Effect of Noncommutative Geometry on Accretion Disks around RGI-Schwarzschild Black Hole

TL;DR

The paper addresses quantum-gravity corrections to accretion disks by combining κ-deformed non-commutative geometry with RG-improved gravity in a Schwarzschild background. It constructs the κ-deformed RGI-Schwarzschild metric with line element dŝ_RGI^2 and analyzes geodesics, the effective potential, ISCO, and disk thermodynamics, all expressed with explicit dependencies on the deformation parameter a and running parameter ṽ ω. Key results include an inward-shifted ISCO to x_{ISCO} ≈ 5.24 for ṽ ω = 0.39, a horizon-existence threshold ṽ ω_c = (16/27) e^{−4 a p^0}, and enhanced inner-disk flux and temperature due to the interplay of non-commutativity and scale-dependent gravity. These findings suggest potential observational signatures in disk spectra and encourage future work on rotating (RGI-Kerr) black holes and comparisons with X-ray binaries and active galactic nuclei.

Abstract

In this study, we explore the combined effects of quantum gravity induced by non-commutativity and scale-dependent gravitational coupling on the thermal properties of the thin accretion disks around a Schwarzschild black hole. We consider a $κ$-deformed Renormalization Group Improved (RGI) Schwarzschild black hole, where the classical Schwarzschild black hole geometry is modified by the $κ$-deformation of space-time and the running Newton's coupling constant $G(r)$. Using the modified metric, we derive the geodesic motion of massive particles, the effective potential, and the thermal properties such as the radiated energy flux, luminosity, and the temperature profile of the accretion disk around the $κ$-deformed RGI-Schwarzschild black hole. Our study shows that when non-commutativity is combined with the RGI framework, the effects produce a noticeable deviation from the classical Schwarzschild case. In particular, for small values of the deformation parameter, we observe an increase in the peak energy flux and the temperature of the accretion disk. This suggests that quantum gravity corrections enhance the disk's radiative efficiency, especially in the inner regions closer to the black hole.

Figures (9)

• Figure 1: The plot shows the comparison of the improved Schwarzschild metric $\hat{f}(r)$ for three different values of the running parameter $\tilde{\omega}$, (a) $\tilde{\omega} > \tilde{\omega}_{c}$ (solid cyan), (b) $\tilde{\omega} = \tilde{\omega}_{c}$ (dashed magenta), and (c) $\tilde{\omega} < \tilde{\omega}_{c}$ (dashed-dotted green), along with the classical Schwarzschild metric $f_{0}(r)$ (dotted red). The corresponding vertical lines indicate the locations of the event horizon for cases (b) and (c), showing that the horizon vanishes for case (a).
• Figure 2: The plot shows the geodesic motion of a massive test particle around the Schwarzschild black hole in different space-time backgrounds. The dotted line (red color) shows the classical Schwarzschild case, the dashed-dotted line (cyan color) corresponds to the RGI-Schwarzschild case, the solid line (magenta color) represents the $\kappa$-deformed RGI-Schwarzschild case.
• Figure 3: The plot shows the variation of the effective potential $\hat{\text{V}}_{\text{eff}}^{\text{RGI}}$ with respect to the dimensionless radial distance $x=r/M$ for massive particles around Schwarzschild black hole in different space-time backgrounds. The dashed line (red color) represents the classical Schwarzschild case, the dashed-dotted line (green color) corresponds to the RGI-Schwarzschild case in commutative space-time, the solid line (cyan color) shows the $\kappa$-deformed Schwarzschild case, and the solid line (magenta color) represents the $\kappa$-deformed RGI-Schwarzschild case. The $\kappa$-deformation and running gravitational coupling modify the depth and location of the potential well, leading to noticeable shifts in the stability and position of circular orbits, particularly the ISCO, as shown by black circles.
• Figure 4: The plot shows the variation of the specific angular momentum $\bar{h} = \hat{h}/M$ with respect to the dimensionless radial distance $x = r/M$ for massive particles around a Schwarzschild black hole in different space-time backgrounds. The dashed line (red color) represents the classical Schwarzschild case, the dashed-dotted line (green color) corresponds to the RGI-Schwarzschild case in commutative space-time, the solid line (cyan color) shows the $\kappa$-deformed Schwarzschild case, and the solid line (magenta color) represents the $\kappa$-deformed RGI-Schwarzschild case.
• Figure 5: The plot shows the variation of the angular velocity $\hat{\Omega} (x)$ with respect to the dimensionless radial distance $x = r/M$ for massive particles around a Schwarzschild black hole in different space-time backgrounds. The dashed line (red color) represents the classical Schwarzschild case, the dashed-dotted line (green color) corresponds to the RGI-Schwarzschild case in commutative space-time, the solid line (cyan color) shows the $\kappa$-deformed Schwarzschild case, and the solid line (magenta color) represents the $\kappa$-deformed RGI-Schwarzschild case.