⚛️ general relativity

Conserved charges and the first law of black holes for field dependent symmetry generators in the covariant phase space formalism

This paper generalizes the covariant phase space formalism to incorporate field-dependent symmetry generators that combine diffeomorphisms and internal gauge transformations, demonstrating that the first law of black hole thermodynamics naturally emerges in generally invariant gravitational theories and validating the approach through applications to torus-like and charged AdS black holes in Einstein-Maxwell and Einstein-Euler-Heisenberg theories.

Original authors: Hai-Feng Ding

Published 2026-09-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hai-Feng Ding

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, symmetry is the silent architect of order. It is the principle that if you shift a system in time or space, or rotate it, the fundamental laws governing it remain unchanged. This deep connection between symmetry and conservation was first formalized in the early 20th century, revealing that every symmetry corresponds to a quantity that cannot be created or destroyed, such as energy or momentum. When physicists turn their gaze to the most extreme objects in the universe—black holes—these principles become the primary tools for understanding their nature. A black hole is not just a gravitational trap; it is a thermodynamic system with a temperature and an entropy, a measure of its hidden information. The relationship between these properties is codified in the "first law of black hole thermodynamics," a rule that links changes in a black hole's mass to changes in its entropy, spin, and electric charge. For decades, calculating these quantities has required complex mathematical machinery, often forcing scientists to make simplifying assumptions about the nature of the symmetries involved.

A researcher named Hai-Feng Ding has now refined this machinery, offering a more flexible and robust way to calculate these conserved charges and verify the first law. In a recent study, Ding extended a powerful framework known as the covariant phase space formalism to handle a specific complication: situations where the rules of symmetry themselves change depending on the state of the fields they act upon. In many standard calculations, the "symmetry generators"—the mathematical instructions that define how a system transforms—are treated as fixed, unchanging constants. However, in theories involving electromagnetic fields or other internal forces, these generators can actually depend on the configuration of the fields around them. Ding's work demonstrates that when you properly account for this field dependence, the first law of black hole thermodynamics emerges naturally and directly, without needing to force the system into a specific shape or rely on linear approximations.

The core of this achievement lies in how the researcher treats the "symmetry generator." Imagine a set of instructions for moving a fluid. In a simple case, the instructions might be fixed: "move everything one meter to the right." But in a more complex scenario, the instructions might change based on how dense the fluid is at any given moment. In the physics of black holes with electric fields, the instructions for symmetry (which include both moving through space and shifting the electric field) can depend on the strength of the gravity and the electric field present. Previous methods often struggled with this dependency, or required the symmetry instructions to be treated as fixed to make the math work. Ding's new approach embraces this dependency. By allowing the symmetry generators to vary with the field configuration, the researcher derived a general formula for the conserved charges that works for a wide range of gravitational theories, including those with complex electromagnetic interactions.

To prove that this new method works, the study applied it to two specific, challenging examples. The first was a "torus-like" black hole in a universe with a cosmological constant, a theoretical object shaped more like a ring or a donut than a sphere. The second was a charged black hole in a theory of nonlinear electrodynamics known as Einstein-Euler-Heisenberg, which describes how light interacts with itself in extreme conditions. In both cases, the researcher calculated the mass, electric charge, and entropy of these black holes using the new field-dependent formalism. The results were striking: the calculated entropy matched the area of the black hole's event horizon divided by four, a fundamental result in black hole physics. Furthermore, the relationship between the changes in mass, charge, and entropy perfectly satisfied the first law of thermodynamics.

A key finding of the study is that while the mass and electric charge of these black holes remained consistent with previous calculations, the treatment of entropy was where the new method shone. In the standard approach, the symmetry generator for entropy is often treated as a fixed constant. However, Ding showed that when the generator is allowed to depend on the fields, the calculation of entropy becomes a direct consequence of the theory's structure. This means the first law is not just a rule that happens to work; it is a direct mathematical result of the theory itself, provided the field dependence is handled correctly. The study confirms that this approach is independent of where you choose to measure the charges, whether at the edge of the universe or right next to the black hole's horizon, as long as the chemical potentials like temperature and electric potential are evaluated at the horizon.

The implications of this work are that the first law of black hole thermodynamics is more universal than previously demonstrated. It holds true even in theories where the rules of symmetry are dynamic and change with the environment. The researcher also noted that this formalism can be easily extended to other complex theories, such as those involving non-Abelian gauge fields, which are central to the Standard Model of particle physics. By removing the need for restrictive assumptions about fixed symmetries, this work provides a cleaner, more general path to understanding the thermodynamic nature of black holes. It suggests that the deep connection between gravity, thermodynamics, and symmetry is robust, surviving even when the mathematical rules governing them are allowed to shift and adapt to the physical reality they describe.

Technical Summary: Conserved Charges and the First Law of Black Holes for Field-Dependent Symmetry Generators

Problem Statement
The paper addresses a specific limitation within the Covariant Phase Space Formalism (CPSF), also known as Wald's formalism, regarding the treatment of symmetry generators. In standard applications of CPSF to gravitational theories with gauge fields (e.g., Einstein-Maxwell or Einstein-Yang-Mills), symmetry generators are typically treated as fixed vectors or parameters independent of the dynamical fields. However, in many physical scenarios, symmetry generators $\epsilon$ depend on the field configuration $\Phi$ (i.e., $\epsilon = \epsilon(\Phi)$ ), leading to non-vanishing variations $\delta\epsilon \neq 0$ .

