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Ricci Flow Gravity
PMC Physics A volume 1, Article number: 3 (2007)
Abstract
A theory of gravitation is proposed, modeled after the notion of a Ricci flow. In addition to the metric an independent volume enters as a fundamental geometric structure. Einstein gravity is included as a limiting case. Despite being a scalartensor theory the coupling to matter is different from JordanBransDicke gravity. In particular there is no adjustable coupling constant. For the solar system the effects of Ricci flow gravity cannot be distinguished from Einstein gravity and therefore it passes all classical tests. However for cosmology significant deviations from standard Einstein cosmology will appear.
PACS Codes: 04.50.+h, 04.20.Cv, 02.40.Hw
1 Introduction
A generalization of Einstein's theory of gravity is developed. It has a purely geometric foundation, including in addition to a metric an independent volume. Although related to scalartensor theories à la JordanBransDicke and to string cosmology, it is nevertheless different: the basic field equations are in the form of Ricci flow equations, generalized to include matter. Einstein's theory is included as the limiting case of no flow. The volume scalar has two interpretations: geometrically, it is responsible for volumepreservation and physically it obeys a masszero real scalar wave equation. This is also the main difference with JordanBransDicke theories, where the scalar couples to the trace of the energymomentum tensor. As a consequence, in general the energymomentum tensor is not anymore "conservative" in the ordinary sense of ∇·T = 0, and point particles do not move anymore on geodesics, having a NewtonNordström potential. But from Noether's fundamental relation conservation still follows from symmetry. Nevertheless, in "ordinary" solarsystem and astrophysical settings, the corrections are negligible: the theory cannot be distinguished from Einstein's and therefore passes all the standard tests. However in a cosmological setting, deviations from standard Einstein gravity are to be expected. This will be the subject of a forthcoming paper.
We will proceed as follows: after this brief introduction, in section 2 the motivations for this kind of extension are discussed. In section 3 a short introduction to volumetric manifolds is given, emphasizing the notion of volumepreservation in section 4. Before establishing the definitive field equations of Ricci flow gravity in section 6, several other choices are discussed in section 5, with emphasis on the main differences with respect to Einstein's and in particular to JordanBransDicke theories. Section 7 refers to the physical interpretation of the volume scalar in Ricci flow gravity. Finally, in section 8 the viability of the theory with respect to the standard tests is discussed. The conclusion in section 9 ends this paper.
2 Motivation and Inputs
The present work is principally motivated by the conviction that the notion of "volume" has an existence independent from any metric – in fact, it must be considered to be a premetric concept. Curiously, such an independent volume had not been taken into consideration in physical theories until relatively recently. Even in differential geometry it is almost ignored. Perhaps the reason for this neglect is that in most circumstances there is a canonical volume element, based on other geometric structures considered to be more basic. For example, in Riemannian geometry the volume element density is defined in terms of the metric. In particular, the important operation of Hodge dual for differential forms is conventionally based on such a Riemannian volume element.
However, from the gravitational sector of the lowenergy limit of string theory (i.e., compactification to dimension n = 4) there comes the suggestive hint (cmp. Garfinkle, Horowitz and Strominger [1]) that when both a dilaton scalar and a twoform are present, the dilaton scalar enters the expression for the volume element density when defining a "natural" Hodge dual operator.^{1} This was taken as the starting point to develop a theory of geometric dilaton gravity (Graf [4]). Although the particular coupling does not exactly correspond to the coupling suggested by string theory, wormholelike solutions were obtained.
Recently a breakthrough on Ricci flow methods was achieved by Perelman [5–7], developing the decisive tools to solve the famous Poincaré conjecture on the topological characterization of the threesphere. Based on 3d (compact and positivedefinite) Riemann spaces, smoothly deformed by a Ricci flow (RF), the "basic" RF equation was originally defined by
∂_{ t }g_{ ik }= 2 R_{ ik },
where R_{ ik }is the Riccitensor corresponding to a "timedependent" threemetric g_{ ik }. Also a special class of diffeomorphisms was considered, with vector $\overrightarrow{v}$ which is essentially the gradient of a scalar φ in the sense of v^{i}= g^{ik}∂_{ k }φ. The so generalized RF equation then becomes
∂_{ t }g_{ ik }= 2 (R_{ ik }+ ∇_{ i }∇_{ k }φ).
