Spacetimes with continuous local isotropies

Abstract

General relativistic spacetimes in which at every point [Formula: see text] in some neighbourhood [Formula: see text] the tangent space [Formula: see text] is invariant under the same continuous subgroup [Formula: see text] of the Lorentz group are considered. Some previously open problems for such spacetimes are discussed and solved, and a comprehensive survey of such spacetimes is provided. For most cases, invariance of the curvature and its first covariant derivative under [Formula: see text] imply the existence of an isometry group in [Formula: see text] containing [Formula: see text] as an isotropy group. In the remaining cases, invariance of second and third derivatives may be required for this implication, and in particular, the necessity of invariance of the third derivative is proved for the locally rotationally symmetric spacetimes discussed by Ellis and by Stewart and Ellis.