Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications

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 set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer Anomaly is addressed. The periastron shift and its post-Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space-times.

Hyperelliptic functions and geodesic equations

“…As a result, the position in a space-time with and without cosmological constant differ only in the scale of 10 −3 m at a heliocentric distance of about 60AU, which is insufficient to explain the anomaly [4]. Another application is that the periastron advance of bound orbits can be calculated in a post-Schwarzschild approximation.…”

“…In the context of string theory and the AdS/CFT correspondence, also higher dimensional space-times are of interest. The geodesic equations in Schwarzschild and Schwarzschild-de Sitter space-times up to a dimension of 11 can be reduced to the same type of equation which appears in four-dimensional Schwarzschild-de Sitter space-time, [4][5][6].…”

“…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.…”

Study of geodesic motion in a (2+1)-dimensional charged BTZ black hole

“…For this purpose, we need to solve geodesic equations that describe the motion of particles and light. The analytical solutions for many famous spacetimes (such as Schwarzschild [14], four-dimensional Schwarzschildde-Sitter [15], higher-dimensional Schwarzschild, Schwarzschild-(anti)de Sitter, ReissnerNordstrom and Reissner-Nordstrom-(anti)-de Sitter [16], Kerr [17], Kerr-de Sitter [18], A black hole in f(R) gravity [19]) have been found previously. The solutions are given in terms of Weierstrass ℘-functions and derivatives of Kleinian sigma functions.…”