The Smarr Formula is Gauss's Law: A Kerr-Schild Single-Copy Perspective

TL;DR

This paper reveals a deep connection between black hole thermodynamics and gauge theory by showing the Smarr formula's structural similarity to Gauss's law within the Kerr-Schild double copy framework, extending to AdS spacetimes.

Contribution

It demonstrates that the Smarr formula corresponds to Gauss's law in the double copy approach and extends this to include thermodynamic pressure-volume terms in AdS spacetimes.

Findings

Smarr formula is structurally identical to Gauss's law for certain black holes.

Pressure-volume terms naturally emerge from gauge-theoretic background subtraction.

Establishes a novel link between double copy and black hole thermodynamics.

Abstract

In the Kerr-Schild double copy, static and spherically symmetric black hole solutions of general relativity are mapped to purely electric solutions of Maxwell's theory in flat spacetime. We demonstrate that, for these configurations, the thermodynamic Smarr formula is structurally identical to the single-copy Gauss's law. Extending this to asymptotically anti-de Sitter spacetimes, we prove that the thermodynamic pressure-volume term ($PV$) naturally emerges from a gauge-theoretic background subtraction. This relationship establishes a novel connection between the classical double copy and black hole thermodynamics.