The GL(4 R) Yang-Mills Theory of Gravity Predicts An Inflationary Scenario For The Evolution of The Primordial Universe
aa r X i v : . [ phy s i c s . g e n - ph ] M a r The GL (4 R ) Yang-Mills Theory of Gravity Predicts AnInflationary Scenario For The Evolution of The PrimordialUniverse
Yi Yang ∗ and Wai Bong Yeung † The Ohio State University, Columbus, OH, USA Institute of Physics, Academia Sinica, Taipei, Taiwan, ROC (Dated: August 6, 2018)
Abstract
We show that the GL (4 R ) gauge theory of gravity will admit an exponentially expandingsolution for the primordial evolution of the Universe. This inflationary phase of the Universe isdriven by a rapidly changing GL (4 R ) gauge field configuration. No cosmological constant and norolling scalar field is required. This inflationary expansion will then be shown to slow down to amuch slower expansion rate during the late time evolution of the Universe. Keywords: gravitation, Yang-Mills, Einstein, cosmology, inflation ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Recently we have proposed a vector theory of gravity [1]. In this theory the four dimen-sional Spacetime is assumed to be equipped with a non-dynamical metric g µν which is thebackground metric to give the Spacetime the notion of world length. Any given metric islegitimate, but only those metrics that extremize our proposed action are the metrics thatwill be observed classically.Our proposed action comes from our observation that some fundamental laws of Nature,namely the Law of Inertia and the Causality Principle, are preserved under a family oftransformation of reference frames called the affine transformations GL (4 R ), on every localpatch of the Spacetime. Therefore we believe that it is natural for us to assume that physicsshould be invariant under these local transformations.These local transformations form a Lie group, and hence we are tempted to construct alocal Yang-Mills theory [1] based on this local group. This Yang-Mills theory is, of course, avector theory, and has 16 gauge vector boson A mnµ . Two sets of equations will follow whenwe vary the metric g µν and the gauge potentials A mnµ independently so as to extremize theYang-Mills action when we are looking for classical solutions to the theory. The first set ofequations, which will be called the Stephenson Equation [2], gives algebraic relations amongthe various components of the Yang-Mills strength tensor. The second set of equationsis the Yang-Mills equation in the presence of a background metric, and will be called theStephenson-Kilmister-Yang Equation [3].The Yang-Mills action can be transformed into a form that might look familiar to uswhen we make a variable substitution of the 64 variables A mnµ by a new set of 64 variablesΓ ρτµ through A mnµ = e mρ e τn Γ ρτµ + e mτ ∂ µ e τn , (1)where e mρ are the vierbein fields for the background metric in reference to a local Minkowskianframe (we shall use the Latin and Greek indices to denote, respectively, local Minkowskianand world coordinates. A hat is always put on an index when we want to emphasize thatwe are talking about local coordinates). With the new variables Γ ρτµ , our proposed actionwill look like [1]S YM [ g, A, ∂A ] = S YM [ g, Γ , ∂ Γ] = κ Z √− gd xg µµ ′ g νν ′ ( F mnµν F nmµ ′ ν ′ ) (2)2 κ Z √− gd xg µµ ′ g νν ′ ( R λσµν R σλµ ′ ν ′ ) . where F mnµν = ∂ µ A mnν − ∂ ν A mnµ + A mpµ A pnν − A mpν A pnµ stands for the Yang-Mills fieldstrength tensor, R λσµν = ∂ µ Γ λσν − ∂ ν Γ λσµ + Γ λρµ Γ ρσν − Γ λρν A ρσµ stands for the Riemanniancurvature tensor, and κ is a dimensionless coupling constant for the theory.The corresponding Stephenson and Stephenson-Kilmister-Yang Eequations will be, re-spectively R λσθρ R σ ρλτ − g θτ R λ ξρσ R σλξρ = 12 κ T θτ , (3) ∇ ρ (Γ)( √− gR βρλσ ) = 1 κ √− gS βλσ . The T θτ and S βλσ are respectively the metric energy-momentum tensor and the gauge currenttensor of the matter source [1].Before we will make further discussions, we find it imperative for us to stress that theaction given in Eq. 