Observational Tests of Regular Black Holes with Scalar Hair and their Stability

TL;DR

The paper addresses how regular black holes with phantom scalar hair characterized by a scalar charge $A$ modify geodesic structure and observable signatures. It develops the four-dimensional setup, derives the equations of motion for particles and photons, and analyzes null geodesics to link strong-field dynamics to the optical appearance via Lyapunov exponents and shadow size. Solar System tests bound $A$ to be very small, preserving general relativity in weak fields, while strong-field analysis shows a systematic stabilization of the photon sphere and a larger, smoother shadow as $A$ grows; these findings are further constrained by EHT observations of M87* and Sgr A*. The results offer a coherent framework for testing regular black holes with current and near-future high-resolution observations of near-horizon gravity.

Abstract

We study the geodesic structure and observable properties of asymptotically flat regular black holes sourced by a phantom scalar field characterized by a scalar charge $A$. This parameter removes the central singularity and continuously deforms the Schwarzschild geometry. The equations of motion for test particles and photons are derived, and the resulting null geodesics are analyzed, including the deflection of light, gravitational time delay, and redshift, in order to constrain $A$ using classical Solar System tests. These observations impose stringent limits on the scalar charge, confirming that $A$ must remain extremely small in the weak-field regime to ensure full consistency with general relativity. In the strong-field regime, we compute the Lyapunov exponent $λ$ associated with the photon sphere and establish its exact relations with the critical impact parameter $\mathcal{B}_u$ and the angular size of the shadow $α_{\mathrm{sh}}$, given by $\mathcal{B}_u = 1/|λ|$ and $α_{\mathrm{sh}} = 1/(r_{0}|λ|)$. These correspondences reveal that the dynamical instability of null circular orbits governs the optical appearance of the black hole. Our results show that increasing $A$ reduces the instability of photon trajectories and enlarges the angular size of the shadow, indicating that the regularization scale leaves a distinct observational imprint on the geometry of regular black holes. In addition, constraints derived from Event Horizon Telescope observations of M87* and Sgr A* further restrict the allowed range of the scalar charge, reinforcing the consistency of the model with current astrophysical observations.

Figures (10)

• Figure 1: Plot of the lapse function $b(r)$. Here we have used the value $m=1$. The event horizon is $r_+=2.000$ for $A=0$, $r_+=1.901$ for $A=1$, $r_+=1.610$ for $A=2$, $r_+=1.146$ for $A=3$, $r_+=0.526$ for $A=4$, and $r_+=0$ for $A=4.71$.
• Figure 2: Plot of the effective potential of photons. Here we have used the values $L=1$, $m=1$.
• Figure 3: Plot of the radial acceleration for massless particles. Here, $M=1$, and $L=1$. The graph shows the radial acceleration. The radial acceleration is maximum at the inflection point of the effective potential. Note that for $r_+<r<r_u$, the radial acceleration $a_{r}<0$, for $r=r_u$, the radial acceleration $a_r=0$, for $r_u<r<\infty$, the radial acceleration $a_r>0$.
• Figure 4: Polar plot for deflection of light with $m=1$, and $L = 1$. All trajectories have the same energy $E^2=0.015$.
• Figure 5: The capture zone, trajectories can plunge into the horizon or escape to infinity. Here, $m = 1$, $L = 1$, and $E^2=0.04$.