https://doi.org/10.1140/epjc/s10052-025-13957-w
Regular Article - Theoretical Physics
Complex degenerate metrics in general relativity: a covariant extension of the Moore–Penrose algorithm
1
LAMFA, University of Amiens, Amiens, France
2
Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044, Frascati, Italy
a ebattista@lnf.infn.it, emmanuelebattista@gmail.com
Received:
31
December
2024
Accepted:
13
February
2025
Published online:
12
March
2025
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.
© The Author(s) 2025
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