Exact Recession Velocity and Cosmic Redshift Based on Cosmological Principle and Yang-Mills Gravity
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Chinese Physics C Vol. xx, No. x (201x) xxxxxx
Exact Recession Velocity and Cosmic Redshift Based onCosmological Principle and Yang-Mills Gravity
Jong-Ping Hsu and Leonardo Hsu Department of Physics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747-2300, USA College of Education and Human Development, University of Minnesota, Minneapolis, MN 55455, USADepartment of Chemistry and Physics, Santa Rosa Junior College, Santa Rosa, CA 95401, USA
Abstract:
Based on the cosmological principle and quantum Yang-Mills gravity in the super-macroscopic limit,we obtain an exact recession velocity and cosmic redshift z, as measured in an inertial frame F ≡ F ( t, x, y, z ) . For amatter-dominated universe, we have the effective cosmic metric tensor G µν ( t ) = ( B ( t ) , − A ( t ) , − A ( t ) , − A ( t )) , A ∝ B ∝ t / , where t has the operational meaning of time in F frame. We assume a cosmic action S ≡ S cos involving G µν ( t ) and derive the ‘Okubo equation’ of motion, G µν ( t ) ∂ µ S∂ ν S − m = 0, for a distant galaxy with mass m . Thiscosmic equation predicts an exact recession velocity, ˙ r = rH/ [1 / p / r H /C o ] < C o , where H = ˙ A ( t ) /A ( t )and C o = B/A , as observed in the inertial frame F . For small velocities, we have the usual Hubble’s law ˙ r ≈ rH forrecession velocities. Following the formulation of the accelerated Wu-Doppler effect, we investigate cosmic redshiftsz as measured in F . It is natural to assume the massless Okubo equation, G µν ( t ) ∂ µ ψ e ∂ ν ψ e = 0, for light emittedfrom accelerated distant galaxies. Based on the principle of limiting continuation of physical laws, we obtain atransformation for covariant wave 4-vectors between and inertial and an accelerated frame, and predict a relationshipfor the exact recession velocity and cosmic redshift, z = [(1+ V r ) / (1 − V r ) / ] −
1, where V r = ˙ r/C o <
1, as observed inthe inertial frame F . These predictions of the cosmic model are consistent with experiments for small velocities andshould be further tested. Key words:
Yang-Mills gravity; recession velocity; speed limit of recession velocity; redshift.
PACS:
In previous work, Yang-Mills gravity with the princi-ple of gauge symmetry (i.e., the spacetime translational( T ) gauge symmetry) was formulated in general refer-ence frames (both inertial and non-inertial) based on flatspacetime. In the geometric-optics limit, the wave equa-tions of quantum particles in Yang-Mills gravity lead toHamilton-Jacobi type equations of motion with an effec-tive metric tensor G µν ( x ) for classical objects and lightrays. We have shown that classical Yang-Mills grav-ity is consistent with experiments such as the perihe-lion shift of Mercury, deflection of light by the sun, red-shifts and gravitational quadrupole radiation.[1–3] Yang-Mills gravity has also been quantized in inertial framesand gravitational Feynman-Dyson rules and the S-matrixhave been obtained.[2, 4] This theory has brought grav-ity back to the arena of gauge field theory and quantummechanics. It has also provided a solution to difficultiesin physics such as the lack of an operational meaning ofspace and time coordinates[5] in the conventional model of cosmology based on general relativity and the incom-patibility between ‘Einstein’s principle of general coordi-nate invariance and all the modern schemes for quantummechanical description of nature.’[6] This ‘most glaringincompatibility of concepts in contemporary physics’ wasdiscussed in details by Dyson in the 1972 Josiah WillardGibbs Lecture, given under the auspices of the AmericanMathematical Society.For readers not familiar with Yang-Mills gravity, wewill first briefly summarize the key ideas and main equa-tions. Quantum Yang-Mills gravity is based on an ex-ternal spacetime translational group ( T ) and involves(Lorentz) vector gauge functions Λ µ ( x ) in flat space-timewith inertial frames.