Although such equations have already been studied since the early eighties starting with the seminal works of Hamilton [8] and DeTurck [9], an essential insight of Perelman was to recognize that the r.h.s. of this equation^{2} can be expressed as the gradient of an appropriate functional. This functional involves a "measure" given in terms of an independent volume element. The gradient property allows to apply a series of standard analytical tools. And the introduction of the measure gives an extra flexibility, analogous to a choice of gauge.^{3}
Since the works of Hamilton and DeTurck just mentioned, geometric flows have been applied to a variety of geometric, topological and analytical problems.^{4} Flowlike equations are also not unfamiliar to physicists, the earliest and most wellknown being the renormalization group equations in quantum field theory (for an introduction, see Mitter [14]), and also the nonlinear σmodel (e.g., Lott [15], Oliynyk, Suneeta and Woolgar [16], Tseytlin [17]). After Ellis [18] called attention to the cosmological "fitting problem", the usefulness of the Ricci flow to deal with volumeaveraged inhomogeneities was immediately recognized and continues to be an active area of research (e.g., Carfora and Marzuoli [19], Buchert and Carfora [20], and the recent review by Buchert [21]). An overview of flow techniques in physics is given in Bakas [22].
Another motivation came however with the insight, that the basic equations derived from the lowenergy limit of string theory can be put into a form suprisingly similar to Ricci flow equations when besides the metric only a dilaton scalar is kept. The main formal difference is the number of dimensions and the signature of the corresponding Riemann spaces: whereas the "classic" RF equations refer to a parameterdependent truly Riemannian three space evolved by an extrinsic "time" parameter, the reformulated string theory equations refer to a fourdimensional Lorentzian spacetime, which is evolved along the directions of an intrinsic vector field.
Neither the "classic" RF approach nor string theory suggest any hints about the coupling of geometry to external matter fields. Therefore we will spend some time to prepare the field in order to include other external matter. As not only geodesy of the motion of "test particles" will in general be violated, but also "conservation" (in the sense of ∇·T = 0), we will be especially careful to lay a coherent and stringent foundation. The Noether identities will be our main guide. As result we will get Ricci flow gravity (RFG).
For the history of scalartensor theories and their current status, we refer to Brans [23], and to the recent monographs of Fujii and Maeda [24] and Faraoni [25].
3 Volumetrical Manifolds
In Graf [4] we already introduced the notion of a volume manifold and its specialization for the case a nondegenerate metric exists. Let us briefly recapitulate the main notions. First, we introduced the fundamental concept of a volume structure, which has to be considered as independent from any metric. This is just a nonnegative nform density ω, and makes the manifold a volume manifold. Secondly, we will need of course a metric structure. However, it does not need to be compatible with the volume structure. This difference is encoded by means of the volume scalar φ by ω= ωe^{φ},^{5} where ω := det g^{1/2}dx^{1} ∧ dx^{2} ∧ ... ∧ dx^{n}is the usual metrical (i.e., Riemannian) volume element density. Furthermore, for the metric derivative along a vector X we have ∇_{ X }ω= (X·∂φ)ω(the dot denoting a contraction) as a measure of incompatibility.
Such a manifold, endowed both with an independent volume and a metric structure, we will denote by volumetrical manifold. Whereas the manifold is considered to be smooth, both metric and volume element density are allowed to diverge or to be degenerate, when they are not locally smooth.
Already in a volume manifold the Gauss theorem for a vector ξ can be expressed very compactly in terms of differential form densities as
where bΩ is a twosided hypersurface bounding the ndimensional region Ω. The scalar factor div ξ in the relation d(ξ·ω) = (div ξ) ωis also better known under the name of divergence of the vector ξ. Evidently the divoperator only depends on the particular choice of ωand not on any metric.
4 Volume Preserving Lie Flows
In a differentiable manifold, the thing coming closest to an autonomous first order differential equation for a "vector" x(t),
is the notion of a Lie equation
£_{ ξ }F = G,
where F and G are geometric objects (e.g., tensors), ξ is some vector field and £_{ ξ }F denotes the Lie derivative of F along ξ. In the simplest case ξ and G are considered as given and F to be determined. However in the applications we have in mind, all elements of the equation will be dynamically determined, G depending nonlinearly on F and its partial derivatives, and even ξ will become dynamical.
In the theory of ordinary differential equations, such systems of firstorder equations which guarantee the longterm existence both to the future and the past, are also called flows and can be characterized by the oneparameter Abelian group property of their solutions. As wellknown, Lie operators share exactly the same oneparameter Abelian group property (at least locally) by means of the exponential map. We can therefore speak of a Lie flow.
