Gravitational aspects of a new bumblebee black hole

TL;DR

This work analyzes a recently proposed static bumblebee black hole characterized by a Lorentz-violating parameter $\chi$, focusing on how spontaneous Lorentz symmetry breaking reshapes geodesics, shadows, perturbations, and lensing. By reformulating the metric to reveal a global-conical asymptotic structure, the authors compute null and timelike geodesics, photon-sphere properties, quasinormal modes across spins, and both weak- and strong-field lensing, including time delays. A key finding is that the photon-sphere radius $r_c$ and the shadow radius $R_{sh}$ at infinity remain at Schwarzschild values (e.g., $r_c=3M$, $R_{sh}=3\sqrt{3}M$) and are largely insensitive to $\chi$, while LV modifies the perturbation spectra and the energetics of particle motion, with higher $\chi$ generally softening potentials and lengthening relaxation times. The paper also derives finite-observer shadow sizes, presents a topological analysis of the photon sphere, and places Solar-System bounds on $\chi$ from Mercury’s perihelion shift, light bending, and Shapiro delay, finding constraints at the $10^{-11}-10^{-5}$ level. Overall, the LV bumblebee black hole provides a consistent framework to test Lorentz violation in strong gravity, yielding observable signatures in lensing and dynamics while remaining aligned with classical solar-system tests within current limits.

Abstract

In this paper, we examine the physical consequences of a recently introduced black hole solution in bumblebee gravity [1]. The geometry is first presented and then reformulated through suitable coordinate adjustments, which make its global conical character evident. We then study the propagation of particles by solving the geodesic equations for null and timelike trajectories. The associated critical orbits (or photon spheres) are obtained, and shadow radius are computed and compared with other Lorentz-violating configurations in bumblebee and Kalb-Ramond models, including their charged and cosmological extensions. Massive particle motion is analyzed separately, followed by the construction of the effective potentials for scalar, vector, tensor, and spinor perturbations. These potentials allow the calculations of quasinormal frequencies and the corresponding time-domain evolution. Gravitational lensing phenomena are investigated in the weak and strong deflection regimes, and the light-travel time delay is also evaluated. The study concludes with bounds on the Lorentz-violating parameter based on classical Solar System experiments.

Figures (32)

• Figure 1: Light--like geodesics for different values of the Lorentz--violating parameter $\chi$. As $\chi$ increases, the trajectories exhibit a contraction effect.
• Figure 2: Time--like geodesics for different values of the Lorentz--violating parameter $\chi$. As $\chi$ increases, the trajectories exhibit a contraction effect.
• Figure 3: The potential $\mathcal{H}(r,\chi)$ for various values of $\chi$, showing a maximum at the photonic radius $r_{\text{ph}} = 3$. The mass is fixed at $M = 1$.
• Figure 4: The normalized vector field in the $(r,\theta)$ plane with encircled photon sphere point at $r_{\text{ph}} = 3$. The other parameters are fixed at $M = 1$, $\chi = 0.1$.
• Figure 5: The effective potential and the corresponding effective force on a test particle for variation of the Lorentz--violating parameter $\chi$. The mass and angular momentum are set at $M = 1$ and $L = 5$.