Non-Minimal Maxwell Coupling in Black Hole Thermodynamics

• Non-Minimal Maxwell Coupling is defined by incorporating curvature or scalar-dependent terms into the electromagnetic action, leading to modified black hole thermodynamics and universal critical ratios.
• It enables analysis of generalized Hawking–Page transitions by introducing invariant entropy, temperature, and curvature constants that mirror known universality classes.
• The framework offers insights into extended phase spaces and dualities in Einstein–scalar gravity, bridging gravitational dynamics with nonextensive Rényi statistical mechanics.

Non-Minimal Maxwell Coupling refers to gravitational theories in which the electromagnetic (Maxwell) sector is coupled to curvature or scalar degrees of freedom in a manner that deviates from minimal coupling, introducing novel thermodynamic and critical phenomena in black hole spacetimes. This mechanism is central to the analysis of generalized black hole phase transitions—particularly the Hawking–Page transition—including the emergence of universality classes, critical constants, and connections to nonextensive statistical mechanics such as Rényi statistics. In these frameworks, the extended thermodynamics of black holes subject to non-minimal Maxwell or scalar couplings exhibits universal ratio structures and criticality absent in the standard minimally coupled Einstein-Maxwell theory.

1. Frameworks and Definitions

Non-minimal Maxwell coupling is realized through the inclusion of curvature or scalar-dependent terms in the electromagnetic action, frequently appearing in Einstein-Maxwell-dilaton or Einstein–scalar gravity theories. The action typically takes the form

$$S = \int \sqrt{-g} \left[ R - \frac{1}{2} (\partial \phi)^2 + V(\phi) - f(\phi) F_{\mu\nu} F^{\mu\nu} \right] d^{d+1}x ,$$

where $f(\phi)$ encapsulates the non-minimality. Scalar potentials $V(\phi)$ exhibiting exponential asymptotics, $V(\phi) \sim e^{4\phi/(d-1)}$, admit continuous Hawking–Page transitions with vanishing horizon area, a distinguishing signature absent for constant $V(\phi)$ (Gursoy, 2010). Charged AdS black holes with non-trivial Maxwell–scalar coupling display characteristic thermodynamic invariants and phase structure, including universality of critical ratios and non-trivial scaling exponents.

2. Thermodynamics and Universal Ratios

Thermodynamic analysis of black holes under non-minimal Maxwell coupling reveals universal dimensionless ratios characterizing the Hawking–Page transition. For AdS black holes, the entropy ($S$), temperature ($T$), and other quantities at the minimum ($\min$) and transition ($\mathrm{HP}$) points provide the basis for universal ratios such as

$$C_S = \frac{S_{\mathrm{HP}} - S_{\min}}{S_{\min}}, \qquad C_T = \frac{T_{\mathrm{HP}} - T_{\min}}{T_{\min}} .$$

For large classes of non-rotating black holes, $C_S = 2$ and $C_T = (2 - \sqrt{3})/\sqrt{3}$, invariant under variations in the AdS scale, charge, and under small rotations (Belhaj et al., 2020).

In the Rényi thermodynamic regime, which generalizes Gibbs–Boltzmann statistics, the entropy is defined as

$$S_\lambda = \frac{1}{\lambda} \ln(1 + \lambda S_{\mathrm{BH}}),$$

with Rényi parameter $\lambda$. The transition points are characterized by radii $r_{\min}$ and $r_{\mathrm{HP}}$, leading to universal horizon, temperature, and entropy ratios,

$$U_r = \sqrt{3}, \qquad U_T = \frac{2}{\sqrt{3}}, \qquad U_S = 3,$$

which persist for arbitrary $\lambda$, charge $Q$, or cosmological constant $\Lambda$ (Barzi et al., 2022).

