Exact solution of vacuum field equation in Finsler spacetime

Abstract

We suggest that the vacuum field equation in Finsler spacetime is equivalent to vanishing of Ricci scalar. Schwarzschild metric can be deduced from a solution of our field equation if the spacetime preserve spherical symmetry. Supposing spacetime to preserve the symmetry of "Finslerian sphere", we find a non-Riemannian exact solution of the Finslerian vacuum field equation. The solution is similar to the Schwarzschild metric. It reduces to Schwarzschild metric while the Finslerian parameter $\epsilon$ vanishes. It is proved that the Finslerian covariant derivative of the geometrical part of the gravitational field equation is conserved. The interior solution is also given. We get solutions of geodesic equation in such a Schwarzschild-like spacetime, and show that the geodesic equation returns to the counterpart in Newton's gravity at weak field approximation. The celestial observations give constraint on the Finslerian parameter $\epsilon<10^{-4}$. And the recent Michelson-Morley experiment requires $\epsilon<10^{-16}$. The counterpart of Birkhoff's theorem exist in Finslerian vacuum. It shows that the Finslerian gravitational field with the symmetry of "Finslerian sphere" in vacuum must be static.