Consequences of the noncompactness of the Lorentz group

TL;DR

This paper discusses the implications of the noncompactness of the Lorentz group on various geometric and topological properties of spacetime, clarifying phenomena that appear counterintuitive due to assumptions based on compact rotation groups.

Contribution

It clarifies how the noncompactness of the Lorentz group explains several counterintuitive features of spacetime geometry and topology.

Findings

Curvature invariants of gravitational waves vanish despite nonflat spacetime.

Eigennullframe components do not represent true curvature scalars.

Euclidean topology in Minkowski spacetime lacks Lorentz-invariant neighborhoods.

Abstract

The following four statements have been proven decades ago already, but they continue to induce a strange feeling: - All curvature invariants of a gravitational wave vanish - in spite of the fact that it represents a nonflat spacetime. - The eigennullframe components of the curvature tensor (the Cartan ''scalars'') do not represent curvature scalars. - The Euclidean topology in the Minkowski spacetime does not possess a basis composed of Lorentz--invariant neighbourhoods. - There are points in the de Sitter spacetime which cannot be joined to each other by any geodesic. We explain that our feeling is influenced by the compactness of the rotation group; the strangeness disappears if we fully acknowledge the noncompactness of the Lorentz group. Output: Imaginary coordinate rotations from Euclidean to Lorentzian signature are very dangerous.