Black hole thermodynamics is gauge independent (2512.11196v1)

Abstract: Black hole thermodynamics provides a rare window into the elusive quantum nature of gravity. In the first-order formalism for gravitational theories, where torsion and gauge freedom are present, it has been suggested that the first law of black hole thermodynamics requires a specific gauge choice, which would undermine its fundamental character. By using principal fiber bundles, it has been shown that the first law is independent of this gauge choice. The present work introduces an alternative method that establishes this independence in a more direct manner, thereby reinforcing the status of the first law as a guide toward quantum gravity. This method also facilitates explicit computations of the first law and helps resolve several ambiguities that commonly appear in such analyses.

• The paper proves that the first law of black hole thermodynamics is inherently gauge independent, overcoming ambiguities arising from torsion and gauge choices.
• It constructs a gauge-independent pre-symplectic current and corresponding Noether charge that cancel gauge-dependent terms in the formulation.
• The approach simplifies previous methods and reinforces the universality of black hole thermodynamics across various gravitational theories.

Gauge Independence in Black Hole Thermodynamics: A First-Order Formalism Approach

Introduction

The thermodynamic properties of black holes are key to probing quantum gravitational effects and understanding the microscopic structure of spacetime. The identification of analogies between black hole laws and classical thermodynamics, initiated by Bekenstein and later cemented by Hawking’s demonstration of black hole radiation, suggests that gravitational dynamics possess an underlying statistical interpretation. Modified gravity theories, including those allowing nontrivial torsion, are often best described using the first-order formalism via differential forms—a framework that introduces increased gauge freedom due to independent variations of the vielbein and spin connection. A critical question has been whether the first law of black hole thermodynamics in these settings depends on gauge choices and if so, whether this undermines its universality.

Background and Motivation

Traditionally, the first law of black hole thermodynamics and Wald’s entropy construction have been formulated in metric gravity, where the invariance under diffeomorphisms is straightforward. In first-order formalism, the dynamical fields—the vielbein and spin connection—also transform under local Lorentz gauge transformations. The presence of torsion further complicates the situation, making the analysis of covariance and gauge independence nontrivial. Prior work indicated potential gauge dependence of the first law, with proposals to resolve this using principal fiber bundle constructions [Prabhu_2017]. This paper supersedes those approaches by presenting a systematic, theory-independent method that directly establishes gauge independence, thereby reinforcing the fundamental character of black hole thermodynamics.

Main Results and Formulation

Pre-symplectic Structure and Noether Charges

A general gravitational action in $n$ dimensions is formulated as an $n$-form Lagrangian $\mathcal{L}(\phi)$. On-shell, the boundary contributions define the pre-symplectic structure and conserved Noether currents. In first-order formalism, the symmetry generator combines diffeomorphisms with gauge transformations, producing distinct Noether charges. The central technical advance is the construction of a gauge-independent pre-symplectic current

$$\hat{\Omega} (\delta_1,\delta_2) = \int_\Sigma \delta_1 \hat{\theta}(\delta_2) - \delta_2 \hat{\theta}(\delta_1) - \hat{\theta}([\delta_1,\delta_2]),$$

where $\hat{\theta}(\delta)$ extends the conventional boundary term $\theta(\delta)$ by inclusion of an exact differential $d\gamma(\delta)$, with $\gamma(\delta)$ defined in terms of variations of the vielbein.

The corresponding gauge-independent Noether charge is

$$\hat{Q}(\xi) = Q(\xi) - Q_{\mathrm{GT}}(\lambda_\xi),$$

where $\lambda_\xi$ encodes the gauge transformation required to realize a Killing symmetry in the presence of torsion.

Proof of Gauge Independence

The main argument establishes that for any transformation combining a diffeomorphism generated by $\xi$ with an arbitrary gauge transformation (with field-dependent parameter $\Lambda$),

$$\delta(\Lambda) = \delta_{\mathrm{Diff}(\xi)} - \delta_{\mathrm{GT}(\Lambda)},$$

the first law derived via the generalized pre-symplectic current is independent of $\Lambda$. Specifically, terms depending on the gauge parameter $\Lambda$ cancel identically in the construction of $\hat{Q}(\xi)$ and $\hat{\theta}(\delta)$, ensuring that all physical charges and currents are invariant under changes in gauge fixing.

The proposal recovers the first law in its conventional form

$$\oint_{\partial\Sigma} \delta \hat{Q}(\xi) - \text{i}_\xi \hat{\theta}(\delta) = 0,$$

demonstrating equivalence with approaches using principal bundles, but with substantial simplification and improved computational transparency.

Technical Consequences and Resolution of Ambiguities

The gauge-independent pre-symplectic current $\hat{\Omega}$ possesses two notable properties:

• It vanishes for gauge transformations: $$\hat{\Omega}(\delta, \delta_{\mathrm{GT}}(\lambda)) = 0$$.
• It is strictly independent of the gauge, as demonstrated by the linearity arguments.

All known ambiguities in the construction of the boundary terms and Noether charges—those associated with exact shifts in the Lagrangian, boundary terms, and conserved charges—are shown to be either irrelevant or exactly canceled in the quantification of $\hat{\Omega}$ and related charges. When fixing the Noether charge using invariance under diffeomorphisms (i.e., demanding $$\text{i}_{f\xi} \mathcal{L} = f\, \text{i}_\xi \mathcal{L}$$), the first law is rendered completely free of such ambiguities.

Analysis of Lorentz--Lie Transformations

The Lorentz--Lie transformation, which combines a diffeomorphism and a field-dependent Lorentz gauge transformation associated with a Killing vector, is shown to be equivalent to a zero transformation for Killing vectors. Thus, the method recovers the correct first law using the familiar mathematical structure, but the analysis clarifies that no special gauge choice is required—contrary to previous claims in the literature [Jacobson].

Implications and Future Directions

The principal implication is that the first law of black hole thermodynamics retains its universality in any vacuum gravitational theory admitting a first-order formalism, including those with nontrivial torsion. The results do not require the machinery of principal bundles or ad hoc modifications and are applicable regardless of the gravitational dynamics, demonstrating deep geometric robustness. The outcome strengthens the theoretical connections between gravity and thermodynamics and supports the notion that black hole laws probe the microscopic degrees of freedom underlying spacetime.

Going forward, extending these results to settings with matter fields—and analyzing their effect on the Noether charges and entropy formulas—remains a significant research direction. The gauge-independent formalism developed here is well-suited to address such generalizations, which are crucial for advancing quantum gravity models and possibly for formulating black hole thermodynamics in non-vacuum, realistic scenarios.

Conclusion

This work provides a rigorous, direct proof that black hole thermodynamics—specifically, the first law—is completely gauge independent in the first-order formalism. The formalism applies to arbitrary vacuum theories of gravity, including those with torsion, and resolves longstanding ambiguities in the derivation of conserved charges and thermodynamic identities. The results preserve the fundamental status of black hole thermodynamics as a probe of quantum gravitational phenomena and establish a robust baseline for future explorations involving matter, alternative gravitational theories, and statistical mechanical interpretations of spacetime microstructure (2512.11196).