The Ancient Greeks presented to philosophers both material space (Plato first said in Timaeus “space and matter are the same”) and Euclidean 3‐geometry for this space. Developing Aristotelian ideas of the spatial plenum‐continuum, Descartes formulated “Philosophiae Naturalis” for his vortex matter‐extension in 1644, well before the 1676 concept of point matter in “Philosophiae Naturalis Principia Mathematica” of Newton. Later, the Newtonian model of material points in immaterial empty 3‐space was adopted by the special (1905) and general (1915) theories of relativity that led to curved 3‐space, non‐physical singularities and sophisticated metric constructions with black holes.
The Euclidean geometry of 3‐space is one of the premier examples of the synthetic a priori knowledge of Immanuel Kant. He consistently suggested the inner core of massive stars in the Euclidean Milky Way and in external nebulae, including the Andromeda nebula. However, today the relativistic physics of point masses became unable to describe their interaction in flat space. Leading experts insist that the Schwarzschild metric with curved 3‐space is a robust benchmark for General Relativity (GR) due to precise measurements of predicted post‐Newtonian corrections. Despite the fact that Euclidean space and Kant’s cosmology are very unpopular with modern relativists, I try to remind by this book that any experiments can only falsify theoretical calculations but never justify them before competing theories (thanks to Karl Popper). Cartesian matter‐extension can be described in the same metric terms of Einstein’s theory but without the Schwarzschild solution and singularities. Moreover, the curved space‐ time with Euclidean 3‐space of extended masses can quantitatively explain all known GR tests, as well as the absence of SQUID accelerometers and gravitational analogues of the Aharonov‐Bohm effect. Instead of a positive (measurable) matter‐extension of Descartes, Newtonian space is filled everywhere by negative (immeasurable) gravitational energy. And this negative (non‐existing in reality) energy‐potential still controls the motion of positive kinetic energies in contemporary textbooks as a “divine action‐at‐a‐distance”. Such a palliative of empty space with postulated negative energies does not a more advanced ontology than Ptolemy’s model with postulated epicentres of planetary motion. The Plato‐Aristotle‐Descartes continuum of kinetic space‐matter with positive energy densities together with quantum nonlocality of material extensions in Euclidean 3‐space may provide more reliable references for relativistic geometrisation of material fields, including the nonlocal unity of the quasi‐equilibrium solar system.
These training chapters for advanced students reiterate the introductory math formalism for extended masses (DOI: 10.4236/jmp.2012.310181) and precede the next level tutorial “Pure field electrodynamics of continuous complex charges” for the 4th‐ and 5th‐year students at the Moscow Institute of Physics and Technology (Moscow: MIPT, 2015, ISBN 978‐5‐7417‐0554‐4, https://search.rsl.ru/ru/record/01007979504). Below we will discuss quantitatively that the curved space‐time 4‐interval of any probe particle does not contradict the flat non‐empty 3‐space, which, in turn, assumes the global material overlap of continuous masses or the nonlocal Universe with universal Euclidean geometry. Particle’s time is a chain function of particle’s displacement or the physical velocity and this time differs from the proper time of a motionless local observer. Equal passive and active relativistic energy‐charges are used to comply with universal free fall and the Principle of Equivalence in non‐empty (material) space, where continuous radial densities of elementary energy‐charges obey local superpositions and the nonlocal organization. The known precession of planetary perihelion, radar echo delay, and gravitational light bending can be quantitatively explained by the singularity‐free metric without deviating from Euclidean spatial geometry. The flat‐space precession of non‐point orbiting gyroscopes is non‐Newtonian one due to the Einstein dilation of local time within the Earth’s radial energy‐charge, and not due to unphysical warping of Euclidean space.