Masking singularities in Weyl gravity and Ricci flows
aa r X i v : . [ g r- q c ] F e b Masking singularities in Weyl gravity and Ricci flows
Vladimir Dzhunushaliev
1, 2, 3, ∗ and Vladimir Folomeev
2, 3, 4, † Department of Theoretical and Nuclear Physics,Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan Institute of Experimental and Theoretical Physics,Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan Academician J. Jeenbaev Institute of Physics of the NAS of theKyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan International Laboratory for Theoretical Cosmology,Tomsk State University of Control Systems and Radioelectronics (TUSUR), Tomsk 634050, Russia (Dated: February 16, 2021)Within vacuum Weyl gravity, we obtain a solution by which, using different choices of the con-formal factor, we derive metrics describing (i) a bounce of the universe; (ii) toroidal and sphericalwormholes; and (iii) a change in metric signature. It is demonstrated that singularities occurringin these systems are “masked”. We give a simple explanation of the possibility of masking thesingularities within Weyl gravity. It is shown that in the first and third cases the three-dimensionalmetrics form Ricci flows. The question of the possible applicability of conformal Weyl gravity assome phenomenological theory in an approximate description of quantum gravity is discussed.
PACS numbers: 04.50.Kd, 04.60.BcKeywords: Weyl gravity; universe bounce; wormholes; change in metric signature; masking singularities;Ricci flows
I. INTRODUCTION
Weyl gravity is a conformally invariant theory of gravity where classes of conformally equivalent metrics serve as aphysical object [1]. For a host of reasons, such theory is not a fundamental theory of gravity, at least on cosmologicaland solar system scales. Nevertheless, there is a point of view that such theory can be useful in studying variousgravitational effects near singularities. For example, in Refs. [2–5], the idea was proposed according to which classes ofconformally equivalent metrics become a physically important object near a singularity. If this is the case, then suchclasses may contain both singular and regular metrics. By the singular metric we mean a metric which makes singularsuch scalar invariants like the scalar curvature and the squares of the Ricci and Riemann tensors. The existence bothof regular and singular metrics within one class of conformally equivalent metrics results in the fact that, despite thedivergence of the aforementioned scalar invariants, the conformally invariant tensors, and hence their squares, will beregular. For instance, we will demonstrate below that, within vacuum Weyl gravity, there exist solutions possessingsuch properties.In Refs. [6, 7], G. ’t Hooft considers the idea that “the conformal symmetry could be as fundamental as Lorentzinvariance, and guide us towards a complete understanding of physics at the Planck scale.” In other words, theconformal invariance may be very important in quantum gravity in describing quantum gravity effects in high cur-vature regions. This means that Weyl gravity may serve as a phenomenological theory that approximately describesquantum gravity effects in high curvature regions, just as the Ginzburg-Landau theory is a phenomenological theoryof superconductivity.In Ref. [8], the idea is proposed that gravity is responsible for breaking the fundamental conformal invariance. Thiscan be treated as follows: in high curvature regions, the conformal invariance is not violated, but it is violated ingoing to low enough curvature regions.Concerning Ricci flows, in differential geometry, they are used in studying the topology of differentiable manifolds.Ricci flows define the occurrence of singular points on a manifold; this would lead us to expect that they can be used ingravitational theories when studying such singularities. For example, in Ref. [9], the field equations are postulated inthe form of the Ricci flow equations, and Einstein’s theory is included as the limiting case where the flow is absent. InRef. [10], the solutions to the equations for Ricci flows are given, and it is shown that these solutions contain metricsdescribing a change in metric signature. In Ref. [11], Ricci flows are under investigation, and the connection with apath integral in quantum gravity is demonstrated. In Ref. [12], the idea that the occurrence of quantum wormholes in ∗ [email protected] † [email protected] spacetime foam can be described using Ricci flows is discussed. In Ref. [13], Ricci flows are used to study transitionsbetween the AdS and warped AdS vacuum geometries within Topologically Massive Gravity.Summarizing the above ideas, one can suppose that Weyl gravity may serve as some phenomenological theory thatapproximately describes quantum gravity effects in high curvature regions, and in low curvature regions the conformalinvariance is violated. In the present