Kerr geodesics in terms of Weierstrass elliptic functions
Adam Cieślik, Eva Hackmann, Patryk Mach
Abstract
We derive novel analytical solutions describing timelike and null geodesics in the Kerr spacetime. The solutions are parametrized explicitly by constants of motion---the energy, the angular momentum, and the Carter constant---and initial coordinates. A single set of formulas is valid for all null and timelike geodesics, irrespectively of their radial and polar type. This uniformity has been achieved by applying a little-known result due to Biermann and Weierstrass, regarding solutions of a certain class of ordinary differential equations. Different from other expressions in terms of Weierstrass functions, our solution is explicitly real for all types of geodesics. In particular, for the first time the so-called transit orbits are now expressed by explicitly real Weierstrass functions.