ModMax-AdS Black Hole with Global Monopole as Source in Kalb-Ramond Gravity

TL;DR

This work analyzes a static ModMax-AdS black hole sourced by a global monopole in Kalb-Ramond gravity, focusing on thermodynamics in extended phase space and the optical properties of photon orbits. The authors derive the metric, identify the effective quantities $\Lambda_{\mathrm{eff}}$ and $Q_{\mathrm{eff}}$, and compute enthalpy $M$, temperature $T$, entropy $S$, Gibbs free energy $G$, and $C_P$, confirming the first law $dM=T dS+V dP+\Phi dQ_{\mathrm{eff}}$ and Smarr relation $M=2TS-2PV+\Phi Q_{\mathrm{eff}}$, with $V=(4\pi/3)r_h^3$ and $\Phi=e^{-\gamma}Q_{\mathrm{eff}}/r_h$. The paper shows Van der Waals-like $P$-$v$ criticality, derives the Joule–Thomson inversion curve and its minimum temperature, and analyzes Hawking radiation sparsity and entropy corrections from thermal fluctuations. Optical analysis yields the photon sphere and shadow radius, connecting thermodynamic parameters to observable strong-field signatures and using Sgr A* shadow measurements to bound model parameters. Overall, the KR and ModMax sectors provide tunable control over both thermodynamic behavior and black-hole optics, suggesting extensions to rotating solutions and quasinormal-mode constraints.

Abstract

In this work, we investigate in detail the thermodynamic properties of a spherically symmetric ModMax-AdS black hole sourced by a global monopole within the Kalb-Ramond gravity. We derive the key thermodynamic quantities, including the Hawking temperature, Gibbs free energy, and specific heat capacity, and analyze how the geometric parameters influence these physical quantities. The first law of thermodynamics and the corresponding Smarr formula are explicitly verified. Furthermore, we study the thermodynamic criticality of the system by deriving the critical points and examining the effects of the space-time geometric parameters. We also obtain the inversion temperature and demonstrate that the minimum inversion temperature is modified by the space-time parameters. In addition, the sparsity of Hawking radiation and thermal fluctuations of the system are investigated, highlighting the effects of the parameters on the entropy corrections. Finally, we analyze the optical properties of the black hole, in particular the photon sphere and shadow radius, showing how these parameters influence these features.

Figures (17)

• Figure 1: Radial profile of the metric function $f(r)$ [Eq. (\ref{['function']})] for $M=1$, $Q=0.5$, $k=1$, and $P=0.003$. In each panel, one parameter is varied continuously over a prescribed interval and encoded by the color gradient (see the vertical colorbar), while the remaining parameters are fixed. Panel (a): $\ell$ varied continuously (with $\eta=0.1$ and $\gamma=0.1$ fixed). Panel (b): $\eta$ varied continuously (with $\ell=0.1$ and $\gamma=0.1$ fixed). Panel (c): $\gamma$ varied continuously (with $\ell=0.1$ and $\eta=0.1$ fixed). The gray dotted horizontal line indicates $f(r)=0$.
• Figure 2: Black hole mass $M$ as a function of the event-horizon radius $r_h$ [Eq. (\ref{['dd2']})], for $Q=0.5$, $k=1$, and $P=0.003$. In each panel, one parameter is varied continuously and encoded by the color gradient (see colorbar), while the remaining parameters are fixed. Panel (a): $\ell$ varied continuously (with $\eta=0.1$ and $\gamma=0.1$ fixed). Panel (b): $\eta$ varied continuously (with $\ell=0.1$ and $\gamma=0.1$ fixed). Panel (c): $\gamma$ varied continuously (with $\ell=0.1$ and $\eta=0.1$ fixed).
• Figure 3: Hawking temperature $T$ as a function of the event-horizon radius $r_h$ [Eq. (\ref{['dd4']})], for $Q=0.5$, $k=1$, and $P=0.003$. In each panel, one parameter is varied continuously and encoded by the color gradient (see colorbar), while the remaining parameters are fixed. Panel (a): $\ell$ varied continuously (with $\eta=0.1$ and $\gamma=0.1$ fixed). Panel (b): $\eta$ varied continuously (with $\ell=0.1$ and $\gamma=0.1$ fixed). Panel (c): $\gamma$ varied continuously (with $\ell=0.1$ and $\eta=0.1$ fixed). The gray dotted horizontal line indicates $T=0$.
• Figure 4: Gibbs free energy $G$ as a function of the event-horizon radius $r_h$ [Eq. (\ref{['dd6']})], for $Q=0.5$, $k=1$, and $P=0.003$. In each panel, one parameter is varied continuously and encoded by the color gradient (see colorbar), while the remaining parameters are fixed. Panel (a): $\ell$ varied continuously (with $\eta=0.1$ and $\gamma=0.1$ fixed). Panel (b): $\eta$ varied continuously (with $\ell=0.1$ and $\gamma=0.1$ fixed). Panel (c): $\gamma$ varied continuously (with $\ell=0.1$ and $\eta=0.1$ fixed). The gray dotted horizontal line indicates $G=0$.
• Figure 5: Specific heat at constant pressure $C_{P}$ as a function of the event-horizon radius $r_h$ [Eq. (\ref{['dd7']})], for $Q=0.5$, $k=1$, and $P=0.003$. In each panel, one parameter is varied continuously and encoded by the color gradient (see colorbar), while the remaining parameters are fixed. Panel (a): $\ell$ varied continuously (with $\eta=0.1$ and $\gamma=0.1$ fixed). Panel (b): $\eta$ varied continuously (with $\ell=0.1$ and $\gamma=0.1$ fixed). Panel (c): $\gamma$ varied continuously (with $\ell=0.1$ and $\eta=0.1$ fixed). Divergences of $C_{P}$ (at fixed $P$) indicate second-order phase transitions separating small, intermediate, and large black-hole branches.