Geodesic stability and quasinormal modes of non-commutative Schwarzschild black hole employing Lyapunov exponent

TL;DR

This paper investigates the stability of circular geodesics and quasinormal modes of a non-commutative Schwarzschild black hole using Lyapunov exponents, revealing how non-commutativity affects black hole perturbations and orbit stability.

Contribution

It introduces a method to analyze geodesic stability and quasinormal modes in non-commutative black holes using Lyapunov exponents, highlighting the impact of non-commutativity.

Findings

Lyapunov exponent varies with orbit radius and non-commutative parameter.

Null circular orbits exhibit specific instability characteristics.

Quasinormal modes are related to null geodesic parameters and affected by non-commutativity.

Abstract

We study the dynamics of test particle and stability of circular geodesics in the gravitational field of a non-commutative geometry inspired Schwarzschild black hole spacetime (NCSBH). The coordinate time Lyapunov exponent ($\lambda_{c}$) is crucial to investigate the stability of equatorial circular geodesics of massive and massless test particles. The stability or instability of circular orbits are discussed by analysing the variation of Lyapunov exponent with radius of these orbits for different values of non-commutative parameter ($\alpha$). In the case of null circular orbits, the instability exponent is calculated and presented to discuss the instability of null circular orbits. Further, by relating parameters corresponding to null circular geodesics (i.e. angular frequency and Lyapunov exponent), the quasinormal modes (QNMs) for a massless scalar field perturbation in the eikonal approximation are evaluated, and also visualised by relating the real and imaginary parts. The nature of scalar field potential, by varying the non-commutative parameter ($\alpha$) and angular momentum of perturbation ($l$), are also observed and discussed.