|The Cosmic Microwave Background temperature fluctuations from the 5-year Wilkinson Microwave Anisotropy Probe data seen over the full sky.|
Could Earth be in one of its two centers?
The two sentences, 'the sun is at rest and the earth moves,' or 'the sun moves and the earth is at rest,' would simply mean two different conventions concerning two different CS.
I. Introduction - About Geocentric and Heliocentric systems
What this article does NOT
II. Nothing wrong with Changing Coordinate Systems
First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars are accelerating, we can determine their positions by the following formulas, where x1(t) is the position in meters of car one after time t seconds and x2(t) is the position of car two after time t.
Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which x1 = x2.
Therefore we set x1 = x2 and solve for t, that is:
Difference between geocentric and heliocentric frames of referece
The "elegance" principle
In the Ptolemaic system of astronomy, the epicycle (literally: on the circle in Greek) was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. It was designed by Apollonius of Perga at the end of the 3rd century BC. In particular it explained the retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from Earth.
When Copernicus transformed Earth-based observations to heliocentric coordinates , he was confronted with an entirely new problem. The Sun-centered positions displayed a cyclical motion with respect to time but without retrograde loops in the case of the outer planets. In principle, the heliocentric motion was simpler but with new subtleties due to the yet-to-be-discovered elliptical shape of the orbits. Another complication was caused by a problem that Copernicus never solved: correctly accounting for the motion of the Earth in the coordinate transformation. In keeping with past practice, Copernicus used the deferent/epicycle model in his theory but his epicycles were small and were called “epicyclets”.
Whether or not Copernicus’ models were simpler than Ptolemy’s is moot. Copernicus eliminated Ptolemy’s somewhat-maligned equant but at a cost of additional epicycles. Various 16th-century books based on Ptolemy and Copernicus use about equal numbers of epicycles. The idea that Copernicus used only 34 circles in his system comes from his own statement in a preliminary unpublished sketch called the Commentariolus. By the time he published De revolutionibus orbium coelestium, he had added more circles. Counting the total number is difficult, but estimates are that he created a system just as complicated, or even more so. The popular total of about 80 circles for the Ptolemaic system seems to have appeared in 1898. It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro, who used either 77 or 79 orbs in his system inspired by Eudoxus of Cnidus. 
Copernicus’ theory was at least as accurate as Ptolemy’s but never achieved the stature and recognition of Ptolemy’s theory. In scarcely more than a hundred years, Copernicus would be overcome by events set in motion by Johannes Kepler and Galileo Galilei. Copernicus’ work provided explanations for phenomena like retrograde motion, but really didn’t prove that the planets actually orbited the Sun.
The first planetary model without any epicycles was that of Ibn Bajjah (Avempace) in 12th century Andalusian Spain, but epicycles were not eliminated in Europe until the 17th century, when Johannes Kepler's model of elliptical orbits gradually replaced Copernicus' model based on perfect circles.
Newtonian or Classical Mechanics eliminated the need for deferent/epicycle methods altogether and produced theories many times more powerful. By treating the Sun and planets as point masses and using Newton’s law of universal gravitation, equations of motion were derived that could be solved by various means to compute predictions of planetary orbital velocities and positions. Simple two-body problems, for example, can be solved analytically. More-complex n-body problems require numerical methods for solution.
The power of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune. Analysis of observed perturbations in the orbit of Uranus produced estimates of the suspected planet’s position within a degree of where it was found. This could not have been accomplished with deferent/epicycle methods.
Important Questions that arise
The Observational Approach to Cosmology
The famous astronomer Edwin Hubble published on 1937 a study on the cosmological model of the universe, under the title "The Observational Approach to Cosmology". In the data published in that study it was evident that Earth apperared like having a "unique" position in the cosmos, i.e. that it was in the center or very close to it. However Hubble chose not to accept that unique position based on philosophical propositions (principles) that be believed in.
Excerpt from "The Observational Approach to Cosmology"
|Plate VI - Remote clusters studied by Hubble|
Relativistic cosmology requires that
The curvature of space is demonstrated and measured by the postulated recession of the nebulae. To the observer the procedure seems artificial. He has counted the nebulae to various limits, applied only the corrections that are necessarily required (energy-corrections), and derived the quite plausible result of uniform distribution. Now, in testing the relativistic theory, he introduces a new postulate, namely, recession of the nebulae, and it leads to discrepancies. Therefore, he adds still another postulate, namely, spatial curvature, in order to compensate the discrepancies introduced by the first. The accumulation of assumptions is uneconomical, and the justification must be sought in the general background of knowledge. The outstanding argument is the fact that velocity-shifts remain the only permissible interpretation of red-shifts that is known at the present time. 