paper, we would like to demonstrate that, within Weyl gravity, there exists suchan interesting feature like masking of some singularities, that in fact is a consequence of quantum gravity, but in thecase under consideration this effect is approximately described by Weyl gravity. By the “masking of singularities”we mean the fact that, in Weyl gravity, there can exist the following unusual situation: some tensors (and hencethe corresponding scalar invariants) are singular (for example, the Ricci and Riemann tensors), while at the sametime there are tensors which are not singular at the same points. As such tensors, there can be, for example, theWeyl and Bach tensors. From the mathematical point of view, this means that the Weyl tensor is constructed sothat the combination of the Riemann and Ricci tensors and of the metric is such that the corresponding singularitieseliminate each other. From the physical point of view, this means that Weyl gravity can be treated as an approximatedescription of quantum gravity effects describing the behavior of spacetime near some singularities.The paper is organized as follows. In Sec. II, we introduce the Lagrangian and show the corresponding fieldequations, as well as the conformally invariant class of metrics which are the solution in Weyl gravity. For suchclass of metrics, in Sec. III, we discuss a cosmological bounce solution, singularities, and Ricci flows; in Sec. IV, wedemonstrate the existence of a solution describing a change in metric signature, discuss the corresponding singularities,and show that the three-dimensional spatial metric is a Ricci flow; in Sec. V, we obtain toroidal, T , and spherical, S , wormholes and study the corresponding singularities. Finally, in Sec. VI, we discuss and summarize the resultsobtained. II. WEYL GRAVITY
In this section we introduce the Lagrangian and write down the corresponding field equations in Weyl gravity.The action can be written in the form [hereafter we work in natural units ~ = c = 1 and the metric signature is(+ , − , − , − )] S = − α g Z d x √− gC αβγδ C αβγδ , (1)where α g is a dimensionless constant, C αβγδ = R αβγδ + ( R αδ g βγ − R αγ g βδ + R βγ g αδ − R γδ g αβ ) is the Weyl tensor.The action (1) and hence the corresponding theory are invariant under the conformal transformations g µν → f ( x α ) g µν , where the function f ( x α ) is arbitrary.The corresponding set of equations in Weyl gravity is B µν ≡ C α βµν ; αβ + C α βµν R αβ = 0 , (2)where B µν is the Bach tensor.In what follows we will work with the solutions obtained in Ref. [14] and consider in detail how such solutions candescribe the structure of spacetime near singularities. To do this, let us consider the following metric: ds = f ( t, χ, θ, ϕ ) (cid:26) dt − r h ( dχ − cos θdϕ ) + dθ + sin θdϕ i(cid:27) = f ( t, χ, θ, ϕ ) (cid:0) dt − r dS (cid:1) . (3)Here, dS is the Hopf metric on the unit sphere; f ( t, χ, θ, ϕ ) is an arbitrary function; r is a constant; 0 ≤ χ, ϕ ≤ π and 0 ≤ θ ≤ π . The Bach tensor for such metric is zero; this means that this metric is a solution of Eq. (2) for Weylgravity.In the following sections we will consider some interpretations of this solution and give the analysis of its singularpoints. III. COSMOLOGICAL BOUNCE SOLUTION AND INFLATION
Consider the case with the conformal factor f ( t, χ, θ, ϕ ) = f ( t ) . Introducing the new time coordinate dτ = f ( t ) dt , we have the following expression for the metric (3): ds = dτ − r f ( τ ) h ( dχ − cos θdϕ ) + dθ + sin θdϕ i . (4)If the function f ( τ ) is chosen so that f ( τ ) is an even function and f (0) = f = const, the metric (4) will describea universe with a bounce at time τ = 0. If, in addition, one chooses f ( τ ) so that asymptotically f ( τ ) ∼ e τ , one willhave a cosmological bounce solution with a subsequent inflationary expansion. This can be done by choosing, forexample, f ( t ) = 1 / cos ( t/t ), as was done in Ref. [14].Consider the behavior of the metric at the bounce point t = 0. Since the function f ( τ ) is assumed to be even, itcan be expanded in a Taylor series as f ( t ) = f + t ˜ f ( t ) ≡ f + t (cid:18) f f t + . . . (cid:19) , where ˜ f ( t ) is an even function. We are interested in the behavior of the metric (4) at the bounce point t = 0, wherethe scalar invariants have the following expansions: C αβγδ C αβγδ = 0 , B αβ B αβ = 0 , (5) R = − f f − r f ∝ f f → −−−→ ∞ , R αβ R αβ = 34 f (4) (0) f f → −−−→ ∞ , R αβγδ R αβγδ = 32 f (4) (0) f f → −−−→ ∞ . (6)If one chooses the function f ( t ) so that f = 0, an interesting situation takes place at this point: there is a singularity,since the scalar invariants R, R αβ R αβ , and R αβγδ R αβγδ diverge here. Nevertheless, the Weyl and Bach tensors areconstructed so that they are just equal to zero, and hence the corresponding invariants C αβγδ C αβγδ and B αβ B αβ arealso zero. This means that, in Weyl gravity, such singularities are masked!
