Lorentz-violating modifications to particle dynamics, thermodynamics and vacuum energy in bumblebee gravity

TL;DR

This work investigates how spontaneous Lorentz symmetry breaking in bumblebee gravity alters particle dynamics, thermodynamics, and vacuum energy around a static black hole. By deriving a modified dispersion relation via an optical–mechanical correspondence, the authors compute optical properties (refractive index, group velocity, time delay), interparticle potentials, and electron-scattering cross sections under Lorentz violation. They then develop a massless-boson ensemble framework to obtain analytic expressions for local thermodynamic quantities—pressure, mean energy, entropy, and heat capacity—in three regimes (near horizon, photon sphere, and asymptotic region)—finding that LIV generally enhances these observables and yields finite asymptotic plateaus. Finally, they analyze the curved-background vacuum state, deriving the regularized zero-temperature Casimir energy and finite-temperature thermal effects, revealing a LIV dependence in horizon tension and vacuum-induced thermodynamics. Overall, the results highlight robust LIV signatures in kinematics, interactions, and vacuum phenomena, with potential observational implications near black holes.

Abstract

We investigate how spontaneous Lorentz symmetry breaking in bumblebee gravity modifies particle dynamics, thermodynamics, and vacuum energy around a static black hole background. Starting from the optical-mechanical correspondence, we derive a modified dispersion relation that encodes the influence of the Lorentz-violating parameter $λ$ on the propagation of massive and massless modes. We analyze the resulting optical properties, including the effective refractive index, group velocity, and energy-dependent time delay, and show how the non-asymptotically flat geometry reshapes signal propagation. From the same dispersion relation, we construct the interparticle potential for massive and massless excitations and evaluate the electron scattering cross section within the Born approximation, identifying characteristic Lorentz-violating corrections. We then develop a statistical-ensemble description based on the deformed energy-momentum relation and obtain analytic expressions for the thermodynamic observables of a massless bosonic gas. The pressure, mean energy, entropy, and heat capacity are examined in three representative regimes -- extremely close to the horizon, near the photon sphere, and in the asymptotic region -- where Lorentz violation systematically increase the magnitude of these quantities and leads to finite asymptotic plateaus. Finally, we analyze the vacuum state in the curved background and compute the regularized Casimir energy at zero and finite temperature.

Figures (33)

• Figure 1: The behavior of the refractive index $\mathrm{n}(r)$ is illustrated as a function of the radial coordinate for different choices of the parameter $\lambda$, while the Schwarzschild limit is displayed for reference. Throughout the analysis the mass has been fixed to $M=1$. In addition, the curves approach the asymptotic constant value $-\sqrt{1+\lambda}$ at large distances.
• Figure 2: The behavior of the group velocity $v_{g}$ is shown for different choices of the parameter $\lambda$, alongside the Schwarzschild solution, which is plotted for reference.
• Figure 3: In the massless regime, the group velocity $v_{g}(m \to 0)$ is plotted for various values of $\lambda$, with the Schwarzschild solution included as a benchmark.
• Figure 4: The dependence of the time delay $\Delta t$ on the parameters $d_{1}$, $d_{2}$, and $\lambda$ is illustrated, showing how different choices of these quantities modify the resulting curves.
• Figure 5: The plot displays how the interparticle potential $V(r)$ behaves for massive modes across different values of the Lorentz--violating parameter $\lambda$, showing its dependence on the radial coordinate $r$. To highlight deviations induced by $\lambda$, the conventional Schwarzschild configuration is also represented for direct comparison.