Black Holes in Lorentz-Violating Gravity: Thermodynamics, Geometry, and Particle Dynamics

TL;DR

This paper analyzes black holes in Lorentz-violating gravity with Kalb–Ramond couplings, deriving static solutions with linear and quadratic LIV terms and exploring how a perturbative LIV parameter reshapes horizon structure and thermodynamics. It establishes a universal extremality relation linking mass, entropy, and LIV perturbations, and develops a thermodynamic-topology framework via Duan’s $\varphi$-mapping to classify phase structure and photon-sphere defects. The authors quantify microstructure interactions using Ruppeiner geometry in extended (AdS) thermodynamics, finding a persistent, ensemble-independent negativity of the curvature, which signals attraction-dominated microstructures enhanced by LIV. They also study timelike geodesics in the LIV spacetime, showing LIV couplings alter ISCOs, photon-sphere radii, and orbit stability, with numerical simulations revealing bound, precessing, and plunging trajectories shaped by $\ell_{1,2}$ and $\Lambda$. Overall, LIV effects enrich the black-hole thermodynamic and geometric landscape while maintaining a consistent, universal framework that could inform observations and quantum-gravity consistency conditions such as the Weak Gravity Conjecture.

Abstract

We investigate the thermodynamics, topology, and geometry of black holes in Lorentz-violating gravity. Modifications in the theory by perturbative parameter lead to coupled changes in horizon structure and thermodynamic behaviour, allowing us to derive generalized universal relations and explore implications for the Weak Gravity Conjecture. The thermodynamic topology reveals distinct topological charges, with photon spheres identified as robust topological defects. Our analysis shows that the Ruppeiner curvature remains universally negative across thermodynamic ensembles, indicating dominant attractive interactions among microstructures. This ensemble-independent behaviour highlights a fundamental thermodynamic universality in Lorentz-violating settings. Together, these results provide a consistent and rich framework for understanding black hole microphysics and gravitational consistency in modified theories. We further study the motion of timelike test particles in these black hole spacetimes by analyzing the effective potential shaped by the Lorentz-violating couplings. The resulting dynamics reveal the existence of bound orbits and stable circular trajectories, with the location of the innermost stable circular orbit and turning points significantly influenced by the parameters $\ell_{1,2}$, and the cosmological constant. Numerical simulations of trajectories in the $x-y,\,x-z$, and 3D planes show precessing, bounded, and plunging orbits, depending on the particle's specific energy and angular momentum. These results highlight how Lorentz-violating effects alter the structure of geodesic motion and provide potential observational signatures in the dynamics of massive particles near black holes.

Figures (11)

• Figure 1: The $(\tau \text{ vs. } r_h)$ diagram, depicted in Figs. (\ref{['3a']}), (\ref{['3c']}), (\ref{['3e']}), and (\ref{['3g']}), illustrates the variations in free parameters for a black hole with a nonlinear electromagnetic field in the presence of a phantom global monopole. This representation elucidates the dependence of thermodynamic quantities on the horizon radius $r_h$, highlighting critical transition points within the phase structure. Additionally, the normal vector field $n$ in the $(r_h - \Theta)$ plane is presented, demonstrating the distribution of Zero Points (ZPs) at specific coordinates $(r_h, \Theta)$. These ZPs correspond to parameter values $\ell_1 = 0.1, 1$ and $\ell_2 = 0.1, 1$, serving as key indicators of stability properties and topological characteristics within the thermodynamic framework.
• Figure 2: The $(\tau \text{ vs. } r_h)$ diagram, as illustrated in Figs. (\ref{['4a']}), (\ref{['4c']}), and (\ref{['4e']}), depicts the variation of free parameters associated with a black hole characterized by a nonlinear electromagnetic field in the presence of a phantom global monopole. This graphical representation provides insight into the dependence of thermodynamic quantities on the horizon radius $r_h$, highlighting crucial transition points within the phase structure. Furthermore, the normal vector field $n$ in the $(r_h - \Theta)$ plane is introduced, illustrating the spatial distribution of Zero Points (ZPs) at specific coordinates $(r_h, \Theta)$. These ZPs, corresponding to parameter values $\ell_1 = 0.1, 0.5, 1$, play a fundamental role in revealing the stability properties and topological characteristics of the thermodynamic system.
• Figure 3: The photon spheres (PSs) for different parameter configurations with $\ell_1 = 0.1, \ell_2=0.1$, $M = 0.1, 1$.
• Figure 4: The photon spheres (PSs) for different parameter configurations with $\ell_1 = 0.1, 0.5$, $M = 1$.
• Figure 5: The behavior of black hole temperature $T$ with entropy $S$ for fixed values of $l_1, l_2$ and thermodynamic pressure $P$.