[1–3] As a result, the difficult prob-lem of the quantization of the gravitational field, as dis-cussed by Dyson, disappears. The T gauge fields aremassless spin-2 symmetric tensor fields φ µν . They areassociated with the T group and the generator p µ = i∂ µ in inertial frames. The T gauge covariant derivatives are Received 16 July 20181) E-mail: [email protected] c (cid:13) obtained through the following replacement,[1, 2] ∂ µ → ∂ µ − igφ νµ p ν = J νµ ∂ ν , J νµ = ( δ νµ + gφ νµ ) , (1)in inertial frames with the metric tensor η µν =(1 , − , − , −
1) and in natural units with c = ~ = 1.As usual, the T gauge curvature C µνα is derived fromthe commutator of the gauge covariant derivative J λµ ∂ λ ,[ J λµ ∂ λ , J σν ∂ σ ] = C µνα ∂ α , (2) C µνα = J µλ ( ∂ λ J να ) − J νλ ( ∂ λ J µα ) , J µλ = J βµ η βλ . (3)The action S φψ of Yang-Mills gravity for the tensorfield φ µν and a charged fermion field ψ in an inertialframe is quadratic in the gauge curvature C µνα , S φψ = Z L φψ d x, L φψ = L φ + L ψ , (4) L φ = 14 g (cid:0) C µνα C µνα − C αµα C µββ (cid:1) , (5) L ψ = + ψiγ µ ( ∂ µ + gφ νµ ∂ ν − ieA µ ) ψ − mψψ. (6)The action S φψ is invariant under local T gauge trans-formations, although the Lagrangian density L φ by itselfis not invariant due to the presence of a total deriva-tive term, which does not contribute to the gravitationalfield equations.[3] The structure of the Yang-Mills actionin (4), (5) and (6) for the spin-2 tensor field φ µν coupledto the charged fermion field ψ are different from those inteleparallel gravity and Einstein-Cartan gravity. Thesetheories of gravity are not formulated in inertial framesand, hence, cannot be quantized to derive Feynman-Dyson rules for the calculation of the S matrix.[3]The wave equation of the gravitational tensor field φ µν can be derived from the Lagrangians (5) and (6), H µν = g S µν , where H µν ≡ ∂ λ ( J λρ C ρµν − J λα C αββ η µν )+ ..... and S µν is the source tensor of the fermion matterfield.[1, 2] It is interesting to observe that its linearizedequation takes the form, ∂ λ ∂ λ φ µν − ∂ µ ∂ λ φ λν − η µν ∂ λ ∂ λ φ + η µν ∂ α ∂ β φ αβ + ∂ µ ∂ ν φ − ∂ ν ∂ λ φ λµ − gS µν = 0 , φ = φ σσ , (7)which is formally the same as the linearized Einsteinequation in general relativity.[3]In quantum Yang-Mills gravity, the symmetric tensorfield φ µν is a massless spin-2 gauge boson. The gravita-tional quadrupole radiation has been discussed with theusual gauge condition ∂ µ φ µν = ∂ ν φ λλ /
2. This gauge con-dition and the linearized equation (7) lead to the usualretarded potential, which can be expressed in terms ofthe polarization tensor e µν . For the symmetric polariza-tion tensor of the massless tensor field in flat space-time,there are only two physical states with helicity ± x / | x | and that radiated by a bodyrotating around one of the principal axes of the ellipsoidof inertia. The results to the second order approximationare the same as that obtained in general relativity andconsistent with experiments.[1, 3]The wave equation of the fermion field ψ in Yang-Mills gravity can be derived from the Lagrangian L ψ . Inthe geometric-optics limit,[1, 2] the fermion wave equa-tion reduces to a Hamilton-Jacobi type equation, G µν ( ∂ µ S )( ∂ ν S ) − m = 0 , G µν = η αβ J αµ J βν . (8)This equation of motion for macroscopic objects in flatspace-time involves a new effective Riemannian metrictensor G µν , which is actually a function of the T gaugefield φ µν in Yang-Mills gravity. It is formally the sameas the corresponding equation of motion for macroscopicobjects in general relativity and we have named it the‘Einstein-Grossmann’ equation of motion.[3] This equa-tion is crucial for Yang-Mills gravity to be consistent withthe perihelion shift of the Mercury, the deflection of lightby the sun and the equivalence principle.