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


Fischer, Elliot (1977) Exact solutions and transformation properties of nonlinear partial differential equations from general relativity. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PXKA-1861.


Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mathematics, Exact Solutions
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Applied Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Lagerstrom, Paco A.
Thesis Committee:
  • Unknown, Unknown
Defense Date:27 May 1977
Non-Caltech Author Email:eafischer (AT)
Record Number:CaltechTHESIS:03212013-145828508
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7546
Deposited On:25 Mar 2013 16:37
Last Modified:09 Nov 2022 19:20

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