Study of Null Geodesics and their Stability in Kalb-Ramond Black Holes

Abstract

We investigate null geodesics and their stability in four-dimensional charged Kalb-Ramond black holes with a cosmological constant. The non-vanishing vacuum expectation value of the background antisymmetric tensor field induces spontaneous Lorentz symmetry breaking, controlled by a dimensionless parameter $l$, and deforms the Reissner-Nordström-(A)dS geometry. We derive the effective potential for massless particles and obtain analytic expressions for the photon-sphere radius, critical impact parameter, capture cross section, and deflection trajectories. The limiting cases $l\rightarrow0$ recover the RN, RN-dS and RN-AdS geodesic structures, while finite values of $l$ shift the turning points, modify the bending of light and deform both first-kind and second-kind photon trajectories. In the AdS branch, we find a special limiting limaçon-type null geodesic whose angular structure is modified by the Kalb-Ramond parameter. We also compute the Lyapunov exponent of the unstable circular null orbit and show that the cosmological constant changes the instability timescale, whereas the Lorentz-violating parameter produces a non-trivial deformation of the photon-sphere instability. Our results identify null geodesics as sensitive probes of Lorentz-violating effects in charged black hole spacetimes.