Spherically Symmetric Quantum-Corrected Black Holes with String Clouds: A Multi-Observable Analysis

TL;DR

This work probes quantum-corrected Letelier black holes threaded by a cloud of strings using two distinct quantum-correction implementations, examining photon spheres, shadows, ISCOs, QPOs, perturbations, gravitational lensing, and thermodynamic topology. By combining geodesic analysis, numerical plasma simulations, perturbation theory, Gauss-Bonnet lensing, and topological thermodynamics, it shows that Model-I and Model-II exhibit opposite dependences on the quantum parameter $\zeta$ in several observables, especially in deflection angles, enabling unambiguous discrimination between the models. The horizon structure is governed by $r_h = \frac{2M}{1-\alpha}$, while photon-sphere and ISCO properties reveal that CoS effects ($\alpha$) generally strengthen gravitational lensing and orbital radii, whereas quantum corrections ($\zeta$) have model-dependent effects on null and timelike orbits. The results provide multiprobe observational discriminants for quantum gravity phenomenology in strong-field regimes, with implications for EHT/VLBI imaging, X-ray timing, and gravitational-wave spectroscopy as future testing grounds.

Abstract

We present an investigation of quantum-corrected black hole spacetimes coupled with clouds of strings, examining two distinct theoretical models that incorporate quantum gravitational effects through different implementations of correction terms. Our study explores the geodesic structure, focusing on photon sphere properties, black hole shadows, and innermost stable circular orbits of test particles around these exotic geometries. The analysis reveals fundamental modifications to particle trajectories that distinguish quantum-corrected solutions from their classical counterparts, with observable implications for high-energy astrophysics. We investigate quasi-periodic oscillations arising from test particle motion, deriving frequency relationships that could serve as observational probes of quantum gravity effects in accreting black hole systems. Through rigorous gravitational lensing analysis using the Gauss-Bonnet theorem, we calculate weak-field deflection angles and identify distinctive signatures that enable discrimination between the two quantum correction models. The gravitational lensing study reveals opposite dependencies on quantum parameters between the models, providing unambiguous observational discriminants. Additionally, we analyze the thermodynamic properties including temperature, entropy, and heat capacity modifications, exploring topological characteristics and phase transition behavior in these quantum-corrected systems. The investigation reveals that the two models exhibit contrasting behaviors across multiple observational channels, from gravitational lensing deflection angles to quasi-periodic oscillation frequencies, providing remarkable discriminants for testing quantum gravity theories.

Figures (25)

• Figure 1: 3D diagrams of the quantum-corrected BHs (Model-I&II) with metric function $g(r) = (1 - \alpha) - \frac{2M}{r} + \frac{\zeta^2}{r^2} \left( (1 - \alpha) - \frac{2M}{r} \right)^2$, where $M = 1$ is fixed, $\alpha \in \{0.0, 0.1, 0.2, 0.5, 0.8\}$ varies the Letelier modification, and $\zeta \in \{0.0, 4.0, 5.0\}$ is the quantum correction parameter. Each diagram features a turquoise surface representing the embedding from the event horizon $r_h$ to $r = 15$, a black falling trajectory, and a red ring at the event horizon. The transition from an extremal or single-root BH at $\zeta = 0$ to non-extremal BHs with two horizons for $\zeta > 0$ reflects the modification of the spacetime's structure. $M=1$
• Figure 2: Three-dimensional plots of the photon sphere radius $r_{\rm ph}$ (left) and shadow radius $R_s$ (right) as functions of $(\alpha, \zeta)$ for Model-I quantum-corrected BHs with CoS. Both quantities increase with the CoS parameter $\alpha$ and decrease with the quantum correction parameter $\zeta$, demonstrating the competing effects of topological defects and quantum gravity on photon orbits. Mass is set to $M = 1$.
• Figure 3: BH shadow profiles for quantum-corrected BHs with CoS. Left (i): Model-I shadows for varying $\alpha$ with fixed $\zeta = 0.3$. Middle (ii): Model-I shadows for varying $\zeta$ with fixed $\alpha = 0.2$. Right (iii): Model-II shadows for varying $\alpha$ with fixed $\zeta = 0.3$. The shadow size increases with $\alpha$ in both models due to enhanced CoS effects, while in Model-I it decreases with $\zeta$ due to quantum corrections. Model-II shadows are independent of $\zeta$ as quantum corrections appear only in $g(r)$. Mass is set to $M = 1$.
• Figure 4: Potential function $H(r, \pi/2)$ for Model-I quantum-corrected BHs with CoS, for different values of the CoS parameter $\alpha$, while keeping fixed $\zeta = 0.1$ and $M = 1$. The peak of the potential shifts to larger radii as $\alpha$ increases, corresponding to the outward movement of the photon sphere due to enhanced gravitational effects from the CoS.
• Figure 5: The arrows represent the unit vector field $\mathbf{n}_H$ on a portion of the $r$--$\Theta$ plane for Model-I of Letelier BHs in quantum gravity with $M = 1$ and $\zeta = 0.1$. The photon ring (PR), marked with a black dot, is at $(r, \theta) = (3.35, \pi/2)$ for $\alpha = 0.05$ (left); and $(r, \theta) = (3.5, \pi/2)$ for $\alpha = 0.10$ (right). The red contour $C_i$ is a closed loop enclosing the photon ring. The topological charge of the photon ring is $Q = -1$, corresponding to a standard unstable light ring.