Light rings and shadows of static black holes in effective quantum gravity II: A new solution without Cauchy horizons

Abstract

Among the three known types of static solutions proposed within the Hamiltonian constraint approach to effective quantum gravity (EQG), the first two have been extensively investigated, whereas the third type-which preserves general covariance, is free of Cauchy horizons, and was only recently obtained-remains relatively unexplored. This solution can describe a black hole with an event horizon for certain parameter ranges, or a horizonless compact object beyond those ranges. In this paper, we focus on the third type and show that its light rings feature both stable and unstable branches, and that the black hole shadow size grows with the quantum parameter-unlike in the first two types. However, when we account for both the shadow and the lensing ring, the overall behavior closely resembles that of the second type, in which an increasing quantum parameter leads to a larger portion of the lensing ring being occupied by the shadow. This feature can serve as a hallmark of black holes in EQG, offering a potential way to distinguish them from their GR counterparts. Remarkably, the parameter ranges under which the solution remains a black hole are highly consistent with the current observational constraints on black hole shadows, lending strong support to the classification of the third type of compact object in EQG as a black hole endowed with an event horizon.

Figures (7)

• Figure 1: The relationship between the quantum parameter and the black hole event horizon is analyzed in a third-type spacetime in EQG, with $n=0$.
• Figure 2: The arrows represent the unit vector field $n$ on a portion of the $r-\Theta$ plane for the this type of static black holes in EQG with (a) $M = 1$, $\zeta = 1$ and (b) $M = 1$, $\zeta = \pi^{3/2}/\sqrt{2}$, respectively. The blue contour $C_i$ are closed loops enclosing the light ring. Obviously, the topological charge of the light ring is always $Q = -1$ in both cases.
• Figure 3: Shadow contours under different quantum parameters for the third type of static compact objects in EQG.
• Figure 4: The estimation of the angular radius of the supermassive black holes Sgr A* and M87* using the metric of the third type of compact object in EQG. The black dashed line represents the critical threshold of parameters that determine the existence of the compact object's event horizon.
• Figure 5: The shadows and lensing rings of the third type of static compact object in EQG, characterized by a quantum parameter $\zeta$ for the case $n=0$, as seen by an observer at $\theta_0 = \pi/2$.