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

Spacetimes 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 spacetime and compare the results with those obtained for the Reissner-Nordstr\"om 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 spacetime metrics with spontaneous symmetry breaking. These findings provide useful insights into how models of spontaneous Lorentz symmetry breaking influence gravitational dynamics in the spacetimes of charged black holes.