Retrolensing by a charged black hole

Compact objects with a light sphere such as black holes and wormholes can reflect light rays like a mirror. This gravitational lensing phenomenon is called retrolensing and it is an interesting tool to survey dark and compact objects with a light sphere near the solar system. In this paper, we calculate the deflection angle analytically in the strong deflection limit in the Reissner-Nordström spacetime without Taylor expanding it in the power of the electric charge. Using the obtained deflection angle in the strong deflection limit, we investigate the retrolensing light curves and the separation of double images by the light sphere of Reissner-Nordström black holes.

Figure 1

The quantities $b_c/M$, $r_m/M$, $\overline{a}$, and $\overline{b}$ in the Reissner-Nordström spacetime as the functions of $Q/M$. The (red) solid, (green) dashed, (blue) dash-dotted, and (purple) dotted curves denote $b_c/M$, $r_m/M$, $\overline{a}$, and $\overline{b}$, respectively.

Figure 2

Lens configuration. The ray of the sun $S$ is reflected by a light sphere $L$ of a black hole with a deflection angle $\alpha$ and reaches the observer $O$. The observer sees an image $I$ with an image angle $\theta .\beta$ is the source angle $\angle OLS$. $\overline{\theta }$ is the angle between the line $LS$ and the light ray at $S$.

Figure 3

The sun moves on the source plane with the orbital velocity.

Figure 4

The motion of the sun with the closest separation ${\beta }_{\min }$ between the center of the sun disk and the axis $\beta =0$ on the source plane. The (red) first, (green) second, (blue) third, and (purple) fourth right arrows from the top to bottom denote the cases for ${\beta }_{\min }=0$, $0.5{\beta }_s$, ${\beta }_s$, and $1.5{\beta }_s$, respectively.

Figure 5

Retrolensing light curves by a noncharged black hole with the mass $M=10M_{\odot }$ at $D_{OL}=0.01\mathrm{pc}$. The (red) solid, (green) dash-dotted, (blue) dashed, and (purple) dotted curves denote the light curves with the closest separation ${\beta }_{\min }=0$, $0.5{\beta }_s$, ${\beta }_s$, and $1.5{\beta }_s$, respectively.

Figure 6

Retrolensing light curves reflected by a noncharged black hole ($Q=0$) at $D_{OL}=0.01\mathrm{pc}$ in the perfect-aligned case (${\beta }_{\min }=0$). The (red) solid, (green) dash-dotted, and (blue) dashed curves denote the light curves lensed by black holes with the mass $M=60M_{\odot }$, $30M_{\odot }$, and $10M_{\odot }$, respectively.

Figure 7

Retrolensing light curves in the perfect-aligned case (${\beta }_{\min }=0$) lensed by a black hole with the mass $M=60M_{\odot }$ at $D_{OL}=0.01\mathrm{pc}$. The (red) solid and (green) dash-dotted curves denote the light curves lensed by the noncharged black hole ($Q=0$) and the maximal charged black hole ($Q=M$), respectively.

Figure 8

The apparent magnitude $m$ of the light curves at the peak in the perfect-aligned case (${\beta }_{\min }=0$) by a black hole at $D_{OL}=0.01\mathrm{pc}$ as a function of $Q/M$. The (red) solid, (green) dash-dotted, and (blue) dashed curves denote the apparent magnitude $m$ of the peak of the light curve lensed by black holes with the mass $M=60M_{\odot }$, $30M_{\odot }$, and $10M_{\odot }$, respectively.

Figure 9

The relative magnitude $\mathrm{\Delta }m$ by a black hole at $D_{OL}=0.01\mathrm{pc}$ in the perfect-aligned case (${\beta }_{\min }=0$) as a function of $Q/M$. The relative magnitude $\mathrm{\Delta }m$ for the black hole with the mass $M=60M_{\odot }$ is plotted but the curves with $M=10M_{\odot }$ and $M=30M_{\odot }$ are very similar to the one with $M=60M_{\odot }$.

Figure 10

The separation ${\theta }_{+}-{\theta }_{-}$ of a double image as a function of $Q/M$. The (red) solid, (green) dash-dotted, and (blue) dashed curves denote the separation ${\theta }_{+}-{\theta }_{-}$ of a double image lensed by black holes at $D_{OL}=0.01\mathrm{pc}$ with the mass $M=60M_{\odot }$, $30M_{\odot }$, and $10M_{\odot }$, respectively.

Figure 11

The relative separation $\mathrm{\Delta }({\theta }_{+}-{\theta }_{-})$ as a function of $Q/M$. The (red) solid, (green) dash-dotted, and (blue) dashed curves denote the relative separation $\mathrm{\Delta }({\theta }_{+}-{\theta }_{-})$ by black holes at $D_{OL}=0.01\mathrm{pc}$ with the masses $M=60M_{\odot }$, $30M_{\odot }$, and $10M_{\odot }$, respectively.