Spacetimes with continuous linear isotropies III: null rotations

TL;DR

This paper demonstrates that local null rotation invariance of curvature derivatives often guarantees the existence of a 3-dimensional isometry group with null rotation symmetry in various spacetime classes, under specific derivative invariance conditions.

Contribution

It establishes conditions under which null rotation invariance of curvature derivatives implies a larger isometry group in certain spacetimes, extending previous classifications.

Findings

Null rotation invariance of curvature and first derivatives implies a 3D isometry group in many cases.

Additional invariance of second derivatives is needed for Petrov type N Einstein spacetimes.

Results apply to spacetimes with pure radiation and conformally flat spacetimes with specific Ricci tensor types.

Abstract

It is shown that in many cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure there is an isometry group G with dimension at least 3 acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. Invariance of the second derivatives is additionally required to ensure this conclusion in Petrov type N Einstein spacetimes, spacetimes containing "pure radiation" (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a "tachyon fluid").