[1, 3]A satisfactory theory of gravity should be able to ex-plain why the gravitational force is attractive rather thanrepulsive. It is gratifying that, in quantum Yang-Millsgravity, this property is embedded in the coupling be-tween the gravitational tensor field φ µν and the matterfermion field ψ at the quantum level in the Lagrangian(6). Let us consider the gravitational ( T ) tensor field φ µν ( x ) and the electromagnetic potential field A µ ( x ) inthe gauge covariant derivative and its complex conjugatein the fermion Lagrangian (6), ∂ µ − igφ νµ p ν − ieA µ + .... = ∂ µ + gφ νµ ∂ ν − ieA µ + .... (9)( ∂ µ − igφ νµ p ν − ieA µ + .... ) ∗ = ∂ µ + gφ νµ ∂ ν + ieA µ + .... (10)The gauge covariant derivative (9) and its complex conju-gate (10) appear respectively in the wave equations of theelectron (i.e., particle with charge e <
0) and the positron(i.e., antiparticle with charge − e ). The electric force be-tween two charged particles is due to the exchange of avirtual photon. In quantum electrodynamics, this canbe pictured in the Feynman diagrams with two verticesconnected by a photon propagator. The key propertiesof the electric force F e ( e − , e − ) (i.e., between electron andelectron) and the force F e ( e − , e + ) (i.e., between electronand positron) are given by the third terms in (9) and in(10), i.e., F e ( e − , e − ) : ( − ie ) × ( − ie ) = − e , repulsive,F e ( e − , e + ) : ( − ie ) × ( ie ) = + e , attractive, and the force F e ( e + , e + ) is the same as F e ( e − , e − ). Thus,we have experimentally established attractive and repul-sive electric forces, which are due to the presence of i in the electromagnetic coupling. The Yang-Mills grav-itational force F Y Mg ( e − , e − ) (i.e., between electron andelectron) and the force F Y Mg ( e − , e + ) (i.e., between elec-tron and positron) are respectively given by the secondterms in (9) and in (10). Because the gravitation cou-pling terms in (9) and (10) do not involve i , we have onlyan attractive gravitational force, F Y Mg ( e − , e − ) : ( g ) × ( g ) = + g , attractive,F Y Mg ( e − , e + ) : ( g ) × ( g ) = + g , attractive, and F Y Mg ( e + , e + ) is the same as F Y Mg ( e − , e − ). Notethat these qualitative results for forces F e ( e − , e − ) and F Y Mg ( e − , e − ) are independent of the signs of the cou-pling constants e and g . Furthermore, the gravitationalcoupling constant g in (9) and (10) has the dimensionof length (in natural units), in contrast to all other cou-pling constants of fields associated with internal gaugegroups, so that g is related to Newtonian constant G by g = 8 πG .[3, 7] These important qualitative resultsrevealed through the coupling of the tensor field φ µν andthe fermion matter field ψ in the Lagrangian (4) appearto indicate that the space-time translation gauge groupof Yang-Mills gravity is just right for gravity.It is intriguing that the T gauge transformations[1, 2]with infinitesimal vector gauge function Λ µ ( x ) in quan-tum Yang-Mills gravity turn out to be identical to theLie derivatives in the coordinate expression. This in-dicates an intimate relation between the mathematicaltheory of Lie derivatives in coordinate expression andthe physical gauge field theory with the external space-time translation T gauge symmetry. The vanishing ofthe Lie derivative of the action S φψ turns out to be thesame as the invariance of the action in Yang-Mills gravityunder the T gauge transformations. ∗ In the usual macroscopic (i.e., geometric-optics)limit, the wave equations of quantum particles withmass m in Yang-Mills gravity reduces to the Einstein-Grossmann (EG) equation, G µν ( x )( ∂ µ S )( ∂ ν S ) − m = 0.Thus, the apparent curvature of macroscopic spacetimeappears to be a manifestation of the flat spacetime trans-lational gauge symmetry for the wave equations of quan-tum particles in the geometric-optics limit.