Quantum Modiﬁed Gravity at Low Energy in the Ricci Flow of Quantum Spacetime M.J.Luo Department of Physics Jiangsu University Zhenjiang 212013 Peoples Republic of China

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Quantum Modiﬁed Gravity at Low Energy in the Ricci Flow of Quantum Spacetime

M.J.Luo

Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China∗

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 New-

tonian 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 character-

istic 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 a0introduced in MOND is related to the low characteristic

energy scale of the theory. The cosmological constant gives a universal leading order contribution to

a0and the ﬂow eﬀect gives the next order scale dependent contribution, which equivalently induces

the “cold dark matter” to the theory. a0is consistent with galaxian data when the “dark matter” is

about 5 times the baryonic matter.

I. INTRODUCTION

A wealth of astronomical observations indicate the presence of missing masses or acceleration discrepancies in the

universe based on the classical gravity theory (general relativity) although the theory is well tested within solar

system very precisely. One possible approach to solve the problem is by separately introducing the missing masses

components into the universe, for instance, the dark energy (DE or the cosmological constant Λ(CC) ) (Equation Of

State w=−1) and cold dark matter (CDM) (w≈0) in the so called ΛCDM-model. Another approach is by modifying

the law of gravity, within which the problem should be more appropriately reconsidered as a gravity/acceleration

discrepancy between the (cosmic or galactic) long distance scale and the (solar system or laboratory) short distance

scale. There are some phenomenological supports for the latter approach, since both the acceleration expansion of

universe (corresponding to the DE or Λ) and galactic rotation/acceleration anomalies (corresponding to the CDM)

empirically manifest a particular acceleration scale a0≈1.2×10−10m/s2≈√Λ

(6∼8) , ﬁrst proposed in the MOdiﬁed

Newtonian Dynamics (MOND) by Milgrom [1] (see reviews [2, 3] and references therein, or long publication list of

Milgrom’s). The baryonic Tully-Fisher law [4, 5] and an amazing “mass discrepancy-acceleration relation” [6] with

little scatter are also observed, which do not occur naturally in the ΛCDM-model. Although the modiﬁed gravity

approach might face its own diﬃculties (e.g. MOND without CDM is failed in ﬁtting the third and subsequent acoustic

peaks in the Cosmic Microwave Background (CMB)), this line of thinking might lead us to a more ambitious and

uniﬁed view to our universe. The internal relation between the cosmological constant and MOND has been generally

conjectured, and varieties of underpinning proposals and possible relativistic generalizations of MOND are suggested

in literature, they are still more or less similar with the Kepler’s law as a phenomenological description, there is

no ﬁrst principle to determine the exact form of the interpolating function between the standard gravity limit and

the modiﬁed one, thus lacking a fundamental underlying principle and theoretical framework remains its essential

weakness.

Recent years the author based on the quantum treatment of physical reference frame, proposed a framework of

quantum spacetime and gravity [7–14]. The basic idea of the theory is that when quantum theory is reformulated

on the new foundation of relational quantum state (an entangled state) describing the “relation” between a state of a

∗Electronic address: mjluo@ujs.edu.cn

arXiv:2210.06082v2 [gr-qc] 26 Apr 2023

under-studied quantum system and a state of a quantum spacetime reference frame system, a gravitational theory is

automatically contained in the quantum framework. Gravitational phenomenon is given by a relational quantum state

describing a relative motion of the under-studied quantum system with respect to the material quantum reference

frame system. And the 2nd order central moment of quantum ﬂuctuations of the quantum reference system introduces

the Ricci ﬂow to the quantum spacetime,

∂gµν

∂t =−2Rµν ,(1)

where gµν and Rµν are the metric and Ricci curvature of spacetime, and tis the ﬂow parameter.

The Ricci ﬂow is historically invented independently from the physics and mathematics points of views. From

the physics angle, the Ricci ﬂow was introduced by Friedan [15, 16] as a renormalization ﬂow of a non-linear sigma

model in 2 + dimension. From the mathematics angle, the Ricci ﬂow was introduced ﬁrst by Hamilton [17] as

a useful tool to deform an initial Riemannian manifold into a more and more “simple” and “good” manifold whose

topology is conserved and ﬁnally can be easily recognized in order to proof geometric theorems (like the Poincare

conjecture). But certain singularity may develop during the Ricci ﬂow and becomes the stumbling block of Hamilton’s

program. Around 2003, Perelman introduced several monotonic functionals [18–20] to control the singularity during

the Ricci ﬂow. What Perelman treated is in fact a density manifold (MD, g, u)with density uas a generalization of

the Riemanian manifold (MD, g), in which the density udescribes a local density of the Riemannian manifold and

physically coming from the quantum ﬂuctuation or uncertainty at each point of the manifold. The Ricci-DeTurck

ﬂow

∂gµν

∂t =−2 (Rµν − ∇µ∇νlog u)(2)

of the density manifold (equivalent to the Ricci ﬂow (1) up to a diﬀeomorphism given by the gradient of the udensity)

is shown to be the gradient ﬂow of Perelman’s functionals, so that he could overcome the stumbling block by using

his functionals and ﬁnally complete the Hamilton’s program.

