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

Abstract

A curved `two-dimensional' absolute intrinsic metric spacetime (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)) on the vertical intrinsic spacetime hyperplane; its invariantly projected flat ‘two-dimensional' absolute proper intrinsic metric spacetime (\(\varnothing\)\({\rho}^\prime\)ab,\(\varnothing\)\({c}\)sab\(\varnothing\)\({t}^\prime\)ab)  and a flat `two-dimensional' absolute proper metric spacetime (\({\rho}^\prime\)ab,\({c}\)sab\({t}^\prime\)ab) as the outward manifestation of the latter, evolve from a flat `four-dimensional' absolute metric spacetime (\(\hat{\mathbb{E}}^3\),\(\hat{c}_s\)\(\hat{t}\)) and its underlying flat `two-dimensional' absolute intrinsic metric spacetime (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)), in all finite neighborhood of the source of a long-range metric force field. The flat four-dimensional relative proper metric spacetime (\(\varnothing{\mathbb{E}}^\prime\)3, \({c}_s\)\(t^\prime)\) and its underlying flat two-dimensional relative proper intrinsic metric spacetime (\(\varnothing\)\({\rho}^\prime\),\(\varnothing\)c\(_s\)\(\varnothing\)\({t}^\prime\)), remain unchanged within the field. The geometry is valid with respect to 3-observers located in the relative proper Euclidean 3-space \({\mathbb{E}}^\prime\)3. A pair of absolute intrinsic metric tensor equations derived on the curved (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)) are solved algebraically to obtain the absolute intrinsic metric tensor and absolute intrinsic Ricci tensor on the curved (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)) in terms of an isolated absolute intrinsic geometrical parameter, referred to as absolute intrinsic `static flow' speed, which the source of a long-range absolute intrinsic metric force field causes to be established on the extended curved (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)) from its location. This third part of this paper is the conclusion of the development of absolute intrinsic Riemann geometry on the curved `two-dimensional' (\(\varnothing\)\(\hat{\rho}\),\(\varnothing\)\(\hat{c}_s\)\(\varnothing\)\(\hat{t}\)) at the first stage of evolutions of spacetime and intrinsic spacetime in long-range metric force fields, started in the first and second parts. The first stage shows up as a numerical evolution. Extension to the second stage shall be done in the fourth and final part of this paper. Particularization to the gravitational field shall then follow in another article.