This is very interesting result, and onemight reasonably suppose that it happens due to the fact that Weyl gravity is an approximate description of quantumgravity effects in high curvature regions, i.e., near some singularities.Thus, the result obtained enables us to say that for small f (notice that this parameter corresponds to the size ofthe universe at the bounce point) the transition from a contraction stage to expansion is a quantum gravity effect,and it may be approximately described by Weyl gravity.Next, when the size of the universe increases, the spacetime becomes less curved; correspondingly, quantum gravityeffects become negligible and the dynamics of the universe is no longer adequately described by Weyl gravity; that is,the spacetime becomes classical and it should be described by general relativity.There is a very simple explanation why Weyl gravity “does not see” the singularity: the reason is that the singularityarises because of the factor f ( τ ) in the spatial part of the metric (4). Since this factor tends to zero as f →
0, thevolume of the space also goes to zero, thereby the singularities in the scalar invariants
R, R αβ R αβ , and R αβγδ R αβγδ appear. But since Weyl gravity is conformally invariant, it “does not see” this change in the volume.Consider now a sequence of spatial metrics from (4), dl = r f ( τ ) h ( dχ − cos θdϕ ) + dθ + sin θdϕ i , (7)for f →
0. This sequence describes the occurrence of the singularity where the invariants (6) go to infinity. Indifferential geometry, this process is described by Ricci flows, ∂γ ij ∂λ = − R ij , (8)where λ is some parameter. The spatial metric tensor γ ij from (7) is defined as γ ij = r f ( τ, λ )˜ γ ij , (9)where ˜ γ ij is the metric on the unit three-dimensional sphere in the Hopf coordinates χ, θ, ϕ ; R ij is the correspondingthree-dimensional Ricci tensor with the spatial indices i, j = 1 , ,
3; the conformal factor f also depends on theparameter λ .The Ricci tensor for the metric (7) takes the form R ij = 2˜ γ ij . (10)Substituting (9) and (10) in (8), we get ∂f ( λ ) ∂λ = − r with the solution f ( λ ) = λ − λr , where λ is an integration constant. This means that the parameter f , starting from the value λ , reaches zerovalue f = 0 when λ = r λ /
4. For this value, there appears a singularity for the scalar invariants
R, R αβ R αβ , and R αβγδ R αβγδ .Thus, in this section, we have shown that there are cosmological bounce solutions in Weyl gravity. When the size ofthe universe decreases, at the bounce point, the scalar invariants R, R µν R µν , and R µνρσ R µνρσ diverge but the Weylinvariants C αβγδ C αβγδ and B αβ B αβ remain finite and equal to zero. This enables us to say that, in Weyl gravity,there is a kind of masking of singularities. It is also shown that the family of such solutions numbered by the size ofthe universe at the bounce time form the Ricci flow.The idea that near singularities a conformal invariance and conformal transformations may be important has beenconsidered in Refs. [2–5]. The main idea of those papers is that near a singularity the conformal, but not the metric,structure of spacetime is of importance. And if there exists a conformal factor transferring a metric with a singularityinto a metric without a singularity, from the physical point of view, there is no singularity in such a spacetime.From our point of view, this means that quantum gravity comes into play, and Weyl gravity is just an approximatedescription of quantum gravity effects in such a situation. IV. PASSING A SINGULARITY WITH A CHANGE IN METRIC SIGNATURE
Another interesting example of ignoring a singularity in Weyl gravity is its masking with a change in metricsignature. To demonstrate this, consider the metric ds = dτ − r h ( τ ) h ( dχ − cos θdϕ ) + dθ + sin θdϕ i (11a)= h ( τ ) (cid:26) dτ h ( τ ) − r h ( dχ − cos θdϕ ) + dθ + sin θdϕ i(cid:27) (11b)= h ( t ) (cid:26) dt − r h ( dχ − cos θdϕ ) + dθ + sin θdϕ i(cid:27) . (11c)In (11c), we have introduced dt = dτ / √ h for τ > τ and dt = dτ / √− h for τ < τ . We choose the function h ( τ ) sothat it changes its sign at some τ : h ( τ ) = ( > τ > τ ,< τ < τ . (12)Thus, at τ > τ , the metric signature is Lorentzian, (+ , − , − , − ), and at τ < τ it is Euclidean, (+ , + , + , +). Tosatisfy the conditions (12) and simplify calculations, let us choose the function h ( τ ) in the form h ( τ ) = τ ˜ h ( τ ) = τ (cid:18) h + h τ
