Phantom scalar charge as a control parameter for optical geometry and photon orbit stability around black holes

Abstract

A phantom scalar field substantially alters the strong-field behavior of black hole spacetimes. We systematically investigate how the phantom scalar charge \(\alpha \) influences null geodesics through the geometry of the optical manifold and the photon dynamics. By examining the Gaussian and geodesic curvatures of the optical geometry, we demonstrate that the phantom parameter \(\alpha \) governs a smooth crossover from a strongly curved, intense-gravity regime to a stable weak-field regime. The present work also employs phase-portrait analysis using logarithmic and semi-logarithmic plots. We show that the Gaussian curvature encodes a smooth crossover between strong and weak-curvature regimes in the optical manifold. Furthermore, we explore how key observables—such as the effective potential, photon trajectories, the photon-sphere radius, the black hole shadow, and both weak and strong deflection angles—depend on \(\alpha \). We additionally evaluate the Lyapunov exponent \(\lambda \), which quantifies the instability of circular null geodesics in the photon sphere. This exponent attains its maximum at the radius of the photon sphere (\(r_{\text{ph}}\)) and decreases as \(r_{\text{ph}}\) increases. The phantom charge \(\alpha \) systematically reduces the Lyapunov exponent \(\lambda \), smoothing the effective potential and lowering orbital instability. For small \(\alpha \), one recovers a Schwarzschild-like strong instability with a steep potential, while for large \(\alpha \), gravitational focusing is weakened, the optical geometry is reshaped, and null orbits become more stable. Our results indicate that the phantom scalar charge acts as a control parameter governing both the geometry and the stability of black hole photon regions.