Lorentzian Manifolds Close to Euclidean Space

Аннотация

We study the Lorentzian manifolds M₁, M₂, M₃, and M₄ obtained by small changes of the standard Euclidean metric on ℝ⁴ with the punctured origin O. The spaces M₁ and M₄ are closed isotropic space-time models. The manifolds M₃ and M₄ (respectively, M₁ and M₂) are geodesically (non)complete; M₁ are M₄ are globally hyperbolic, while M₂ and M₃ are not chronological. We found the Lie algebras of isometry and homothety groups for all manifolds; the curvature, Ricci, Einstein, Weyl, and energy-momentum tensors. It is proved that M₁ and M₄ are conformally flat, while M₂ and M₃ are not conformally flat and their Weyl tensor has the first Petrov type.