[Paper Review] Gravitational probe of quantum spacetime

Nikola Herceg, Tajron Jurić+3 more|arXiv (Cornell University)|Oct 9, 2023

Noncommutative and Quantum Gravity TheoriesPhysics and Astronomy9 references3 citations

TL;DR

This paper proposes a noncommutative (NC) deformation of the Regge-Wheeler potential in Schwarzschild spacetime using a Drinfeld twist-based NC differential geometry framework. It derives the first-order quantum-corrected potential and shows that axial gravitational perturbations of the NC Schwarzschild black hole remain stable, with quasinormal mode frequencies exhibiting a leading-order NC correction. The results provide a phenomenologically accessible quantum gravity signature in gravitational wave physics.

ABSTRACT

A quest for phenomenological footprints of quantum gravity is among the central scientific tasks in the rising era of gravitational wave astronomy. We study gravitational wave dynamics within the noncommutative geometry framework, based on a Drinfeld twist and newly proposed noncommutative Einstein equation, and obtain the leading quantum correction to Regge-Wheeler potential up to first order in the noncommutativity parameter. By calculating the quasinormal mode frequencies we show that the noncommutative Schwarzschild black hole remains stable under axial gravitational perturbations.

Motivation & Objective

• To investigate quantum gravity effects on gravitational wave dynamics using noncommutative (NC) spacetime geometry.
• To derive a noncommutative generalization of the Regge-Wheeler potential for axial gravitational perturbations.
• To assess the stability of the NC Schwarzschild black hole under linear perturbations.
• To provide a phenomenologically viable quantum gravity correction that appears at first order in the NC parameter, unlike most prior works.
• To establish a framework compatible with R-symmetry and deformed Leibniz rules, avoiding overcompleteness issues in NC Einstein tensor construction.

Proposed method

• Employing a Drinfeld twist to define a Moyal-type noncommutative ⋆-product on spacetime, with deformation parameter encoded in an antisymmetric matrix Θμν.
• Formulating NC differential geometry via a Hopf algebra of deformed diffeomorphisms, ensuring consistency with R-symmetry and deformed Leibniz rules.
• Postulating a new NC Einstein equation based on a deformed Ricci tensor and a modified Einstein tensor that respects R-symmetry.
• Linearizing the NC Einstein equation around a Schwarzschild background metric up to first order in the NC parameter ¯q and gravitational perturbation h.
• Deriving a Schrödinger-like equation for axial gravitational perturbations governed by a ¯q-deformed Regge-Wheeler potential.
• Calculating quasinormal mode (QNM) frequencies numerically to assess stability and extract quantum corrections.

Experimental results

Research questions

• RQ1 What is the form of the Regge-Wheeler potential in a noncommutative spacetime framework with a Drinfeld twist?
• RQ2 How do quantum corrections from noncommutative geometry modify the quasinormal mode spectrum of a Schwarzschild black hole?
• RQ3 Is the NC Schwarzschild black hole stable under axial gravitational perturbations at first order in the NC parameter?
• RQ4 Can the NC deformation lead to observable phenomenological signatures in gravitational wave data, particularly at leading order?
• RQ5 How does the proposed NC framework, with deformed Leibniz rules and R-symmetry, resolve issues of overcompleteness in previous NC Einstein tensor constructions?

Key findings

• The leading quantum correction to the Regge-Wheeler potential appears at first order in the noncommutativity parameter ¯q, in contrast to second-order corrections in most prior studies.
• The quasinormal mode (QNM) frequencies of the NC Schwarzschild black hole exhibit a first-order correction in ¯q, providing a potentially detectable signature in future gravitational wave observations.
• The NC Schwarzschild black hole remains stable under axial gravitational perturbations, as confirmed by the absence of unstable modes in the QNM spectrum.
• The deformed Regge-Wheeler potential shows a characteristic shift in the potential barrier, with a divergence at the horizon that can be regularized via a coordinate transformation to ensure well-defined boundary conditions.
• In the limit of vanishing noncommutativity (¯q → 0), the standard Regge-Wheeler potential and QNM frequencies of classical general relativity are smoothly recovered.
• The proposed NC Einstein equation, based on a deformed Ricci tensor and R-symmetry-preserving Einstein tensor, avoids overcompleteness problems present in earlier formulations.