{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T22:54:29Z","timestamp":1774652069418,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T00:00:00Z","timestamp":1749686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Government of Canada through Industry Canada"},{"name":"Province of Ontario through the Ministry of Research and Innovation (MRI)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker (FLRW) Big Bang cosmology. By extending the relevant coordinates into the complex plane and carefully choosing integration contours, we show that it is possible to regularize these singularities, resulting in physically meaningful, singularity-free solutions when projected back onto real spacetime. The removal of the singularity at the Big Bang allows for a bounce cosmology. The approach offers a potential bridge between classical general relativity and quantum gravity effects, suggesting a way to resolve longstanding issues in gravitational physics without requiring a full theory of quantum gravity.<\/jats:p>","DOI":"10.3390\/axioms14060459","type":"journal-article","created":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T06:42:48Z","timestamp":1749710568000},"page":"459","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Complex Riemannian Spacetime and Singularity-Free Black Holes and Cosmology"],"prefix":"10.3390","volume":"14","author":[{"given":"John W.","family":"Moffat","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1017\/S0305004100031686","article-title":"Generalized Riemann spaces","volume":"52","author":"Moffat","year":"1956","journal-title":"Math. Proc. Camb. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1017\/S030500410003245X","article-title":"The foundations of a generalization of gravitation theory","volume":"53","author":"Moffat","year":"1957","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1017\/S0305004100032461","article-title":"The static spherically symmetric solutions in a unified field theory","volume":"53","author":"Moffat","year":"1957","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1007\/BF02751648","article-title":"On spherically symmetric solutions in Moffat\u2019s unified field theory","volume":"8","author":"Kerr","year":"1958","journal-title":"II Nuovo C."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"169466","DOI":"10.1016\/j.aop.2023.169466","article-title":"Holomorphic general coordinate invariant modified measure gravitational theory","volume":"458","author":"Guendelman","year":"2023","journal-title":"Ann. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Liu, R., Quintin, J., and Afshordi, N. (2025). Conformal Cores of Quantum Black Holes in Quadratic Gravity. arXiv.","DOI":"10.1103\/PhysRevD.111.044031"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Garnier, A., and Battista, E. (2025). Complex degenerate metrics in general relativity: A covariant extension of the Moore-Penrose algorithm. arXiv.","DOI":"10.1140\/epjc\/s10052-025-13957-w"},{"key":"ref_8","unstructured":"Griffiths, P., and Harris, J. (1978). Principles of Algebraic Geometry, Wiley."},{"key":"ref_9","unstructured":"Kobayashi, S., and Nomizu, K. (1963). Foundations of Differential Geometry, Wiley."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Penrose, R. (1972). Techniques of Differential Topology in Relativity, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970609"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1743","DOI":"10.1103\/PhysRev.119.1743","article-title":"Maximal extension of Schwarzschild metric","volume":"119","author":"Kruskal","year":"1960","journal-title":"Phys. Rev."},{"key":"ref_12","first-page":"295","article-title":"On the singularities of a Riemannian manifold","volume":"7","author":"Szekeres","year":"1960","journal-title":"Publ. Mat. Debr."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1103\/PhysRevLett.11.237","article-title":"Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics","volume":"11","author":"Kerr","year":"1963","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1038\/physci229177a0","article-title":"Extraction of rotational energy from a black hole","volume":"229","author":"Penrose","year":"1971","journal-title":"Nat. Phys. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"004","DOI":"10.1088\/1475-7516\/2006\/03\/004","article-title":"Scalar-Tensor-Vector Gravity Theory","volume":"2006","author":"Moffat","year":"2006","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1140\/epjc\/s10052-015-3405-x","article-title":"Black holes in modified gravity (MOG)","volume":"75","author":"Moffat","year":"2015","journal-title":"Eur. Phys. J."},{"key":"ref_17","first-page":"252","article-title":"Gravitational collapse: The role of general relativity","volume":"1","author":"Penrose","year":"1969","journal-title":"Riv. Nuovo Cim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1140\/epjp\/i2013-13022-4","article-title":"On CCC-predicted concentric low-variance circles in the CMB sky","volume":"128","author":"Gurzayan","year":"2013","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.physletb.2010.11.046","article-title":"Black holes in an ultraviolet complete quantum gravity","volume":"695","author":"Modesto","year":"2011","journal-title":"Phys. Lett. D"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/459\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:50:34Z","timestamp":1760032234000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/459"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,12]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060459"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060459","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,12]]}}}