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 𝑝𝜑 is conserved because 𝐻 does not depend on 𝜑,but𝐻 is not conserved unless all three momenta The Standard Cosmological Model entails a space-time met- vanish. If 𝑝𝜑 =0, the polar angular momentum 𝑝𝜃 is con- ric with line element [1–4] served because 𝐻 no longer depends on 𝜃.If𝑝𝜃 =𝑝𝜑 =0, 2 2 2 2 2 2 2 2 2 2 𝑝 𝐻 𝑑𝑠 =𝑐𝑑𝑡 −𝑎 (𝑡) [𝑑𝑟 +𝑟 𝑑𝜃 +𝑟 sin (𝜃) 𝑑𝜑 ], the radial momentum 𝑟 is conserved because no longer depends on 𝑟.Defining𝑝𝑟 ≡𝑝, 𝐻 then reduces to (1) where 𝑐 isthevacuumspeedoflightrelativetoalocalLor- 𝑝2 𝐻 (𝑝, 𝑡)√ =𝑐 𝑚2𝑐2 + , (3) entz frame, 𝑡 is cosmic time, 𝑎(𝑡) is the time-dependent scale 𝑎2 (𝑡) factor of the Universe, and 𝑟, 𝜃, 𝜑 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 𝑚 has the Hamiltonian [5, 6] expanding Universe [5]. 𝐻(𝑟,𝜃,𝑡,𝑝 ,𝑝 ,𝑝 ) Alogicalquestion,then,iswhere does that energy go? A 𝑟 𝜃 𝜑 logicalansweristhatitgoestotheHamiltonianoftheUni- verse, which, in a one-dimensional minisuperspace model, 2 (2) 𝑝2 𝑝2 𝑝 can be expressed in geometrized units as the conserved quan- =𝑐√𝑚2𝑐2 + 𝑟 + 𝜃 + 𝜑 , 𝑎2 𝑟2𝑎2 𝑟2sin2 (𝜃) 𝑎2 tity [7] 2 𝑝 𝑝 𝑝 𝑃 where 𝑟, 𝜃,and 𝜑 are canonical momenta conjugate to H = H (𝑃, 𝑥) = +𝑀U (𝑥) , (4) canonical coordinates 𝑟, 𝜃,and𝜑, respectively. The azimuthal 2𝑀 2 Journal of Gravity with canonical momentum 𝑃, canonical coordinate 𝑥,effec- Using an overdot to denote 𝑑/𝑑𝑡 and substituting 𝑃=𝑀𝑥̇ tive mass 𝑀,andpotential in (6), give

2/3 2 1 3 9𝑘 𝑥 3Λ𝑥 ̇2 2 √ 2 −2/3 2 −4/3 U (𝑥) =( )( ) − 𝑀𝑥 =𝐾+ 𝑀Λ𝑥 − ∑ 𝑚𝑛 +𝛼 𝑝𝑛𝑥 , (9) 8 𝛼 8 2 8 𝑛 (5) 𝑥 2/3 indicating that 𝑥̇ is a monotonically increasing function of 𝑥, −3𝜋𝜌 𝑥2 −3𝜋𝜌 𝑥2( 𝑜 ) , 𝑚𝑜 𝑜 𝑟𝑜 𝑜 𝑥 and 𝑥̈ is positive definite. But this mathematical acceleration does not necessarily imply physical acceleration. Expressing where 𝑘 is the spatial curvature constant (±1or0),𝛼 is a (9) in terms of the scale factor 𝑎 through the relation 𝑥= 1/3 2/3 −1/2 3/2 constant such that 𝑎=𝛼 𝑥 , Λ is Einstein’s cosmological 𝛼 𝑎 gives constant, 𝜌𝑚𝑜 and 𝜌𝑟𝑜 are energy densities of matter and 9 2 −1 3 2 −1 radiation, respectively, at some initial time 𝑡𝑜,and𝑥𝑜 =𝑥(𝑡𝑜). √ 2 2 −2 𝑀𝑎̇ = 𝐾𝛼𝑎 + 𝑀Λ𝑎 −𝛼𝑎 ∑ 𝑚𝑛 +𝑝𝑛𝑎 , The first term of U is eliminated by metric (1), for which 𝑘=0. 