Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry

Details Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry

2011-12-22

arxiv.org/abs/1112.5481v2

Class. Quantum Grav. 29 (2012) 065010

Physics

Mathematical Physics

28 pages

Scientific paper

10.1088/0264-9381/29/6/065010

In the search for vacuum solutions, with or without a cosmological constant, of the Einstein field equations of Petrov type N with twisting principal null directions, the CR structures to describe the parameter space for a congruence of such null vectors provide a very useful tool. A work of Hill, Lewandowski and Nurowski has given a good 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 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 by Nurowski, is shown to be a special case in this class. Cartan's method for determining equivalence of CR manifolds is used to show that this class is indeed much more general. 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.

Affiliated with

Also associated with

No associations

Rating

Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-191647