Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Applied Mathematics, Exact Solutions

Fischer, Elliot ; Lagerstrom, Paco A.

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: Applied Mathematics, Exact Solutions;   

Others  :  https://thesis.library.caltech.edu/7546/1/Fischer%201977.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

Various families of exact solutions to the Einsteinand Einstein-Maxwell field equations of General Relativityare treated for situations of sufficient symmetry that onlytwo independent variables arise. The mathematical problemthen reduces to consideration of sets of two coupled nonlineardifferential equations.

The physical situations in which such equations ariseinclude: a) the external gravitational field of an axisymmetric,uncharged steadily rotating body, b) cylindricalgravitational waves with two degrees of freedom, c) collidingplane gravitational waves, d) the external gravitationaland electromagnetic fields of a static, charged axisymmetricbody, and e) colliding plane electromagnetic and gravitationalwaves. Through the introduction of suitable potentialsand coordinate transformations, a formalism ispresented which treats all these problems simultaneously.These transformations and potentials may be used to generatenew solutions to the Einstein-Maxwell equations from solutionsto the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a toolfor generation of similarity solutions and generalized similaritysolutions. It is further used to find the invariancegroup of the equations; this in turn leads to various finitetransformations that give new, physically distinct solutionsfrom old. Some of the above results are then generalized tothe case of three independent variables.