In a volume manifold, a Lie flow with vector ξ is called volumepreserving (or VP)^{6} if
£_{ ξ }ω= 0, or equivalently, div ξ = 0.
In the rest of this paper we will try to make plausible a particular scalartensor extension of Einstein gravity in terms of a volumepreserving Ricci Lie flow in a volumetric manifold.
5 A choice of ScalarTensor Field Equations
Assuming that the total Lagrangian (or at least the field equations) can be uniquely split into a pure geometrical part and the "physical" part, we can already draw important conclusions about both the algebraic and the differential properties of the "physical" energymomentum tensor just from examining the purely geometrical part. Note that whereas Perelman's analysis is "metriccentered", with an auxiliary scalar, in the following physical applications this scalar will play a role at the same conceptual level as the metric. Therefore the "classical" Lagrangian approach is appropriate.
Let us start with the "geometrical" Lagrangian living on a volumetric manifold M,
where ω:= e^{φ}ω and (∇φ)^{2} := g^{ij}∂_{ i }φ∂_{ j }φ and λ is a constant parameter. Despite its simple form it not only includes the one used initially by Perelman and in the lowenergy limit of string theory (when ignoring the axion and the other moduli fields), but which also is essentially the JordanBransDicke Lagrangian.
Defining the volume factor Φ := e^{φ}, we then have as variational derivatives (up to volume element, gdualizations of P and a common sign 1)
where G denotes the Einstein tensor G_{ ik }:= R_{ ik }– $\frac{1}{2}$Rg_{ ik }and Δ := ∇^{2} the d'Alembertian. For the above Lagrangian the Noether identity can be written compactly as
with some tensor ${\tilde{P}}_{k}^{i}$ to be determined by it. More conventionally,
As this identity must hold for any smooth vector ξ, we get separately
Note that from equation (11) follows conservation in the proper sense, if $\overrightarrow{\xi}$ is a simultaneous Killing vector both of the metric and of the scalar, even if Q ≠ 0.
The following algebraicodifferential relations evidently hold:
symmetry: ${\tilde{P}}_{ik}={\tilde{P}}_{ki}$, and
balance: ∇_{ i }(Φ${\tilde{P}}_{k}^{i}$) = Q∂_{ k }Φ.
Up to this point we only made use of identities, but not of any field equations. In particular if we equate P and Q to their corresponding physical quantities, these algebraicodifferential relations will be "impressed" on them. In fact, it is not even necessary that they follow from a Lagrangian.
But note that there is a dependency not only on λ but also on the number n of dimensions, making n = 4 and n = 3 (for λ = 1) somewhat special. This is most evident in the relation
where P is the "trace" P := g^{ik}P_{ ik }. Assuming from now on n = 4, this simplifies to
P + Q = (3  2λ)Φ^{1}ΔΦ.
Let us define the geometrical energymomentum tensor P_{ ik }:= 1/2 ${\tilde{P}}_{ik}$ and more closely examine the corresponding balance relation
The following cases can be distinguished, when equating the geometrical quantities P and Q to their "physical" counterparts P_{ m }and Q_{ m }:

a)
Pure Einstein, φ = 0

b)
Conformally Einstein, λ = 3/2

c)
"Conservative", Q = 0

d)
VP flow, ΔΦ = 0

e)
Fully dynamical.
Each of these choices will now be discussed individually.
5.1 Pure Einstein
This is just the "compatibility mode", or "Einsteinlimit" φ → 0 (if it exists). It is thus a volumetric theory only in the trivial sense of φ = 0.
5.2 Conformally Einstein
The system of equations is underdetermined. This becomes evident by going to the Einsteinframe by means of the conformal transformation ${{g}^{\prime}}_{ik}={e}^{\phi}{g}_{ik}$, where the scalar field drops out completely.
