Testing linear-quadratic GUP modified Kerr Black hole using EHT results

TL;DR

The paper investigates how the linear-quadratic GUP (LQG) modifies Kerr black holes by adjusting the ADM mass to $\mathcal{M}=M\left(1-\frac{\alpha}{4M}+\frac{\alpha^2}{8M^2}\right)$ and analyzes the resulting horizon structure, null geodesics, and shadows. It derives the LQG-corrected metric, identifies a critical spin value $a=\frac{7M}{8}$, and shows a forbidden $\alpha$ interval for $a>\frac{7M}{8}$, along with extremal configurations at the interval endpoints. Using Hamilton-Jacobi formalism, the study computes photon orbits and shadow shapes, revealing a minimum shadow size at $\alpha=1.0M$ for subcritical spins and explicit inclination dependence. By confronting the predictions with EHT measurements of $M87^*$ and $SgrA^*$, it extracts upper bounds on $\alpha$ (typically a few $M$, depending on source and angle) and constrains $a$ to $[0,M]$, thereby showing that LQKBHs can remain viable astrophysical black holes under current data, with tighter future prospects anticipated from improved EHT results and related observations.

Abstract

The linear-quadratic Generalized uncertainty principle (LQG) is consistent with predictions of a minimum measurable length and a maximum measurable momentum put forth by various theories of quantum gravity. The quantum gravity effect is incorporated into a black hole (BH) by modifying its ADM mass. In this article, we explore the impact of GUP on the optical properties of an LQG modified \k BH (LQKBH). We analyze the horizon structure of the BH, which reveals a critical spin value of $7M/8$. BHs with spin $(a)$ less than the critical value are possible for any real GUP parameter $\a$ value. However, as the spin increases beyond the critical value, a forbidden region in $\a$ values pops up that disallows the existence of BHs. This forbidden region widens as we increase the spin. We then examine the impact of $\a$ on the shape and size of the BH shadow for inclination angles $17^o$ and $90^o$, providing a deeper insight into the unified effect of spin and GUP on the shadow. The size of the shadow has a minimum at $\a=1.0M$, whereas, for the exact value of $\a$, the deviation of the shadow from circularity becomes maximum when the spin is less than the critical value. No extrema is observed for $a\,>\, 7M/8$. The shadow's size and deviation are adversely affected by a decrease in the inclination angle. Finally, we confront theoretical predictions with observational results for supermassive BHs $M87^*$ and $SgrA^*$ provided by the EHT collaboration to extract bounds on the spin $a$ and GUP parameter $\a$. We explore bounds on the angular diameter $þ_d$, axial ratio $D_x$, and the deviation from \s radius $\d$ for constructing constraints on $a$ and $\a$. Our work makes LQKBHs plausible candidates for astrophysical BHs.

Figures (9)

• Figure 1: Parameter space $(a-\alpha)$ for the LQKBH where the parameter values in the colored region permit the BH solution, and the white region is where the BH solution is disallowed.
• Figure 2: Variation of $\Delta$ with $r/M$ for different values of $\alpha$. The left one is for spin $a=0.8M$ and the right one is for spin $a=0.9M$.
• Figure 3: Shadows cast by LQKBH when the inclination angle is $\theta_o=90^o$. The left one is for $a=0.87M$ and the right one is $a=0.90M$.
• Figure 4: Shadows cast by LQKBH when the inclination angle is $\theta_o=17^o$. The left one is for $a=0.87M$ and the right one is $a=0.90M$.
• Figure 5: Variation of angular diameter $\theta_d$ with spin $a$ and GUP parameter $\alpha$. Here, the values along bold lines and sidebars are in $\mu as$. The upper panel is for $\theta_0=90^o$, and the lower panel is for $\theta_o=17^o$. Left plots are for $M87^*$ and right plots are for $SgrA^*$.