The Physical and Mathematical Foundations of the Theory of Relativity

Abstract

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Comparing Equivalent Gravities: common features and differences

“…4) The Riemannian geometry represents the first arena within which Einstein framed his theory, constructed only upon the curvature tensor [76,77].…”

“…5) The Minkowski geometry is obtained by setting curvature, torsion, and non-metricity to zero, where the flat metric η µν , as well as zero affine connections, are adopted. This is the arena of Special Relativity [76,77].…”

“…Therefore, to each point p ∈ M, we can associate its coordinates by (x 0 , x 1 , x 2 , x 3 ) := ϕ(p) ∈ R 4 [77]. Defined the coordinate x µ -axes in R 4 , it is possible to construct the related coordinate curves γ x µ on M via the use of the charts.…”

“…The Einstein theory is essentially based on the following pillar ideas, which can be stated as follows [29,76,77]:…”

First post-Newtonian $N$-body problem in Einstein-Cartan theory with the Weyssenhoff fluid: equations of motion

“…Note that this hypothesis resembles the static equilibrium used in GR. The subsequent calculations, performed "as in the GR static equilibrium case", are not spoiled by the presence of the spin as long as the intrinsic rotation of each fluid element is stationary, i.e., the spin vector associated to each fluid element neither changes direction nor varies in time [47,48]. This relativistic issue presents already at the classical level, when we deal with (nonclosed) micropolar continuous systems, which find physical applications in ferromagnetic substances or liquid crystals [49].…”