2! + . . . (cid:19) , (13)where ˜ h ( τ ) is an even function.At the point τ = 0, we have the following Taylor expansions for the scalar invariants: C αβγδ C αβγδ = 0 , B αβ B αβ = 0 ,R ≈ h τ τ → −−−→ ∞ , R αβ R αβ ≈ h τ τ → −−−→ ∞ , R αβγδ R αβγδ ≈ h τ τ → −−−→ ∞ . Analogous to what was done in the previous section, the Weyl and Bach tensors are nonsingular when passing thepoint τ = 0, while the invariants R, R µν R µν , and R µνρσ R µνρσ are singular. This means that, as well as in the previoussection, at the point τ = 0, where the metric changes its signature, the singularity is masked.The spatial part of the metric from (11a) is dl = r h ( τ ) h ( dχ − cos θdϕ ) + dθ + sin θdϕ i = r h ( τ )˜ γ ij dx i dx j with the corresponding Ricci tensor R ij = 2˜ γ ij . In contrast to the cosmological bounce solution considered in Sec. III where the quantity λ serves as a parameter inthe Ricci flow, here the time coordinate τ may serve as such a parameter. This enables us to write the equation forRicci flows (8) in the form ∂h ( τ ) ∂τ = − r , which gives us the following solution for a Ricci flow: h ( τ ) = − r τ. This solution is a special case of the solution to equations for Weyl gravity (11a) and (13) for ˜ h ( τ ) = const.Thus, in this section, we have shown that, in Weyl gravity, there are solutions describing a change in metricsignature. At the transition point, the scalar invariants R, R µν R µν , and R µνρσ R µνρσ go to infinity, but the Weylinvariants C αβγδ C αβγδ and B αβ B αβ remain finite and equal to zero. As in the case of the solution of Sec. III, thiseffect may be called the masking of singularities in Weyl gravity. Another interesting feature of the solutions witha change in metric signature obtained here is that there exists a special solution which is also the Ricci flow for thecorresponding spatial part of the metric. Notice that the fact of a change in metric signature in passing through asingular point has also been pointed out in Ref. [10]. V. WORMHOLES
In the previous sections we have considered a choice of the conformal factor leading to metrics describing a bounceof the universe and a change in metric signature. In this section we examine metrics describing wormholes possessingdifferent cross-sections: a T torus and a S sphere. In both cases we will obtain the corresponding Ricci flows. A. Toroidal T wormhole In this subsection we consider the case where the conformal factor f ( t, χ, θ, ϕ ) depends only on the spatial coordinate θ , f ( t, χ, θ, ϕ ) = f ( θ ) . Let us introduce the new spatial coordinate dx = − f ( θ ) dθ so that the function f ( θ ( x )) = f ( x ) would have aminimum at x ( θ = π/
2) = 0 and tend to ±∞ as x → ±∞ : f (0) = min , f ( x = ±∞ ) = ±∞ . (14)In this case we get the following metric: ds = f ( θ ( x )) dt − r n dx + f ( θ ( x )) h ( dχ − cos ( θ ( x )) dϕ ) + sin ( θ ( x )) dϕ io . (15)The area of a torus spanned on the coordinates χ, ϕ is defined by the determinant of the two-dimensional metric inthe square brackets in Eq. (15): dl = f ( θ ( x )) h ( dχ − cos ( θ ( x )) dϕ ) + sin ( θ ( x )) dϕ i . Consistent with the conditions (14) for the function f , it is seen that the area of the torus S = r f ( θ ( x )) / θ = π/ x → ±∞ . Also, we require that f ( θ ( x )) sin( θ ( x )) → const as θ → π/ T wormhole.Taking into account that dx = − f ( θ ) dθ and using the condition f ( θ ) sin( θ ) = C = const , (16)we have the following solution for the function f ( x ), see Ref. [14]: f ( x ) = C cosh x. Then the metric (15) takes the form ds = C cosh xdt − r (cid:26) dx + C cosh x (cid:20) ( dχ − tanh xdϕ ) + 1cosh x dϕ (cid:21)(cid:27) , (17)and the coordinate x covers the range −∞ < x < + ∞ .Consider now Ricci flows for this case. In Secs. III and IV, we have considered three-dimensional Ricci flows for thespatial parts of four-dimensional metrics. The argument was that the singularities occurred because of the fact thatthe spatial volume vanishes. In this subsection we consider the case where the area of the wormhole throat goes tozero; therefore, we will consider two-dimensional Ricci flows defined on two-dimensional tori which are cross-sectionsof a wormhole.Two-dimensional