8 8 𝑛 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 (𝐾𝛼𝑎 ) is a monotoni- 2/3 𝑎 third term gives 𝑥∝𝑡,and𝑎∝𝑡 , the Einstein-de Sitter cally decreasing function of , while the second and third 𝑎̈ scale factor of a matter dominated Universe. The fourth term terms are monotonically increasing, so can be positive or 1/2 gives 𝑎∝𝑡 in a radiation dominated Universe. The negative. Mathematics is simpler in terms of the canonical 𝑥 constraint H =0gives the same differential equation for the coordinate [7], but physics is clearer in terms of the 𝑎 expansion factor 𝑎(𝑡) 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- Λ𝑎 4𝛼 𝑚2 +2𝑝2𝑎−2 4𝐾𝛼 𝜌 𝑚 𝑎=̈ + ∑ 𝑛 𝑛 − . ical Model is defective. 𝑚𝑜 only includes the rest mass of 2 2 (11) 3 9𝑀𝑎 𝑛 2 2 −2 9𝑀𝑎 an object whose relativistic mass is Hamiltonian (3)with √𝑚𝑛 +𝑝𝑛𝑎 momentum 𝑝. This deficiency is corrected here by revising H to the form Defining the conserved quantities 󵄨 󵄨 2 2 2 M ≡ ∑𝑚 , P ≡ ∑ 󵄨𝑝 󵄨 𝑃 𝑝𝑛 3𝑀Λ𝑥 𝑛 󵄨 𝑛󵄨 (12) 𝐾(𝑃,𝑥,𝑝)= + ∑√𝑚2 + − , (6) 𝑛 𝑛 𝑛 2 2𝑀 𝑛 𝑎 8 conserved because the 𝑚𝑛 and 𝑝𝑛 are constant; Hamiltonian 2 2/3 4/3 where 𝑎 =𝛼 𝑥 in terms of the cosmic coordinate 𝑥, (6) allows a momentum dominated epoch at very early thereby conserving 𝐾 by making 𝜕𝐾/𝜕𝑡 =0.Thesumover times, a matter dominated epoch at intermediate times, and 𝑛 is taken over all objects 𝑚𝑛, whose momenta 𝑝𝑛 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. 𝑝 denotes all the 𝑝𝑛,whichremainconstant positive again in the vacuum era. because 𝐾 is independent of the radial coordinates 𝑟𝑛. 3. Momentum Dominated Epoch 2. Canonical Equations Momentum domination occurs when Λ and the 𝑚𝑛 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 𝑀𝑎2̇ = 𝐾𝛼𝑎−1 − P𝛼𝑎−2, 8 (13) 𝑑𝑟 𝜕𝐾 𝑎−1𝑝 𝑛 = = 𝑛 , 9 −3 −2 (7) 𝑀𝑎=2̈ P𝛼𝑎 − 𝐾𝛼𝑎 . (14) 𝑑𝑡 𝜕𝑝𝑛 2 2 2 4 √𝑎 𝑚𝑛 +𝑝𝑛

𝑎̈ vanishes at scale factor 𝑎1 =2P/𝐾, the first inflection point, 𝑑𝑥/𝑑𝑡 = 𝜕𝐾/𝜕𝑃 =𝑃/𝑀 ℓ ≡𝑎𝑟 .Defining 𝑛 𝑛 as the proper with 𝑎>0̈ for 𝑎<𝑎1 and deceleration for 𝑎>𝑎1. 𝑎̇ vanishes 𝑛 or physical distance of object from a real or hypothetical at scale factor 𝑎𝑜 = P/𝐾 and is positive definite for 𝑎>𝑎𝑜 but 𝑛 observer and dropping the subscript ,givetherecession imaginary for 𝑎<𝑎𝑜, which is thereby forbidden by (13)as velocity physically impossible. Thus, the momentum factor P rules 𝑑ℓ 𝑑𝑎 𝑝 out a singularity by requiring the Universe to be created =𝑟 + , 𝑎 >0 𝑑𝑡 𝑑𝑡 (8) spatially flat with initial scale factor 𝑜 —the ultimate √𝑎2𝑚2 +𝑝2 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 𝑝 and decreased by erated expansion continues for 𝑎𝑜 <𝑎<𝑎1, after which decel- negative 𝑝. eration sets in. Journal of Gravity 3

