Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields II

Abstract

The geometry of curved ‘three-dimensional’ absolute intrinsic metric space (an absolute intrinsic Riemannian metric space) \(\varnothing\hat{\mathbb{M}}^3\), which is curved (as a curved hyper-surface) toward the absolute time/absolute intrinsic time ‘dimensions’ (along the vertical), and projects a flat three-dimensional absolute proper intrinsic metric space \(\varnothing\hat{\mathbb{E}}^\prime\)ab3 and its outward manifestation namely, the flat absolute proper 3-space \(\hat{\mathbb{E}}^\prime\)ab3, both as flat hyper-surfaces along the horizontal, isolated in the first part of this paper, is subjected to graphical analysis. Two absolute intrinsic metric tensor equations, one of which is of the form of Einstein free space field equations and the other which is a tensorial statement of absolute intrinsic local Euclidean invariance (A\(\varnothing\)LEI) on \(\varnothing\hat{\mathbb{M}}^3\), are derived. Simultaneous (algebraic) solution to the equations yields the absolute intrinsic metric tensor and the absolute intrinsic Ricci tensor of absolute intrinsic Riemann geometry on the curved \(\varnothing\hat{\mathbb{M}}^3\), in terms of a derived absolute intrinsic curvature parameter. A superposition procedure that yields the resultant absolute intrinsic metric tensor and the resultant absolute intrinsic Ricci tensor, when two or a larger number of absolute intrinsic Riemannian metric spaces co-exist (or are superposed) is developed. The fact that a curved ‘three-dimensional’ absolute intrinsic metric space \(\varnothing\hat{\mathbb{M}}^3\) is perfectly isotropic and is consequently contracted to a ‘one-dimensional’ isotropic absolute intrinsic metric space, denoted by \(\varnothing\hatρ^3\), which is curved toward the absolute time/absolute intrinsic time ’dimensions’ (\(\hat{c}\)\(\hat{t}\)/\(\varnothing\)\(\hat{c}\)\(\varnothing\)\(\hat{t}\)) along the vertical is derived.