[End of excerpt from "The Observational Approach to Cosmology"]
The full text of the abovementioned reserach paper by Hubble can be found online at The Observational Approach to Cosmology archived at the respective site of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology (CalTech), under contract with the National Aeronautics and Space Administration (NASA).
Cosmology has not proved or disproved any model
1. Isotropy- the universe looks the same in any direction (and from any place)
2. Homogeneity- The make-up of the universe is more or less the same everywhere.
These two points taken together are called the Cosmological Principle. Since it is a "principle", this means that pretty much all cosmologists, astronomers, etc. will make this assumption. Wikipedia simplifies that Cosmological Principle "on a large scale the universe is pretty much the same everywhere". What does Wikipedia call on for support?
Observed isotropy of the cosmic microwave background radiation (CMB), combined with the Copernican principle
In cosmology, the Copernican Principle, named after Nicolaus Copernicus, states the Earth is not in a central, specially favoured position. More recently, the principle is generalised to the simple statement that humans are not privileged observers. In this sense, it is equivalent to the mediocrity principle, with significant implications in the philosophy of science. 
Many of the above point towards Milne's metaphysical views, which were based in positivism, most especially in operationalism: only those objects whose properties could be directly revealed by some observational procedure, or operation, were to be counted among the real. His ‘cosmological principle’, namely, that every cosmologist in the universe should look out upon the same cosmos, was the prime example of such thinking. Later, Bondi would found Steady-state Cosmology on an even more general version of Milne's principle, which he called the ‘perfect cosmological principle.’ Dingle is here arguing against the view, Milne's view, that authentic science may be rationalist in epistemology (scientific knowledge is founded in pure theoretical reasoning apart from sense perception), and hypothetico-deductive in method (general principles are justified by their deductively implying correct observations). According to Milne's principle, every observer in the universe should get the same world picture, that is, should make precisely the same observations of the universe at the same moment as any other observer.  Uniformity over spatial slices is guaranted by Milne's invoking of the principle. Yet Milne's universe evolves, it changes its form over time. Hence it has no temporal uniformity. Bondi felt that this raised the possibility that physics itself might change over time. Because of this risk, Bondi generalized Milne's cosmological principle into what he called the perfect cosmological principle [=PCP]. According to this principle, all observers at all places and at all times will look out upon the same unchanging, unevolving, universe. Such a universe is a universe in a steady-state—hence the name. Although the principle (and the theory which results from it—steady state cosmology) is, as McVittie remarked, “much more restrictive than general relativity”;  it is this very restrictiveness which satisfies Bondi's Popperian wishes: For the correct argument has always been that the steady state model was the one that could be disproved most easily by observation, Therefore, it should take precedence over other less disprovable ones until it has been disproved.  But PCP and the theory which it engendered were exactly as described: eminently falsifiable. No matter the extent of Dingle et al's disdain, Steady State theory stayed right out in front, ready for whatever empirical observations might be slung at it. As Bondi said “Show me some fossils from an evolving universe, and I'll give it up.” In 1965, the fossils arrived, courtesy of the observations of the 3° K remnant microwave radiation. And Bondi, true to his philosophy, gave it up. 
He does provide and alternative view, though, as he continues: "There is, however, an alternate explanation: the universe might look the same in every direction as seen from any other galaxy, too. This, as we have seen, was Friedmann’s second assumption. We have no scientific evidence for, or against, this assumption.
We believe it only on grounds of modesty: it would be most remarkable if the universe looked the same in every direction around us, but not around other points in the universe." 
IV. Modern cosmology and Earth positioning
Ellis has proposed that we live on a planet that is near one of the two centers the universe has and - according to his calculations - that geocentric model removes the need to "invent" terms like "dark energy" or "dark matter" to explain how galaxies in the cosmos move.
Mathematics, Department, University of Cape Town
In particular George Ellis proposed a "semi-geocentric" model universe that contains a naked singularity as a recycling mechanism, which he claims gives almost as good a description of the real universe as the conventional model.  In an article in Nature  Ellis proposes that the universe is like a cylinder-shaped universe with two "centers", with the Earth is located on one side and a naked singularity on the other.
