Mathematical Aspects of Paradoxes in Cosmology: Can Mathematics Explain the Contemporary Cosmological Crisis?

This book provides a mathematical and numerical analysis of many problems which lead to paradoxes in contemporary cosmology, in particular, the existence of dark matter and dark energy. It is shown that these hypothetical quantities arise from excessive extrapolations of simple mathematical models to the whole physical universe. Written in a completely different style to most books on General Relativity and cosmology, the important results take the form of mathematical theorems with precise assumptions and statements. All theorems are followed by a corresponding proof, or an exact reference to the proof.

Some nonstandard topics are also covered, including violation of the causality principle in Newtonian mechanics, a critical mathematical and numerical analysis of Mercury's perihelion shift, inapplicability of Einstein's equations to the classical two-body problem due to computational complexity, non-uniqueness of the notion of universe, the topology of the universe, various descriptions of a hypersphere, regular tessellations of hyperbolic spaces, local Hubble expansion of the universe, neglected gravitational redshift in the detection of gravitational waves, and the possible distribution of mass inside a black hole.

The book also dispels some myths appearing in the theory of relativity and in contemporary cosmology. For example, although the hidden assumption that Einstein's equations provide a good description of the evolution of the whole universe is considered to be obvious, it is just a null hypothesis which has not been verified by any experiment, and has only been postulated by excessive extrapolations of many orders of magnitude.

Michal Křížek is a senior researcher at the Institute of Mathematics of the Czech Academy of Sciences and a full professor at Charles University in Prague. For many years he was Editor-in-chief of the journal Applications of Mathematics and the Czech journal Advances of Mathematics, Physics and Astronomy. He is a member of the Czech Learned Society and the International Astronomical Union. His research focuses on mathematical and functional analysis, the numerical solution of partial differential equations, mathematical physics, and astrophysics.

Lawrence Somer is a professor emeritus of mathematics at the Catholic University of America in Washington, D.C. He obtained his B. A. degree in mathematics from Cornell University in Ithaca, New York, and his PhD. from the University of Illinois. He is a member of the Editorial board of the journal Fibonacci Quarterly. His research interests include number theory, combinatorics, algebra, geometry, and cosmology.