Kerr-Newman black hole lensing of relativistic massive particles in the weak-field limit

The gravitational lensing of relativistic neutral massive particles caused by a Kerr-Newman black hole is investigated systematically in the weak-field limit. Based on the Kerr-Newman metric in Boyer-Lindquist coordinates, we first derive the analytical form of the equatorial gravitational deflection angle of a massive particle in the third post-Minkowskian approximation. The resulting bending angle, which is found to be consistent with the result in the previous work, is adopted to solve the popular Virbhadra-Ellis lens equation. The analytical expressions for the main observable properties of the primary and secondary images of the particle source are thus obtained beyond the weak-deflection limit, within the framework of standard perturbation theory. The observables include the positions, magnifications, and gravitational time delays of the individual images, the differential time delay, and the total magnification and centroid position. The explicit forms of the correctional effects induced by the deviation of the initial velocity of the massive particle from the speed of light on the observables of the lensed images are then achieved. Finally, serving as an application of the formalism, the supermassive black hole at the Galactic Center, Sagittarius ${\mathrm{A}}^{*}$, is modeled to be a Kerr-Newman lens. The magnitudes of the velocity-induced correctional effects on the practical lensing observables as well as the possibilities to detect them in this scenario are also analyzed.

Figure 1

The lens diagram of a Kerr-Newman black hole. The positions of the source, lens, observer, and image are given by $S$, $L$, $O$, and $I$, respectively. All of them are situated in the equatorial plane ($x\text{−}y$ plane) of the lens. The $x$ axis is assumed to be the optic axis $OL$ which joins the lens and observer. $d_S$ and $d_L$ are the angular diameter distances of the source and lens from the observer, respectively, and $d_{LS}$ is the angular diameter distance of the source from the lens. $\mathcal{B}$ and $\vartheta$ denote, respectively, the angular source and image positions. $\widehat{\alpha }$ is the gravitational deflection angle of the massive particle. $b(=d_L\sin \vartheta )$ denotes the impact parameter. Without loss of generality, the intrinsic angular momentum vector $\mathit{J}=J{\mathit{e}}_3$ of the gravitational lens is assumed to be along the positive $z$ axis ($J>0$).

Figure 2

${\theta }_0^{+}$, ${\theta }_1^{+}$, ${\theta }_2^{+}$, ${\theta }_0^{+}+{\theta }_0^{-}$, ${\theta }_1^{+}\pm {\theta }_1^{-}$, and ${\theta }_2^{+}\pm {\theta }_2^{-}$ plotted as the functions of $\beta (\in [0.01,10])$ for prograde ($s^{+}=+1$) or retrograde ($s^{+}=-1$) motion of the particle. Here and in the following figures of this section, we assume $w=0.1$, $\widehat{a}=0.9,\widehat{Q}=0.01$, and $D=0.5$, as an example of our scenario of the KN lensing of massive particles. Additionally, the cases of the KN lensing of light ($w=1$) as well as the Schwarzschild lensing $(a=Q=0)$ of light and massive particles are also presented for comparison.

Figure 3

${\mu }_0^{+}$, ${\mu }_1^{+}$, ${\mu }_2^{+}$, ${\mu }_0^{+}-{\mu }_0^{-}$, ${\mu }_1^{+}\pm {\mu }_1^{-}$, and ${\mu }_2^{+}\pm {\mu }_2^{-}$ plotted as the functions of $\beta$ for prograde ($s^{+}=+1$) or retrograde ($s^{+}=-1$) motion of the massive particle.

Figure 4

${\mathrm{\Theta }}_{\mathrm{cent},0}$, ${\mathrm{\Theta }}_{\mathrm{cent},1}$, and ${\mathrm{\Theta }}_{\mathrm{cent},2}$ plotted as the functions of $\beta$ for the particle’s prograde or retrograde motion.

Figure 5

$\mathrm{\Delta }{\widehat{\tau }}_0$ and $\mathrm{\Delta }{\widehat{\tau }}_1$ plotted as the functions of $\beta$ for the particle’s prograde or retrograde motion.

Figure 6

$\delta {\vartheta }_0^{+}$, $\delta {\vartheta }_1^{+}ϵ$, $\delta {\vartheta }_2^{+}{ϵ}^2$, $\delta ({\vartheta }_0^{+}+{\vartheta }_0^{-})$, $\delta ({\vartheta }_1^{+}\pm {\vartheta }_1^{-})ϵ$, and $\delta ({\vartheta }_2^{+}\pm {\vartheta }_2^{-}){ϵ}^2$ displayed in color-indexed form as the bivariate functions of $w$ and $\beta$, for the particle’s prograde ($s^{+}=+1$) or retrograde ($s^{+}=-1$) motion. The values of the related parameters are given in Sec. 6a. Note that we do not show $\delta {\vartheta }_2^{+}{ϵ}^2$ and $\delta ({\vartheta }_2^{+}+{\vartheta }_2^{-}){ϵ}^2$ for the case of $s^{+}=-1$, since it is hard to distinguish them from the corresponding results for the case of $s^{+}=+1$, respectively. Note also that $\delta ({\vartheta }_1^{+}-{\vartheta }_1^{-})ϵ$ and $\delta ({\vartheta }_2^{+}-{\vartheta }_2^{-}){ϵ}^2$ may take negative values for $0.05\lesssim w<1$ and $0.01\le \beta \le 10$. Here and thereafter, a white region of a figure indicates the value domain where the magnitude of the velocity effect is too large or too small to be shown properly, and we do not fill them by adjusting the value range for the convenience of display.

Figure 7

$\delta F_0^{+}/F_{\mathrm{s}}$, $\delta F_1^{+}ϵ/F_{\mathrm{s}}$, $\delta F_2^{+}{ϵ}^2/F_{\mathrm{s}}$, $\delta (F_0^{+}+F_0^{-})/F_{\mathrm{s}}$, $\delta (F_1^{+}\pm F_1^{-})ϵ/F_{\mathrm{s}}$, and $\delta (F_2^{+}\pm F_2^{-}){ϵ}^2/F_{\mathrm{s}}$ plotted as the color-indexed functions of $w$ and $\beta$ for the particle’s prograde or retrograde motion. We do not show $\delta F_2^{+}{ϵ}^2/F_{\mathrm{s}}$, $\delta (F_1^{+}+F_1^{-})ϵ/F_{\mathrm{s}}$, and $\delta (F_2^{+}-F_2^{-}){ϵ}^2/F_{\mathrm{s}}$ for the case of $s^{+}=-1$ due to the same reason given in Fig. 6 or the symmetry.

Figure 8

$\delta {\mathrm{\Xi }}_{\mathrm{cent},0}$, $\delta {\mathrm{\Xi }}_{\mathrm{cent},1}ϵ$, $\delta {\mathrm{\Xi }}_{\mathrm{cent},2}{ϵ}^2$, $\delta \mathrm{\Delta }{\tau }_0$, and $\delta \mathrm{\Delta }{\tau }_1ϵ$ plotted as the color-indexed functions of $w$ and $\beta$.