Gauge Independence in Black Hole Thermodynamics

• Gauge independence is the invariance of black hole thermodynamic relations under local and global symmetries, ensuring observable quantities like entropy and mass remain robust.
• The topic explores methods such as covariant phase space and principal bundle techniques that cancel gauge-dependent contributions to derive a consistent first law.
• It highlights how boundary term adjustments and phase-space redefinitions resolve ambiguities, reinforcing thermodynamic consistency in classical and semiclassical gravity.

Gauge independence in black hole thermodynamics refers to the property that all physically meaningful thermodynamic relations—specifically the first law and related quantities—are invariant under the local and global gauge symmetries present in gravitational theories. This invariance is critical to the physical interpretation of entropy, mass, angular momentum, and other conserved charges, particularly in formulations such as the first-order (tetrad + connection) formalism, gauge/gravity theories with internal symmetries, and generalized gravity models with variable couplings. The gauge-invariant structure establishes the robustness of black hole thermodynamics as a guiding principle for quantum gravity and constrains ambiguities arising from gauge choices in both classical and semiclassical analyses.

1. Formalism and Fundamental Structures

Black hole thermodynamics is analyzed in both metric and first-order (coframe + connection) formalisms. In the first-order formalism, the primary fields are:

• The coframe (tetrad), $e^I$, an $\mathrm{so}(1,3)$-valued 1-form mapping internal Lorentz indices to spacetime vectors.
• The Lorentz spin connection, $\omega^I{}_J$, encoding parallel transport and curvature.

Dynamical symmetries include:

• Diffeomorphisms: Generated by vector fields $\xi$, acting as Lie derivatives $\mathcal{L}_\xi$ on all fields.
• Internal (local Lorentz or Yang-Mills) gauge transformations: Parameterized by $\lambda^{IJ}=-\lambda^{JI}$, acting as

$$\delta_\mathrm{GT}(\lambda) e^I = -\lambda^I{}_J e^J,\quad \delta_\mathrm{GT}(\lambda)\omega^I{}_J = D\lambda^I{}_J.$$

Both field types are naturally unified via the principal bundle $P\to M$ construction, with automorphisms encoding joint diffeomorphism and internal gauge operations (Prabhu, 2015). In this setting, the fields are horizontal, equivariant forms on $P$.

Noether currents $J_X$ and charges $Q_X$ are associated to infinitesimal automorphisms $X$ of $P$. The variational bicomplex produces the pre-symplectic potential $\theta$, from which conserved currents and charges are constructed. The first law emerges from on-shell identities relating fluxes of these charges across spatial infinity and the black hole horizon (Ramirez et al., 12 Dec 2025, Prabhu, 2015, Hajian et al., 2022).

2. Gauge-Invariant Derivation of the First Law

In theories possessing local gauge freedom, naive applications of Noether’s theorem may yield charges or fluxes with explicit gauge dependence, spoiling physical interpretation. The resolution requires a systematic treatment of the gauge variations:

• Ramírez & Bonder method: By considering variations $\delta$ combined with an arbitrary, field-dependent gauge transformation $\Lambda$, the would-be gauge-dependent contributions $\Omega(\delta, \delta_\mathrm{GT}(\Lambda))$ in the pre-symplectic form cancel algebraically due to linearity (Ramirez et al., 12 Dec 2025).
• Corrected boundary data: The introduction of a boundary 2-form $\gamma(\delta)$, leading to gauge-covariant definitions

$$\widehat{\theta}(\delta) = \theta(\delta) + d\gamma(\delta),\quad \widehat{Q}(\xi) = Q(\xi) - Q_\mathrm{GT}(\lambda_\xi),$$

ensures that the integrals

$$\oint_{\partial\Sigma}\bigl[ \delta\widehat{Q}(\xi) - i_\xi\widehat{\theta}(\delta) \bigr]= 0$$

are manifestly gauge independent and equivalent to the standard first law, $$\delta M = \frac{\kappa}{8\pi}\delta A + \Omega_H \delta J + \Phi_H \delta Q$$.

• Phase space formalism: In Einstein–Maxwell and related models, the Lee–Wald–Iyer covariant phase space construction achieves gauge-invariant charges by augmenting the standard diffeomorphism generator with a field-dependent gauge parameter (e.g., $\lambda = -\xi\cdot A$), neutralizing all explicit gauge dependence in thermodynamic relations (Hajian et al., 2022).

3. Principal Bundle and Horizon Potentials

Formulating the field space on principal bundles reveals a key principle: the potentials ($\Phi^I$) conjugate to boundary or horizon charges are covariantly constant along the horizon generator. In the formulation of (Prabhu, 2015), the lift of a bifurcate Killing field $K^\mu$ to the principal bundle allows the identification of horizon potentials,

$K\lrcorner A = \Phi^I \, h_I,$

where $\{h_I\}$ spans a Cartan subalgebra of the gauge group $G$. The generalized zeroth law follows: these potentials are constant along the horizon. The first law is then

$$\delta E_\mathrm{can} - \Omega_H \delta J_\mathrm{can} = ( \frac{\kappa}{2\pi} ) \delta S + \Phi^I \delta Q_I,$$

where all quantities are defined covariantly, unaffected by bundle automorphism ambiguities.

