The shadows and photon rings of two minimal deformations of Schwarzschild black holes

TL;DR

The paper investigates two minimal deformations of the Schwarzschild black hole, KS and GK, and analyzes their optical appearances under spherical and prograde thin-disk accretion. It derives exact metrics, null geodesics, and shadow-related observables, showing KS enlarges the shadow while GK contracts it, with photon rings becoming narrower for KS and broader for GK. Using EHT data, the authors constrain the deformation parameters, demonstrating the potential to distinguish these hair-induced deformations through shadow imaging and disk-model dependent features. The work highlights that, in thin-disk models, direct emission dominates the observed intensity and that shadow radii are sensitive to the disk's inner edge, providing practical diagnostics for testing GR and alternative theories of gravity.

Abstract

This paper primarily investigates the optical characteristics of two minimal Schwarzschild black hole deformations, the Kazakov-Solodukhin and Ghosh-Kumar black holes, under different accretion models. The event horizon, photon sphere, and critical impact parameter of the former increase compared with the Schwarzschild black hole, but those of the latter decrease. The data from the Event Horizon Telescope Collaboration are used to constrain the parameter ranges of the two black holes. In the case of spherical accretion, the quantum correction of Kazakov-Solodukhin black hole leads to the increase of black hole shadow size and the decrease of integrated intensity, while the shadow size of magnetically charged Ghosh-Kumar black hole decreases and the integrated intensity increases. The shadow radius of the black hole is independent of the spherical accretion models. For an optically and geometrically thin accretion disk, the integrated intensity is mainly contributed by direct emission, and the contributions of photon rings and lensed rings are very small. In addition, the photon rings and lensed rings of Kazakov-Solodukhin black hole are narrower, while those of Ghosh-Kumar black hole are wider. Whereas the Kazakov-Solodukhin black hole exhibits higher brightness, the Ghosh-Kumar black hole shows lower brightness. Additionally, a disk closer to the black hole correlates with a smaller shadow radius. This paper proposes a method to distinguish different black holes in a specific thin disk model.

Figures (14)

• Figure 1: The Hawking temperature $T_{\mathrm{Hawking}}$ as a function of the radial coordinate $r$ for three BHs. The black, red, and blue solid lines correspond respectively to the SC BH, the KS BH with ${a^{\mathrm{KS}}}/M=1$, and the GK BH with ${a^{\mathrm{GK}}}/M=1$. Here, ${a^{\mathrm{KS}}}$ and ${a^{\mathrm{GK}}}$ are the "hair" parameters induced by the deformations.
• Figure 2: The effective potential $V_{\mathrm{eff}}$ as a function of the radial coordinate $r$ for three BHs. The black, red, and blue solid lines correspond respectively to the SC BH, the KS BH with ${a^{\mathrm{KS}}}/M=1$, and the GK BH with ${a^{\mathrm{GK}}}/M=1$.
• Figure 3: Left panel: Dependence of the photon sphere radius $r_p/M$ on the BH hair ${a^{\mathrm{KS}}/M}$ and ${a^{\mathrm{GK}}/M}$. As the parameter ${a^{\mathrm{KS}}/M}$ and ${a^{\mathrm{GK}}/M}$ increases, the photon sphere radius of the KS BH increases rapidly, but that of the GK BH decreases more rapidly, with the photon sphere disappearing at ${a^{\mathrm{GK}}}=2M$. Right panel: Dependence of the critical impact parameter $b_p/M$ on the BH hair ${a^{\mathrm{KS}}/M}$ and ${a^{\mathrm{GK}}/M}$. After averaging the Keck and VLTI mass-to-distance ratio priors for Sgr A*, the yellow and white regions are consistent with the EHT angular size images of Sgr A* at $1\sigma$ and $2\sigma$ levels, respectively. Conversely, the cyan region is excluded by the same observations at more than $2\sigma$.
• Figure 4: Photon trajectories for three BHs in the equatorial plane. From left to right: SC BH, KS BH with ${a^{\mathrm{KS}}}/M=1$, and GK BH with ${a^{\mathrm{GK}}}/M=1$. In each trajectory plot, the red and green lines represent light rays with $b<b_p$ and $b>b_p$, respectively, but the blue dashed line denotes the photon sphere. The BH is depicted as a black disk. The coordinates are scaled by $M$.
• Figure 5: Distribution of the integrated intensity $F_{\mathrm{obs}}$ as a function of the impact parameter $b$ for three BHs under static spherical accretion. The black, red, and blue solid lines correspond respectively to the SC BH, the KS BH with ${a^\mathrm{KS}}/M=1$, and the GK BH with ${a^\mathrm{GK}}/M=1$. The maximum value of the integrated intensity always occurs at the critical impact parameter $b_p$ for each BH.