Free-falling test particles in a charged Kalb-Ramond black hole: gravitational Doppler effect and tidal forces

TL;DR

This work analyzes the dynamics of free-falling test particles in a static, spherically symmetric charged Kalb-Ramond black hole incorporating spontaneous Lorentz symmetry breaking via a non-minimal KR–Maxwell coupling. By deriving the metric, geodesic equations, Doppler signals, and tidal-force equations, the authors show how the Lorentz-violating parameter $l$ and charge $q$ modify horizon structure, radial and angular tides, and geodesic deviation relative to Reissner–Nordström and Schwarzschild spacetimes. They provide analytic solutions for geodesic deviation under two initial-condition regimes and demonstrate RN-like behavior with LV corrections that are most pronounced near the singularity. The paper also compares KR results with other LV metrics (RN-like and bumblebee models), illustrating how different Lorentz-violating implementations shape strong-field gravitational dynamics and offering potential observational discriminants via horizons, time dilations, and tidal responses.

Abstract

Space-times exhibiting spontaneous Lorentz symmetry-breaking have recently attracted much attention, with Kalb-Ramond (KR) gravity providing a notable example. In this context, we examine the free-fall motion of a test particle toward an electrically charged black hole arising from the coupling of the KR field with the Maxwell one in General Relativity. We investigate how the Lorentz symmetry-breaking parameter affects the free-fall velocity of the particle as it approaches black hole inner regions. Additionally, we analyze the influence of this parameter on the emission and detection of signals by observers in different frames. We furthermore explore modifications to the radial and angular components of tidal forces in this space-time and compare the results with those obtained for the Reissner-Nordström black hole. Finally, we analytically solve the geodesic deviation equation under two different conditions, identifying a subtle effect of the Lorentz symmetry breaking parameter in the charged KR metric, and compare it with two other space-time metrics with spontaneous symmetry breaking. These findings provide useful insights into how models of spontaneous Lorentz symmetry-breaking influence gravitational dynamics in the space-times of charged black holes.

Figures (15)

• Figure 1: Velocities reached by the test particle in free fall, outside the outer horizon $(v)$ towards the center of the black hole and between the horizons $(V)$. We took $q=0.6$.
• Figure 2: Ratio between the frequencies of electromagnetic signals emitted by a free-falling test particle and received by a fixed observer between the $r_{\pm}$ horizons. Again $q=0.6$.
• Figure 3: (a) Radial tidal forces with $q=0.3$ and different $l$. (b) Radial tidal forces with $l=1.24\times 10^{-1}$ for different $q$. Here we have not used the negative values of $l$ because within the range obtained in KReletrico its effect is very small and its effects are not evident.
• Figure 4: (a) Angular tidal forces with $q=0.3$ and different $l$. (b) Angular tidal forces with $l=1.24\times 10^{-1}$ for different $q$. Note that we did not used negative values of $l$ because their effects can be neglected within the range obtained in KReletrico.
• Figure 5: Event horizon $r_{\pm}$, $R^{\rm rad}_0$, $R^{\rm ang}_0$ and $R^{\rm stop}$ plotted in (a) for $l<0$ as functions of $q/M$. The vertical dashed line marks the position of the critical charge $q_c=1.00689 M$ for $l$. In (b) $r_{\pm}$, $R^{\rm rad}_0$, $R^{\rm ang}_0$ and $R^{\rm stop}$ are plotted for $l>0$ as functions of $q/M$ and the critical charge indicated by the vertical dashed line at $q_c= 0.819891 M$.