Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits

TL;DR

The paper addresses strong-deflection gravitational lensing near photon spheres, focusing on marginally unstable photon spheres where standard analyses can fail. It extends the Bozza/Eiroa–Romero–Torres framework to light rays just inside and outside the marginally unstable sphere in general static, spherically symmetric, and asymptotically flat spacetimes, applying it to Reissner–Nordström and Hayward spacetimes. It demonstrates deflection-angle behavior in the strong-deflection limit with coefficients $\bar{c}_{\pm}$ and constants $\bar{d}_{\pm}$, computes associated lensing observables such as $\theta_{\infty}$ and $\theta_{\mathrm{E}n\pm}$, and resolves previous mismatches in semi-analytic results. These results enhance modeling of extreme gravitational lensing and aid in distinguishing black holes from mimickers with future high-precision imaging and space-based observations.

Abstract

It is believed that rays bent inside and outside photon spheres could affect partially the black hole shadow images by the Event Horizon Telescope and the rays near photon spheres would be detected by near-future space observations. The features of the rays near the photon spheres in not only black hole spacetimes but also exotic spacetimes would be important since one will need them to exclude black-hole mimickers. The deflection angles of the rays deflected by photon spheres diverge logarithmically and we can treat them by a strong-deflection-limit analysis. The error of the strong-deflection-limit analysis becomes large if antiphoton spheres exist in the spacetimes and the analysis breaks down when the photon spheres and the antiphoton spheres degenerate to form a marginally unstable photon sphere. This is because the deflection angles of the rays bent by the marginally unstable photon sphere diverge in powers. In this paper, we extend Eiroa, Romero, and Torres's method to gravitational lensing of rays inside and outside of the marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits and we apply it to a Reissner-Nordström spacetime and a Hayward spacetime with the marginally unstable photon sphere. We have also confirmed that the deflection angles in the strong deflection limits by the method converge correctly to the deflection angle without approximations, while there are the mismatches of the coefficient of the power-divergent term of the deflection angles of the rays deflected just outside of the marginally unstable photon sphere in a semi-analytic calculation by the author previously.

Figures (7)

• Figure 1: The effective potential $V(r/M,b)$ in the Reissner-Nordström spacetime with the marginally unstable photon sphere is shown. The broken (green), solid (red), and dotted (magenta) curves denote $V(r/M,b)$ for $b=0.99 b_{\mathrm{m}}$, $b_{\mathrm{m}}$, and $1.01 b_{\mathrm{m}}$, respectively.
• Figure 2: The deflection angles $\alpha$ in the Reissner-Nordström spacetime by using Eq. (2.17) is denoted by the dot-dashed (magenta) curve. The one by using Eq. (3.12) with $\bar{c}_+$ of Eq. (\ref{['eq:cRN1']}), $\bar{d}_+$ of Eq. (\ref{['eq:dRN1']}), $\bar{c}_-$ of Eq. (\ref{['eq:cRN4']}), and $\bar{d}_-$ of Eq. (\ref{['eq:dRN4']}) in our numerical method, with $\bar{c}_+$ of Eq. (\ref{['eq:cRN2']}), $\bar{d}_+$ of Eq. (\ref{['eq:dRN2']}), $\bar{c}_-$ of Eq. (\ref{['eq:cRN5']}), and $\bar{d}_-$ of Eq. (\ref{['eq:dRN5']}) by Sasaki Sasaki:2025web, and with $\bar{c}_+$ of Eq. (\ref{['eq:cRN3']}) and $\bar{d}_+$ of Eq. (\ref{['eq:dRN3']}) by Tsukamoto Tsukamoto:2020iez are denoted by the dashed (green), dotted (black), and solid (red) curves, respectively. Notice that the dashed (green) and dotted (black) curves are overlapped.
• Figure 3: The percent errors of deflection angles $\alpha$ of Eq. (3.12) against the deflection angle $\alpha$ of Eq. (2.17) in the Reissner-Nordström spacetime are plotted in top and bottom panels for rays producing images outside and inside of the marginally unstable photon sphere, respectively. In the top pnnel, dashed (green), dotted (black), solid (red) curves denote the percent errors of deflection angles $\alpha$ of Eq. (3.12) with $\bar{c}_+$ of Eq. (\ref{['eq:cRN1']}) and $\bar{d}_+$ of Eq. (\ref{['eq:dRN1']}) in our numerical method, with $\bar{c}_+$ of Eq. (\ref{['eq:cRN2']}) and $\bar{d}_+$ of Eq. (\ref{['eq:dRN2']}) by Sasaki Sasaki:2025web, and with $\bar{c}_+$ of Eq. (\ref{['eq:cRN3']}) and $\bar{d}_+$ of Eq. (\ref{['eq:dRN3']}) by Tsukamoto Tsukamoto:2020iez, respectively. In the bottom panel, dashed (green) and dotted (black) curves denote the percent errors of deflection angles $\alpha$ of Eq. (3.12) with $\bar{c}_-$ of Eq. (\ref{['eq:cRN4']}) and $\bar{d}_-$ of Eq. (\ref{['eq:dRN4']}) in our numerical method, and with $\bar{c}_-$ of Eq. (\ref{['eq:cRN5']}) and $\bar{d}_-$ of Eq. (\ref{['eq:dRN5']}) by Sasaki Sasaki:2025web, respectively. Note that the dashed (green) and dotted (black) curves are overlapped.
• Figure 4: The effective potentials $V(r/M,b)$ for $b=0.99 b_{\mathrm{m}}$, $b_{\mathrm{m}}$, and $1.01 b_{\mathrm{m}}$ in the Hayward spacetime with the marginally unstable photon sphere are shown as the broken (green), solid (red), and dotted (magenta) curves, respectively.
• Figure 5: The deflection angles $\alpha$ in the Hayward spacetime with the marginally unstable photon sphere are shown. The dot-dashed (magenta) curve denotes the deflection angle $\alpha$ of Eq. (2.17). The dashed (green), dotted (black), and solid (red) curves denote the deflection angles $\alpha$ of Eq. (3.12) with $\bar{c}_+$ of Eq. (\ref{['eq:cH1']}), $\bar{d}_+$ of Eq. (\ref{['eq:dH1']}), $\bar{c}_-$ of Eq. (\ref{['eq:cH4']}), and $\bar{d}_-$ of Eq. (\ref{['eq:dH4']}) in our numerical method, with $\bar{c}_+$ of Eq. (\ref{['eq:cH2']}) and $\bar{d}_+=0$ by Chiba and Kimura Chiba:2017nml, and with $\bar{c}_+$ of Eq. (\ref{['eq:cH3']}) and $\bar{d}_+$ of Eq. (\ref{['eq:dH3']}) by Tsukamoto Tsukamoto:2020iez, respectively.