Maxwell-dilaton-dRGT Massive Gravity

• Maxwell-dilaton-dRGT massive gravity is a theory that couples Einstein gravity with a scalar dilaton and Maxwell field, augmented by dRGT mass terms to break spatial diffeomorphisms.
• The model offers rich black hole and black brane solutions, including Lifshitz branes and spherically symmetric black holes with modified thermodynamics and reverse reentrant phase transitions.
• It produces finite DC transport coefficients and observable effects such as altered black hole shadows, providing a bridge between theoretical predictions and astrophysical observations.

Maxwell-dilaton-dRGT-like Massive Gravity refers to a class of gravitational theories that couple Einstein gravity to both a scalar dilaton and a Maxwell $U(1)$ gauge field, and introduce graviton mass terms of de Rham–Gabadadze–Tolley (dRGT) type. These models are constructed to interpolate ultraviolet (dilaton) and infrared (massive gravity) corrections, allowing both for rich black hole and black brane solutions and for controlled breaking of bulk diffeomorphism invariance, with direct implications for holography, black hole thermodynamics, phase transitions, and observable black hole shadows.

1. Gravitational Action and Field Content

The Maxwell-dilaton-dRGT-like massive gravity model is governed by the action in four dimensions

$$S = \frac{1}{16\pi}\int d^4x \sqrt{-g} \left[ \mathcal{R} - 2(\nabla\varphi)^2 - V(\varphi) - F_{\mu\nu}F^{\mu\nu} + e^{-2\beta\varphi} \, m_g^2 \sum_{i=1}^4 \eta_i u_i(g,h) \right]$$

where:

• $\mathcal{R}$: Ricci scalar of $g_{\mu\nu}$,
• $\varphi$: dilaton field with potential $V(\varphi)$,
• $F_{\mu\nu}$: Maxwell tensor,
• $m_g^2 \sum_i \eta_i u_i$: dRGT-type mass term, $u_i(g,h)$ being the $i$-th symmetric polynomial of the eigenvalues of $K^\mu{}_\nu=\sqrt{g^{\mu\rho}h_{\rho\nu}}$,
• $h_{\mu\nu}$: fixed (reference) metric,
• $\beta$: dilaton–massive gravity coupling constant,
• $\eta_i$: massive gravity parameters, with in 4D only $u_1$ and $u_2$ generally nonzero (Heidari et al., 3 Dec 2025, Yue et al., 20 Mar 2024, Kuang et al., 2017).

By a suitable choice of $h_{\mu\nu}$ (typically diagonal with vanishing $tt$ and $rr$ components), only spatial diffeomorphisms are broken, enabling controlled IR momentum relaxation in the boundary theory (Kuang et al., 2017, Zhou et al., 2015). The coupling $e^{-2\beta\varphi}$ multiplies the graviton mass term and represents a nonminimal interaction between the dilaton and the massive gravity sector.

2. Solutions: Black Branes and Spherically Symmetric Black Holes

The model admits both planar black brane and spherically symmetric black hole solutions:

• Lifshitz black branes with metric ansatz

$$ds^2 = -r^{-2z}f(r)dt^2 + r^{-2}(dx^2 + dy^2) + \frac{dr^2}{r^2 f(r)}, \quad A=A_t(r)dt, \quad \varphi = \varphi(r)$$

support arbitrary dynamical exponent $z$, with the dilaton and massive gravity coupling encoded in exponential or two-term ansätze for $Z(\varphi), \beta(\varphi), V(\varphi)$ (Kuang et al., 2017).

• Spherically symmetric black holes with

$$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 R^2(r)d\Omega_{(2)}^2, \quad R(r) = e^{\alpha \varphi(r)}$$

admit explicit analytic solutions for $\varphi(r)$, determined by the parameter $\alpha$ (dilaton coupling), and a metric function $f(r)$ which consists of a sum of power laws in $r$, each corresponding to vacuum, cosmological constant, charge, and dRGT-massive gravity terms (Yue et al., 20 Mar 2024, Heidari et al., 3 Dec 2025).

The gauge field acquires a distinctly dilaton-modulated profile, e.g., for $\alpha<1$,

$$A_t(r) = \frac{1+\alpha^2}{\alpha^2-1}\,Q\,\delta^{-\frac{2\alpha^2}{1+\alpha^2}}\,r^{-\frac{1-\alpha^2}{1+\alpha^2}}$$

where $Q$ is the electric charge and $\delta$ an integration constant.