2, though looks very similar to the ones found in many occasions inthe discussion of the theory of gravity, is fundamentally different from them because thereexists no prior relationship between g µν and Γ ρτµ in our theory. The geometric objects here,such as the connections and the Riemann curvature tensor, are in fact, the Yang-Mills gaugepotentials and the Yang-Mills field strength tensors of GL (4 R ) in disguise [1]. We haveconverted the original Yang-Mills theory into a theory with geometric languages because wewant to make use of the many works done in the past decades by people working in thegeometric theory of gravity. The theory here is not a higher derivative theory of the metricand the only dynamical variables are the Yang-Mills potentials which satisfy a second orderdifferential equation.The theory given in Eqs. 1, 2 and 3 are shown to be able to comply with the gravitationaltests in the solar system because the Yang-Mills potentials induce the Schwarzschild metricas one of the admissible metrics [1]. Furthermore there are some other physical phenomenathat are predicted by the above theory which are not predicted by the theory of GeneralRelativity. These new predictions include the existence of two gravitational copies of matterwhich reproduce the effects of Dark Matter as observed in astronomy and also the existenceof a primordial torsion which mimics the effects of Dark Energy in the late time evolutionof the Universe [1]. 3 I. COSMOLOGY FROM THE POINT OF VIEW OF THE GL (4 R ) YANG-MILLSTHEORY
In order to be sure that this vector theory of gravity, basing on the local gauge theory of GL (4 R ) with a non-dynamical world metric, is a viable theory of gravity, we must put thistheory on test with the well-known thermal history of the Universe.In this article, we are going to show that this vector theory of gravity can accommodatean inflationary expansion of the Universe during it’s early phase of evolution, and thisinflationary expansion will then slow down and will take up a much slower acceleratingexpansion during its late time evolution.To this end, we will start by searching for solutions in which the Yang-Mills potentialswith symmetric gauge indices are vanishing, and that the cosmic metric will be taken as thespatially flat Friedmann-Lematre-Robertson-Walker (FLRW) form ds = − dt + a ( t )( dr + r dθ + r sin θdφ ) . (4)Furthermore we will make some assumptions on the form of our cosmic Yang-Mills poten-tial A mnµ . Because of the fact that the Yang-Mills potentials are related to the connectionsΓ ρτµ through Eq. 1, an assumption on the form of the cosmic Yang-Mills potentials will beequivalent to an assumption on the form of the cosmic connections, which are now compati-ble with the metric because the Yang-Mills potentials that we are seeking are anti-symmetricin their gauge indices.There are totally 24 components for the torsion tensor τ ραβ , which are the anti-symmetricparts of the connections. The connections will then have two parts, namely the Christoffelsymbols and terms composing of the torsions and the metric.We choose to describe the cosmic torsion tensor by what are observed by a localMinkowskian observer. This observer will also see 24 local components. These 24 localcomponents will fall into categories in accordance with their parity signatures under thelocal spatial parity operations (which consist of either one spatial inversion, two spatialinversions or three spatial inversions) of the local Minkowkian frame. Of all these 24 com-ponents, only 3 are invariant under the above said spatial parity operations. They are τ ˆ0ˆ1ˆ1 , τ ˆ0ˆ2ˆ2 , and τ ˆ0ˆ3ˆ3 . We have use a hat for each index because we want to emphasize that theseare the components measured by a local Minkowskian observer. We shall then assume that4nly the torsion components that are invariant under local parity operations will show upin the cosmic evolution of the Universe.Local isotropy will also play a role in the determination of the form of the local torsioncomponents