There is no cosmic inflation – the galaxies are arranged very unevenly, with a great deal of material crowded round the singularity, and very little near the Earth. The effect of such a distribution of matter is to produce a red shift of light that, at the Earth, has the same characteristics as if the galaxies were receding.  In particular, the Universe seems to be expanding ever faster — a phenomenon generally ascribed to the influence of 'dark energy'. But Ellis suggests that the observed acceleration be a trick of the light in an inhomogeneous Universe and if we accept his geocentric model, we could do away with dark energy by imagining that we lived at the center of a spherical  inhomogeneity.
Possible objections to the model of Ellis
Moreover, modern cosmological models describe that the Big Bang happened at a singularity, or in other words: it happened at what was at then "everywhere". That point the created the universe inflation we observe now. So in that way we are all now at the universe "center".
VI. Conclusions - Where do we stand after all?
(1) The Newtonian theory of cosmology is not yet adequately resolved. Newtonian theory is only a good theory of gravitation when it is a good approximation to General Relativity; obtaining this limit in non-linear cosmological situations raises a series of questions and issues that still need clarifying, particularly relating to boundary conditions in realistic Newtonian cosmological models.
(2) We have some understanding of how the evolution of families of inhomogeneous models relates to that of families of higher symmetry models. It has been indicated that a skeleton of higher symmetry models seems to guide the evolution of lower symmetry models in the state space (the space of cosmological space-times). This relation needs further elucidation. Also, anisotropic and inhomogeneous inflationary models [note: inhomogenity is closely related to the cosmological model proposed by Ellis] are relatively little explored and problems remain.
(3) It must be noted that it was Kepler and his idea of elliptical orbits which eliminated the need for epicycles - and not the heliocentric system idea by itself. Some people (wrongly) claim that the heliocentric model is more "correct" (whatever that means) because it is more simple than geocentric model. But is an ellipsis more "simple" than a circle? And what about the explanation of us seeing all galaxies getting away from us (Hubble) or the elimination of the need for dark energy if we consider Earth at the center (Ellis)? Does that mean that the geocentric model is in general more simple in explaining the observations? Even simple words like "simple" could be hard and complicated to define after all... And what about the fact that we are here (at Earth) looking at planets going around us? Is it really more "simple" to think that we stand somewhere else?!?!? If you are in a car and see other cars running, do you make your observations as if you were in *another* car? Would it be logical to do something like that?
In another of his papers, Ellis has postulated the following principle: Scientific exploration can tell us much about the universe but not about its ultimate nature, or even much about some of its major geometrical and physical characteristics. Some of this uncertainty may be resolved, but much will remain. Cosmological theory should acknowledge this uncertainty. 
Taking into account all of the above, we must be very careful when we speak as scientists or when we read as intelligent beings. When formulating / reading a theory we must clearly state / know what are the preassumptions that we make and the axioms or propositions that we use. 
- The Evolution of Physics, Einstein and Infeld, p.212 (p.248 in original 1938 ed.).
- Frame of Reference [Wikipedia]
- Geocentric model [Wikipedia]
- Deferent and epicycle [Wikipedia]
- The Observational Approach to Cosmology, Edwin Hubble, Oxford University Press, 1937.
- Copernican Principle [Wikipedia]
- Milne (E.A. Milne), 1934b.
- McVittie 1990, p. 45.
- Bondi and Kilmister 1959, p. 55-6.
- Cosmology: Methodological Debates in the 1930s and 1940s [Stanford Encyclopedia of Philosophy]
- A Brief History of Time, Stephen Hawking, Bantam Dell Publishing Group, 1988.
- "Thinking Globally, Acting Universally", Scientific American, George Ellis, October 1995.
- Dark matter and dark energy proposals: maintaining cosmology as a true science?, George F. R. Ellis, CRAL-IPNL conference "Dark Energy and Dark Matter", Lyon 2008.
- George Ellis [Wikipedia]
- Cosmology: Patchy solutions, George Ellis, Nature 452, 158-161 (13 March 2008) | doi:10.1038/452158a; Published online 12 March 2008.
- ALTERNATIVES TO THE BIG BANG, G.F.R. Ellis, Annu. Rev. Astron. Astrophys. 1984. 22: 157-184.
- Dark Energy Has Long Been Dark-Energy-Like [DISCOVER]
- COSMOLOGICAL MODELS, CARG`ESE LECTURES 1998, George F R Ellis and Henk van Elsy, Cosmology Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa, September 2, 2008.
- Issues in the Philosophy of Cosmology, George F R Ellis, Mathematics Department and Applied Mathematics, University of Cape Town, February 5, 2008.
- The Limits of Science, Knol, Spiros Kakos [knol.google.com]
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