Spacetimes with continuous linear isotropies I: spatial rotations

2021

Gen Relativ Gravit

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Conditions are found which ensure that local boost invariance (LBI), invariance under a linear boost isotropy, implies local boost symmetry (LBS), i.e. the existence of a local group of motions such that for every point P in a neighbourhood there is a boost leaving P fixed. It is shown that for Petrov type D spacetimes this requires LBI of the Riemann tensor and its first derivative. That is also true for most conformally flat spacetimes, but those with Ricci tensors of Segre type [1(11,1)] may require LBI of the first three derivatives of curvature to ensure LBS.

Spacetimes with continuous linear isotropies II: boosts

2021

Gen Relativ Gravit

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Spacetimes with continuous linear isotropies III: null rotations

2021

Gen Relativ Gravit

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It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group $$G_r$$ G r with $$r\ge 3$$ r ≥ 3 acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are 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”).

Spacetimes with continuous local isotropies

2022

Int. J. Mod. Phys. D

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.