Relativity and Cosmology

   

Evolutionary Sequence of Spacetime/intrinsic Spacetime and Associated Sequence of Geometry in a Metric Force Field. Part II.

Authors: Akindele J. Adekugbe

Graphical analysis of the geometry of a curved 'three-dimensional' absolute intrinsic metric space, (an absolute intrinsic Riemannian metric space) φM3, which is curved onto the absolute time/absolute intrinsic time 'dimensions' (along the vertical), as a curved hyper-surface, and projects a flat three-dimensional proper intrinsic metric space φE'3 underlying its outward manifestation namely, the flat proper physical Euclidean 3-space φE'3, both as flat hyper-surfaces along the horizontal, isolated in part one of this paper, is done. Two absolute intrinsic tensor equations, one of which is of the divergenceless form of Einstein free-space field equations and the other which is a tensorial statement of local Euclidean invariance on φM3, are derived. Simultaneous (algebraic) solution of the equations yields the absolute intrinsic metric tensor and absolute intrinsic Ricci tensor of absolute intrinsic Riemann geometry on the curved absolute intrinsic metric space φM3, in terms of an isolated absolute intrinsic curvature parameter. Relations for absolute intrinsic coordinate projections into the underlying flat proper intrinsic space are derived. A superposition procedure that yields resultant absolute intrinsic metric tensor and resultant absolute intrinsic Ricci tensor, as well as resultant absolute intrinsic coordinate projection relations when two or a larger number of absolute intrinsic Riemannian metric spaces co-exist, are developed. Finally the fact that a curved 'three-dimensional' absolute intrinsic metric space φM3 is perfectly isotropic (that is, all directions are perfectly the same) and is consequently contracted to a 'onedimensional' absolute intrinsic metric space denoted by φρ, which is curved onto the absolute time/absolute intrinsic time 'dimensions' along the vertical and that the underlying projective three-dimensional flat proper intrinsic metric space φE'3 is perfectly isotropic and is consequently contracted to a straight line one-dimensional isotropic proper intrinsic metric space φρ' along the horizontal, with respect to observers in the physical proper Euclidean 3-space φE'3 that overlies φρ', are deduced.

Comments: 20 pages. Submitted to Progress in Physics

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Submission history

[v1] 30 Nov 2010

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