Paper · Jan 26, 2022
Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime
As on arXiv
Abstract
It is known that the conformal instability or bottomless problem rises in the path integral method in quantizing the general relativity. Does quantum spacetime itself really suffer from such conformal instability? If so, does the conformal instability cause the collapse of local spacetime region or even collapse the whole spacetime? The problems are studied in the framework of the Quantum Spacetime Reference Frame (QSRF) and induced spacetime Ricci flow. We find that if the lowest eigenvalue of an operator, associated with the F-functional in a local compact (closed and bounded) region, is positive, the local region is conformally unstable and will tend to volume-shrinking and curvature-pinching along the Ricci flow-time t; if the eigenvalue is negative or zero, the local region is conformally stable up to a trivial rescaling. However, the local non-collapsing theorem in the Ricci flow proved by Perelman ensures that the instability will not cause the local compact spacetime region collapse into nothing. The total effective action is also proved positive defined and bounded from below keeping the whole spacetime conformally stable, which can be considered as a generalization of the classical positive mass theorem of gravitation to the quantum level.
Comments: 14 pages. v2:introduction and references added, appendix moved to section-II, generalization of the Ricci flow and the monotonic functionals to the Lorentzian 4-spacetime emphasized. Final version published in Annals of Physics. arXiv admin note: text overlap with arXiv:2112.00218
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.