too. Local isotropy will require that τ ˆ0ˆ1ˆ1 , τ ˆ0ˆ2ˆ2 and τ ˆ0ˆ3ˆ3 are all equal. Homo-geneity will require that they are functions of time only. Hence we have arrived at theconclusion that we will concern ourselves only with cosmos which have the following localtorsion components τ ˆ0ˆ1ˆ1 = τ ˆ0ˆ2ˆ2 = τ ˆ0ˆ3ˆ3 ≡ ξ ( t )2 . (5)See also the works of Ramaswamy and Yasskin [4], and Baekler and Hehl [5] and Chen,Hsu and Yeung [6] for the selection of the local torsion components in their discussions ofcosmologies in the Poincare Gauge Theory of Gravity.With the torsion components given in Eq. 5 and the metric given in Eq.4, we can calculatethe connections, and, in turn, the Riemann curvature tensor. We will have the followingnon-vanishing local components: R ˆ0ˆ1ˆ0ˆ1 ≡ D = ¨ aa − ξ ˙ aa − ˙ ξ,R ˆ0ˆ2ˆ0ˆ2 = R ˆ0ˆ3ˆ0ˆ3 ≡ E = − ¨ aa + ξ ˙ aa + ˙ ξ, (6) R ˆ1ˆ2ˆ1ˆ2 = R ˆ1ˆ3ˆ1ˆ3 ≡ M = ( ˙ aa − ξ ) ,R ˆ2ˆ3ˆ2ˆ3 ≡ N = − ( ˙ aa − ξ ) . Using these local curvature components, we can write the Stephenson and the Stephenson-Kilmister-Yang equations as( D + 2 E − M − N ) = 12 κ T ˆ0ˆ0 , (2 E + N − D − M ) = 12 κ T ˆ1ˆ1 , (7)( D − N ) = 12 κ T ˆ2ˆ2 = 12 κ T ˆ3ˆ3 ,
2( ˙ aa ) E + ˙ E + 2( ˙ aa − ξ ) M = 0 . (8)In Eq. 8 and in the subsequent discussions, we will assume that the cosmic gauge currenttensor S βλσ of the matter field source will be averaged out during the course of evolution ofthe Universe. 5ote the fact that Eq. 8 is the Yang-Mills Equation for the GL (4 R ) group while Eq. 7is the algebraic equations for the various components of the Yang-Mills strength tensor.See also the works of Refs. [4], [5] and Chen, Hsu and Yeung [6] for the discussions ofcosmic expansion under different situations.There is a hidden constraint built in Eq. 7 if we are going to impose T ˆ1ˆ1 = T ˆ2ˆ2 = T ˆ3ˆ3 because of isotropy requirement. Namely that the matter metric energy-momentum tensorcomponents should satisfy the algebraic relation of T ˆ1ˆ1 = T ˆ2ˆ2 = T ˆ3ˆ3 = 13 T ˆ0ˆ0 . (9)If we use ρ to denote T ˆ0ˆ0 and identify it as the local energy density and use p to denote T ˆ i ˆ i , where i = 1, 2 or 3, and identify it as the pressure, then the above relation is just theequation of state for the matter fields, p = 13 ρ. (10)During the expansion of the Universe, the Universe is supposed to undergo an adiabaticprocess, and hence satisfies the First Law of Thermodynamics and the equation of state ofEq. 10 can be integrated to give ρ = 6 κA a , (11)where 6 κA is an integration constant.Hence the cosmic equations that we are going to solve are( ¨ aa − ξ ˙ aa ) − ( ˙ aa − ξ ) = ( Aa ) , (12)2( ˙ aa )( − ¨ aa + ξ ˙ aa + ˙ ξ ) + ddt ( − ¨ aa + ξ ˙ aa + ˙ ξ ) + 2( ˙ aa − ξ ) = 0 . (13)These two equations are far more complicated than the Friedmann Equations. And in thefollowing we shall investigate the implications of these two equations on the evolution of theUniverse. III. THE EVOLUTION OF THE EARLY UNIVERSE
Now consider the situation in which ξ is an extremely small constant , say ξ = 10 − sec − .And also consider the case in which the initial condition for a ( t ) at t = 0 (i.e. a (0)) of our6osmic equation is an extremely small number . With an extremely small a ( t ), the matterin the Universe will be highly relativistic and we will have p = ρ as the equation of sate,and hence Eq. 11 will be satisfied. At the moment when t ≈
0, the right-hand-side of theEq. 12 will be huge for the reason that a ( t ) is extremely small. Then every term in Eq. 12and Eq. 13 will be a very large number, and as a result the extremely small number ξ willbe of no importance in determining the evolution of a ( t ). Therefore these two equations willbe simplified to ( ¨ aa ) − ( ˙ aa ) = ( Aa ) , (14)2( ˙ aa )( − ¨ aa ) + ddt ( − ¨ aa ) + 2( ˙ aa ) = 0 . (15)Eq. 14 and Eq. 15 will then describe the evolution of the early Universe.It is interesting to point out that Eq. 14 and Eq. 15 will admit a simultaneous solutionof the form of a ( t ) = a (cosh 2 βt ) , (16)where a and β are two integration constants related