP brane by promoting the brane tension $\sigma$ to a thermodynamic variable, introducing a brane-pressure $P_\sigma=-\sigma$ and a conjugate volume $V_\sigma$. On the normal (ghost-free) branch, the brane is asymptotically flat, enabling a clean Iyer–Wald construction with a flat-brane path that preserves asymptotic structure while capturing induced-gravity corrections as $\mathcal{O}(r_h/r_c)$. The authors derive the extended first law and Smarr relation on the brane for static and rotating black holes, with a key result that $V_\sigma$ reduces to the geometric volume $\dfrac{4\pi}{3} r_h^3$ in the static case and generalizes to $V_\sigma=\dfrac{4\pi}{3}(r_+^3+a^2 r_+)$ for Kerr seeds. Maintaining flatness along the variation requires co-varying $\Lambda_5$, yielding a consistent, AdS-free realization of black hole chemistry on a flat braneworld. These findings broaden the scope of black hole thermodynamics beyond AdS, linking localized brane vacuum energy to thermodynamic variables and providing a platform for future explorations in higher curvature and holographic contexts.

Abstract

We develop extended black-hole thermodynamics on a Dvali--Gabadadze--Porrati (DGP) brane by promoting the brane tension \(σ\) to a thermodynamic variable within the extended Iyer--Wald framework. The brane tension acts as a localized vacuum energy with pressure \(P_σ\equiv -σ\), yielding a new work term \(V_σ\,\mathrm{d}P_σ\) 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 \(V_σ=\tfrac{4π}{3}r_h^3\); for stationary, axisymmetric solutions it admits a covariant, slice-independent definition and evaluates to \(V_σ=\tfrac{4π}{3}\!\left(r_+^3+a^2 r_+\right)\). 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.