Theorems of Birkhoff type in Finsler spaces

In this paper we give a summary of basic definitions of Finsler geometry as well as a possible extension of General Relativity, in the vacuum case, for these spaces, followed by a brief account of symmetry imposition for Finsler metrics. We then make use of these results in order to generalise, for up to first order departures from the Riemannian setting, the well-known Birkhoff's theorem from General Relativity, thereby establishing a perturbative, first order version of this theorem for Finsler spaces, where some simplifying assumptions have been made. This result has been fully accomplished by means of computer algebra — actually, the very problem of explicitly determining non-Riemannian solutions for some generalised theory of gravity has only been made possible by computer algebra. The process is described in detail within the paper.