[Paper Review] Reissner-Nordstrom black hole in noncommutative spaces

S. A. Alavi|arXiv (Cornell University)|Sep 9, 2009

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper investigates the Reissner-Nordström black hole in noncommutative spacetime by replacing the point-like mass with a Gaussian-smeared distribution to resolve singularities. It derives a modified metric and Hawking temperature, showing a minimal non-zero mass and finite maximum temperature, with charge increasing the minimal mass and reducing the maximum temperature. The noncommutativity parameter is bounded above by √θ ≈ 10⁻³⁴ cm.

ABSTRACT

We investigate the behaviour of a non-commutative radiating Reissner-Nordstrom(Re-No)black hole. We find some interesting results : a). the existence of a minimal non-zero mass to which the black hole can shrink. b). a finite maximum temperature that the black hole can reach before cooling down to absolute zero. c) compared to the neutral black holes the effect of charge is to increase the minimal non-zero mass and lower the maximum temperature. d) the absence of any curvature singularity. We also derive some essential thermodynamic quantities from which we study the stability of the black hole. Finally we find an upper bound for the non-commutativity parameter $θ$.

Motivation & Objective

• To resolve curvature singularities and UV divergences in Reissner-Nordström black holes using noncommutative geometry.
• To investigate how noncommutativity modifies black hole thermodynamics, particularly Hawking temperature and mass evolution.
• To derive a consistent noncommutative version of the Reissner-Nordström metric that avoids inconsistencies in prior approaches.
• To establish an upper bound for the noncommutativity parameter θ based on physical constraints from black hole evaporation.
• To ensure coordinate independence and tensorial consistency in the modified energy-momentum tensor formulation.

Proposed method

• Implementing noncommutativity via Gaussian smearing of the point-like mass distribution, replacing δ(r) with a nonlocal Gaussian source.
• Preserving the standard Einstein tensor while modifying the right-hand side of the field equations through a nonlocal energy-momentum tensor.
• Using coherent states as the closest approximation to position eigenstates in noncommutative space, enabling mean-value calculations of coordinates.
• Solving the modified Einstein equations with the smeared source to derive a new metric that reduces to the standard Reissner-Nordström form at large distances.
• Calculating thermodynamic quantities such as Hawking temperature and entropy from the modified metric and mass function.
• Applying regularization via the noncommutative minimal length scale √θ to eliminate divergences in temperature and curvature invariants.

Experimental results

Research questions

• RQ1 How does noncommutativity affect the existence and value of a minimal non-zero mass for a Reissner-Nordström black hole?
• RQ2 Does the introduction of noncommutativity eliminate the curvature singularity at r=0 in the Reissner-Nordström solution?
• RQ3 What is the behavior of the Hawking temperature in the noncommutative regime, and does it exhibit a finite maximum value?
• RQ4 How does electric charge Q influence the minimal mass and maximum temperature in the noncommutative framework?
• RQ5 What upper bound can be placed on the noncommutativity parameter θ based on consistency with black hole thermodynamics and Planck-scale physics?

Key findings

• A minimal non-zero mass M₀ ≈ 0.5√(πθ) + 0.2Q²/√(πθ) exists, preventing complete evaporation to zero mass.
• The black hole reaches a finite maximum temperature T_H^max ≈ 1.5×10⁻²/√θ before cooling to absolute zero, resolving the divergence in standard Hawking radiation.
• The presence of charge Q increases the minimal mass and decreases the maximum temperature compared to neutral black holes.
• The curvature invariants remain finite at r=0, indicating the absence of a curvature singularity due to nonlocal smearing.
• An upper bound on the noncommutativity parameter is derived: √θ ≈ 10⁻³⁴ cm, consistent with suppression of back-reaction effects.
• The modified metric asymptotically reduces to the classical Reissner-Nordström solution at large distances (r/√θ → ∞), ensuring consistency with general relativity in the classical limit.