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Journal of Gravity
Volume 2013 (2013), Article ID 617142, 3 pages
Nonsingular Einsteinian Cosmology: How Galactic Momentum Prevents Cosmic Singularities
6400 N. Sheridan No. 2604, Chicago, IL 60626-5331, USA
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.
The Standard Cosmological Model entails a space-time metric with line element [1–4] where is the vacuum speed of light relative to a local Lorentz frame, is cosmic time, is the time-dependent scale factor of the Universe, and , , are spherical coordinates of a spatially flat (Euclidean) 3-space. A particle of proper masshas the Hamiltonian [5, 6] where , , and are canonical momenta conjugate to canonical coordinates , , and , respectively. The azimuthal angular momentum is conserved because does not depend on , but is not conserved unless all three momenta vanish. If , the polar angular momentum is conserved because no longer depends on . If , the radial momentum is conserved because no longer depends on . Defining , then reduces to showing how a particle with momentum loses energy to the expanding Universe .
A logical question, then, is where does that energy go? A logical answer is that it goes to the Hamiltonian of the Universe, which, in a one-dimensional minisuperspace model, can be expressed in geometrized units as the conserved quantity  with canonical momentum , canonical coordinate , effective mass , and potential where is the spatial curvature constant (±1 or 0), is a constant such that , is Einstein’s cosmological constant, and are energy densities of matter and radiation, respectively, at some initial time , and . The first term of is eliminated by metric (1), for which . The second term gives exponentially accelerated expansion of a vacuum dominated Universe, with attributed to dark energy due to vacuum fluctuations, a quantum effect. The third term gives , and , the Einstein-de Sitter scale factor of a matter dominated Universe. The fourth term gives in a radiation dominated Universe. The constraint gives the same differential equation for the expansion factor as is obtained from the Standard Cosmological Model [4, 7].
However, is defective, because the Standard Cosmological Model is defective. only includes the rest massof an object whose relativistic mass is Hamiltonian (3) with momentum . This deficiency is corrected here by revising to the form where in terms of the cosmic coordinate , thereby conserving by making . The sum over is taken over all objects , whose momenta create an interaction with the Universe , so they are no longer treated like test particles. denotes all the , which remain constant because is independent of the radial coordinates .
2. Canonical Equations
Hamiltonian (6) is an example of scientific induction , from which rigorous mathematical deduction, in the form of Hamilton’s equations, gives the coordinate velocities . Defining as the proper or physical distance of object from a real or hypothetical observer and dropping the subscript , give the recession velocity the Hubble velocity increased by positive and decreased by negative .
Using an overdot to denote and substituting in (6), give indicating that is a monotonically increasing function of , and is positive definite. But this mathematical acceleration does not necessarily imply physical acceleration. Expressing (9) in terms of the scale factorthrough the relation gives where the first term on the right is a monotonically decreasing function of, while the second and third terms are monotonically increasing, so can be positive or negative. Mathematics is simpler in terms of the canonical coordinate , but physics is clearer in terms of the expansion factor . The time derivative of (10) gives the acceleration
Defining the conserved quantities conserved because the and are constant; Hamiltonian (6) allows a momentum dominated epoch at very early times, a matter dominated epoch at intermediate times, and a vacuum dominated epoch at later times. Acceleration (11) is positive in the momentum era, negative in the matter era, and positive again in the vacuum era.
3. Momentum Dominated Epoch
Momentum domination occurs whenand thecan be neglected in (10) and (11), giving vanishes at scale factor , the first inflection point, withfor and deceleration for . vanishes at scale factor and is positive definite for but imaginary for , which is thereby forbidden by (13) as physically impossible. Thus, the momentum factorrules out a singularity by requiring the Universe to be created spatially flat with initial scale factor —the ultimate example of inflation—after which no further inflation is needed to achieve flatness or size. But (14) indicates that accelerated expansion continues for , after which deceleration sets in.
Equation (13) can be integrated to give with the initial time being defined so that when . Equation (15) confirms the impossibility of the scale factor being less than , since that would make imaginary.
The momentum factor, defined in (12), is essential for these results and is independent of the signs of thein (6). This momentum symmetry, together with the time reversal symmetry of Einstein’s equation, allows a contracting Universe to undergo a smooth bounce at the minimum scale factor and then rebound from it as if it had been created at .
The recession velocity (8) reduces to the form indicating thatis assumed to be so large that the motion of the receding object approximates that of a massless particle moving at luminal speed during this era. In this respect, it is like a radiation era but with repulsive radiation, rather than the attractive radiation of Hamiltonian (4). But the end result is the same, as the Universe expands and matter domination sets in.
4. Matter and Vacuum Dominated Epochs
When thecan be neglected for (the starting time of matter domination), (10) gives For , (17) is then readily integrated to give Definingas the transition time from matter domination to vacuum domination, it follows that, for and , (18) gives the Einstein-de Sitter scale factor with and in the matter dominated era. For , (18) gives the de Sitter scale factor, withandin the vacuum dominated era, so called because, which occurs naturally 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 , making it uncertain whether such vacuum fluctuations can explain, since their contributions to positive and negativemay be canceled.
The initial singularity of the Standard Model comes from neglecting the conserved momentain the relativistic mass terms of Hamiltonian (6). When theare included, Einstein’s equation forbids a singularity, thereby disproving the singularity theorems [1, 3, 10]. This quantum leap in cosmology is achieved within the framework of general relativity, through the classical mechanism of momentum, without quantization or any non-Einsteinian effects. It does not improve on Einstein’s theory, but proves that Einstein’s theory is much better than it was thought to be. Other models based on a nonsingular bounce followed by expansion are not strictly Einsteinian, because they invoke other mechanisms  in lieu of the, whereas this galactic momentum is the essential mechanism of nonsingular Einsteinian cosmology.
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