Solution of Vacuum Field Equation Based on Physics Metrics in Finsler Geometry and Kretschmann Scalar

The Lemaître-Tolman-Bondi (LTB) model represents an inhomogeneous spherically symmetric universe filled with freely falling dust-like matter without pressure. First, we have considered a Finsleriananstaz of (LTB) and have found a Finslerian exact solution of vacuum field equation. We have obtained the $R(t,r)$ and $S(t,r)$ with considering establish a new solution of $R_{µν} = 0$. Moreover, we attempt to use Finsler geometry as the geometry of space-time which compute the Kretschmann scalar. An important problem in General Relativity is singularities. The curvature singularities is a point when the scalar curvature blows up diverges. Thus we have determined $K_s$ singularity is at $R = 0$. Our result is the same as Riemannian geometry. We have completed with a brief example of how these solutions can be applied. Second, we have some notes about anstaz of the Schwarzschild and Friedmann- Robertson- Walker (FRW) metrics. We have supposed condition $d\log (F) = d\log (\bar{F})$ and we have obtained $\bar{F}$ is constant along its geodesic and geodesic of $F$. Moreover we have computed Weyl and Douglas tensors for $F^2$ and have concluded that $R_{ijk} = 0$ and this conclude that $W_{ijk} = 0$, thus $F^2$ is the Ads Schwarzschild Finsler metric and therefore $F^2$ is conformally flat. We have provided a Finslerian extension of the Friedmann-Lemaitre-Robertson- Walker metric based on solution of the geodesic equation. Since the vacuum field equation in Finsler spacetime is equivalent to the vanishing of the Ricci scalar, we have obtained the energy-momentum tensor is zero.