Equation (13) can be integrated to give 5. Conclusions −1 (𝑎 + 2𝑎𝑜) √𝑎−𝑎𝑜 =2𝛼𝐾𝑀 𝑡, (15) The initial singularity of the Standard Model comes from 𝑎=𝑎 𝑡=0 neglecting the conserved momenta 𝑝𝑛 in the relativistic with the initial time being defined so that 𝑜 when . 𝑝 Equation (15) confirms the impossibility of the scale factor mass terms of Hamiltonian (6). When the 𝑛 are included, Einstein’s equation forbids a singularity, thereby disproving 𝑎(𝑡) being less than 𝑎𝑜,sincethatwouldmake𝑡 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 𝑝𝑛 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 [𝑎(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 𝑎𝑜 and then rebound from it as if it had been created at 𝑡=0. basedonanonsingularbouncefollowedbyexpansionarenot H 𝐾 strictly Einsteinian, because they invoke other mechanisms Unlike Hamiltonian of (4), Hamiltonian of (6) 𝑝 need not vanish. 𝐾>0is necessary for accelerating expan- [11]inlieuofthe 𝑛, whereas this galactic momentum is the essential mechanism of nonsingular Einsteinian cosmology. sion (𝑎>0̈ ) in the momentum dominated era (𝑎𝑜 <𝑎<𝑎1). The recession velocity (8)reducestotheform ̇ 󵄨 󵄨−1 References ℓ=𝑎𝑟̇ 󵄨 +𝑝 𝑝󵄨 , (16) indicating that |𝑝| 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 𝑝𝑛 can be neglected for 𝑡≥𝜏(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, 𝑀𝑎̇ = (𝐾−M) 𝛼𝑎 + 𝑀Λ𝑎 . (17) 8 8 no. 4, pp. 418–421, 2002. For 𝑡≥𝜏,(17) is then readily integrated to give [8] D. Harriman, TheLogicalLeap:InductioninPhysics,New American Library, New York, NY, USA, 2010. 1/3 (𝐾−M) 𝛼 2/3 √3Λ [9] L. Boi, The Quantum Vacuum, Johns Hopkins University Press, 𝑎 (𝑡) =2[ ] ( 𝑡) . (18) 3𝑀Λ sinh 2 Baltimore, Md, USA, 2011. 𝑇 [10] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Defining as the transition time from matter domination Space-Time, Cambridge University Press, Cambridge, UK, 1973. to vacuum domination, it follows that, for 𝜏≤𝑡≤𝑇and (√3Λ/2)𝑇 ≪ 1 [11] S. E. Perez Bergliaffa, “Nonsingular cosmological models,” http:// ,(18) gives the Einstein-de Sitter scale factor arxiv.org/abs/1105.5424. 2 (𝐾−M) 𝛼 1/3 𝑎 (𝑡) =[ ] 𝑡2/3, (19) 𝑀 with 𝑎>0̇ and 𝑎<0̈ in the matter dominated era. For (√3Λ/2)𝑡 ≫,( 1 18)gives (𝐾−M) 𝛼 1/3 √Λ𝑡 𝑎 (𝑡) = [ ] exp ( ) , (20) 3𝑀Λ √3 the de Sitter scale factor, with 𝑎>0̇ and 𝑎>0̈ in the vacuum dominated era, so called because Λ,whichoccursnaturally in Einstein’s equation, is a classical property of the vacuum, whose quantum fluctuations are not invoked here because virtual particle-antiparticle pairs created spontaneously from the vacuum can have positive or negative energy [9], making it uncertain whether such vacuum fluctuations can explain Λ, since their contributions to positive and negative Λ may be canceled. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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