引力场方程静态球对称解与球极射影理论的研究  

THE STATIC SPHERICAL SOLUTION OF GRAVITATIONAL FIELD EQUATION AND STEREOGRAPHIC PROJECTION

作　　者：赵江峰[1] 姜孟瑞[1] 

机构地区：[1]山东师范大学物理与电子科学学院,济南250014

出　　处：《山东师范大学学报（自然科学版）》2017年第1期112-117,共6页Journal of Shandong Normal University(Natural Science)

摘　　要：采用球极射影的方法描述四维空间中引力场方程的静态球对称解,研究了四维空间中的Schwarzschild空间度量.运用球极射影的方法降低维度,将四维的球对称解的维度降低到三维.在四维超球坐标系中,利用测地线方程,求解了四维超球面的测地线方程,并简明论证了球极射影映射中保形的特性,为将球极射影理论应用到场理论中提供了理论依据.通过球极射影的方法从四维空间向三维空间投影,减少了四维超球坐标中的角度项,再从三维空间向二维空间投影,得到仅含一个角度项的度量表达式,最后得出用球极射影方法描述的四维空间Schwarzschild时空度量.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.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 transformned 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,where there is only one angle component.Finally we get the new Schwarzschild space-time solution in a five-dimensional Pseudo-Euclidean space in physics.

关 键 词：球极射影 静态球对称解 超球面 测地线 

分 类 号：O412[理学—理论物理]