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

TL;DR

This paper introduces a covariant extension of the Moore-Penrose algorithm to handle complex, degenerate metrics in general relativity, enabling unique pseudoinverses and new geometric insights.

Contribution

It presents a novel covariant method for degenerate metrics, ensuring uniqueness and defining covariant derivatives and curvature tensors in complex spacetime geometries.

Findings

Unique pseudoinverse metric via covariant relations

Application to complex black hole and cosmological models

New generalized geodesic concept using autoparallel and extremal curves

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.