Exact solution of vacuum field equation in Finsler spacetime

We suggest that the vacuum field equation in Finsler spacetime is equivalent to the vanishing of the Ricci scalar. The Schwarzschild metric can be deduced from a solution of our field equation if the spacetime preserves spherical symmetry. Supposing that the spacetime preserves the symmetry of the “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 the Schwarzschild metric as the Finslerian parameter $\epsilon$ vanishes. It is proven 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 the geodesic equation in such a Schwarzschild-like spacetime, and show that the geodesic equation returns to its counterpart in Newtonian gravity in the weak-field approximation. Celestial observations give a constraint on the Finslerian parameter $\epsilon <10^{-4}$, and the recent Michelson-Morley experiment requires $\epsilon <10^{-16}$. A counterpart of Birkhoff’s theorem exists in the Finslerian vacuum. This shows that the Finslerian gravitational field with the symmetry of the “Finslerian sphere” in vacuum must be static.