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

O. Akindele Adekugbe Joseph *

Department of Physics, Ondo State University of Science and Technology, Center for the Fundamental Theory and Uniﬁcation, Okitipupa, Nigeria.

*Author to whom correspondence should be addressed.

### 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.

Keywords: Long-range metric force ﬁelds, ﬁrst stage of evolution of spacetime, numerical evolution of spacetime, absolute intrinsic Riemannian spacetime geometry, coexisting absolute intrinsic metric spacetimes, superposition procedure, resultant absolute intrinsic metric tensor, resultant absolute intrinsic Ricci tensor

**How to Cite**

*Physical Science International Journal*25 (10):78-112. https://doi.org/10.9734/psij/2021/v25i1030288.