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Strong gravitational lensing by a Reissner-Nordström naked singularity with a marginally unstable photon sphere

Tadashi Sasaki

TL;DR

This work analyzes strong gravitational lensing by a Reissner-Nordström naked singularity with a marginally unstable photon sphere, focusing on non-logarithmic divergences that arise when the photon sphere is marginal. The authors derive a Picard-Fuchs equation for the deflection angle $\Delta\phi(x)$ with $x=\tfrac{27M^2}{2b^2}$ and solve it in full generality using hypergeometric series, yielding accurate full-order expansions for both outside and inside the photon sphere SDL. They show higher-order terms are essential to match numerical results and provide improved image-position formulas, including relativistic Einstein-ring diameters for winding numbers $N=1,2,3$, while comparing with and correcting prior results (e.g., Tsukamoto 2020) near the marginal limit. The PF-based approach thus offers precise, region-wise approximations of $\Delta\phi$ and observables in RN naked singularity spacetimes, with potential implications for identifying lensing signatures of naked singularities in astrophysical data.

Abstract

We investigate strong gravitational lensing by a marginally unstable photon sphere in a Reissner-Nordström naked singularity spacetime. Using the Picard-Fuchs equation, we derive full-order power series expressions for the deflection angle in various regimes, including the strong deflection limits from both outside and inside the photon sphere. We show that the deflection angle diverges non-logarithmically in both cases, refining existing asymptotic formulae. Comparing truncated approximations with numerical results, we find that higher-order corrections are essential to achieve comparable accuracy to logarithmic divergence cases. Using these improved formulae, we also derive precise approximations for image positions that are not restricted to the almost perfectly aligned cases.

Strong gravitational lensing by a Reissner-Nordström naked singularity with a marginally unstable photon sphere

TL;DR

This work analyzes strong gravitational lensing by a Reissner-Nordström naked singularity with a marginally unstable photon sphere, focusing on non-logarithmic divergences that arise when the photon sphere is marginal. The authors derive a Picard-Fuchs equation for the deflection angle with and solve it in full generality using hypergeometric series, yielding accurate full-order expansions for both outside and inside the photon sphere SDL. They show higher-order terms are essential to match numerical results and provide improved image-position formulas, including relativistic Einstein-ring diameters for winding numbers , while comparing with and correcting prior results (e.g., Tsukamoto 2020) near the marginal limit. The PF-based approach thus offers precise, region-wise approximations of and observables in RN naked singularity spacetimes, with potential implications for identifying lensing signatures of naked singularities in astrophysical data.

Abstract

We investigate strong gravitational lensing by a marginally unstable photon sphere in a Reissner-Nordström naked singularity spacetime. Using the Picard-Fuchs equation, we derive full-order power series expressions for the deflection angle in various regimes, including the strong deflection limits from both outside and inside the photon sphere. We show that the deflection angle diverges non-logarithmically in both cases, refining existing asymptotic formulae. Comparing truncated approximations with numerical results, we find that higher-order corrections are essential to achieve comparable accuracy to logarithmic divergence cases. Using these improved formulae, we also derive precise approximations for image positions that are not restricted to the almost perfectly aligned cases.

Paper Structure

This paper contains 18 sections, 89 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Integral Representation of the Deflection Angle
  3. Picard-Fuchs equation for $\Delta\phi(x)$ and its solutions
  4. Boundary conditions
  5. Solutions for scattering outside the photon sphere: $0<x<1$
  6. Solutions for scattering inside the photon sphere: $1<x$
  7. Accuracy of the series expansions
  8. Lens equation and Image position
  9. Discussion
  10. Data Availability Statement
  11. Analytic continuation of $\Delta\phi_{\rm I}^{(0)}$ and $\Delta\phi_{\rm I}^{(\infty)}(x)$ to $x=1$
  12. Derivation of eq.(\ref{['dphi0_continuation']})
  13. Derivation of eq.(\ref{['dphiinf_continuation']})
  14. Explicit coefficients at each singularity
  15. $x=0$ (WDL)
  16. ...and 3 more sections

Figures (11)

  • Figure 1: WDL $(x\to+0 \Leftrightarrow b/b_{\rm m}\to\infty)$
  • Figure 2: SDL from outside $(x\to1-0 \Leftrightarrow b/b_{\rm m}\to1+0)$
  • Figure 3: Head-on limit $(x\to\infty \Leftrightarrow b/b_{\rm m}\to+0)$
  • Figure 4: SDL from inside $(x\to1+0 \Leftrightarrow b/b_{\rm m}\to1-0)$
  • Figure 6: WDL $(x\to+0 \Leftrightarrow b/b_{\rm m}\to\infty)$
  • ...and 6 more figures