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arxiv: 2605.14707 · v2 · pith:MMC6B44Tnew · submitted 2026-05-14 · ✦ hep-th · gr-qc

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

Gokhan Alkac This is my paper

Pith reviewed 2026-05-25 05:50 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black holesdouble copySmarr formulaGauss's lawKerr-Schildblack hole thermodynamicsAdS spacetimeMaxwell theory
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0 comments X

The pith

The Smarr formula for static spherical black holes is identical to Gauss's law in the single-copy Maxwell theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Kerr-Schild double copy maps static spherically symmetric black hole solutions of gravity to electric solutions of Maxwell theory in flat space, making the integrated Smarr formula the same as the single-copy Gauss law. This holds because the double copy identifies the relevant thermodynamic quantities directly with the gauge field quantities. For asymptotically AdS black holes the pressure-volume term in the Smarr relation arises from subtracting a constant background in the gauge theory alone. A reader would care because the result supplies an explicit dictionary between gravitational thermodynamics and ordinary flat-space electromagnetism.

Core claim

For the Kerr-Schild single copies of static and spherically symmetric black hole solutions, the thermodynamic Smarr formula is structurally identical to the single-copy Gauss's law; extending the construction to asymptotically anti-de Sitter spacetimes shows that the pressure-volume term emerges from a gauge-theoretic background subtraction.

What carries the argument

The Kerr-Schild double copy, which converts gravitational solutions into Maxwell solutions in flat space and thereby equates the Smarr integral to the electric flux integral.

If this is right

  • The Smarr formula follows from the Maxwell equations once the double-copy map is applied.
  • The AdS pressure term has a purely gauge-theoretic origin and does not require separate gravitational input.
  • Black-hole thermodynamic relations acquire a direct interpretation as statements about the single-copy gauge field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structural identity may hold for any solution whose double copy is known, even if the paper restricts attention to the static spherical case.
  • If the double copy extends to time-dependent or non-spherical configurations, the corresponding thermodynamic identities would likewise reduce to gauge-theory statements.
  • The result supplies a concrete route to derive thermodynamic relations from flat-space Maxwell theory rather than from the Einstein equations.

Load-bearing premise

The double copy map lets thermodynamic quantities be read off directly from the single-copy gauge quantities with no extra gravitational contributions required.

What would settle it

An explicit computation for a static spherical black hole in which the numerical value of the Smarr left-hand side differs from the surface integral of the single-copy electric field.

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that in the Kerr-Schild double copy, static spherically symmetric black hole solutions of GR map to electric Maxwell solutions in flat space such that the thermodynamic Smarr formula is structurally identical to the single-copy Gauss's law; it further claims that the PV term for asymptotically AdS solutions emerges from a purely gauge-theoretic background subtraction, establishing a novel link between the classical double copy and black hole thermodynamics.

Significance. If the claimed structural identity holds without additional gravitational input, the result would provide a direct bridge between double-copy constructions and black hole thermodynamics, potentially allowing thermodynamic relations to be read off from gauge theory. The work is notable for focusing on the integral form of Gauss's law matching the Smarr relation for these configurations and for addressing the AdS extension via background subtraction.

major comments (2)
  1. [flat spacetime case] The central claim requires that thermodynamic quantities (M, T, S) are identified with single-copy gauge quantities (Q, E) such that the Smarr relation follows automatically from the single-copy equations. The manuscript must explicitly derive this identification in the flat-space section to show it does not rely on gravitational first-law input beyond the Kerr-Schild map.
  2. [AdS extension] For the AdS extension, the claim that the PV term emerges from a gauge-theoretic background subtraction must be shown to involve no residual gravitational input in the definition of pressure or volume; the current assertion leaves open whether the subtraction is purely Maxwellian or imports horizon integrals from gravity.
minor comments (1)
  1. Notation for the single-copy quantities (e.g., how E is defined relative to the electric field) should be clarified to avoid ambiguity with gravitational energy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results. We address each major point below and will incorporate revisions to strengthen the explicit derivations as requested.

read point-by-point responses

  1. Referee: [flat spacetime case] The central claim requires that thermodynamic quantities (M, T, S) are identified with single-copy gauge quantities (Q, E) such that the Smarr relation follows automatically from the single-copy equations. The manuscript must explicitly derive this identification in the flat-space section to show it does not rely on gravitational first-law input beyond the Kerr-Schild map.

    Authors: We agree that an explicit derivation of the identification is needed for full transparency. In the revised manuscript we will add a dedicated subsection deriving the map from the Kerr-Schild ansatz to the single-copy Maxwell solution, showing that M is identified with the integrated charge Q and TS with the surface integral of E directly from the flat-space Maxwell equations and the single-copy field strength, without any additional gravitational first-law relations. revision: yes

  2. Referee: [AdS extension] For the AdS extension, the claim that the PV term emerges from a gauge-theoretic background subtraction must be shown to involve no residual gravitational input in the definition of pressure or volume; the current assertion leaves open whether the subtraction is purely Maxwellian or imports horizon integrals from gravity.

    Authors: We will revise the AdS section to provide an explicit calculation of the background subtraction performed entirely in Maxwell theory, using the flat-space single-copy solution as the reference configuration. The resulting PV contribution will be shown to arise from the difference in the gauge-field integrals over a surface at infinity, with no horizon integrals or other gravitational quantities entering the definitions of pressure or volume. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on external double-copy framework

full rationale

The paper asserts a structural identity between the Smarr formula and single-copy Gauss's law via the Kerr-Schild mapping, with the AdS PV term emerging from gauge-theoretic subtraction. No quoted equations or steps in the abstract or description reduce the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The mapping is treated as an independent input from prior double-copy literature, and the thermodynamic identification is presented as a consequence rather than an input, rendering the derivation self-contained against the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are inferred from the stated mapping and extension. The central claim rests on the applicability of the Kerr-Schild double copy to thermodynamic quantities and on standard black-hole thermodynamic relations.

axioms (2)
  • domain assumption Kerr-Schild ansatz maps static spherically symmetric GR solutions to Maxwell solutions
    The abstract invokes the Kerr-Schild double copy as the mechanism that produces the structural identity.
  • domain assumption Standard Smarr formula and thermodynamic relations hold for the black holes considered
    The paper treats the Smarr formula as given from black-hole thermodynamics.

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discussion (0)

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Reference graph

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