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Physicists have learned to translate the equations of gravity into the language of electricity. This trick, double copy, works like a dictionary: it replaces the complex geometry of curved spacetime with simple electrical rules. When the method was applied to the simplest black holes, a coincidence emerged: the formula linking their mass, size, and entropy (a measure of disorder) exactly mirrors Gauss's law — the rule by which a charge creates a field around itself. Two different phenomena are described by the same mathematical pattern.

The double copy turns Einstein's equations into Maxwell's equations. Back in the 1970s, Hawking proved that black holes radiate heat. Now, that heat can be 'heard' in the language of electricity.

Original abstract

as written by the authors arXiv:2605.14707v2

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 naturally emerges from a gauge-theoretic background subtraction. This relationship establishes a novel connection between the classical double copy and black hole thermodynamics.