Strong Gravitational Lensing by Compact Object without Cauchy Horizons in Effective Quantum Gravity

TL;DR

This study addresses strong gravitational lensing in the third EQG solution that has no Cauchy horizons, focusing on the $q=0$ case where the spacetime can be a black hole or a horizonless wormhole depending on the ratio $ζ/M$. The authors derive the photon-sphere radius $r_m$ from $r_m^6-9M^2r_m^4-4M^2ζ^4=0$, compute the shadow radius $R_{sh}=b_m$ and the strong-deflection coefficients for the lensing potential, and then obtain observables for relativistic images using Bozza's formalism. By comparing with EHT shadow measurements of SgrA* and M87*, they constrain $ζ/M$ to [0,1.878] (VLTI) or [0,3.035] (Keck) for SgrA* and up to [0,5.485] for M87*, finding that SgrA* disfavors wormholes while M87* allows both BH and wormhole cases. The analysis predicts observable trends in image positions, separations, and time delays, with time delays between the first two relativistic images potentially detectable for M87*, offering a practical route to test EQG in the strong-field regime.

Abstract

In this work, we theoretically investigate strong gravitational lensing effects and evaluate various lensing observables of a static, spherically symmetric solution in the context of effective quantum gravity (EQG). Among the three types of solutions proposed in EQG backgrounds, this is the third type without having Cauchy horizons. This solution gives rise to black hole as well as horizonless wormhole solutions depending on the range of values of the parameters of the theory. Based on the data from SgrA* and M87* observations, possible bounds on the parameter are obtained. It is found that the horizonless wormhole solution is ruled out by SgrA* observations, but is allowed by M87* observations. We analyze and provide estimates of the lensing observables, some of which can potentially be detected by observational tools.

Figures (9)

• Figure 1: Plot of $A^{(0)}(r)$ as a function of $r$ (in units of $M$) for different values of $\zeta$. For $\zeta/M = 0.0,1.0,2.0,3.0$ & $2\left(\frac{\pi}{2}\right)^{3/2}$ event horizons do exist, with $\zeta/M=2\left(\frac{\pi}{2}\right)^{3/2}$ representing the extremal black hole. However, for $\zeta/M=5.0$, event horizon does not exist.
• Figure 2: Combined plot of $r_h$ and $r_0$ (in units of $M$) versus $\zeta$ (in units $M$). It can be seen that $r_h>r_0$ for $\zeta/M<2\left(\frac{\pi}{2}\right)^{3/2}$ representing a black hole. The extremal black hole case is denoted by the red dot having coordinates $\left(2\left(\frac{\pi}{2}\right)^{3/2},\pi\right)$. Beyond this point, $r_h$ does not exist signifying a wormhole spacetime with $r_0$ being the throat.
• Figure 3: Plots of $V_{eff}/L^2$ as functions of (a) $r/M$ (Black hole) and (b) $l/M$ (Wormhole), with $q=0$, for different values of $\zeta$. The maxima of $V_{eff}$ correspond to the locations of photon spheres. In figure (a) for the black hole case, the radii of photon spheres increase with increasing value of $\zeta$. In figure (b) for the wormhole case, $V_{eff}$ is plotted against the proper radial coordinate $l$, such that at the throat $l(r_0)=0$.
• Figure 4: Plots of $R_{\text{sh}}$ (in units of $M$) as a function of $\zeta$ (in units of $M$). The shaded regions represent the observational ranges of $R_{\text{sh}}/M$ for (a) SgrA* and (b) M87*. The values of $\zeta/M$ that fall within these shaded regions should be allowed. In (a), the 'light blue' shaded region represents VLTI and 'light green' region corresponds to Keck observations. Constraints on $\zeta$ have been shown with vertical lines: green dot-dashed for VLTI and red dot-dashed for Keck. In (b), the 'light orange' shaded region represents M87* observational range. Constraints on $\zeta$ have been shown with vertical red dot-dashed line. The black-dashed vertical lines in both the plots correspond to the extremal value of $\zeta$.
• Figure 5: Plots of (a) $\bar{a}$ and (b) $\bar{b}$ as functions of $\zeta$ (in units of $M$) for $q=0$. Markers are provided with vertical lines for $\zeta=1.878M$ (VLTI), $\zeta=3.035M$ (Keck), $\zeta=2\left(\frac{\pi}{2}\right)^{3/2}M$ and $\zeta=5.485M$ (M87*).