While previous works have addressed field-dependent diffeomorphisms, there was a need for a systematic generalization that explicitly incorporates internal gauge transformations alongside diffeomorphisms when the generators are field-dependent. The absence of a unified framework for these "field-dependent symmetry generators" complicates the rigorous derivation of conserved charges and the first law of black hole thermodynamics, particularly for theories where the generators themselves vary across the phase space.

Methodology
The author generalizes the CPSF to accommodate symmetry generators $\epsilon = \{\xi, \lambda\}$ that are functions of the dynamical fields $\Phi$ , where $\xi$ represents the spacetime diffeomorphism and $\lambda$ represents the internal gauge transformation. The core methodological steps include:

Generalized Symmetry Definition: The symmetry generator is defined as $\epsilon(\Phi)$ , implying $\delta\epsilon \neq 0$ . The transformation of fields is given by $\delta_\epsilon \Phi = \{\mathcal{L}_\xi \Phi, \delta_\lambda \Phi\}$ .
Extension of Noether Currents: The standard Noether current $J_\epsilon$ is re-evaluated considering the field dependence. The variation of the Lagrangian under $\epsilon$ includes terms arising from the variation of the generator itself.
Derivation of Surface Charge Density: By analyzing the variation of the Noether current and utilizing the Lee-Wald symplectic potential $\Theta$ $Θ$ , the author derives a modified expression for the surface charge density $k_\epsilon(\delta\Phi, \Phi)$ $k_{ϵ} (δ Φ, Φ)$ . This expression includes an additional term $\Sigma_\epsilon(\delta\Phi, \Phi)$ $Σ_{ϵ} (δ Φ, Φ)$ , which arises specifically from the field dependence of the generator and the presence of Chern-Simons-like terms or gauge field variations.
- The infinitesimal charge variation is defined as $\delta H_\epsilon = \oint_{\partial\Sigma} k_\epsilon$ .
Proof of the First Law: The formalism is applied to stationary, axisymmetric black holes. The entropy generator $\eta_H$ is constructed as a linear combination of mass, angular momentum, and gauge charge generators, weighted by chemical potentials (temperature, angular velocity, electric potential). Crucially, because the chemical potentials depend on solution parameters, the entropy generator $\eta_H$ is field-dependent ( $\delta\eta_H \neq 0$ ).
Validation via Examples: The formalism is tested on two specific solutions:
- A torus-like black hole in Einstein-Maxwell theory.
- A charged Einstein-Euler-Heisenberg (EEH) Anti-de Sitter (AdS) black hole in nonlinear electrodynamics.

Key Contributions

Generalized Formalism: The paper provides a systematic derivation of the CPSF conjugate to field-dependent symmetry generators that include internal gauge transformations. This extends previous work (e.g., Ref. [25]) which focused primarily on diffeomorphisms.
Direct Derivation of the First Law: The author demonstrates that when field dependence is properly accounted for, the first law of black hole thermodynamics ( $\delta M = T_H \delta S + \Omega_H \delta J + \Phi_H \delta Q$ ) emerges as a direct mathematical consequence for generally invariant gravitational theories (up to total derivative terms).
Surface Independence: A significant feature of this formalism is that the calculation of conserved charges and the proof of the first law are independent of the choice of the integration surface $\Sigma$ or the asymptotic region. The integration can be performed on any smooth, closed codimension-2 surface surrounding the singularity, provided the symplectic current vanishes on-shell for the relevant symmetries.
Clarification of Entropy: The work clarifies that while the generators for mass, angular momentum, and electric charge are often field-independent (yielding results consistent with the original Wald formalism), the generator for entropy is inherently field-dependent. Consequently, the field dependence of the formalism manifests primarily in the definition and calculation of black hole entropy.

Results
The application of the generalized formalism to the two test cases yielded the following results:

Torus-like Black Hole (Einstein-Maxwell):
- The conserved mass $M$ and electric charge $Q$ were derived. The mass matched the parameter $m$ , while the electric charge was found to be $Q = 2\sqrt{\pi}q$ , differing from the parameter $q$ in the original literature [34].
- The entropy was calculated as $S = \pi^2 r_H^2$ , which corresponds to one-quarter of the horizon area ( $A/4$ ).
- The first law $\delta M = T_H \delta S + \Phi_H \delta Q$ was verified to hold.
Charged EEH-AdS Black Hole (Einstein-Euler-Heisenberg):
- For this nonlinear electrodynamics model, the mass and electric charge were found to be $M=m$ and $Q=q$ , respectively, consistent with the original Wald formalism because the corresponding generators are field-independent.
- The entropy was derived as $S = \pi r_H^2$ (one-quarter of the horizon area).
- The first law was confirmed to be satisfied.

Significance and Claims
The paper claims that its primary significance lies in unifying the treatment of diffeomorphisms and internal gauge transformations within the CPSF when symmetry generators are field-dependent. The author asserts that this approach removes the necessity for linearizing dynamical fields or restricting calculations to specific horizons or asymptotic regions.

The work posits that the first law of black hole thermodynamics is not merely an empirical observation or a result requiring specific boundary conditions, but a "direct result" for generally invariant gravitational theories when the field dependence of symmetry generators is correctly incorporated. The formalism is presented as a robust tool for investigating black hole thermodynamics in complex theories, such as those involving Chern-Simons terms or nonlinear electrodynamics, where standard methods might face ambiguities regarding the definition of conserved charges. The author notes that the formalism can be straightforwardly extended to non-Abelian gauge theories and higher-form gauge fields.

Technical Summary: Conserved Charges and the First Law of Black Holes for Field-Dependent Symmetry Generators

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