Complex degenerate metrics in general relativity: a covariant extension of the Moore-Penrose algorithm

TL;DR

This work introduces a covariant Moore-Penrose extension to general relativity, enabling a unique covariant pseudoinverse for degenerate metrics and the consistent definition of curvature tensors and a covariant derivative, even in complexified spacetimes. Degeneracy is geometrically tied to a torsion tensor, yielding a Riemann-Cartan-like structure that preserves tensorial curvature objects. The authors apply the formalism to complexified Schwarzschild, RN, RN-dS, and FLRW geometries, deriving complex Friedmann equations and exploring extended notions of geodesics via autoparallels and extremals. These results open avenues for incorporating degenerate metrics into quantum gravity contexts and for exploring arrowless or holomorphic cosmologies within a covariant, coordinate-free framework.

Abstract

The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving complex non-invertible metrics. Unlike the standard technique, this approach guarantees the uniqueness of the pseudoinverse metric through the fulfillment of a set of covariant relations, and it allows for the proper definition of a covariant derivative operator and curvature-related tensors. Remarkably, the degenerate nature of the metric can be given a geometrical representation in terms of a torsion tensor, which vanishes only in special cases. Applications of the new scheme to complex black hole geometries and cosmological models are also investigated, and a generalized concept of geodesics that exploits the notion of autoparallel and extremal curves is presented. Relevance of our findings to quantum gravity and quantum cosmology is finally discussed.