Analytic solutions of equations of motion in General Relativity

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There are only very few analytic solutions known for the geodesic equation in black hole space–times. They are given by elliptic functions. Based on our recent work (1, 2, 3) where we worked out a new method yielding analytic solutions in terms of hyperelliptic functions based on polynomials of 6th order and restricted to the theta–divisor, we intend to find the complete set of analytic solutions in all Plebański–Demiański black hole space–times which includes Kerr–de Sitter, for example, and also to higher dimensional axially symmetric space–times. We also intend to apply this method to the geodesic motion of point particles in axially symmetric mass multipole field and also to particles with spin and mass quadrupole moment and to strings. This new method should be extended to higher order polynomials. Based on these analytical solutions all possible physical effects like Perihelion shift, deflection angles, Lense–Thirring effect, etc. can be given analytically. Analytic solutions of the geodesic equation are important for a physical interpretation of space–times, can be used for solving the effective one–body dynamics of binary systems yielding gravitational wave templates, may serve as test bed for numerical codes, and has practical applications in geodesy and navigation.