Complex degenerate metrics in general relativity: a covariant extension of the Moore–Penrose algorithm
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 standar...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-03-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | https://doi.org/10.1140/epjc/s10052-025-13957-w |
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| Summary: | 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. |
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| ISSN: | 1434-6052 |