Holographic Black Hole Chemistry

TL;DR

This work shows that a universal holographic Smarr relation emerges for large-$N$ gauge theories by exploiting the simple scaling $\Omega(N,\mu,T,R)=N^2 \Omega_0(\mu,T,R)$ of the grand canonical free energy and the conformal equation of state. By carefully mapping bulk and boundary quantities through the holographic dictionary, the authors derive the bulk Smarr formula from the universal boundary relation and demonstrate how boundary pressure and volume arise from variations in the AdS length $L$ (and thus $N$) while keeping other variables fixed. They validate the construction on AdS Reissner–Nordström black holes and extend it to hyperscaling violating Dp-brane geometries, showing universality of the Smarr relation across large classes of theories. The analysis also discusses finite-$N$ corrections, higher-curvature terms, and the resulting richer phase structure, highlighting how a generalized Smarr relation may emerge when new thermodynamic variables are introduced. The results provide a cohesive framework linking holographic thermodynamics to black hole chemistry and offer a route to derive Smarr-like relations in a broad set of large-$N$ theories with potential implications for phase structure and holographic duals beyond conformal fixed points.

Abstract

Thermodynamic quantities associated with black holes in Anti-de Sitter space obey an interesting identity when the cosmological constant is included as one of the dynamical variables, the generalized Smarr relation. We show that this relation can easily be understood from the point of view of the dual holographic field theory. It amounts to the simple statement that the extensive thermodynamic quantities of a large $N$ gauge theory only depend on the number of colors, $N$, via an overall factor of $N^2$.