Applications of symmetry groups and the Wahlquist-Estabrook procedure in general relativity

Phillip Charles Harmsworth
1988 Bulletin of the Australian Mathematical Society  
This thesis examines the application of symmetry groups and the Wahlquist-Estabrook procedure [10, 6] to the Einstein gravitational field equations for spacetimes with a two-parameter Abelian isometry group. The first chapter reviews the geometric formulation of partial differential equations and their symmetries in the language of jet bundles. Some methods are discussed for the construction of pseudopotentials associated with certain systems of equations. A new method is described identifying
more » ... he resulting (prolongation) structures with finite dimensional Lie algebras, which augments that of Dodd and Fordy [4] . The final section outlines the formulation of a system of partial differential equations with an associated set of nontrivial pseudopotentials as a zero-curvature condition on a principal bundle [8]. A discussion of the application of these methods to the Ernst [5] and Cosgrove [1, 2] formulations of the vacuum field equations occupies the second chapter, which reviews the results of Cosgrove [3], Harrison [7] and Neugebauer [9] . It is argued that the existing formulations of the pseudopotentials associated with the Ernst system are incomplete. The third chapter introduces two approaches to this problem. Neugebauer's formulation of the field equations is used to produce the defining equations for the pseudopotentials, and particular solutions to these are considered. The first produces an apparently nontrivial prolongation structure, which can be mapped to solvable Lie algebras of up to twelve dimensions (by the method described in the first chapter), while the second reduces the problem of solving a set of nonlinear partial differential equations (with matrix-valued dependent variables) to the solution of an Euler-Darboux equation (also with matrix-valued dependent variables). It. is shown that the symmetry group of this equation is isomorphic to the non-local Q symmetry of Cosgrove. Harrison's pseudopotential formulation is shown to result from the only similarity solution of this equation that leads to a non-trivial prologation structure.
doi:10.1017/s0004972700027106 fatcat:xyd7jfivovfahajmlojkj4m4cq