Fe b 20 05 A Beltrami ’ s theorem and the geodesics in the Schwarzschild and Kerr space - time

We revisit a theorem, somehow neglected at present, due to E. Beltrami, through which the integration of the geodesic equations of a curved manifold is obtained by means of a method which, even if inspired by Hamilton-Jacobi method, is merely geometric. The application of this theorem to the Schwarzschild and Kerr metrics allows us to obtain in a straightforward and general way the solution of the geodesic equations. This way of dealing with the problem is, in our opinion, well in accordance with the geometric spirit of the Theory of General Relativity. On the contrary, the usually applied methods carry out the integration of the geodesic equations by translating back the geometrical problem into a mechanical one. PACS 02.40.Ky Semi-Riemannian geometry PACS 04.20.-q General relativity