Jacobi and Lyapunov stability analysis of circular geodesics around a spherically symmetric dilaton black hole

TL;DR

This paper investigates the stability of circular geodesics around a charged black hole in string theory using both Lyapunov and Jacobi stability methods, finding consistent results that enhance understanding of black hole spacetime dynamics.

Contribution

It applies and compares Lyapunov and Jacobi stability analyses to geodesics in a string theory black hole spacetime, demonstrating their equivalence in this context.

Findings

Lyapunov and Jacobi stability methods yield consistent results.

Circular geodesics are stable under both stability analyses.

The stability results depend on the second derivative of the effective potential.

Abstract

We analyze the stability of the geodesic curves in the geometry of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi-Cartan-Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that in the particular case of the geodesic motion on circular orbits in the Gibbons-Maeda-Garfinkle-Horowitz-Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.