metric for the spacetime metric (17) is dl = − C r x (cid:20) ( dχ − tanh xdϕ ) + 1cosh x dϕ (cid:21) = γ ij dx i dx j , x = χ, x = ϕ, (18)and Ricci flows should be examined precisely for this two-dimensional metric. In this case a Ricci flow is written as ∂γ ij ∂λ = − R ij , where the indices i, j = χ, ϕ are two-dimensional indices defined on a two-dimensional torus with the metric (18).Since the Ricci tensor for the metric (18) is identically zero, R ij = 0, this means that the two-dimensional metric (18)is unchanged in the Ricci flow, as is obvious if we note that if the condition (16) is satisfied, we have only one solution –the metric (17).One can ignore the condition (16) and consider wormholes without using it. In that case the even function f ( x )has the following Taylor expansion near x = 0: f ( x ) = h ( x ) = h + x ˜ h ( x ) = h + x (cid:18) h + h x
2! + . . . (cid:19) . As h →
0, there will occur a singularity but, apparently, in this case the factor f ( x ) sin x before the term with dϕ in (15) will also go to zero. This means that we will have a spherical S wormhole which will be considered in thenext subsection.Thus, in this subsection, we have considered a toroidal T wormhole and shown that for it the Ricci flow is stationary,and thereby singularities are absent in this case. B. Spherical S wormhole In the above discussion, we have considered the spacetime with the spatial cross-section in the form of a three-dimensional sphere S on which the Hopf coordinates are introduced. In particular, in the previous subsection, wehave shown that, for some special choice of the conformal factor, it is possible to obtain a toroidal T wormhole.Here, we will demonstrate that, by choosing the standard spherical coordinates on a three-dimensional sphere, it ispossible to get a spherical S wormhole with a cross-section in the form of a two-dimensional sphere S .Using the usual spherical coordinates, the spacetime metric can be written in the form ds = f ( t, χ, θ, ϕ ) (cid:8) dt − r (cid:2) dχ + sin χ (cid:0) dθ + sin θdϕ (cid:1)(cid:3)(cid:9) = f ( t, χ, θ, ϕ ) (cid:0) dt − r dS (cid:1) , (19)where 0 χ π, θ π, ϕ π are the angular coordinates on a three-dimensional sphere.Let us define the conformal factor as f ( t, χ, θ, ϕ ) = f ( χ ). Then, introducing the new coordinate dx = − rf ( χ ) dχ ,we have from (19): ds = (cid:18) x + x rx (cid:19) dt − dx − (cid:0) x + x (cid:1) (cid:0) dθ + sin θdϕ (cid:1) , (20)where we have used the function f ( χ ) = x / (cid:0) r sin χ (cid:1) , which gives x = x cot χ with −∞ < x < + ∞ . It is evidentthat this is the metric of a wormhole with the throat radius x .For simplicity, we will consider below a Z symmetric wormhole. This assumes that after introducing the newcoordinate x [see Eq. (20) above], the function f ( x ) will be even. Then the radius of the two-dimensional sphere inthe metric (19) can be expanded in a Taylor series in the vicinity of x = 0 as follows: f ( x ) sin ( χ ( x )) = h ( x ) = h + x ˜ h ( x ) ≡ h + x (cid:18) h + h x
2! + . . . (cid:19) . (21)The parameter h defines the area of a two-dimensional sphere at the center of the wormhole (that is, at the throat).Then the metric (19) takes the form ds = f ( x ) dt − dx − h ( x ) (cid:0) dθ + sin θdϕ (cid:1) . (22)Let us keep track of the behavior of the scalar invariants when a cross-sectional area of the wormhole underconsideration goes to zero: C αβγδ C αβγδ = 0 , B αβ B αβ = 0 ,R ≈ − r h h h → −−−−→ ∞ , R αβ R αβ ≈ r (cid:18) h h (cid:19) h → −−−−→ ∞ , R αβγδ R αβγδ ≈ r (cid:18) h h (cid:19) h → −−−−→ ∞ . It is seen from these expressions that the scalar invariants associated with the conformal tensors remain equal tozero, while the scalar invariants