to the other integration constant A given in Eq. 11 by A = 4 β a . (17)The a here is the initial value of a ( t ) at t = 0, and when we compare Eq. 11 and Eq. 17,we will see immediately that the initial value of the energy density of the Universe at t = 0is related to β through ρ ( t = 0) = 24 κβ . (18)If Eq. 16 is to describe the evolution of the Universe starting at t = 0, and if β is anextremely large number , say β = × sec − , and if the initial value of a ( t ) at t = 0 (whichis a ) is extremely small, then it is evident that the Universe will undergo an inflationaryphase at the very beginning of its evolution. When the Universe evolves from the age of10 − sec to 10 − sec, its radius will increase by a factorexp[2 β (10 − sec)] / exp[2 β (10 − sec)] = 10 . (19) IV. THE EVOLUTION OF THE LATE-TIME UNIVERSE
However the Universe won’t keep on expanding forever like what is given by Eq. 16because the simplified Eq. 14 and Eq. 15 will no longer describe the Universe when a ( t ) is7o longer extremely small. At the time the Universe is getting large enough, the right-hand-side of the cosmic equations of Eq. 12 will no longer be very large. Instead it gets very smallvery fast as time goes by, and at a point when the importance of ξ cannot be neglected,the simplified equations of Eq. 14 and Eq. 15 will no longer describe the Universe. Whatare describing the Universe now are the cosmic equations of Eq. 12 and Eq. 13 with theirright-hand-sides vanishing. Eq. 12 and Eq. 13 with vanishing right-hand-sides have a trivialsimultaneous solution of ˙ aa − ξ = 0 , (20)because (cid:20) ¨ aa − ξ ( ˙ aa ) (cid:21) = (cid:20) ( ˙ aa )( ˙ aa − ξ ) + ddt ( ˙ aa − ξ ) (cid:21) , (21)for a constant ξ .The solution for Eq. 20 is a = r exp( ξt ) , (22)with r as an integration constant. In this solutuion the ρ and p are all equal to zero andhence satisfy p = ρ . V. CONCLUSIONS
According to the GL (4 R ) Yang-Mills theory of gravity, our Universe would inflate like a ( t ) = a (cosh 2 βt ) if it has a very huge matter density ρ = 24 κβ at the very beginningof time. And then it would slow down and will take a much slower rate of expansion of a ( t ) = r exp( ξt ) in its late time evolution, if a torsion of extremely small local value ξ wascreated at the birth of the Universe. This small primordial torsion, however, has no effecton the evolution of the early Universe. VI. DISCUSSIONS
Even though we have carried out most of our discussions by using geometric languagessuch as the connections and the curvature tensor, these geometric objects are, in fact,derived from the gauge ideas of the gauge potentials and the Yang-Mills field strength tensor.The cosmic metric is driven in accordance with the evolution of an underlying gauge field8onfiguration because Eq. 8 is the Yang-Mills Equation for the GL (4 R ) group while Eq. 7is the algebraic equations for the various components of the Yang-Mills strength tensor.We have shown that the Yang-Mills gauge theory of gravity basing on the GL (4 R ) localgroup admits solutions which can give good descriptions of the early-time and late-timeevolution of the Universe. No cosmological constant and no rolling scalar field is required.Because the solution given in Eq. 16 can be extrapolated back in time, it will take an infinitetime for the Universe to evolve to any finite size if it starts at a state with zero size. Soaccording to the GL (4 R ) Yang-Mills gauge theory of gravity, our Universe has existed inthe infinite past, and takes an infinite time to evolve to our present day status, on thecondition that has started at zero size and that we can ignore all the quantum effects duringits evolution.A solution of the form of a ( t ) = a (cos 2 βt ) is also possible for our cosmic equations.This solution doesn’t seem to give a proper description of our Universe, though. [1] Y. Yang and W. B. Yeung, arXiv:1210.0529 (2012); Y. Yang and W. B. Yeung, arXiv:1205.2690(2012).[2] G. Stephenson, Nuovo Cimento
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