3. Thermodynamics and Phase Structure

The model's black holes and branes admit a complete set of thermodynamic quantities derived from geometric and Wald/EUclidean techniques:

• Entropy: $S = \pi r_h^2 R(r_h)^2$ (area law modulated by dilaton)
• Temperature: $T = \frac{f'(r_h)}{4\pi}$
• Mass, charge, potential, and thermodynamic volume: Computed analytically, with modifications from both Maxwell and massive gravity terms
• Equation of state: $T = T(r_h, Q, P; \alpha, \beta, \eta_i)$, with $P$ identified with $-\Lambda/8\pi$

Critical point structure is determined by the conditions

$$\frac{\partial T}{\partial r_h} = 0, \qquad \frac{\partial^2 T}{\partial r_h^2} = 0$$

allowing for multiple real, positive critical radii.

A novel "reverse reentrant" phase transition is triggered when two critical points are present ($\alpha\neq0$), manifesting as an SBH–LBH–SBH transition sequence (small–large–small black hole order) as temperature increases, opposite to the usual reentrant transition. The phase diagram displays a swallowtail in the $G$–$T$ plane and a tricritical point where first- and zeroth-order transition curves meet. These effects are fundamentally controlled by the dilaton–massive gravity coupling $e^{-2\beta\varphi}$ (Yue et al., 20 Mar 2024).

4. DC Transport, Momentum Dissipation, and Holography

In the context of holography, the Maxwell-dilaton-dRGT-like model describes dual quantum systems with nonrelativistic scaling and explicit momentum relaxation:

• Finite DC conductivities: The graviton mass term (with $f_{\mu\nu}$ breaking translation invariance) ensures finite DC electric, thermoelectric, and thermal conductivities:

$$\sigma_{DC} = r_h^{2-2z} + \frac{Q^2 r_h^2}{2(z^2+z-2) + B_1 r_h^2}$$

and similar expressions for $\alpha_{DC}$ and $\bar{\kappa}_{DC}$, where $M_\text{eff} = 2(z^2+z-2)+B_1 r_h^2$ sets the effective graviton mass squared (Kuang et al., 2017).

• Wiedemann–Franz law violation: The nontrivial dependence of Lorenz ratios $L = \bar{\kappa}_{DC}/(\sigma_{DC} T)$ on $z$ and $B_1$ signals breakdown of the Wiedemann–Franz law, interpreted as a marker of strong interactions in the dual theory.
• Momentum dissipation: The reference metric $f_{\mu\nu}$ of the form $\mathrm{diag}(0,0,1,1)$ in the mass term breaks spatial diffeomorphism invariance in the bulk and relaxes momentum in the boundary field theory, regularizing otherwise divergent transport coefficients.

5. Stability, Phase Transitions, and Geometrothermodynamics

Thermodynamic stability is governed by local (heat capacity) and global (Gibbs free energy) analyses:

• Local stability: Heat capacity at fixed charge ($C_Q$) or chemical potential ($C_\mu$) — positive values indicate stability.
• Second-order phase transitions: Divergences of $C_Q$ correspond to continuous (second-order) SBH–LBH transitions; zeros mark physical-limitation points.
• Geometrothermodynamics: The HPEM metric on the $(S,Q)$ space produces a Ricci scalar whose divergences exactly match those of $C_Q$, providing a robust geometric diagnostic for phase transitions (Heidari et al., 3 Dec 2025).

6. Optical Properties: Photon Sphere and Black Hole Shadows

The structure of unstable null orbits determines observable shadows:

• Photon sphere $r_\mathrm{ph}$: Calculated via extrema of the effective potential $V_\mathrm{eff}(r)$, depending on $(\alpha, q, \eta_1, m_g)$.
• Black hole shadow radius:

$$\mathcal{R}_{sh} = b_{crit} = \frac{r_{ph}R(r_{ph})}{\sqrt{f(r_{ph})}}$$

For given EHT measurements (e.g., $Sgr A^*$), the relation $\mathcal R_{sh}/M$ constrains the gravity and dilaton parameters (Heidari et al., 3 Dec 2025).

Typically, increasing $q$ or $\eta_1$ enlarges the shadow, while increasing $m_g$ reduces its size. This provides observationally testable constraints on the allowed parameter space of Maxwell-dilaton-dRGT-like massive gravity.

7. Energy Emission and Observational Signatures

The Hawking emission spectrum at high frequency asymptotes to $\sigma_{lim} \simeq \pi \mathcal{R}_{sh}^2$. The peak of the emission rate

$$\frac{d^2E}{d\omega dt} = \frac{2\pi^2 \sigma_{lim} \omega^3}{e^{\omega/T}-1}$$

is modulated by ($\alpha, q, \eta_1, m_g$): higher values of $\alpha, q, \eta_1$ raise and blue-shift the peak, while higher $m_g$ lowers and red-shifts it (Heidari et al., 3 Dec 2025). These features could in principle be mapped to observational data from black hole shadow and spectrum measurements.

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