The Schwarzschild Gravitational Field And Stereographic Projection

The paper describes the solution of the Einstein field equation in four-dimensional space by the stereographic projection and studies the Schwarzschild space-time solution.First,the paper introduced the principle of Einstein deduced with the gravitational field equations of Lagrange Hamilton,demonstrated the symmetric of Riemann geometry curvature tensor.Then,starting with the generalized symmetry,we derived the Schwarzschild space-time metric clearly,and discussed the particle geodesic and the effective of Schwarzschild gravitational field potential.We discussied Schwarzschild black hole and the wormhole from the special properties of space-time metric at some point and surface.In the end,we proposed the method of stereographic projection,and studied the Schwarzschild space metric in four-dimensional space.Stereographic projection can reduce one dimension and transform the four-dimension static spherical solution to three-dimension static spherical solution.In the four-dimensional hypersphere coordinate,we find the solution of geodesic equation in four-dimension by the differential equations and argue the shape preserving of stereographic projection.The shape preserving of stereographic projection provides the theoretical basis of applying the stereographic projection to field theory.The four-dimensional space is transformed to a three-dimensional space by stereographic projection and reduced an angle component of the four-dimensional hypersphere coordinate.Then the three-dimensional space can also be transformed to a two-dimensional.There is only one angle component.Finally,we get the new Schwarzschild space-time solution in a fivedimensional Pseudo-Euclidean space in physics.This paper provides a method for studying the static spherically symmetric solutions of high dimensional field equations on the basics of relativistic gravity.