Complete Analytic Solution of the Geodesic Equation in Schwarzschild–(Anti-)de Sitter Spacetimes

Abstract

The complete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti) de Sitter space-time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta-divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.

Geodesic motion in Schwarzschild spacetime surrounded by quintessence

“…In General Relativity (GR), the curvature and geometry of the space-time play crucial role as space-time is curved with the presence of matter fields [26,27,28,29]. A number of studies related to the geodesic motion in the background of various spacetimes has been performed time and again due to its astrophysical importance [30,31,32,33,34,35]. In general, the effects of the curvature in a given space-time is studied through the Geodesic Deviation Equations (GDE) [36,37,38], the equations which describe the relative acceleration of two neighbouring geodesics in diversified scenario [28,29,39,40,41,42].…”

Hyperelliptic functions and geodesic equations

“…It can be shown [3,8] that φ ∞ is an element of the theta-divisor, that is, the set of zeroes of the theta function. It follows σ( φ) = 0 and a functional relation between the components φ ∞,1 and φ ∞,2 of φ ∞ .…”

“…Second, the functional relation between the components of φ ∞ can be used to get rid of φ ∞,j = f (φ ∞,i+1 ), where j = i + 1 and i given by (3). The function f has to be chosen such that φ ∞ is an element of the theta-divisor, i.e.…”

“…The analytical solution of the geodesic equation in Schwarzschild space-time was elaborated by Hagihara [2], but for a nonvanishing cosmological constant Λ, the analyical solution of the geodesic equation was only recently found by the authors [3,4]. In the context of string theory and the AdS/CFT correspondence, also higher dimensional space-times are of interest.…”

“…A complete elaboration of this can be found in [3,4,6]. With this method, which is based on the theory of Hyperelliptic functions and Riemannian surfaces, the solution to the geodesic equation can be expressed in terms of derivatives of Kleinian sigma functions.…”