Image of a quantum-corrected black hole without Cauchy horizons illuminated by a static thin accretion disk

TL;DR

The paper investigates a covariant quantum-corrected black hole solution without Cauchy horizons, introducing a quantum parameter $\\zeta$ and studying its impact on observable optics. By deriving the photon geodesics and a transfer-function framework for a static thin-disk emission, the authors show that key scales $r_h$, $r_{ph}$, $b_c$, and $r_{isco}$ grow with $\\zeta/M$, and they establish a theoretical bound $\\zeta/M < 2(\\pi/2)^{3/2} \\approx 3.9374$ for horizon existence, complemented by observational bounds from EHT shadow measurements. The analysis reveals that larger $\\zeta/M$ enlarges the shadow, brightens the outer bright rings, and reduces their spacing, while also modifying the slopes of the first three transfer functions, offering a distinctive optical fingerprint compared to Schwarzschild and prior quantum-corrected BHs. Collectively, the results provide testable predictions for quantum gravity effects in strong gravity regimes and guide future high-resolution observations to probe covariant BH spacetimes without Cauchy horizons.

Abstract

Latest advances in effective quantum gravity propose a quantum-corrected black hole solution that avoids Cauchy horizons. In this paper, we study the image of the black hole and explore the influence of the quantum parameter $ζ$ on its image. First, we investigate the influence of $ζ$ on the event horizon, photon sphere, critical impact parameter, and innermost stable circular orbit associated with the black hole. We find that all these quantities exhibit an increase with increasing $ζ$. Meanwhile, we analyze the allowed range of $ζ$ from both theoretical and observational perspectives. We then derive the photon trajectory equation and analyze briefly the behavior of the trajectories. A detailed analysis shows that as $ζ$ increases, the photon trajectories near the event horizon undergo modifications. Finally, by plotting the optical appearance of the black hole under three emission models, we find that as $ζ$ increases, the quantum-corrected black hole exhibits a larger shadow, along with a brightening of the bright rings and a reduction in the spacing between them near the critical impact parameter. Therefore, we can distinguish the quantum-corrected black hole from the Schwarzschild one by its unique optical appearance.

Figures (13)

• Figure 1: The event horizon $r_{\rm h}$, critical impact parameter $b_{\rm c}$, radius of photon sphere $r_{\rm ph}$, and innermost stable circular orbit $r_{\rm isco}$ as the functions of $\zeta/M$. The vertical red dashed lines mark the location of the maximum allowed value $\zeta/M=3.9374$.
• Figure 2: Variation of $\mu(r)$ with radial coordinate $r$ at three $\zeta/M$ values: 0 (blue), 1.5 (green), and 3.5 (red). The dashed lines mark where the event horizon forms for each $\zeta/M$.
• Figure 3: Deflection rate $\frac{{\rm d}\varphi}{{\rm d}r}$ as a function of $r$ for selected values of $\zeta/M$: 0 (blue), 1.5 (green), and 3.5 (red). The dashed vertical lines indicate the event horizon for each case. The impact parameter is fixed at $b=b_{\rm c}/3$.
• Figure 4: Constraints on $\zeta/M$ from EHT observations of the shadow sizes for M87* (left) and Sgr A* (right). The gray areas show the observed diameters, and the red dashed lines mark the maximum allowed $\zeta/M$.
• Figure 5: Total number of orbits with $\zeta/M=1.5$: $n<3/4$ (black), $3/4<n<5/4$ (orange), and $n>5/4$ (red), corresponding to the direct, lensed, and photon ring trajectories, respectively.