5.3 "Conservative": JordanBransDicke
From Q = 0 there follows "conservation" in the usual sense of ∇_{ i }(Φ${P}_{k}^{i}$) = 0. This assumes the particular relation PΦ = (3  2λ) ΔΦ, which could either be postulated or obtained by a specially tailored Lagrangian. In order to have a more familiar looking equation, P_{ ik }could be equated to the physical quantity T_{ ik }over P_{ ik }= Φ^{1}T_{ ik }, so that in fact "conservation" in the sense of ∇_{ i }${T}_{k}^{i}$ = 0 would result. This kind of "conservation" was considered as absolutely essential in the closely related scalartensor theories of Jordan and BransDicke (in short, JBD theories).^{7} In fact, this can be achieved as follows: their scalars φ (resp. κ) must be identified with Φ, and we must identify their coupling parameters ω (resp. ζ) with λ. Moreover λ ≠ 3/2 has to be assumed, otherwise the conformally Einstein theory would result. Then 1/φ (resp. κ) is interpreted as the (variable) gravitational constant. Both for Jordan's and BransDicke's material energymomentum tensor it is supposed that it does not depend on the scalar φ. However in Jordan's theory it is the product κ^{2} ${T}_{k}^{i}$ which is "conserved".^{8} In particular, for a "dust model" geodesy of $\overrightarrow{u}$ (resp. κ^{2}$\overrightarrow{u}$) and conservation of ρ$\overrightarrow{u}$ (resp. κ^{2} ρ$\overrightarrow{u}$) still follow when staying in the original conformal ("Jordan") frame.
5.4 Volumepreserving Flow
When not a conformally Einstein coupling, from equation (14) the condition P + Q = 0 is equivalent to ΔΦ = 0, which in turn is equivalent to volumepreservation Â£_{ ξ }ω = 0. This translates to the scalar condition
P_{ m }+ Q_{ m }= 0
for the corresponding "material" quantities. Let us call such a coupling to matter a volumepreserving material coupling (VPMC), and assume it to hold troughout this section. Then
Evidently, if the trace P_{ m }of the energymomentum tensor vanishes the VPMC is satisfied if we set Q_{ m }= 0. Then the standard "conservation" continues to hold. This is the case e.g. for the Maxwell field.
As an important example where P ≠ 0, let us take the ideal fluid model, where the material 6 energymomentum tensor is given by ${P}_{m}^{ik}$ := T^{ik}= ρu^{i}u^{k}+ p∏^{ik}, and ${\prod}_{k}^{i}:={\delta}_{k}^{i}+{u}^{i}{u}_{k}$ is the projector orthogonal to the trajectory with (normalized) tangent $\overrightarrow{u}$. Its trace is T = 3p  ρ. To satisfy the VPMC, we must set Q_{ m }= T. Specializing to pure dust we get ∇_{ i }(Φρu^{i}u^{k}) = $\frac{1}{2}$ρg^{ik}∂_{ i }Φ. Splitting into tangential and orthogonal parts, we then get the separate equations
Due to the nonvanishing of the r.h.s. of these equations, both "conservation of matter" and geodesy for "test particles" are broken unless φ = const. And due to the particular form of the equation of motion 18b (i.e., being proportional to a gradient) we have in fact got a NewtonNordströmterm.^{9}
Concerning the divergence expression (18a), it can nevertheless be rewritten as a conservation law, ∇_{ i }(Φ^{1/2}ρu^{i}) = 0. Therefore, for such a theory with volumepreserving flow, both the equation of motion as well as the "conservation of dust matter" are not anymore the wellknown standard expressions from Einstein or JordanBransDicke theory. It can be expected that this will have profound consequences in a cosmological setting.
We will continue the discussion of volumepreserving theories in section 6, where we further specialize to the coupling parameter λ = 1.
5.5 Fully dynamic Scalar Field
Here the scalar φ is dynamically determined by a set of field equations obtained via a suitable Lagrangian, and no case of the previously discussed ones fits. This would normally be the "standard" procedere in physics, where not only the Lagrangian is set up as a linear combination of individual Lagrangians, each one describing a different matter model, but in addition possibly introducing some extra "potential terms" containing φ and ∂φ, or even let λ depend on φ. However this will in general prevent a simple geometrical interpretation in terms of a flow, and in particular will lack the crucial VP property. For example, our geometric dilaton gravity (Graf [4]) belongs to this more general class.
