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

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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.

Author(s): Michal Křížek, Lawrence Somer
Publisher: Springer
Year: 2023

Language: English
Pages: 269
City: Cham

Preface
Contents
Glossary of Symbols
1 Mathematical Modeling
1.1 Introduction
1.2 Kepler's Laws
1.3 Newton's Gravitational Law
1.4 The N-Body Problem
1.5 General Computational Scheme
1.6 Estimation of Modeling and Numerical Errors
2 Paradoxes in the Special Theory of Relativity
2.1 Short Introduction to the Special Theory of Relativity
2.2 Time Dilation
2.3 The Lorentz Transformation Does not Allow Superluminal Velocities
2.4 Length Contraction
2.5 The Twin Paradox
2.6 Correct Solution of the Twin Paradox
2.7 Conclusions
3 Einstein's Equations
3.1 Historical Facts of Importance
3.2 Exterior Schwarzschild Solution
3.3 Interior Schwarzschild Solution
3.4 Composite Schwarzschild Metric Tensor
3.5 Einstein's Equations with General Right-Hand Side
3.6 Black Holes
3.7 Einstein's Equations for Euclidean Space
3.8 Cosmological Constant
4 Numerical Analysis of Mercury's Perihelion Shift
4.1 A Brief Historical Overview
4.2 The Inaccuracy of the Difference Between Two Almost Equally Large Numbers
4.3 The Observed Perihelion Shift of Mercury
4.4 Computed Perihelion Shift of Mercury
4.5 A Method of Albert Einstein
4.6 Further Applications and Critical Remarks
4.7 Conclusions
5 Computational Problems of Einstein's Equations
5.1 On the Explicit Form of the First Einstein Equation
5.2 Difficulties with Initial and Boundary Conditions
5.3 Extreme Computational Complexity of Einstein's Equations
5.4 Further Arguments
6 Friedmann Equation
6.1 Nonuniqueness of the Notion Universe
6.2 How to Imagine the Sphere S3?
6.3 Metric Tensor for the Sphere S3a
6.4 Calculation of the Christoffel Symbols
6.5 Calculation of the Ricci Tensor
6.6 The Friedmann Equation for the Sphere S3a
6.7 Division by Zero in Cosmological Parameters
6.8 Maximally Symmetric Manifolds
6.9 Regular Tessellations of Hyperbolic Manifolds
7 Excessive Extrapolations From the Friedmann Equation
7.1 Typical Examples
7.2 The Origin of Excessive Extrapolations
7.3 Distances in Time
7.4 Classification of Cosmological Distances in Space
7.5 A Simple Form of the Friedmann Equation
7.6 Einstein's Static Universe
7.7 Some Consequences of the Friedmann Equations
7.8 The Age of the Universe
7.9 Flight Around the Universe
7.10 Conclusions
8 Arguments Against the Proclaimed Amount of Dark Matter
8.1 Problem of Missing Matter
8.2 Analysis of Zwicky's Method
8.3 Two Simple Examples
8.4 Analysis of Rotation Curves by Vera Rubin
8.5 Final Remarks
9 Dark Energy and the Local Hubble Expansion
9.1 Local Hubble Expansion
9.2 The Universe Expands Almost as Rapidly as the Moon Recedes From the Earth
9.3 Mars Was Much Closer to the Sun When There Were Rivers
9.4 Orbital Expansion of Titan
9.5 Expansion of the Solar System
9.6 Gravithermal Catastrophe
9.7 Do Single Galaxies Expand?
10 Anthropic Principle and the Hubble-Lemaître Constant
10.1 Weak Formulation of the Anthropic Principle
10.2 The Faint Young Sun Paradox
10.3 The Expansion of the Ecosphere
10.4 Two-Sided Error Estimates
10.5 A Possible Future Development of Earth's Orbit
10.6 Why Does the Law of Conservation of Energy Not Hold in the Physical Universe?
11 Gravitational Waves
11.1 The First Detection of Gravitational Waves
11.2 Emitted Versus Detected Frequencies
11.3 Neglected Gravitational Redshift
11.4 Other Arguments
11.5 Conclusions
12 Possible Distribution of Mass Inside a Black Hole
12.1 A Remarkable Coincidence
12.2 Possible Distribution of Mass Inside a Black Hole
12.3 Other Properties
12.4 Final Conclusions
Bibliography
References to Webpages (Valid as of June 2022)
Index
Author Index