Thermodynamics and Gravitational Signatures of Rotating Black Holes in the Generalized Extended Uncertainty Principle

TL;DR

This work presents a phenomenological study of rotating black holes under the Generalized Extended Uncertainty Principle (GEUP), introducing UV and IR corrections via an effective mass $\mathcal{M}$ and a Kerr-like geometry obtained with the Newman-Janis algorithm. It analyzes two observable windows—black hole thermodynamics and gravitational wave spectroscopy—showing that IR corrections drive $T_H$ to scale as $M^{-3}$ and enlarge the entropy as $S \propto M^6$, while UV corrections imprint equal and opposite shifts on QNM frequencies without breaking isospectrality. The Teukolsky formalism is adapted to the GEUP background, preserving Type D structure and separability, with explicit eikonal QNM shifts: $\omega_R$ blueshifted by UV and redshifted by IR, and $\omega_I$ enhanced by UV and suppressed by IR. Observational data from LIGO/Virgo (GW150914) primarily constrain the UV sector ($\beta$), whereas EHT shadow measurements of M87* place stringent bounds on the IR sector ($\alpha$), illustrating a complementary strategy to probe quantum gravity phenomenology across mass scales. The framework highlights a fundamental scale-invariance breaking, offering a roadmap for population-level tests of GEUP through shadows and ringdowns, and sets the stage for extending analyses to additional perturbations and observational probes.

Abstract

We investigate the phenomenological implications of quantum gravity on rotating black holes within the framework of the Generalized Extended Uncertainty Principle (GEUP), which incorporates both minimal length (ultraviolet) and large-scale (infrared) corrections. Lacking a full non-perturbative formulation of quantum gravity, we adopt a metric-based approach. We construct a stationary, axisymmetric ansatz via the Newman-Janis algorithm to model the kinematic features of a rotating black hole subject to Generalized Extended Uncertainty Principle (GEUP) corrections. The thermodynamic analysis reveals that in the infrared-dominated regime, the Hawking temperature scales as $T_H \sim M^{-3}$, leading to a rapid cooling phase that significantly prolongs the lifetime of supermassive black holes. We derive the modified Teukolsky Master Equation for gravitational perturbations and demonstrate that the background geometry preserves the isospectrality between axial and polar modes. In the eikonal limit, the quasinormal mode (QNM) spectrum exhibits orthogonal shifts: the minimal length parameter $β$ induces a spectral blueshift and enhanced damping, while the large-scale parameter $α$ induces a spectral redshift and suppressed damping. Finally, we constrain the theory using observational data from LIGO/Virgo and the Event Horizon Telescope. We establish that the shadow of M87* is approximately $10^6$ times more sensitive to large-scale corrections than Sgr A*, placing stringent bounds on the EUP parameter, while gravitational wave spectroscopy provides complementary constraints on the GUP sector.

Figures (2)

• Figure 1: Behavior of the heat capacity $C_{J}$ (in units of $M_{Pl}^2$) as a function of the horizon radius $r_{+}$ (normalized by the bare mass $GM$). The vertical asymptotes correspond to Davies points, marking the phase transition. We adopt the standard thermodynamic sign convention where $C_{J} > 0$ indicates stability and $C_{J} < 0$ indicates instability. The GEUP corrections shift the location of these critical points, extending the domain of stability compared to the standard Kerr metric.
• Figure 2: Deformation of the Quasinormal Mode spectrum in the complex frequency plane. The black star indicates the standard General Relativity value for the fundamental $\ell=2, m=2$ mode. The grid illustrates how the frequency shifts under the influence of the GEUP parameters. Red lines indicate increasing EUP strength ($\alpha$) resulting in a spectral redshift and suppressed damping (upward-left shift). Blue lines indicate increasing GUP strength ($\beta$) resulting in a spectral blueshift and enhanced damping (downward-right shift). The near-orthogonality of these grid lines demonstrates that $\alpha$ and $\beta$ represent distinct physical modifications that can be observationally disentangled.