Superenergy and Supermomentum of Goedel Universes

Abstract

We review the canonical superenergy tensor and the canonical angular supermomentum tensors in general relativity and calculate them for space-time homogeneous Gödel universes to show that both of these tensors do not, in general, vanish. We consider both an original dust-filled pressureless acausal Gödel model of 1949 and a scalar-field-filled causal Gödel model of Rebou\c cas and Tiomno. For the acausal model, the non-vanishing components of superenergy of matter are different from those of gravitation. The angular supermomentum tensors of matter and gravitation do not vanish either which simply reflects the fact that Gödel universe rotates. However, the axial (totally antisymmetric) and vectorial parts of supermomentum tensors vanish. It is interesting that superenergetic quantities are {\it sensitive} to causality in a way that superenergy density $_g S_{00}$ of gravitation in the acausal model is {\it positive}, while superenergy density $_g S_{00}$ in the causal model is {\it negative}. That means superenergetic quantities might serve as criterion of causality in cosmology and prove useful.