3. Dualities and Curvature Constants

Non-minimal coupling introduces dual relations in the thermodynamics of black holes. For Schwarzschild–AdS black holes, there exists an exact temperature duality between the minimum temperature in $d+1$ dimensions and the Hawking–Page transition temperature in $d$ dimensions:

$T_0(d+1) = T_{\mathrm{HP}}(d),$

which formalizes a holographic correspondence between bulk and boundary critical temperatures (Wei et al., 2020).

Additionally, the normalized Ruppeiner scalar curvature evaluated at the transition point is a universal constant,

$$R(T_\mathrm{HP}, r_\mathrm{HP}) = -\frac{(d-1)(d-3)}{2},$$

interpreted as a threshold of microstructural interaction strength required for black hole formation. The constancy of this curvature, independent of all continuous parameters, places such transitions alongside conventional universality classes in statistical physics.

4. Phase Transition Order and Criticality

Phase transitions in non-minimally coupled black hole systems cover a spectrum from first-order to higher continuous orders. In Einstein–scalar gravity, the order of the transition is determined by the subleading behavior of $V(\phi)$:

• If $V_{\rm sub}(\phi) \sim e^{-k\phi}$ ($k>0$), the transition is $n$-th order, with scaling $\Delta F \sim t^n$, entropy $S \sim t^{n-1}$, and heat capacity $C \sim t^{n-2}$ as $t \to 0$.
• For power-law $V_{\rm sub}(\phi) \sim \phi^{-a}$ ($a>0$), a BKT-type infinite-order transition emerges (Gursoy, 2010).

Transport coefficients display universality: shear viscosity $\eta/s = 1/4\pi$ and bulk viscosity $\zeta/s = 1/[2\pi(d-1)]$ at criticality, independent of the specific form of $V(\phi)$.

5. Geometric Generalizations and Topology

Generalized Hawking–Page transitions occur in bulk spacetimes with topologies $S^{d_1} \times S^{d_2}$, subject to non-minimal electromagnetic or scalar field coupling. The Einstein equations admit two principal branches, each associated with one sphere shrinking smoothly. Transition points are indexed by critical ratios of the boundary radii. For $d_1 + d_2 < 9$, first-order transitions dominate, but a singular solution at a critical ratio features a universal scaling exponent for the free energy difference $\Delta F \sim |R - R_s|^2$. Near this critical point, an infinite tower of solutions accumulates, defining a geometric universality class with sharp thresholds at $d_1 + d_2 = 9$ (Aharony et al., 2019).

6. Physical Implications and Universality Classes

The existence of universal ratios and constants in the thermodynamics of non-minimally coupled black holes implies a robust universality class structure akin to ordinary critical phenomena, but adapted to the gravitational and gauge context. For Hawking–Page transitions, it is found that entropy and temperature jumps, horizon ratios, and Ruppeiner curvature thresholds remain pure numbers, invariant under parametric changes in coupling and background structure. Nonextensive statistics (Rényi extension) preserve this universality, signaling a deep connection between weakly non-minimal coupling and underlying microphysical organization of black hole degrees of freedom (Barzi et al., 2022, Belhaj et al., 2020).

The interplay between non-minimal coupling, geometric topology, and scalar potentials enriches the phase structure of black holes in AdS and flat backgrounds. Continuous and higher-order transitions provide holographic parallels to superfluid and BKT transitions in condensed matter, while universal transport ratios and critical exponents bridge gravitational thermodynamics to well-established paradigms in critical phenomena.

7. Future Directions and Open Questions

Further research is warranted into the generalization of these universal relations to more complex coupling structures, higher-curvature corrections, extended phase spaces, and novel statistics. The persistence of universal transition ratios and curvature constants for arbitrary non-minimal couplings, charges, and nonextensivity parameters suggests that black hole thermodynamics shares deep structural similarities with conventional statistical systems. Questions remain regarding the microscopic origin of these invariants and their significance in holographic dualities and quantum gravitational ensembles. The possibility of new universality classes in exotic topologies and matter couplings is a compelling direction for ongoing investigation.