Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's "geometrical" method

Dino Boccaletti, Francesco Catoni, Roberto Cannata, Paolo Zampetti

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5 Citations (Scopus)


We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.
Original languageEnglish
Pages (from-to)2261 - 2273
Number of pages13
JournalGeneral Relativity and Gravitation
Issue number12
Publication statusPublished - Dec 2005
Externally publishedYes


All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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