6 Ricci Flow Gravity
The class of volumepreserving volumetric theories can be further refined by requiring the particular value λ = 1 of the coupling, as is common in the lowenergy limit of string theory. With this particular value the field equations can be rearranged into an explicit flowlike form and we get the Ricci flow gravity equations (RFG equations)
describing Ricci flow gravity (RF gravity). Here the flow vector is defined in terms of the volume scalar as $\overrightarrow{\xi}$ = g^{1}∂φ, ^{10} and ${\overline{T}}_{ik}$ := 8π(T_{ ik }– $\frac{1}{2}$Tg_{ ik }).^{11} In contrast to the JBD equations, they have a much simpler structure and an immediate geometric character. Through their particular flowlike form, they exhibit a strong dynamical touch: broadly speaking, the rate of change of the metric is driven by the difference of the geometrical and the physical energy momentum tensors. Evidently, when the flow vector can be ignored (e.g., when it vanishes) equations equivalent to Einstein's are obtained. In this sense Einstein gravity is a special case of Ricci flow gravity.
The RFG vacuum equations are equivalent to JBD's vacuum equations with ω = 1. More remarkably is the fact that they are also equivalent to the equations following from the lowenergy limit of string theory when the standard dilaton coupling with λ = 1 is chosen and besides the metric only the dilaton scalar is kept. And of course there is a strong resemblance to Perelman's Ricci flow equations which can be made more evident as follows. Consider tentatively on M := M_{3} × T the vector $\overrightarrow{\xi}$ := (∂_{ t }+ $\overrightarrow{\nu}$) and the metric g_{ ik }with line element ds^{2} = dσ^{2} – dt^{2}. Then the generalized RF equation (2) can be written as ${\pounds}_{\xi}{g}_{ik}=2\phantom{\rule{0.5em}{0ex}}{R}_{ik}^{(3)}$, which differs (in content, but not in form) only on the r.h.s. from the corresponding RFG vacuum equation.^{12}
This coincidence of seemingly different approaches could signal a deeper raison d'être.
7 On the physical interpretation of the scalar Φ
The scalar Φ was here interpreted geometrically in the context of a volumetrical manifold as the volume factor. In the theories of JBD the corresponding scalar is essentially interpreted as "gravitational constant" κ – more precisely κ = Φ in Jordan's theory, whereas κ = 1/Φ in BransDicke theory. However this physical interpretation cannot be upheld anymore in a volumepreserving theory like RFG where κ is constant.
Due to the fact that the volume factor Φ of a volumepreserving theory obeys the d'Alembertian wave equation ΔΦ = 0 it must therefore be interpreted as a massless real scalar field. By the tenets of relativistic quantum mechanics this corresponds to a totally uncharged massless bosonic particle.^{13} The volumepreservation will also be instrumental to guarantee an almost perfect compliance with the standard solar system tests of gravity. This is a fair return for the price we had to pay for giving up the geodesy of "test particles".
8 Compliance of Ricci flow gravity with the Standard Tests
For the standard solar system tests the corresponding generalization of the Schwarzschild metric is needed. The general asymptotically flat and static spherically symmetric vacuum RFG solution with Φ → 1 for r → ∞ can be written as
ds^{2} = Y^{γσ}dt^{2} + Y^{γσ}(dr^{2} + Z^{2}dΩ^{2}),
with γ^{2} + σ^{2} = 1, and it is assumed that r ≥ r_{+} > r_{} ≥ 0.^{14} Being for r > r_{+} a vacuum RFG solution, it is also the corresponding general JBD vacuum solution in the "Jordan"frame. But whereas in JBD gravity the source of the volume factor Φ for a mass point has to be a certain nonzero distribution supported by r = r_{+}, in RF gravity due to ΔΦ = 0 it must be sourceless. This can be shown to hold even for a compactly supported smooth static spherically symmetric energymomentum tensor as source, if both metric and volume factor are smooth and the manifold is simplyconnected. Therefore for RFG σ = 0, whereas for JBD σ = 1/2 (3 + 2ω)^{1}.