[2, 7] Accord-ing to quantum Yang-Mills gravity, macroscopic objectsmove as if they were in a curved spacetime because theirequation of motion involves the ‘effective metric tensor’ G µν ( x ), which is actually a function of T tensor gaugefields in flat space-time.The conventional FLRW model of cosmology[8, 9]was based on general relativity to discuss expansion dy-namics of the universe. In quantum Yang-Mills grav- ity, the emergence of the effective metric tensor for themotion of macroscopic objects suggests that we can useYang-Mills gravitational field equations to discuss an al-ternate dynamics for the expanding universe (i.e., a newHHK model of particle cosmology[10]). In the super-macroscopic limit with the cosmologi-cal principle of homogeneity and isotropy, the effectivespacetime-dependent metric tensor G µν ( x ) in the EGequation in Yang-Mills gravity further simplifies to atime-dependent effective metric tensor G µν ( t ) with thediagonal form in an inertial frame F ≡ F ( t, x, y, z ) , c = ~ = 1. The T gauge field equation and the cosmologicalprinciple lead to the solution,[10] G µν ( t ) = ( B ( t ) , − A ( t ) , − A ( t ) , − A ( t )) , (11) B = βt / , A = αt / , c = ~ = 1 , for matter dominated cosmos, where β = 3 α / g ρ o and α = (8 g ωρ o / / . The discussions in the cosmic HHKmodel of particle cosmology are based on Yang-Millsgravity within the framework of flat spacetime and theinertial frame F ( t, x, y, z ). One advantage is that cos-mological predictions of the HHK model are based onwell-defined inertial frames of reference, in which spaceand time coordinates have the usual operational mean-ing.The basic equation of motion of a distant galaxy withmass m can be derived from the principle of least ac-tion involving the effective cosmic metric tensor G µν ( t )in (11) and the cosmic action R ( − mds ) , where ds = G µν ( t ) dx µ dx ν . To derive the cosmic equation of motion,we consider the covariant space-time variation δx ν witha fixed initial point and a variable end point and theactual trajectory.[11] We obtain δS cos = δ Z (cid:18) − m q G µν ( t ) dx µ dx ν (cid:19) = m Z T ν δx ν ds + (cid:20) mG µν ( t ) dx µ ds (cid:21) δx ν = p ν δx ν . (12) T ν = (cid:20) dx µ ds dx λ ds ∂G µλ ∂x ν − dds (cid:18) G µν dx µ ds (cid:19)(cid:21) = 0 ,p ν = mG µν ( t ) dx µ ds , where the actual trajectory satisfies the equation T ν =0, and the non-vanishing term in (12) is contributed ∗ To prove the T gauge invariance of the action of Yang-Mills gravity, Cartan’s formula in the theory of Lie derivative facilitatesthe calculation of the change of the volume element (e.g., W ( x ) d x ) under the T gauge transformations. See appendix A in the forth-coming second edition (entitled “Space-Time, Yang-Mills Gravity and Dynamics of Cosmic Expansion” , World Scientific, 2019,) of themonograph in ref. 3. from the variable endpoint. Based on (12), we de-fine the generalized four-momentum of an object mov-ing in the super-macroscopic world as the derivative ∂S cos /∂x ν = p ν .[11] Consequently, we obtain the equa-tion G µν ( t ) p µ p ν = m , where G µν ( t ) G νλ ( t ) = δ µλ and p µ is given by the last equation in (12). As usual, substi-tuting ∂S cos /∂x µ for p µ in G µν ( t ) p µ p ν = m , we find theOkubo equation for the motion of a distant galaxy withmass m : † G µν ( t ) ∂ µ S∂ ν S − m = 0 , S ≡ S cos . (13)Thus, we have seen that this cosmic Okubo equation isthe generalization of the Einstein-Grossmann equation G µν ( x )( ∂ µ S )( ∂ ν S ) − m = 0 in Yang-Mills gravity frommacroscopic world to the super-macroscopic world forthe motion of distant galaxies and for the expansion ofthe universe. Let us explore physical and cosmological implicationsof the effective metric tensor G µν ( t ) and the Okubo equa-tion (13) in the super-macroscopic world ( roughly ≥
