Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime

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.

arXiv comment14 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

References in corpus (24)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Recent Progress on Ricci Solitons
• Gaussian densities and stability for some Ricci solitons
• A Gradient Flow for Worldsheet Nonlinear Sigma Models
• Ricci flow and black holes
• Spacetime Energy Decreases under World-sheet RG Flow
• Fixed Points of Nonlinear Sigma Models in d>2
• Towards physically motivated proofs of the Poincare' and geometrization conjectures
• On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
• On the geometry of Riemannian manifolds with density
• Hidden Superconformal Symmetry of the Cosmological Evolution
• The Cosmological Constant Problem and Re-interpretation of Time
• Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems
• The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons
• The Cosmological Constant Problem and Quantum Spacetime Reference Frame
• Entropy functionals and thermodynamics of relativistic geometric flows, stationary quasi-periodic Ricci solitons, and gravity
• Asymptotic safety in gravity and sigma models
• Investigating Off-shell Stability of Anti-de Sitter Space in String Theory
• Ricci Flow Approach to The Cosmological Constant Problem
• Quantum geometric information flows and relativistic generalizations of G. Perelman thermodynamics for nonholonomic Einstein systems with black holes and stationary solitonic hierarchies
• Dark Energy from Quantum Uncertainty of Distant Clock
• Trace Anomaly, Perelman's Functionals and the Cosmological Constant
• Constructing Lifshitz spaces using the Ricci flow

Cited by in corpus (6)

• A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity
• Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime
• The Ricci Flow and the Early Universe
• Second-Order Moment Quantum Fluctuations and Quantum Equivalence Principle
• Quantum Mechanics Relative to a Quantum Reference System: a Relative State Approach
• Local Short-Time Acceleration induced Spectral Line Broadening and Possible Implications in Cosmology