In fact the underlying physics of Perelman’s formalism is not fully clear for physicist, the quantum spacetime

reference frame picture is proposed by the author to lay the physical foundation. In the framework of the quantum

spacetime reference frame, the spacetime is measured by physical rods and clocks as reference frame system and

hence subject to quantum ﬂuctuation. When the quantum ﬂuctuation of the reference frame system is unimportant

(mean ﬁelds approximation), the quantum framework recovers the standard textbook quantum theory without gravity.

When the 2nd order central moment quantum ﬂuctuation as the quantum correction of the reference frame system

is important and be taken into account (Gaussian approximation), Ricci ﬂow and gravity emerge in the quantum

framework, as if one introduces gravitation into the standard textbook quantum mechanics. The physical reference

frame modeled by the frame ﬁelds system is prepared and calibrated in a laboratory, which is mathematically described

by a non-linear sigma model using lab’s spacetime dimension 4−(a more practical example is a multi-wire chamber

using the electrons as frame ﬁelds to measure the coordinates of events in a laboratory). An under-studied quantum

system (e.g. the events) has physical meaning only with respect to the quantum reference frame system (the multi-wire

chamber). The 2nd order central moments of quantum ﬂuctuations of the reference frame ﬁelds blur the event and

equivalently give quantum variance to the spacetime coordinates. The variance of the coordinates directly modiﬁes

the quadratic form metric of the Riemannian spacetime geometry, making the spacetime vary with the scale of the

quantum ﬂuctuation. Such scale dependent quantum correction to the metric continuously deforms the spacetime

geometry driven by its Ricci curvature, which is exactly the Ricci ﬂow: a renormalization ﬂow of the spacetime. The

tparameter is related to the cutoﬀ energy scale of the Fourier components of the spacetime coordinates promoted to

be quantum frame ﬁelds. As the Ricci ﬂow starts from short distance scale (UV) t→ −∞ and ﬂows to long distance

scale (IR) t→0, or from the astronomical viewpoint, the energies of spectral lines (as the tracers of astronomical

observations) start from short distance laboratory scale and are redshifted to long distance galactic or cosmic scale.

During the process, the spacetime coordinates and metric at a long distance scale tare given by averaged out the

shorter distance ﬁner details which produces an eﬀective correction to them. In a more intuitive picture, as the wave

pocket of the reference frame ﬁelds, such as the spectral lines, gradually Gaussian (2nd order) broaden when they

travel a long distance, at long distance (e.g. cosmic or galactic) scale, the 2nd moment (i.e. the intrinsic spectral

lines broadening) correction to the spacetime coordinate or metric becomes signiﬁcant and hence can not be ignored.

The 2nd order moment quantum ﬂuctuation of spacetime gives rise to correction to the 2nd or quadratic order,

thus quantities like curvature or acceleration as the second spacetime derivative obtain additional coarse-graining

corrections in a natural and rigid way at long distance scale, which is considered as the root of the acceleration

discrepancies in astronomical scale observations and the quantum modiﬁed gravity at low energy.

Further, in the framework, the 2nd order quantum corrections to gravity and acceleration are in a universal way, so

that the correction is not merely the correction to speciﬁc spectral line itself but the correction to the spacetime. In

this sense the Equivalence Principle retains at the quantum level, which lays the physical foundation for the physical

measurement of spacetime geometry and geometric description of gravity. The quantum description of the spacetime

reference frame together with the quantum version of the Equivalence Principle leads to a completely diﬀerent view on

the behavior of quantum modiﬁed gravity: it is at the long distance scale where the quantum correction is signiﬁcant.

Beside the above intuitive “wave pocket broaden” picture of spacetime fuzziness at long distance scale, it is also

reﬂected in the characteristic scale of the gravity theory described by the only input dimensional constant λof the

quantum spacetime reference frame (in fact the only input constant for the d= 4 −non-linear sigma model). As

we will see in the section-II-B that, to recover the standard Einstein gravity the constant must be exactly the critical

density λ=3H2

8πG =Λ

ΩΛ8πG ≈10−3eV4. In contrast to the general believing of quantum gravity, the input constant is

not the single Newton’s constant but the critical density λof the universe, as a combination of the Newton’s constant

Gand Hubble’s constant H0. As a consequence, the characteristic energy scale of the gravity theory is not the Planck

scale, but the low critical density scale which is a long distance cosmic scale. As the tparameter in the framework is

a ratio of the cutoﬀ energy scale k2of the frame ﬁelds over the critical density, t=−1

64π2λk2, when the energy scale

of the frame ﬁelds is highly redshifted by the scale factor a2, i.e. k2∝a2→0, or t→0−, it is at the low energy limit

that gravity is strongly modiﬁed. In this paper, when we mention “scale t” (or later τ) of an astronomical object, it

can be understood physically as the the scale factor t∝ −a2or related redshift in the sense of the standard expanding

universe picture.

