Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917411047866368 |
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| author | Luo, M. J. |
| author_facet | Luo, M. J. |
| contents | This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional compact Riemannian manifolds. The monotonicity of these entropy functionals is utilized to prove the well-posedness of applying Ricci flow to four-dimensional Lorentzian spacetime for a long flow-time, particularly for the timelike modes which would seem blow up and ill-defined. The general idea is that the the Ricci flow of a Lorentzian spacetime metric and the coupled conjugate heat flow of a density on the Lorentzian spacetime as a whole turns out to be the gradient flows of the monotonic functionals for a long flow-time, so the superficial "blow-up" in the individual Ricci flow system or the conjugate heat flow system contradicts the boundedness of the monotonic functionals within finite flow interval, which gives a semi-global control to the whole coupled system. The physical significance and applications of these monotonic entropy functionals in real gravitational systems are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula Luo, M. J. General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional compact Riemannian manifolds. The monotonicity of these entropy functionals is utilized to prove the well-posedness of applying Ricci flow to four-dimensional Lorentzian spacetime for a long flow-time, particularly for the timelike modes which would seem blow up and ill-defined. The general idea is that the the Ricci flow of a Lorentzian spacetime metric and the coupled conjugate heat flow of a density on the Lorentzian spacetime as a whole turns out to be the gradient flows of the monotonic functionals for a long flow-time, so the superficial "blow-up" in the individual Ricci flow system or the conjugate heat flow system contradicts the boundedness of the monotonic functionals within finite flow interval, which gives a semi-global control to the whole coupled system. The physical significance and applications of these monotonic entropy functionals in real gravitational systems are also discussed. |
| title | Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2509.17733 |