Notably, only variations—$\delta S$, $\delta Q_I$—are uniquely defined; absolute entropies and charges can shift by topological terms or constant gauge transformations, but such ambiguities cancel in the first law (Prabhu, 2015).

4. Gauge-Independence in Extended and Higher-Derivative Theories

For gravity theories with dynamical couplings (e.g., cosmological constant $\Lambda$, higher-curvature parameters), or in higher dimensions, the assignment of globally gauge-independent charges becomes subtle. Recent developments exploit the background Killing charge method (Tavlayan et al., 4 Jul 2024) and the extended Iyer–Wald phase space (Campos et al., 4 Jul 2025):

• Background Killing charge method: Charges are defined via fluxes of conserved currents constructed from background (A)dS Killing fields, yielding manifest gauge-independence for mass, angular momentum, and generalized thermodynamic quantities in any theory with maximally symmetric vacua (Tavlayan et al., 4 Jul 2024). This approach obviates ad hoc subtractions or normalization ambiguities encountered in Komar integrals.
• Extended Iyer–Wald formalism and isohomogeneous transformations: When treating couplings as thermodynamic variables, exact isohomogeneous transformations (EITs) connect different gauge-choices—specifically, rescalings of horizon generators and shifts of the Killing 2-form potential—that all yield integrable and gauge-consistent first laws. The standard Kerr–AdS thermodynamics appears as a specific EIT-induced gauge fixing, but the formalism encompasses an entire equivalence class of such gauge-related first laws (Campos et al., 4 Jul 2025).

Approach	Gauge-Independence Mechanism	Main Applications
Principal bundle/noether lift	Covariant automorphisms, horizon constancy	First-order gravity, gauge fields
Covariant phase space	Field-dependent symmetry generator	Einstein–Maxwell, U(1) gauge sector
Background Killing charges	Conserved currents, background symms	Higher-curvature, dS/AdS theories
EITs in extended Iyer–Wald	Contactomorphism in phase space	Variable coupling, thermodynamic volume

5. Semiclassical and Statistical Interpretations

Gauge independence arises not only in the classical variational principle but also in semiclassical entropy derivations:

• Background-invariant action principle: By fixing only scalar, coordinate-invariant data at the boundary (e.g., mass, areal radius), and abstaining from imposing gauge or coordinate-dependent conditions such as the lapse or three-geometry, the semiclassical action for black hole formation becomes fully diffeomorphism invariant (Bachlechner, 2018). The resulting imaginary part of the on-shell action delivers the Bekenstein–Hawking area law,

$S_{BH} = \frac{A}{4G\hbar},$

independently of gauge-fixing choices.

• Quantized phase-space volume: The quantization of black hole microstates, derived from uncertainty relations and recurrence time arguments, also respects gauge invariance, as only scalar invariants fixed at the boundary enter the microcanonical sum (Bachlechner, 2018).

6. Modified Gravity and Mixed Symmetries

In more exotic settings, such as Hořava–Lifshitz gravity, additional mixed gauge-global symmetries arise, with conserved charges that are neither purely local nor purely global. These "leaf-reparameterization" symmetries introduce new charges (e.g., $Q_{leaf}$) tied to the foliation structure, leading to extended first laws:

$$\delta M = T_{UH} \delta S_{UH} + V_{Thermo} \delta \Lambda + X \delta Q_{leaf},$$

with $X$ determined by the variation structure. Crucially, only the inclusion of all true symmetry charges ensures gauge invariance of the thermodynamic laws; omission leads to apparent first-law violations. When mixed symmetries and their charges are properly incorporated, observables such as black hole temperature and entropy remain invariant under the enlarged gauge group (Martin et al., 16 Aug 2024).

7. Resolution of Ambiguities and Physical Significance

The unambiguous, gauge-invariant formulation of black hole thermodynamics has several critical implications:

• No special gauge fixing is necessary; algebraic cancellations and corrected boundary terms subsume would-be gauge dependencies (Ramirez et al., 12 Dec 2025).
• Ambiguities in the metric, connection, or potential normalizations are resolved by cohomological (total derivative) adjustments of boundary terms or by identification of equivalence classes under principal bundle automorphisms (Prabhu, 2015, Campos et al., 4 Jul 2025).
• Thermodynamic consistency (first and Smarr laws) is maintained across a wide class of theories, including those with higher-curvature corrections, non-trivial topology, and dynamical couplings (Tavlayan et al., 4 Jul 2024).
• Physical observables—including mass, entropy, surface gravity, and chemical potentials—are robust under all gauge freedom, and quantization conditions in semiclassical gravity do not rely on coordinate or gauge-fixing conventions (Bachlechner, 2018).

In effect, the gauge independence of black hole thermodynamics is structurally guaranteed by the interplay between variational principles, symplectic geometry, and group-theoretic analysis of gauge symmetries. This underpinning gives the first law and associated relations a universal status across gravitational theories, making them a central guide for any consistent theory of quantum gravity.