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 to decouple the r-and θ-component of the geodesic equation. The solutions of the geodesic equation can be classified according to the particle's 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 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.

We study the geodesic equations 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 these equations for massive and massless particles, respectively. The solutions of the geodesic equations can be classified according to the particle's 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.

Another approach to the problem of inversion of integrals of the second and third kind that is based on the generalized θ-function goes back to Clebsch and Gordan [CG866] and was developed in [Fed99,BF08]. Here we do not discuss generalized Jacobians, and we plan to make a comparison between these two methods in another publication. B. Physical motivationThe mathematical results described in this paper have direct applications to the solution of the geodesic equation in certain Hořava-Lifshitz black hole space-times. The Hořava-Lifshitz theory [H09a, H09b] is an alternative gravity theory that is powercountable renormalizable. The basic idea is that only higher spatial derivative terms are added, while higher temporal derivatives which would lead to ghosts are not considered. This leads unavoidably to the breaking of Lorentz invariance at short distances. Static and spherically symmetric black hole solutions have been studied in this theory [KS09, LMP09, P09]. Considering the Hořava-Lifshitz theory as a modification of General Relativity, one can study the solutions of the geodesic equation in the Hořava-Lifshitz black hole space-times. In this paper, we are mainly interested in one of the black hole space-times given in [LMP09].The mathematical techniques described in this paper can not only be used to solve analytically the geodesic equation in the Hořava-Lifshitz space-time considered here. The differentials of the first and third kind with underlying polynomial curves of arbitrary genus appear in the geodesic equations in many general relativistic space-times. The holomorphic differentials appear in the equations for the r-and ϑ-coordinates, while the differentials of the third kind appear in the equations for the ϕ-and t-coordinates. This is also the case e.g. in the geodesic equations for neutral particles in Taub-NUT [KKHL10] space-times and in the space-times of Schwarzschild and Kerr black holes pierced by a cosmic string [HHLS10], as well as for charged particles in the Reissner-Nordström [GK10] space-time, where elliptic integrals of the first and third kind appear. They also appear in the Schwarzschild-de Sitter [HL08a, HL08] and Kerr-de Sitter space-times [HKKL09a] as well as in generalized black hole Plebański-Demiański space-times in 4 dimensions [HKKL09] with underlying hyperelliptic curves of genus two in the geodesic equations. Also in the higher dimensional space-times of Schwarzschild, Schwarzschild-de Sitter, Reissner-Nordström and Reissner-Nordström-de Sitter [HKKL08, EHKKL11] this powerful mathematics of the theory of hyperelliptic functions of higher genera is successfully applicable. Geodesics in higher dimensional axially symmetric space-times, the Myers-Perry space-times, are integrated by the hyperelliptic functions of arbitrary genus as well [EHKKL11]. In [EHKKL11] the integration of holomorphic integrals for any genus of the underlying hyperelliptic polynomial curve has been presented. Here we expand our considerations and present the solution for the integrals of the third kind for ...

We study the geodesic equation in the space-time of an Abelian-Higgs string and discuss the motion of massless and massive test particles. The geodesics can be classified according to the particles energy, angular momentum and linear momentum along the string axis. We observe that bound orbits of massive particles are only possible if the Higgs boson mass is smaller than the gauge boson mass, while massless particles always move on escape orbits. Moreover, neither massive nor massless particles can ever reach the string axis for non-vanishing angular momentum. We also discuss the dependence of light deflection by a cosmic string as well as the perihelion shift of bound orbits of massive particles on the ratio between Higgs and gauge boson mass and the ratio between symmetry breaking scale and Planck mass, respectively.Comment: 20 pages including 14 figures; v2: references added, discussion on null geodesics extended, numerical results adde

We study the particle motion in the space-time of a Kehagias-Sfetsos (KS) black hole. This is a static spherically symmetric solution of a Hořava-Lifshitz gravity model that reduces to General Relativity in the IR limit and deviates slightly from detailed balance. Taking the viewpoint that the model is essentially a (3+1)-dimensional modification of General Relativity we use the geodesic equation to determine the motion of massive and massless particles. We solve the geodesic equation exactly by using numerical techniques. We find that neither massless nor massive particles with non-vanishing angular momentum can reach the singularity at r = 0. Next to bound and escape orbits that are also present in the Schwarzschild space-time we find that new types of orbits exist: manyworld bound orbits as well as two-world escape orbits. We also discuss observables such as the perihelion shift and the light deflection.