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 universefilledwithfreelyfallingdustlikematterwithoutpressure. First,wehaveconsideredaFinslerian anstaz 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 Rµν = 0. Moreover, we attempttouseFinslergeometryasthegeometryofspacetimewhichcomputetheKretschmannscalar. 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 Ks singularity is at R = 0. Our result is the same as Reimannian 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 dlog(F) = dlog( ¯ F) and we have obtained ¯ F is constant along its geodesic and geodesic of F. Moreover we have computed Weyl and Douglas tensors for F2 and have concluded that Rijk = 0 and this conclude that Wijk = 0, thus F2 is the Ads Schwarzschild Finsler metric and therefore F2 is conformally flat. We have provided a Finslerian extention of 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.