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 uni-verse 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 ob-tained theR(t, r)andS(t, r)with considering establish a new solution ofRμν= 0. Moreover, weattempt to use Finsler geometry as the geometry of spacetime which compute the Kretschmann scalar.An important problem in General Relativity is singularities. The curvature singularities is a pointwhen the scalar curvature blows up diverges. Thus we have determinedKssingularity is atR= 0.Our result is the same as Reimannian geometry. We have completed with a brief example of howthese solutions can be applied. Second, we have some notes about anstaz of the Schwarzschild andFriedmann- Robertson- Walker (F RW) metrics. We have supposed conditiondlog(F) =dlog( ̄F)and we have obtained ̄Fis constant along its geodesic and geodesic ofF. Moreover we have com-puted Weyl and Douglas tensors forF2and have concluded thatRijk= 0and this conclude thatWijk= 0, thusF2is the Ads Schwarzschild Finsler metric and thereforeF2is conformally flat. Wehave provided a Finslerian extention of Friedmann- Lemaitre- Robertson- Walker metric based onsolution of the geodesic equation. Since the vacuum field equation in Finsler spacetime is equivalentto the vanishing of the Ricci scalar, we have obtained the energy- momentum tensor is zero.