Exact solutions and transformation properties of nonlinear partial differential equations from general relativity
<p>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 nonli...
| Summary: | <p>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. </p>
<p>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. </p>
<p>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.</p>
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