300 million lightyears[8]). Using spherical coordinates,we choose a specific radial direction with specific angles θ and φ to express (13) in the form, B − ( ∂ t S ) − A − ( ∂ r S ) − m = 0 . Since r is cyclic in cosmic Okubo equation (13) because G µν ( t ) does not involve r ,[11] we have the conserved‘generalized’ momentum p = ∂ r S = ∂S/∂r . As usual,to solve eq. (13), we look for an S in the form[3, 11] S = − f ( t ) + pr . We have f ( t ) = Z p p C o + m B dt, C o = B/A = const. (14)The trajectory of a distant galaxy is determined by theequation ∂S/∂p = constant. [3, 11] Therefore, we have ‡ r − Z pC o p p C o + m B dt = constant. (15)Since B = βt / , equation (15) leads to˙ r = drdt = pC o p p C o + m β t = C o √ t , Ω = mβpC o , (16) r = 2 pC o m β p p C o + m β t = 2 C o Ω √ t. (17) In the low velocity approximation, i.e., m >> p , we have f ( t ) ≈ (2 mβ/ t / , r ≈ t / (2 pC o /mβ ) . Thus, we obtainthe usual Hubble laws (linear in r ) as the low velocity ap-proximation of the solution to the Okubo equation (13),˙ r = pC o mβt / = H ( t ) r, H ( t ) = ˙ AA . (18)Suppose we do not take the low velocity approxima-tion. We have to solve for Ω in terms of r and t by using(17), Ω = 2 p t + r /C o − t , (19)and use this result (19) in (16). Thus, we derive theexact recession velocity ˙ r in terms of rH ( t ),˙ r = rH / p / r H /C o < C o , (20) C o = BA = constant. The upper limit C o of recession velocity in (20) can beseen as follows: When the velocity rH is very large, ˙ r approaches C o , as shown in (20). We note that such alimiting velocity C o is the recession velocity at time t = 0,as one can see in (16) with t = 0. Thus, the HHK modelpredicts that the exact recession velocity is given by (20)with the upper limit, C o = βα = (3 ω ) / , c = ~ = 1 , (21)where ω = P/ρ < / P to energydensity ρ of macroscopic bodies.[8, 10] There is little ex-perimental data for the parameter ω . Presumably, ω ismuch smaller than 1/3. For example, suppose (3 ω ) is10 − , the limiting speed C o is 0 . × m/s ).It is natural to interpret (20) as the exact ‘non-relative’ recession velocity as measured in an inertialframe, according to quantum Yang-Mills gravity: The recession velocity of a distant galaxy, as mea-sured in an inertial frame, is dictated by the cosmicOkubo equation (13) with the effective metric tensor (11).It is exactly given by (20) at a given time and has an up-per limit C o in (21) for matter dominated cosmos. Wemay interpret C o as an ‘effective speed of light’ becauseit is associated with (11) through the vanishing effectivemetric, ds = G µν ( t ) dx µ dx ν = 0 . † This equation of motion (13) for a distant galaxy with mass m derived from the quantum Yang-Mills gravity, together with the cos-mological principle, was called the ‘cosmic Okubo equation’ of motion, in memory of his endeavor in particle physics and his ‘departure... to the black hole.’ ‡ We neglect the constant associated with r in (15) because we are dealing with extremely large distances r in the super macroscopicworld and, so far, we have no data to determine its value. To see the next correction term for ˙ r in (18), we ex-pand the square-root in (20) and obtain the approximaterecession velocity,˙ r ≈ rH ( t ) (cid:20) − r H ( t ) C o (cid:21) . (22)The results (16)-(22) are for the matter dominated cos-mos in the HHK model.For radiation dominated (rd) cosmos, we have differ-ent scale factors A rd ( t ) and B rd ( t ),[10] A rd ( t ) = α ′ t / , α ′ = (cid:18) g ωρ o × (cid:19) / , (23) B rd ( t ) = β ′ t / , β ′ = 24 α ′ g ρ o , g = 8 πG, (24)where G is the Newtonian gravitational constant. Asusual, we follow previous steps of calculations with S = − f ( t ) + pr , the recession velocity ˙ r rd of the radiationdominated (rd) cosmos is˙ r rd = pC ′ o p p C ′ o + m β ′ t / < C ′ o , C ′ o = B rd /A rd ; (25)˙ r rd ≈ Hr (cid:20) − H r C ′ o (cid:21) , H = ˙ A rd A rd = 25 t , m >> p. Based on (23)-(25) for the radiation dominated cosmos,the HHK model with Yang-Mills gravity predicts the lim-iting recession speed C ′ o to be C ′ o = β ′ /α ′ = (6 ω ) / , (26)which is the maximum Hubble recession speed at time