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100 years of the Friedmann equation
Křížek, Michal
In 1922, Alexander Friedmann applied Einstein’s equations to a three-dimensional sphere to describe the evolution of our universe. In this way he obtained a nonlinear ordinary differential equation (called after him) for the expansion function representing the radius of that sphere. At present, the standard cosmological ΛCDM model of the universe is based just on the Friedmann equation. It needs a significant amount of dark matter, about six times that of the usual baryonic matter, besides an even larger amount of dark energy to be consistent with the real universe. But to date, both dark matter and dark energy have remained without concrete evidence based on direct physical measurements. We present several arguments showing that such a claimed amount of dark matter and dark energy can only be the result of vast overestimation, incorrect extrapolations, and that it does not correspond to the real universe. The spatial part of our universe seems to be locally flat and thus it can be locally modeled by the Euclidean space. However, Friedmann did not consider the flat space with zero curvature. Therefore, in the second part of this paper we will derive a general form of the corresponding metric tensor satisfying Einstein’s equations with zero right-hand side.
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Excessive extrapolations of Einstein's equations
Křížek, Michal ; Somer, L.
The standard cosmological model is surprisingly quite thoroughly investigated even though it possesses many paradoxes. We present several arguments indicating why excessive extrapolations of Einstein's equations to cosmological distances are questionable. First, we show how to express explicitly the first of Einstein's 10 partial differential equations to demonstrate their extremely large complexity. Therefore, it would be very difficult to find their solution for two or more bodies to model, e.g., the evolution of the Solar system. Further, we present some unexpected failures of the Schwarzschild and Friedmann solution of these equations. Then we explain why application of Einstein's equations to the whole universe represents incorrect extrapolations that lead to dark matter, dark energy, and several unrealistic situations. Finally, we give 10 further arguments showing why celebrated Einstein's equations do not describe reality well.
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On the Friedmann equation for the three-dimensional hypersphere
Křížek, Michal ; Mészáros, A.
The present standard cosmological model of the evolution of our universe, is based on the Friedmann equation, which was published by Alexander Friedmann in 1922. He applied Einstein’s equations to an expanding threedimensional sphere which enabled him to avoid boundary conditions. However, his description was very brief. Therefore, the main objective of this article is to detailed a derivation of the Friedmann equation for an unknown expansion function a = a(t) representing the radius of the universe. Furthermore, we present serious arguments showing why the validity of Einstein’s equations should not be extrapolated to the entire universe.
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