Time Like Geodesics of Regular Black Holes with Scalar Hair

TL;DR

This paper studies the motion of particles around regular black holes with scalar hair, revealing how scalar charge influences orbits, perihelion precession, and scattering, with implications for observational constraints.

Contribution

It provides a detailed analysis of timelike geodesics in regular black holes with scalar hair, including orbit classifications and corrections to perihelion precession.

Findings

Scalar charge shifts the location of circular and stable orbits.

Perihelion precession receives scalar charge-dependent corrections.

The scalar hair affects the threshold for scattering and capture.

Abstract

We investigate timelike geodesics in asymptotically flat regular black holes supported by a phantom scalar field characterized by a scalar charge $A$. This parameter removes the central singularity and continuously deforms the Schwarzschild geometry while preserving asymptotic flatness. We derive the equations of motion for massive test particles and classify bounded and unbounded trajectories in terms of the conserved energy and angular momentum. We determine circular and critical orbits, including the innermost stable circular orbit (ISCO), and analyze the transition between capture and scattering. We show that the scalar charge modifies the location of the unstable and stable circular orbits, the ISCO, and the threshold angular momentum for scattering, exhibiting a nontrivial dependence on the radial coordinate. Their physical scales are naturally described in terms of the invariant areal radius $R(r)=\sqrt{r^2+A^2}$. In the weak-field regime, we compute the perihelion precession and obtain corrections proportional to the scalar charge, allowing us to constrain the scalar charge from Solar System observations. We also analyze the motion with vanishing angular momentum and show that, while the qualitative structure of the trajectories remains connected to the Schwarzschild limit $A\to 0$, the quantitative deviations encode the geometric effects of the scalar hair.