CONSEQUENCES OF THE NONCOMPACTNESS OF THE LORENTZ GROUP

The following four statements for indefinite metrics of Lorentz signat ure do not possess an analogous statement in the definite Euclidean si gnature case: (1) All curvature invariants of a gravitational wave van ish, in spite of the fact that it represents a nonflat spacetime. (2) The eigennullframe components of the curvature tensor (the Cartan ''sc alars'') do not represent curvature scalars. (3) The Euclidean topolog y in the Minkowski spacetime does not possess a basis composed of Lore ntz-invariant neighborhoods. (4) There are points in the de Sitter spa cetime which cannot be joined to each other by any geodesic. We show t hat these four statements all follow from the noncompactness of the Lo rentz group. We conclude that the popular (and often useful) imaginary -coordinate rotation from Euclidean to Lorentzian signature (called Wi ck rotation) is not an isomorphism.