Physics & Astronomy ETDs


Xuefeng Zhang

Publication Date



In the search for vacuum solutions, with or without the cosmological constant Λ, of the Einstein field equations for Petrov type N with twisting principal null directions, the CR structures, which describe the parameter space for the geodesic congruence tangent to such null vectors, provide a useful invariant approach. Work of Hill, Lewandowski and Nurowski has laid a solid foundation for this, reducing the field equations to a set of differential equations for two functions, one real, one complex, of three variables. Under the assumption of the existence of one special Killing vector, the infinite-dimensional classical symmetries of those equations are determined and group-invariant solutions are considered. This results in a single ODE of the third order which may easily be reduced to one of the second order. A one-parameter class of power series solutions, g(w), of this second-order equation is realized, holomorphic in a neighborhood of the origin and behaving asymptotically as a simple quadratic function plus lower-order terms for large values of w, which constitutes new solutions of the twisting type N problem. The solution found by Leroy, and also later by Nurowski, is shown to be a special case in this class. Cartan’s method for determining local equivalence of CR manifolds is used to show that this class is indeed much more general. Also for the general metrics determined by this second-order ODE, two Killing vectors, including the one already assumed, can be found, both of which are inherited from symmetries of the underlying CR structures. In addition, for a special choice of a parameter, this ODE may be integrated once, to provide a first-order Abel equation. It can also determine new solutions to the field equations although no general solution has yet been found for it.

Degree Name


Level of Degree


Department Name

Physics & Astronomy

First Advisor

Finley, Daniel

First Committee Member (Chair)

Finley, Daniel

Second Committee Member

Boyer, Charles

Third Committee Member

Allahverdi, Rouzbeh

Fourth Committee Member

Duan, Huaiy




Gravitational waves--Mathematical models, Einstein field equations--Numerical solutions, CR submanifolds, Abel integral equations.

Document Type