An important feature of the framework is that the critical density λis the characteristic scale of the quantum

gravity, as a consequence, the cosmological constant problem appearing in the naive quantum general relativity is

more readily understood. Since the natural scale of the cosmological constant is no longer the Planck scale, which

is 10120 times the observed value, but of order of the critical density λ. And the fraction in the critical density

ΩΛ=Λ

8πGλ ≈0.7of order one is given by the counter term to the spacetime volume ﬂow, which is related to a Ricci

ﬂow of the late epoch isotropic and homogeneous spacetime. Phenomenologically speaking, the Ricci ﬂow and its

counter term blurs the spacetime coordinate and equivalently universally broadens the spectral lines (as the universe

expanding tracers). The broadening contributes a universal variance to the redshift, thus the redshift-distance relation

is modiﬁed at second order in Taylor’s series expanding the distance in powers of the redshift, which gives rise to an

equivalent accelerating expansion of the universe as the quantum version Equivalence Principle asserts [8–11]. Since

the redshift variance (over the redshift mean squared) is independent to the speciﬁc energies of the spectral lines, so

they are seen universally accelerating “free-falling” (in fact expanding), and the uniform acceleration now is not merely

a speciﬁc property of the spectral lines, but measures and be interpreted as the universal property of the quantum

spacetime.

When a distant earth observer measures the rotation velocities of spiral galaxies at galactic long distance scale, the

mechanism works in a similar way. What the observer measures is not directly the rotation velocity of the spiral galaxy

but its Doppler (red and blue) shifts induced broadening of the spectral lines (as the rotation tracers) with respect to

the ones in laboratory (as the starting reference). As the galaxy is suﬃciently redshifted at long distance scale, the

spectral lines themselves are intrinsically quantum broadened, interpreted as the quantum variance or ﬂuctuation of

the distant spacetime coordinates based on the quantum Equivalence Principle. Thus it gives additional correction

to the Doppler broadening and enhances the rotation velocities.

In treating the quantum ﬂuctuation correction eﬀect to spacetime by the Ricci ﬂow, it is useful to introduce an

important and special solution of the Ricci ﬂow (or more general the Ricci-DeTurck ﬂow (2) for a density manifold)

which only shrinks the local size or volume of a manifold but its local shape unchanged, named the Ricci Soliton (or

more general Gradient Shrinking Ricci Soliton of a density manifold) [18]. Its Ricci curvature is proportional to the

metric Rµν =1

2τgµν (or more general a gradient normalized Ricci curvature is proportional to the metric)

Rµν − ∇µ∇νlog u=1

2τgµν (3)

where τ=t∗−tis a backwards Ricci ﬂow parameter from a limit scale t∗(see section-II-A for details). The Gradient

Shrinking Ricci Soliton is a (temporary t∗6= 0 or ﬁnal t∗= 0) limit spacetime conﬁguration or a (local or global) ﬁxed

point in the RG-sense, it (locally or globally) maximizes the Perelman’s monotonic functionals (at ﬁnite scale t∗6= 0

or IR limit scale t= 0). In some simple cases, including for examples, the homogeneous and isotropic late epoch

spacetime [10, 11], the spatial inﬂationary early universe [14], and static thermal equilibrium black hole [13], and the

topics concerned in the paper about the local galaxies, for all these limit (or nearly limit) spacetime, the Gradient

Shrinking Ricci Soliton equation is more useful as simple examples.

Further more, at the fundamental level, the proposed new framework of quantum spacetime and related gravity

seem to avoid several fundamental diﬃculties that other approaches to quantum gravity typically face. For examples,

the renormalizability of the quantum gravity is the renormalizability of the d= 4 −non-linear sigma model, or

at the Gaussian level correlates to the mathematical problem of the convergence of the Ricci ﬂow. The problem is

solved based on the works of Hamilton, Perelman and further developed by many other mathematicians, especially

after the discoveries of several monotonic functionals for the Ricci ﬂow in general dimensions and the generalization

of techniques to the non-compact and pseudo-Riemannian (Lorentzian) spacetime. The Hilbert space of quantum

gravity correlates to the classiﬁcation of spacetime geometries by using the Ricci ﬂow approach, which is fully solved

in 3-space, and can be full understood in 4-spacetime by using the Ricci ﬂow approach without fundamental obstacle.

