Holonomy groups and spacetimes

Abstract

A study is made of the possible holonomy group types of a spacetime for which the energy-momentum tensor corresponds to a null or non-null electromagnetic field, a perfect fluid or a massive scalar field. The case of an Einstein space is also included. The techniques developed are also applied to vacuum and conformally flat space-times and contrasted with already known results in these two cases. Examples are given.

Конформно Плоские Лоренцевы Многообразия Со Специальными Группами Голономии

“…Стан-дартное пространство-время Фридмана-Робертсона-Волкера является конфор-мно плоским и имеет алгебру голономии so(1, 3) (см. [6]). …”

“…Возможные алгебры голономии конформно плоских лоренцевых многооб-разий размерности 4 найдены в [6]. Метрика 1 размерности 4 получена в [7].…”

“…Метрика 1 размерности 4 получена в [7]. В статье [6] была обозначена проблема построения примера конформно плоской метрики с алгеброй голономии sim(2) (эта алгебра в [6] обозначена через R 14 ). Попытка построить такую метрику была сделана в [8], где была построена сле-дующая метрика:…”

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…The binary operation on L is then that of matrix commutation. Such a representation of L is well-known and has been classified into fifteen convenient types [12] (for details of the possible holonomy types most relevant for the physics of general relativity see [13,10]). This representation is given in the first three columns of table 1 using either a null tetrad l, n, x, y or an orthonormal tetrad u, x, y, z to describe a basis for each subalgebra.…”

On the compatibility of Lorentz metrics with linear connections on four-dimensional manifolds

“…It then follows from [5,4] that, on W , g ′ = cg (0 = c ∈ R) and hence the contradiction that Γ ′ = Γ on W . Now consider the situation when V is a space-time of curvature class B and hence of holonomy type R 7 [4,23] and thus the metric g is of the Bertolli-Robinson type (see, e.g. [24]).…”