t = 0 for the radiation dominated cosmos.The limiting recession speed (26) for distant galax-ies appears to be in harmony with the 4-dimensionalspace-time framework of quantum Yang-Mills gravity.Both limiting speeds (21) and (26) are ‘effective speedsof light’ given by the vanishing effective line element ds = G µν ( t ) dx µ dx ν = 0, where the scale factors are givenby (11) for matter dominated universe and by (23)-(24)for radiation dominated universe. For cosmic redshifts of lights emitted from distantgalaxies with non-constant recession velocities (20) or(22) can be treated similar to the Doppler effect or, tobe more specific, accelerated Wu-Doppler effects.[3, 12]The exact Doppler shift can be obtained by the transfor-mation of two wave 4-vectors k ′ µ (emitted from a sourceat rest in a moving inertial frame F ′ ) and k µ (observed in an inertial frame F ), which are related by the in-variant law η µν k ′ µ k ′ ν = η µν k µ k ν . Similarly, the wave 4-vector k eµ emitted from a distant galaxy is associatedwith an eikonal equation or the massless Okubo equa-tion G µν ( t ) ∂ µ ψ e ∂ ν ψ e = 0 with ∂ µ ψ e = k eµ . This isconsistent with the (massive) Okubo equation (13) re-lated to distant galaxies. It is also the generalizationof the usual eikonal equation in the macroscopic worldto the super-macroscopic world with the effective time-dependent metric tensor G µν ( t ) in Yang-Mills gravity.The wave 4-vector of light as measured by observersin the inertial frame F satisfies the usual eikonal equa-tion, η µν ∂ µ ψ∂ ν ψ = 0 with ∂ µ ψ = k µ .[11] The distantgalaxy moves with (negative) acceleration and the ob-server is in an inertial frame F . Based on the principle oflimiting continuation of physical laws, the laws of physicsin a reference frame F with an acceleration a must re-duce to those in a reference F with an acceleration a in the limit where a approaches a .[3, 12] In the spe-cial case a = 0, this principle of limiting continuation ofphysical laws reduces to the principle of relativity in thezero acculeration limit. Therefore, it is natural to treatthe observed redshift in analogy to the accelerated Wu-Doppler effect with the ‘covariant’ eikonal equation[3, 12] G µν ( t ) ∂ µ ψ e ∂ ν ψ e = η µν ∂ µ ψ∂ ν ψ, (27) ∂ µ ψ e = k eµ , ∂ µ ψ = k µ . For simplicity, we choose a specific radial direction in aspherical coordinate with k eµ = ( k e , k e , ,
0) and k µ =( k , k, , B − ( t ) k e − A − ( t ) k e = k − k = 0 , (28)based on the principle of limiting continuation of physi-cal laws. The recession velocity of a distant light sourceis the velocity V r and, hence, the frequency k observedin F decreases. The relation (28) leads to the transfor-mations for these covariant wave 4-vectors: k e B = Γ [ k + V r k ] , k e A = Γ [ k + V r k ] , (29)Γ = 1 p − V r . One can verify that the covariant law (28) is preservedby the transformations (29) with a velocity function V r .Since a distant galaxy moves with a non-relative veloc-ity (20), by dimensional analysis, it is natural to identify V r < V r = ˙ r/C o < k = + k for the recession of theradiation source in the radial direction.For convenience of explanation, let us introduce an‘auxiliary expansion frame’ F e associated with the effec-tive metric tensor G µν ( t ) and all distant galaxies are, by definition, at rest[11] in F e . Thus, the frequency shift fora source at rest in an expansion galaxy is given by (cid:20) k e B (cid:21) at rest in F e = k (1 + V r ) p − V r , V r = ˙ r ( t ) C o , (30)where k = [ k ] observed in F and V r = ˙ r ( t ) /C o is the non-relative recession velocity of the distant galaxy at themoment when the light was emitted. The frequency shift(30) is formally the same as the accelerated Wu-Dopplershift involving constant-linear-acceleration source.