Examples of Einstein spacetimes with recurrent null vector fields

Abstract

The Einstein Equation on 4-dimensional Lorentzian manifolds admitting recurrent null vector fields is discussed. Several examples of a special form are constructed. The holonomy algebras, Petrov types and the Lie algebras of Killing vector fields of the obtained metrics are found.

On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines

“…H 1 = 0, then consider the coordinates as in Theorem 2. Equation (14) shows that H 1 is a family of harmonic functions on the family of the Riemannian manifolds with metrics h(x − ). Fixing any such H 1 we get Equations (12) and (13) on the family of Ricci-flat Riemannian metrics h(x − ).…”

“…For most of the other functions f Equations (40) and their solutions become much more difficult. Further examples are considered in [14]. In particular, in [14] there are obtained examples such that the Riemannian part h depends non-trivially on the parameter x − .…”

Holonomy groups of Lorentzian manifolds

“…This topic is motivated by the paper of theoretical physicists Gibbons and Pope [76]. The results of this section are published in [60], [61], [62], [74].…”

“…Next we show that on each Walker manifold there exist special coordinates allowing to simplify appreciably the Einstein equation [74]. Examples of Einstein metrics from [60], [62] are given.…”

Holonomy groups of Lorentzian manifolds

“…Далее, мы показываем, что на многообразии Волкера существуют специальные координаты, позволяющие существенно упростить уравнение Эйнштейна [74]. Даны примеры метрик Эйнштейна из [60], [62].…”