Research Article Nonsingular Einsteinian Cosmology: How Galactic Momentum Prevents Cosmic Singularities

Hindawi Publishing Corporation Journal of Gravity Volume 2013, Article ID 617142, 3 pages http://dx.doi.org/10.1155/2013/617142 Research Article Nonsingular Einsteinian Cosmology: How Galactic Momentum Prevents Cosmic Singularities Kenneth J. Epstein 6400 N. Sheridan No. 2604, Chicago, IL 60626-5331, USA Correspondence should be addressed to Kenneth J. Epstein; [email protected] Received 5 June 2013; Accepted 27 June 2013 Academic Editor: Sergei Odintsov Copyright © 2013 Kenneth J. Epstein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is shown how Einstein’s equation can account for the evolution of the universe without an initial singularity and can explain the inflation epoch as a momentum dominated era in which energy from matter and radiation drove extremely accelerated expansion of space. It is shown how an object with momentum loses energy to the expanding universe and how this energy can contribute to accelerated spatial expansion more effectively than vacuum energy, because virtual particles, the source of vacuum energy, can have negative energy, which can cancel any positive energy from the vacuum. Radiation and matter with momentum have positive but decreasing energy in the expanding universe, and the energy lost by them can contribute to accelerated spatial expansion between galactic clusters, making dark energy a classical effect that can be explained by general relativity without quantum mechanics, and, as (13) and (15) show, without an initial singularity or a big bang. This role of momentum, which was overlooked in the Standard Cosmological Model, is the basis of a simpler model which agrees with what is correct in the old model and corrects what is wrong with it. 1. Introduction angular momentum is conserved because does not depend on ,but is not conserved unless all three momenta The Standard Cosmological Model entails a space-time met- vanish. If =0, the polar angular momentum is con- ric with line element [1–4] served because no longer depends on .If = =0, 2 2 2 2 2 2 2 2 2 2 = − () [ + + sin () ], the radial momentum is conserved because no longer depends on .Defining ≡, then reduces to (1) where isthevacuumspeedoflightrelativetoalocalLor- 2 (, ) =√22 + , (3) entz frame, is cosmic time, () is the time-dependent scale 2 () factor of the Universe, and , , are spherical coordinates of a spatially flat (Euclidean) 3-space. A particle of proper showing how a particle with momentum loses energy to the mass has the Hamiltonian [5, 6] expanding Universe [5]. (,,, , , ) Alogicalquestion,then,iswhere does that energy go? A logicalansweristhatitgoestotheHamiltonianoftheUni- verse, which, in a one-dimensional minisuperspace model, 2 (2) 2 2 can be expressed in geometrized units as the conserved quan- =√22 + + + , 2 22 2sin2 () 2 tity [7] 2 where , ,and are canonical momenta conjugate to H = H (, ) = +U () , (4) canonical coordinates , ,and, respectively. The azimuthal 2 2 Journal of Gravity with canonical momentum , canonical coordinate ,effec- Using an overdot to denote / and substituting =̇ tive mass ,andpotential in (6), give 2/3 2 1 3 9 3Λ ̇2 2 √ 2 −2/3 2 −4/3 U () =( )( ) − =+ Λ − ∑ + , (9) 8 8 2 8 (5) 2/3 indicating that ̇ is a monotonically increasing function of , −3 2 −3 2( ) , and ̈ is positive definite. But this mathematical acceleration does not necessarily imply physical acceleration. Expressing where is the spatial curvature constant (±1or0), is a (9) in terms of the scale factor through the relation = 1/3 2/3 −1/2 3/2 constant such that = , Λ is Einstein’s cosmological gives constant, and are energy densities of matter and 9 2 −1 3 2 −1 radiation, respectively, at some initial time ,and =(). √ 2 2 −2 ̇ = + Λ − ∑ + , The first term of U is eliminated by metric (1), for which =0. 8 8 The second term gives exponentially accelerated expansion (10) of a vacuum dominated Universe, with Λ attributed to dark −1 energy due to vacuum fluctuations, a quantum effect. The where the first term on the right ( ) is a monotoni- 2/3 third term gives ∝,and∝ , the Einstein-de Sitter cally decreasing function of , while the second and third ̈ scale factor of a matter dominated Universe. The fourth term terms are monotonically increasing, so can be positive or 1/2 gives ∝ in a radiation dominated Universe. The negative. Mathematics is simpler in terms of the canonical constraint H =0gives the same differential equation for the coordinate [7], but physics is clearer in terms of the expansion factor () as is obtained from the Standard Cos- expansion factor .Thetimederivativeof(10) gives the accel- mological Model [4, 7]. eration H However, is defective, because the Standard Cosmolog- Λ 4 2 +22−2 4 =̈ + ∑ − . ical Model is defective. only includes the rest mass of 2 2 (11) 3 9 2 2 −2 9 an object whose relativistic mass is Hamiltonian (3)with √ + momentum . This deficiency is corrected here by revising H to the form Defining the conserved quantities 2 2 2 M ≡ ∑ , P ≡ ∑ 3Λ (12) (,,)= + ∑√2 + − , (6) 2 2 8 conserved because the and are constant; Hamiltonian 2 2/3 4/3 where = in terms of the cosmic coordinate , (6) allows a momentum dominated epoch at very early thereby conserving by making / =0.Thesumover times, a matter dominated epoch at intermediate times, and is taken over all objects , whose momenta create an a vacuum dominated epoch at later times. Acceleration (11)is interaction with the Universe [7], so they are no longer treated positive in the momentum era, negative in the matter era, and like test particles. denotes all the ,whichremainconstant positive again in the vacuum era. because is independent of the radial coordinates . 3. Momentum Dominated Epoch 2. Canonical Equations Momentum domination occurs when Λ and the can be Hamiltonian (6) is an example of scientific induction [8], neglected in (10)and(11), giving from which rigorous mathematical deduction, in the form of 9 Hamilton’s equations, gives the coordinate velocities 2̇ = −1 − P−2, 8 (13) −1 = = , 9 −3 −2 (7) =2̈ P − . (14) 2 2 2 4 √ + ̈ vanishes at scale factor 1 =2P/, the first inflection point, / = / =/ ℓ ≡ .Defining as the proper with >0̈ for <1 and deceleration for >1. ̇ vanishes or physical distance of object from a real or hypothetical at scale factor = P/ and is positive definite for > but observer and dropping the subscript ,givetherecession imaginary for <, which is thereby forbidden by (13)as velocity physically impossible. Thus, the momentum factor P rules ℓ out a singularity by requiring the Universe to be created = + , >0 (8) spatially flat with initial scale factor —the ultimate √22 +2 example of inflation—after which no further inflation is needed to achieve flatness or size. But14 ( ) indicates that accel- the Hubble velocity increased by positive and decreased by erated expansion continues for <<1, after which decel- negative . eration sets in. Journal of Gravity 3 Equation (13) can be integrated to give 5. Conclusions −1 ( + 2) √− =2 , (15) The initial singularity of the Standard Model comes from = =0 neglecting the conserved momenta in the relativistic with the initial time being defined so that when . Equation (15) confirms the impossibility of the scale factor mass terms of Hamiltonian (6). When the are included, Einstein’s equation forbids a singularity, thereby disproving () being less than ,sincethatwouldmake imaginary. The momentum factor P, defined in (12), is essential the singularity theorems [1, 3, 10]. This quantum leap in cosmology is achieved within the framework of general for these results and is independent of the signs of the in (6). This momentum symmetry, together with the time relativity, through the classical mechanism of momentum, reversal symmetry of Einstein’s equation, allows a contracting without quantization or any non-Einsteinian effects. It does Universe to undergo a smooth bounce [(0)̇ = 0] at the mini- not improve on Einstein’s theory, but proves that Einstein’s theory is much better than it was thought to be. Other models mum scale factor and then rebound from it as if it had been created at =0. basedonanonsingularbouncefollowedbyexpansionarenot H strictly Einsteinian, because they invoke other mechanisms Unlike Hamiltonian of (4), Hamiltonian of (6) need not vanish. >0is necessary for accelerating expan- [11]inlieuofthe , whereas this galactic momentum is the essential mechanism of nonsingular Einsteinian cosmology. sion (>0̈ ) in the momentum dominated era ( <<1). The recession velocity (8)reducestotheform ̇ −1 References ℓ=̇ + , (16) indicating that || isassumedtobesolargethatthemotion [1] G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum, Relativistic Cosmology, Cambridge University Press, Cambridge, UK, 2012. of the receding object approximates that of a massless particle moving at luminal speed during this era. In this respect, it is [2] S. Weinberg, Cosmology, Oxford University Press, Oxford, UK, 2008. like a radiation era but with repulsive radiation, rather than the attractive radiation of Hamiltonian (4). But the end result [3]R.M.Wald,General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984. is the same, as the Universe expands and matter domination sets in. [4]C.W.Misner,K.S.Thorne,andJ.A.Wheeler,Gravitation, Freeman, New York, NY, USA, 1973. [5] K. J. Epstein, “Hamiltonian geometrodynamics,” Physics Essays, 4. Matter and Vacuum Dominated Epochs vol.5,no.1,pp.133–141,1992. [6] K. J. Epstein, “Hamiltonian approach to frame dragging,” Gen- When the can be neglected for ≥(the starting time of eral Relativity and Gravitation, vol. 40, no. 7, pp. 1367–1378, matter domination), (10)gives 2008. 9 2 −1 3 2 [7] K. J. Epstein, “The quantum connection,” Physics Essays,vol.15, ̇ = (−M) + Λ . (17) 8 8 no. 4, pp. 418–421, 2002.

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