R, R αβ R αβ , and R αβγδ R αβγδ diverge. This means that, in Weyl gravity, when thecross-section of the wormhole decreases, nothing special happens, since the corresponding invariants do not diverge.From the physical point of view, this process of decrease (or of increase) of the cross-section can be interpreted as anannihilation (or creation) process of a quantum wormhole in spacetime foam.As in the case of the toroidal wormhole from Sec. V A, here, we will consider two-dimensional Ricci flows, definednow not on two-dimensional tori but on two-dimensional spheres, which are cross-sections of the wormhole underconsideration. The corresponding two-dimensional metric follows from the spacetime metric (22), dl = − h ( x ) (cid:0) dθ + sin θdϕ (cid:1) = γ ij dx i dx j = h ( x )˜ γ ij dx i dx j , x = θ, x = ϕ. (23)For it, a Ricci flow is ∂γ ij ∂λ = − R ij , (24)where the indices i, j = θ, ϕ are defined on a two-dimensional sphere. The Ricci tensor for the metric (23) is R ij = 2˜ γ ij . Taking this expression into account and substituting γ ij and ˜ γ ij from (23) and h ( x ) from (21) in Eq. (24), we get anequation describing the Ricci flow, ∂h ∂λ = − , with the solution h = λ − λ. Thus, in this subsection, we have demonstrated that, in Weyl gravity, there is a family of solutions describing S wormholes parameterized by the throat size h . It is shown that when h goes to zero, there occur singularitiesfor such invariants like R, R µν R µν , and R µνρσ R µνρσ . At the same time, the scalar invariants associated with theconformally invariant tensors like C αβγδ C αβγδ and B αβ B αβ remain regular. This means that, in Weyl gravity, suchsingularities are masked. It is also shown that for the S wormholes under investigation there are the Ricci flows whosepresence can be physically interpreted as the description of the creation/annihilation process of quantum wormholesin spacetime foam. VI. DISCUSSION AND CONCLUSIONS
The main purpose of the present paper is to demonstrate that, in Weyl gravity, there is an interesting phenomenon –the masking of singularities. This means that there are solutions for which the scalar invariants
R, R αβ R αβ , and R αβγδ R αβγδ are singular but the tensors employed in Weyl gravity (the Weyl and Bach tensors) remain regular.Perhaps this happens because Weyl gravity can be actually treated as an approximate theory describing quantumgravity effects near singularities, just as the Ginzburg-Landau theory is a phenomenological theory of superconduc-tivity. Notice also an interesting connection between the solution obtained here within Weyl gravity and “the Weylcurvature hypothesis ” proposed in Ref. [5]: in both cases, the Weyl tensor is equal to zero.An unexpected result of the present study is that we have found the connection between the solutions obtainedwithin Weyl gravity and Ricci flows. We have shown that for the cosmological bounce solution there is the familyof solutions γ ( τ, λ ) indexed by the size of the universe r f (0 , λ ) at the bounce time. The element of the family isthe metric γ ( τ, λ = const) (the spatial part of the four-dimensional metric) which is the solution of the gravitationalWeyl equations. In any such family, the metrics γ ( τ = 0 , λ ) are a Ricci flow with the Ricci parameter λ . The solutionfound in Sec. IV, which describes a change in metric signature, is a Ricci flow where the Ricci parameter coincideswith the time coordinate τ .Another interesting result is that all the solutions discussed here refer to one conformally equivalent class ofmetrics. There, they describe different physical situations: a bounce of the universe from a singularity with a possiblesubsequent exponential expansion, toroidal, T , and spherical, S , wormholes, and a change in metric signature.A possible physical explanation of the fact that the metrics under discussion mask the singularities is that Weylgravity is a phenomenological approximation for microscopic quantum gravity, just as the Ginzburg-Landau theoryis a phenomenological description of superconductivity.Thus, summarizing the results obtained: • Within Weyl gravity, there are obtained four types of solutions which are conformally equivalent each other butdescribe different physical situations. • It is shown that for all these solutions the singularities are masked in the sense that, even though such scalarinvariants like the scalar curvature and the squares of the Ricci and Riemann tensors are singular, the squaresof the Weyl and Bach tensors (which are employed in Weyl gravity) remain regular. • It is shown that for these solutions the three/two-dimensional spatial metrics are simultaneously Ricci flows. • A possible interpretation of Weyl gravity as a phenomenological theory which approximately describes quantumgravity effects is discussed.
ACKNOWLEDGMENTS
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