Generating Matrices of Rotations in Minkowski Spaces using the Lie Derivative

Abstract

This paper aims to generate materices of rotations in Minkowski using the Lie Derivative. The calculus on manifolds in Lorentzian spaces are used to generate matrices of rotation in three-dimensional Lorentz-Minkowski space which includes one axis in timelike and the other two are spacelike axes. The findings showed that the manifolds and their calculus dramatically increased the use of Lie derivative in many branches of mathematics and physics, The findings also revealed that matrices ( of rotation) leave one line ( axis) fixed and these matrices of rotation are used widely in differential geometry in physics. Furthermore, the findings demonstrated that any surfaces of revolution inside this space must be invariant under one of these matrices. The main result of this paper is a new procedure of creating rotational matrices explicitly using the Lie derivative and deriving it into a linear system of ordinary differential equaion. Solving this system leads to matrices of rotation that leaves one axis fixed in Minkowski space.