Analytical solution of the geodesic equation in Kerr-(anti-) de Sitter space-times

Abstract

The complete analytical solutions of the geodesic equations in Kerr-de Sitter and Kerr-anti-de Sitter space-times are presented. They are expressed in terms of Weierstrass elliptic ℘, ζ, and σ functions as well as hyperelliptic Kleinian σ functions restricted to the one-dimensional θ-divisor. We analyse the dependency of timelike geodesics on the parameters of the space-time metric and the test-particle and compare the results with the situation in Kerr space-time with vanishing cosmological constant. Furthermore, we systematically can find all last stable spherical and circular orbits and derive the expressions of the deflection angle of flyby orbits, the orbital frequencies of bound orbits, the periastron shift, and the Lense-Thirring effect.PACS numbers:

Campanelli-Lousto and veiled spacetimes

“…It had to be so because, in vacuo, the Jebsen-Birkhoff theorem requires that the only static spherically symmetric solution of R ¼ 0 with zero cosmological constant be the Schwarzschild solution. This is also in agreement with a weak version of the Jebsen-Birkhoff theorem in scalar-tensor gravity [22]. …”

Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions

“…In [24] and [25], the geodesic equations are analytically solved in the background of Schwarzschild-(anti) de Sitter spacetimes, where the solutions are expressed in terms of Kleinian sigma functions. In [26], the investigation of the analytic solutions has been extended to Kerr-(anti) de Sitter spacetimes where in this case the solutions are presented in terms of Weierstrass elliptic functions. In a similar fashion, geodesic equations are solved in the spacetime of Kerr black hole pierced by a cosmic string [27], where the perihelion shift and the Lense-Thirring effect have also been investigated for bound orbits.…”

Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime

“…The geodesic equations (11) can be solved as functions of Mino time in closed form using elliptic functions [19,[23][24][25][26]. We here repeat the explicit solutions for bound geodesics given by Fujita and Hikida [19], in a much simplified form.…”