- The paper establishes a formalism that promotes the gravitational constant G to a conserved charge in black hole thermodynamics.
- It applies a modified Einstein-Hilbert action with auxiliary scalar and gauge fields alongside the ADT formalism to derive quasi-local conserved charges.
- This approach extends the thermodynamic first law and Smarr formula by treating G on par with mass and the cosmological constant, with implications for holography and quantum gravity.
Gravitational Constant as a Conserved Charge in Black Hole Thermodynamics
Introduction
This paper presents a formalism in which the gravitational constant $G$ is promoted to a conserved charge within the context of black hole thermodynamics. Building on recent developments that allow coupling constants in the action to be interpreted as conserved charges via auxiliary scalar and gauge fields, the authors construct a modified Einstein-Hilbert action in four dimensions. The approach leverages the quasi-local off-shell Abbott-Deser-Tekin (ADT) formalism to derive explicit expressions for the conserved charges associated with mass, cosmological constant, and gravitational constant. The resulting framework is shown to be consistent with the extended thermodynamic first law and the Smarr formula.
Modified Action and Gauge Symmetries
The starting point is a modified Einstein-Hilbert action:
$S = \int d^4x \sqrt{-g} \, \alpha \left[ R + \beta(1 - \nabla_\mu B^\mu) - \nabla_\mu A^\mu \right]$
where $\alpha$ and $\beta$ are scalar fields, and $A^\mu$, $B^\mu$ are gauge fields. This construction, inspired by [Hajian et al., (2303.00000)], enables the promotion of both $G$ and $\Lambda$ to integration constants, which become conserved charges associated with underlying gauge symmetries.
The action is invariant under diffeomorphisms and local gauge transformations of $A^\mu$ and $B^\mu$. The corresponding off-shell Noether current and potential are derived, and the ADT formalism is applied to obtain quasi-local conserved charges. The explicit spherically symmetric solution is given in ingoing Eddington-Finkelstein coordinates, with integration constants $\alpha_0$, $\beta_0$, $\gamma_0$, $C$, and $D$.
Quasi-local Conserved Charges
By constructing a one-parameter family of solutions interpolating between a reference background and the target solution, the authors compute the quasi-local ADT charges associated with the Killing vector and global gauge transformations. The identifications
$\alpha_0 = \frac{1}{16\pi G}, \quad \beta_0 = 2\Lambda, \quad \gamma_0 = 2GM$
lead to the following conserved charges:
- $Q[\xi,0,0] = M$ (mass)
- $Q[0,\lambda,0] = G^{-1}$ (inverse gravitational constant)
- $Q[0,0,\chi] = \frac{\Lambda}{8\pi G}$ (cosmological constant)
This result demonstrates that $G$ can be interpreted as a quasi-local conserved charge, analogous to mass and cosmological constant, and is sourced by the global part of the gauge symmetry associated with $A^\mu$.
The ADT formalism is used to derive the extended thermodynamic first law:
$\delta M = T \delta S + \Phi \delta G^{-1} - V \delta P$
where $T$ is the Hawking temperature, $S$ is the entropy, $\Phi$ is the conjugate potential to $G^{-1}$, $V$ is the thermodynamic volume, and $P$ is the pressure (related to $\Lambda$). The explicit expressions for these quantities are:
- $T = \frac{1}{4\pi}\left( \frac{1}{r_h} + \Lambda r_h \right)$
- $S = \frac{\pi r_h^2}{G}$
- $\Phi = \frac{1}{4}(r_h - \Lambda r_h^3)$
- $V = \frac{4}{3}\pi r_h^3$
- $M = \frac{1}{2G}\left( r_h + \frac{\Lambda}{3} r_h^3 \right)$
- $P = -\frac{\Lambda}{8\pi G}$
The Smarr formula is then obtained via scaling arguments:
$M = TS + \Phi G^{-1} - PV$
This is consistent with the extended first law and places $G$ on equal footing with other thermodynamic variables.
Physical Interpretation and Implications
The formalism implies that the gravitational constant, typically a fixed parameter, can be treated as a thermodynamic variable and a conserved charge. The charge is concentrated at $r=0$, analogous to a point source, and is not radiative in the sense of black hole hair. Unlike mass, charge, or angular momentum, $G$ does not correspond to a degree of freedom that can be radiated away or measured at infinity. This distinction is important for the interpretation of $G$ as black hole hair and for understanding its role in black hole thermodynamics.
The approach provides a consistent principle for varying $G$ and $\Lambda$ in the thermodynamic first law, which is relevant for studies in black hole chemistry and holography, where these constants are related to the number of degrees of freedom in the dual CFT.
Future Directions
The formalism opens several avenues for further research:
- Investigating the physical realization of the gravitational constant as a charge carrier and its implications for quantum gravity.
- Extending the framework to other coupling constants and higher-dimensional theories.
- Exploring the consequences for black hole phase transitions and holographic interpretations.
- Examining the role of $G$ in dynamical processes and its possible measurement in physical space.
Conclusion
The paper establishes a covariant framework in which the gravitational constant is promoted to a conserved charge via a scalar-gauge pair in a modified Einstein-Hilbert action. This construction yields an extended thermodynamic first law and Smarr formula, placing $G$ on the same footing as mass and cosmological constant in black hole thermodynamics. The results have implications for the interpretation of coupling constants as thermodynamic variables and for the broader understanding of black hole thermodynamics in quantum gravity and holography.