Journal of Theoretical, Experimental, and Applied Physics
Lie Symmetries and Conserved Quantities of Four-Dimensional Static Spacetime in Spherical Coordinate Systems
Abstract
Jing Li Fu, Xiao Fan Sun, Yong Xin Guo and Hui Dong Cheng
In this study, the Lagrange equation for static four-dimensional spacetime in spherical coordinate system was presented; Introducing the Lie group of transformations about four-dimensional static spacetime, we constructed an infinitely small generator vector field and its first-order and second-order extensions for four-dimensional static spacetime; Based on the invariance of the Lagrange equation for four-dimensional static spacetime in spherical coordinate system under the transformation of Lie group, the Lie symmetry determination equation of the system and its corresponding Lie symmetry partial differential equation system are given; Solving a series of partial differential equations and obtaining a definite transformation Lie group; The Lie symmetry theorem for four-dimensional static spacetime in spherical coordinate system was proposed, and the conservation quantity (first integral) determined by the metric coefficient of the system was obtained; It has been discovered that any value of the metric coefficient in four-dimensional static spacetime in a spherical coordinate system corresponds to a conserved quantity, which means that there can be countless conserved quantities corresponding to the metric coefficient in four-dimensional static spacetime in a spherical coordinate system. This article systematically establishes the Lie symmetry theory of four-dimensional static spacetime in spherical coordinate system.