This can also be expressed more conveniently in terms of the "dilaton charge" D, which in the context of the lowenergy limit of string theory is defined for a static solution with Killing vector η (normalized to η^{2} = 1 at infinity) as
where the integral is taken over a closed and externally orientable 2sphere at spatial infinity.^{15} For RF gravity the twoform density χ := η·ξ·ωis even closed, dχ = 0, for any stationary solution with Killing vector η so that the above integral only depends on the homology class of the closed externally orientable 2sphere. In particular it vanishes if this 2sphere bounds. With the flow vector $\overrightarrow{\xi}$ = σ (r_{+} – r_{})Y^{γ+σ}Z^{2} ∂_{ r }for the above solution this results in D = σ (r_{+}  r_{}). For vanishing dilaton charge the Schwarzschild solution is evidently reobtained after substituting r by r + r_{}, setting m = (r_{+}  r_{})/2 and assuming m > 0. Thus for the standard solarsystem tests the flow vector vanishes and we have full compatibility with Einstein gravity, which passes these tests with ever increasing accuracy (cf. Will [32]).^{16}
Of course where the flow vector does not vanish, Ricci flow gravity and Einstein gravity will lead to different answers. Using heuristically the term "charge" as introduced above (possibly without stationarity) we note that differently from the "mass charge" m, the "dilaton charge" D can have any sign.^{17} Therefore the contributions to the total charge of a collection of "charged regions" can still sum up to zero, so as to make the NewtonNordströmterms of the equation of motion insignificant for sufficiently big distances.^{18} This should be considered to be in fact the case for "ordinary matter" building up planets, stars and perhaps, galaxies. Significant differences are however to be expected in a cosmological setting, where the "big bang" will affect the behaviour of the volume scalar φ.
Although for the "compliant mode" φ = const evidently it makes no difference if the metric is interpreted in the geometric frame or in the Einstein frame, this is not so in the general case where even the equations of motion for a point particle are modified. We have to chose the particular conformal frame, where the field equations find their "most natural expression". This is the geometrical frame with an independent volume element density.
9 Conclusion
Motivated by the neglect of the notion of an independent volume and led by the appeal of Perelman's approach to solve the Poincaré conjecture, as well as by the equations following from the lowenergy limit of string theory, we developed the equations of Ricci flow gravity as a natural extension of Einstein gravity. The main differences with regard to other scalartensor theories were worked out in the framework of volumetric manifolds. The volumepreservation of the flow turned out to be of decisive importance for the theory and allowed it to essentially agree with Einstein's under noncosmological settings and not too small distances in the case of vanishing "total dilatonic charge".
Notes
^{1}for the dilaton general concept, see Sundrum [2]; for the connection of gravity to strings, see Ortín [3]
^{2}the expression in parenthesis in the r.h.s. of equation (2) has its own measuretheoretic meaning and is also known under the name "BakryÉmery" tensor (cmp. Lott [10])
^{3}in particular he envisages volumepreserving flows and certain diffeomorphic images thereof
^{4}see the recent monograph of Chow and Knopf [11] on Ricci flows (not covering Perelman's contributions), the introduction by Topping [12], and the lecture notes by Morgan and Tian [13]
^{5}the factor 1 of φ is purely conventional – here we follow Perelman [5], in contrast to string theory, where 2 is preferred
^{6}this is a local concept in contrast to the much weaker global definition of Huisken [26]
^{7}cmp. Jordan [27], Weinberg [28], part II, ch. 7, §3 and Fujii and Maeda [24]
^{8}this is suggested by his interpretation of the KaluzaKlein decomposition
^{9}recall that around 1912–13 Nordström developed a precursor relativistic gravitational theory, where the gravitational potential φ obeys a Minkowskian potential equation, Δφ = 0. This was shown in 1914 by Einstein and Fokker to admit a conformally Minkowskian formulation
^{10}the arbitrary minussign is taken in view of cosmological applications
^{11}we use troughout the sign and units conventions of Misner, Thorne and Wheeler [29]
^{12}the connection between some solutions of the Ricci flow equations for n = 3 and solutions of the Einstein equations for n = 4 was further elaborated by Bleecker [30] and by List [31]
^{13}except for a ''dilaton charge''; see the discussion in next section
^{14}in the ''degenerate'' case r_{+} = r_{} the metric is locally flat and the volume factor constant
^{15}cmp. Garfinkle, Horowitz and Strominger [1]
^{16}to compare, for JBD gravity to pass the current tests ω > 4 × 10^{4} must be assumed
^{17}this allows the dilaton scalar to act ''repulsively'', as shown in Graf [4]
^{18}e.g., for a ''multipole charge'' when the distance is much bigger than the individual ''charges''
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Acknowledgements
I thank the members of the relativity group of the University of Vienna for discussions and useful suggestions, and in particular Peter C. Aichelburg. Partial financial support from the Fundación Federico is also acknowledged.
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Graf, W. Ricci Flow Gravity. PMC Phys A 1, 3 (2007). https://doi.org/10.1186/1754041013
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Keywords
 Einstein Gravity
 Ricci Flow
 Relativistic Gravitational Theory
 Dilaton Charge
 Dilaton Scalar