[3]We stress that the expansion frame F e is only an‘auxiliary frame’, which does not have well-defined spaceand time coordinates over all space. Therefore, all ex-periments and observations must be carried out in theinertial frame F = F ( t, x, y, z ) with operationally definedspace and time coordinates. For experimental test ofredshift based on (30), we cannot have data of k e mea-sured in F e and, furthermore, the non-inertial auxiliaryframe F e and the inertial frame F are not equivalentbecause F e with the metric tensor G µν ( t ) is not an in-ertial frame in flat space-time. Therefore, we must ex-press [ k e /B ] at rest in F e in (30) in terms of the same kindof radiation source at rest and observed in the inertialframe F . Here, the situation is again similar to the ac-celerated Wu-Doppler effects, in which the acceleratedfrequency shift of a radiation source is at rest in an ac-celerated frame and observed in an inertial frame. Thus,we treat F e as a non-inertial frame and assume weakequivalence of non-inertial frames on the basis of theprinciple of limiting continuation for physical law.[3, 13]Such a weak equivalence has been supported by Davise-Jennison’s two-laser experiments, which involve orbitinglaser sources and are observed in inertial laboratory. Ac-cording to weak equivalence, we have[13] (cid:20) k e √ G (cid:21) at rest in F e = (cid:20) k √ η (cid:21) at rest in F , (31) p G = B, where [ ] at rest in F should be understood as that “thesource is at rest in F and its emitted frequencies aremeasured in the F frame.” Therefore, (30) and (31)with η = 1 lead to the following redshift of frequency k = ω as measured in the inertial frame F ,[ k ] at rest in F ≡ k ( emission )= k ( observed ) (1 + V r ) p − V r . (32)The cosmic redshift z is defined by[8] k ( emission ) k ( observed ) = 1 + z. (33) It follows from (32) and (33) that the exact law of theredshift z is related to the recession velocity V r of a dis-tant galaxy as follows: z = 1 + V r p − V r − , V r = ˙ r/C o < . (34)This is the prediction of Yang-Mills gravity for redshiftof light emitted by a distant galaxy in terms of its reces-sion velocity. Note that V r = ˙ r ( t ) /C o is the non-constantrecession velocity of the distant galaxy at the momentwhen the light was emitted.Only for a small recession velocity V r <<
1, we havethe usual approximate relation[8] with a second-ordercorrection term, z ≈ V r + 12 V r . (35)For matter dominated cosmos, the result (34) enables usto express the recession velocity ˙ r/C o = V r of a distantgalaxy in terms of its directly measurable redshift z , V r = ˙ rC o = (1 + z ) − z ) + 1 = 2 z + z z + z < , (36)as observed in the inertial frame F . One can also verifythat z = 0 and z → ∞ correspond to V r = 0 and V r → V r = ˙ r/C o is given by 0.5 and 0.8, onehas z=0.8 and 2 respectively. Thus, according to the present HHK model[10] of par-ticle cosmology based on Yang-Mills gravity, the Hubblerecession velocity V r of a distant galaxy, as measured inan inertial frame, can only take the values between 0 and C o . However, the cosmic redshift z can take the valuesbetween 0 and infinity in the matter dominated cosmos.The Okubo equation of motion for a distant galaxyleads to exact solutions for r ( t ) and ˙ r ( t ) in (17) and (16)for the matter dominated universe with the effective met-ric tensor in (11). These solutions sketch a total cosmichistory from the beginning to the end of the universe:(i) In the beginning t = 0, we have the following features:initial mass run away velocity ˙ r = C o ,initial non-vanishing radius r = 2 p C o / ( m β ) ≡ r o ,initial Hubble recession velocity V r = ˙ r/C o = 1 , initial cosmic frequency red-shift given by (32), z = k o ( emission ) k o ( observed ) − ∞ , f or V r = 1 . (ii) At the end t → ∞ , we have the following features:final velocity of galaxies ˙ r → , final radius r → ∞ , final Hubble recession velocity V r → z → , where we have used (11), (16), (17) and (34). Theseproperties will be modified when the quantum nature ofYang-Mills gravity and other new long-range and short-range forces § in particle physics are taken into account.Of course, a more realistic model will not be completelydominated by matter. It may be dominated by a com-bination of matter, radiation and some sort of effective‘vacuum energy.’ One can imagine that the universe hasbeen doing extremely complicated multi-tasks during itsevolution. ¶ Interestingly, it appears that the Okubo equationwith m > k because CPT invariance implies the maximum symme-try between particles and antiparticles regarding theirmasses, lifetimes and interactions.[10, 14]It may be interesting to observe that we assume somelocal properties of the spacetime in (11) and (13) toderive some properties about the behavior of point-likegalaxies at super-macroscopic distances. This result maynot be surprising because the ‘local’ effective metric ten-sor (11) embodies the super-macroscopic properties ofhomogeneity and isotropy. Thus, such a treatment ofthe physical system of distant galaxies in the super-macroscopic world seems to resemble ‘Riemann geom-etry in the large.’[15] In this sense, Riemann geometryin the large may play a role as the mathematical basefor the expanding cosmos in the HHK model of particlecosmology with quantum Yang-Mills gravity.In the conventional FLRW model with generalrelativity, one has the Hamilton-Jacobi equation g µν ( t ) ∂ µ S∂ ν S − m = 0 with the metric tensor g µν ( t ) =(1 , − a − ( t ) , − a − ( t ) , − a − ( t )) , where the scale factoris given by a ( t ) = a o t / for the matter dominateduniverse.[8] Following similar calculations from (13) to(18), one obtains the recession velocity ˙ r = ( ˙ a/a )( r/
2) forlow velocity approximation ( m >> p ). However, there isno constant upper limit for the recession velocity ˙ r atlarge momenta ( p >> m ). Since these results are notobtained in an inertial frame, it is difficult to have a satisfactory comparison between these results and thoseobtained by Yang-Mills gravity in an inertial frame.In summary, within Yang-Mills gravity, the effectivecosmic metric tensor G µν ( t ) = ( B − , − A − , − A − , − A − )appears to play a more basic and useful role thanthat of g µν ( t ) = (1 , − a − , − a − , − a − ) in the conven-tional theory. The reason is that the Okubo equations G µν ( t ) ∂ µ S∂ ν S − m = 0, with m > , and m = 0 can de-scribe completely recession velocities, 0 ≤ ˙ r/C o <
1, andcosmic redshift z for 0 ≤ z < ∞ without making low ve-locity approximations. Furthermore, Yang-Mills gravitysuggests new views of the universe: (A) the linear Hub-ble law is the low-velocity approximate solution to theOkubo equation for distant galaxies, and (B) the reces-sion velocity has an upper limit, whose numerical valuedepends on the equation of state, P = ρω . Thus, cosmicOkubo equations (13) with m ≥ References (15) 403 (2012).3 L. Hsu and J. P. Hsu, Space-Time Symmetry and QuantumYang-Mills Gravity (World Scientific, 2013), Part I, pp.23-30,pp. 38-42, pp.109-123 and pp. 134-141, pp. 153-161.4 S. H. Kim, Ph.D. Thesis, Univ. of Mass. Dartmouth (2012).5 E. P. Wigner,
Symmetries and Reflections, Scientific Essays ,(The MIT Press, 1967), pp. 52-53.6 F. Dyson, in
100 Years of Gravity and Accelerated Frames,The Deepest Insight of Einstein and Yang-Mills , eds. Jong-Ping Hsu and Dana Fine, (World Scientific, 2005) pp. 347-351.7 S. H. Kim and J. P. Hsu, Eur. Phys. J. Plus, , 146 (2012).8 S. Weinberg,
Cosmology (Oxford Univ. Press, 2008), pp. 10-11,p. 40, p. 62.9 D. Katz, Int. J. Mod. Phys. A , 1550119 (2015).10 J. P. Hsu, L. Hsu and D. Katz, Mod. Phys. Letters A, ,1850116 (2018).11 L. Landau and E. Lifshitz, The Classical Theory of Fields ,Addison-Wesley, 1951, pp. 268-271, pp. 312-313, pp. 336-344and p. 350.12 L. Hsu and J. P. Hsu, Nuovo Cimento, , 1147 (1997).Appendix.13 J. P. Hsu and L. Hsu, Eur. Phys. J. Plus, , 74 (2013). Ap-pendix A.14 T. D. Lee,
Particle Physics and Introduction to Field Theory (New York, Hardwood Academic, 1981), pp. 320-333,15 W. H. Huang,