Eva Hackmann

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

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.

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

All observations in the gravitational domain can be explained by means of Einstein’s General Relativity. While for small scale gravitational effects (e.g. in the solar system) the standard Einstein field equations are sufficient, a consistent description of large scale obervations like the accelerated expansion of the universe can be achieved by the introduction of a cosmological term into the Einstein field equation

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

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.

Charged particle motion in Kerr-Newmann space-times

The motion of charged test particles in the gravitational field of a rotating and electromagnetically charged black hole as described by the Kerr-Newman metric is considered. We completely classify the colatitudinal and radial motion on the extended manifold -∞≤r≤∞, including orbits crossing the horizons or r=0. Analytical solutions of the equations of motion in terms of elliptic functions are presented which are valid for all types of orbits.

The Complete set of solutions of the geodesic equations in the space-time of a Schwarzschild black hole pierced by a cosmic string

We study the geodesic equation in the space-time of a Schwarzschild black hole pierced by an infinitely thin cosmic string and give the complete set of analytical solutions of this equation for massive and massless particles, respectively. The solutions of the geodesic equations can be classified according to the particles energy and angular momentum, the ratio between the component of the angular momentum aligned with the axis of the string and the total angular momentum, the deficit angle of the space-time and as well the horizon radius (or mass) of the black hole. For bound orbits of massive test particles we calculate the perihelion shift, we discuss light deflection and comment on the Newtonian limit.

Test particle motion in the space-time of a Kerr black hole pierced by a cosmic string

We study the geodesic equation in the space-time of a Kerr black hole pierced by an infinitely thin cosmic string and give the complete set of analytical solutions of this equation for massive and massless particles in terms of Mino time that allows one to decouple the r and {theta} components of the geodesic equation. The solutions of the geodesic equation can be classified according to the particles energy and angular momentum, the mass and angular momentum per mass of the black hole. We give examples of orbits showing the influence of the cosmic string. We also discuss the perihelion shift and the Lense-Thirring effect for bound orbits and show that the presence of a cosmic string enhances both effects. Comparing our results with experimental data from the LAGEOS satellites we find an upper bound on the energy per unit length of a string piercing the earth which is approximately 10{sup 16} kg/m. Our work has also applications to the recently suggested explanation of the alignment of the polarization vector of quasars using remnants of cosmic string decay in the form of primordial magnetic field loops.

Motion of test particles in a regular black hole space–time

We consider the motion of test particles in the regular black hole space-time given by Ayon-Beato and Garcia [Phys. Rev. Lett. 80, 5056 (1998)]. The complete set of orbits for neutral and weakly charged test particles is discussed, including for neutral particles the extreme and over-extreme metric. We also derive the analytical solutions for the equation of motion of neutral test particles in a parametric form and consider a post-Schwarzschild expansion of the periastron shift to second order in the charge.

Motion of spinning test bodies in Kerr spacetime

Extreme mass ratios in astrophysical situations, for example as found in the galactic center, allow for an approximate analytic description of the motion in certain parameter regimes. The steadily improving observational situation of the galactic center [1–3] may soon enable us to test different competing theoretical approaches to model the motion of astrophysical objects in the theory of General Relativity. In this work we study the motion of extended spinning test bodies in a Kerr background. Our starting point is an explicit velocity formula based on the multipolar description [4–8] of pole-dipole test bodies, with the help of which we classify the orbital motion in the equatorial plane of a Kerr black hole for aligned and anti-aligned test body spin. An exact expression for the periastron shift is given and compared with corresponding post-Newtonian results. We provide an estimate of the test body spin corrections for orbits around the black hole in the galactic center. The structure of the paper is as follows. In section II we provide the equations of motion for spinning test bodies and derive a general formula which relates the momentum and the velocity of the test body. The motion of spinning test bodies is then studied in a Kerr background in section III. These equations of motion are of a mathematical structure which allows for an analytic solution [9, 10] and a systematic classification of different orbit types in section IV. In section V a general formula for the periastron shift is given and compared to corresponding post-Newtonian results. Our conclusions are drawn in section VI. In the appendices A and B we pro

Analytic solutions of the geodesic equation in axially symmetric space-times

The complete sets of analytic solutions of the geodesic equation in Taub-NUT-(anti-) de Sitter, Kerr-(anti-)de Sitter and also in general Plebanski-Demianski space-times without acceleration are presented. The solutions are given in terms of the Kleinian sigma functions.

Generalized gravitomagnetic clock effect

In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of