Strong field gravitational lensing of particles by a black-bounce-Schwarzschild black hole

TL;DR

The paper analyzes strong-field gravitational lensing of neutral massive particles in the regular black-bounce-Schwarzschild spacetime, deriving the particle-sphere radius $r_c$ and the velocity-dependent strong-field deflection angle $\alpha(\vartheta,w) \approx -\bar{a}\ln\left(\dfrac{d_L\vartheta}{u_c}-1\right) + \bar{b}$, with a velocity-variant critical impact parameter $u_c(w)$. It then computes observable lensing quantities for relativistic images, namely $\vartheta_{\infty}=u_c/d_L$, image separation $s$, and magnitude difference $\mathcal{R}_m$, and introduces explicit velocity corrections $\Delta\vartheta_{\infty}$, $\Delta s$, and $\Delta\mathcal{R}_m$ that capture departures from the lightlike case. An application to Sgr A$^*$ assesses detectability with future multimessenger and astrometric capabilities, showing that velocity effects can reach detectable levels for certain $w$ and bounce parameter $\hat{\eta}$. The results offer a framework to test regular black-hole geometries through strong-field lensing of massive particles and highlight the prospects and challenges for multimessenger probes of such spacetimes.

Abstract

The gravitational lensing of relativistic and nonrelativistic neutral massive particles in the black-bounce-Schwarzschild black hole spacetime is investigated in the strong deflection limit. Beginning with the explicit equations of motion of a massive particle in the regular spacetime, we achieve the equation of the particle sphere and thus the radius of the unstable timelike circular orbit. It is interesting to find that the particle sphere equation can reduce to the well-known photon sphere equation, when the particle's initial velocity is equal to the speed of light. We adopt the strong field limit approach to calculate the black-bounce-Schwarzschild deflection angle of the particle subsequently, and obtain the strong-deflection lensing observables of the relativistic images of a pointlike particle source. The observables mainly include the apparent angular particle sphere radius, the angular separation between the outermost relativistic image and the other ones which are packed together, and the ratio between the particle-flux magnification of the outermost image and that of the packed ones (or equivalently, their resulted magnitudelike difference). The velocity effects induced by the deviation of the initial velocity of the particle from light speed on the corresponding strong-field lensing observables of the images of a pointlike light source in the regular geometry, along with these on the strong deflection limit coefficients and the critical impact parameter of the lightlike case, are then formulated. Serving as an application of the results, we finally concentrate on evaluating the astronomical detectability of the velocity effects on the lensing observables and analyzing their dependence on the parameters, by modeling the Galactic supermassive black hole (i.e., Sgr A$^{\ast}$) as the lens.

Figures (3)

• Figure 1: Geometrical configuration of lensing of the massive particle in an asymptotically flat black-bounce-Schwarzschild black hole spacetime, adopted from HXJL2024.
• Figure 2: The critical impact parameter $u_c$ and the color-indexed strong field limit coefficients $\bar{a}$ and $\bar{b}$, along with the particle sphere radius $\vartheta_{\infty}$ and the color-indexed observables $s$ and $\mathcal{R}_m$, plotted as the functions of $w$ (and $\hat{\eta}$). As an example, here we assume the black-bounce-Schwarzschild lens to be the galactic supermassive black hole which has a mass $M=4.2\times10^6M_{\odot}$BG2016Parsa2017 and a distance $d_L=8.2$ kpc BG2016 from us, with $M_{\odot}$ ($=4.8\times10^{-17}$ kpc) being the rest mass of the Sun.
• Figure 3: The velocity effects $\Delta\vartheta_{\infty}$ and $\Delta u_c$, together with the color-indexed velocity effects $\Delta s$, $\Delta\mathcal{R}_m$, $\Delta\bar{a}$, and $\Delta\bar{b}$, plotted as the functions of $w$ (and $\hat{\eta}$) in the lensing scenario of Sgr A$^{\ast}$. Our attention is focused on the absolute values of these velocity effects in the discussion of their detectability.