https://neckpinch.com/authors/m-j-luo/New arXiv papers by M. J. Luo, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2509.17733arxiv:2509.17733Mon, 22 Sep 2025 00:00:00 GMTM. J. Luo — 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 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.Ricci flowSingularitiesHeat kernel3-manifolds4-manifoldsBoundaries & non-compactPhysics https://arxiv.org/abs/2302.08651arxiv:2302.08651Fri, 17 Feb 2023 00:00:00 GMTM. J. Luo — The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a d=4-ε quantum non-linear sigma model, interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density matrix is deduced. In this framework, the diffeomorphism anomaly of the theory has a deep thermodynamic interpretation. The trace anomaly is related to a Shannon entropy in terms of the density matrix, which monotonically flows and achieves its maximal value at the flow limit, called the Gradient Shrinking Ricci Soliton (GSRS), corresponding to a thermal equilibrium state of spacetime. A relative Shannon entropy w.r.t. the maximal entropy gives a statistical interpretation to Perelman's partition function, which is also monotonic and gives an analogous H-theorem to the statistical frame fields system. A temporal static 3-space of a GSRS 4-spacetime is also a GSRS in lower 3-dimensional, we find that it is in a thermal equilibrium state, and Perelman's analogies between his formalism and the thermodynamics of the frame fields in equilibrium can be explicitly given in the framework. Extending the validity of the Equivalence Principle to the quantum level, the quantum frame fields theory at low energy gives an effective theory of gravity, a scale dependent Einstein-Hilbert action plus a cosmological constant is recovered. As a possible underlying microscopic theory of gravity, the theory is also applied to understand the thermodynamics of the Schwarzschild black hole.Ricci flowRicci solitons3-manifoldsHarmonic maps & DeTurck trickPhysics https://arxiv.org/abs/2210.06082arxiv:2210.06082Wed, 12 Oct 2022 00:00:00 GMTM. J. Luo — Quantum treatment of physical reference frame leads to the Ricci flow of quantum spacetime, which is a quite rigid framework to quantum and renormalization effect of gravity. The theory has a low characteristic energy scale described by a unique constant: the critical density of the universe. At low energy long distance (cosmic or galactic) scale, the theory modifies Einstein's gravity which naturally gives rise to a cosmological constant as a counter term of the Ricci flow at leading order and an effective scale dependent Einstein-Hilbert action. In the weak and static gravity limit, the framework gives rise to a transition trend away from Newtonian gravity and similar to the MOdified Newtonian Dynamics (MOND) around the characteristic scale. When local curvature is large, Newtonian gravity is recovered. When local curvature is low enough to be comparable with the asymptotic background curvature corresponding to the characteristic energy scale, the transition trend produces the baryonic Tully-Fisher relation. For intermediate general curvature around the background curvature, the interpolating Lagrangian function yields a similar transition trend to the observed radial acceleration relation of galaxies. When the baryonic matter density is much lower than the critical density at the outskirt of a galaxy, there may be a universal "acceleration floor" corresponding to the acceleration expansion of the universe, which differs from MOND at its deep-MOND limit. The critical acceleration constant a₀ introduced in MOND is related to the low characteristic energy scale of the theory. The cosmological constant gives a universal leading order contribution to it and the flow effect gives the next order scale dependent contribution, which equivalently induces the "cold dark matter" to the theory. a₀ is consistent with galaxian data when the "dark matter" is about 5 times the baryonic matter.Ricci flowPhysics https://arxiv.org/abs/2201.10732arxiv:2201.10732Wed, 26 Jan 2022 00:00:00 GMTM. J. Luo — 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.Ricci flowPerelman entropy(Non)collapsingCurvature pinchingStability & uniquenessEigenvalues & spectrumPhysics https://arxiv.org/abs/2112.00218arxiv:2112.00218Wed, 01 Dec 2021 00:00:00 GMTM. J. Luo — A framework of quantum spacetime reference frame is proposed and reviewed, in which the quantum spacetime at the Gaussian approximation is deformed by the Ricci flow. At sufficient large scale, the Ricci flow not only smooths out local small irregularities making the universe a homogeneous and isotropic Friedmann-Robertson-Walker metric, but also develops a local singularity at the physical-time origin. Due to the phenomenological suppression of the non-Gaussian primordial perturbations, we assume the validity of the Ricci flow applying to the high curvature region near the local singularity of the early universe. The no-local-collapsing theorem of Perelman ensures the existence of a canonical neighborhood around the large curvature pinching point, which resembles a gradient shrinking Ricci soliton (GSRS) solution of the Ricci flow. Without any inflaton field, the GSRS naturally reproduces an exact inflationary deSitter universe near the singularity at the leading order. Without any rolling-down behavior of inflaton, the deviation from exact deSitter described by the "slow roll parameters" can be calculated by a small deviation from the singular flow-time via the Ricci flow, and the primordial perturbations can also be studied on the GSRS background, the power spectrum of the scalar perturbation agrees with present observations, and the one of the tensor perturbation is predicted too small to be detectable than the standard inflation. The previous treatment of the cosmological constant and the effective gravity are also briefly reviewed in the framework. So we argue that the Ricci flow provides us a possible unified view and treatment of the late epoch accelerating expansion and early epoch inflation of the universe without introducing dark energy or inflaton (dark energy of the second kind).Ricci flowRicci solitonsSingularities(Non)collapsingCurvature pinchingHomogeneous spacesEigenvalues & spectrumPhysics