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

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. 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 is discussed by analyzing 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 for a massless scalar field perturbation in the eikonal approximation are evaluated and also visualized 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), is also observed and discussed.