The unitarity of quantum gravity correlates to the problem of intrinsic diﬀemorphism anomaly of the spacetime [13],

which is given by the functional integral method for the quantum spacetime reference frame and found deeply related

to the thermodynamic nature of the quantum spacetime. The local conformal stability of quantum spacetime [12]

correlates to the sign of the lowest eigenvalue related to the F-functional of Perelman, and the collapsibility of the

quantum spacetime correlates to the ﬁniteness of the W-functional of Perelman. The background independence of

quantum gravity correlates to the initial (metric) condition independence of the Ricci ﬂow, i.e. the Ricci ﬂow and

quantum ﬂuctuations about all general initial background are on equal footings. The problem is considered fully

solved by the Perelman’s formalism of the Ricci ﬂow for general initial condition of manifold (not be restricted on

some special initial condition). In the sense that the 2nd order quantum ﬂuctuation of spacetime has been important

at low energy, it also has a completely diﬀerent view on the “graviton” w.r.t. the ﬂat background. The “graviton” as the

low energy excitation degrees of freedom of metric have been averaged out in the Ricci ﬂow and eﬀectively contribute

to the general curved spacetime at certain scale, and hence it seems not to be a good signature of missing-energy in

particle collision, unlike some high energy modiﬁed versions of quantum gravity.

As the quantum framework modiﬁes the gravity at long distance scale in such a very tight and rigid way, the primary

objective of this paper is to explore the weak and static gravity limit of the theory, and to determine if MOND can

be derived from the theory, and if so, whether there is anything diﬀerent or beyond MOND in the framework.

To avoid overlong pages, the general background of the quantum reference frame and the Ricci ﬂow is given in

the introduction section, in the next section, we skip the detail of the quantum reference frame and direct starting

from the partition function derived from it, which can be found from the previous works [7–14]. In the section II, we

derived the low energy eﬀective action of gravity from the partition function. And in the subsequent sections, several

phenomenological consequences, e.g. the baryonic Tully-Fisher Relation in section III, the radial acceleration relation

in section IV and “missing matter” in section V are discussed. Finally we discuss the relation between the theory and

MOND and conclude the paper.

II. EFFECTIVE GRAVITY AT LOW ENERGY

A. Partition Function of Quantum Reference Frame and Pure Gravity

By using the quantum frame ﬁelds described by a d= 4 −non-linear sigma model, in [11, 12] the author has

derived the partition function of a pure gravity in terms of the relative Shannon entropy ˜

Nof the spacetime 4-manifold

MD=4

Z(MD) = eλ˜

N(MD)−D

2−ν=eλN(MD)−D

2−ν

eλN∗(MD)(4)

where D≡4and λ=3H2

8πG ≈10−3eV4is the critical density of the universe. Up to a constant multiple, it is

in fact the inverse of Perelman’s partition function [18] (the inverse is not physical important) he used to deduce

his thermodynamics analogous functionals, although the underlying physical interpretation and relation to gravity is

unclear in his seminal paper. The partition function is a proper starting point to pure gravity. Let us explain some

quantities appearing in the partition function as follows.

(1) Nand N∗terms

The relative Shannon entropy ˜

N(M4, t)is the Shannon entropy N(M4, t)w.r.t. the extreme value N∗(M4, t∗)given

by a Gradient Shrinking Ricci Soliton (3) as the Ricci ﬂow limit t→t∗, i.e. ˜

N=N−N∗. The quantum reference

frame theory is described by a non-linear sigma model in d= 4 −, the trivialness of the homotopy group πd(M4)of

the mapping of the non-linear sigma model makes the Ricci ﬂow globally singularity free, simply giving t∗= 0, even

though local singularities may developed at ﬁnite t∗6= 0 during the Ricci ﬂow for some general initial spacetime. The

Shannon entropy is given by the udensity

N(M4, t) = −ZM4

d4Xu log u, (5)

in which

u(X, t) = d4x

d4X(6)

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QuantumModiedGravityatLowEnergyintheRicciFlowofQuantumSpacetimeM.J.LuoDepartmentofPhysics,JiangsuUniversity,Zhenjiang212013,People'sRepublicofChinaQuantumtreatmentofphysicalreferenceframeleadstotheRicciowofquantumspacetime,whichisaquiterigidframeworktoquantumandrenormalizationeectofgravity.Theth...

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Quantum Modiﬁed Gravity at Low Energy in the Ricci Flow of Quantum Spacetime M.J.Luo Department of Physics Jiangsu University Zhenjiang 212013 Peoples Republic of China

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