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

Abstract

The flat two-dimensional relative proper intrinsic metric spacetime (\(\varnothing \rho^{\prime}, \varnothing c_s \varnothing t^{\prime}\)) underlying the flat four-dimensional relative proper metric spacetime \(\left(\mathbb{E}^{\prime 3}, c_s t^{\prime}\right)\), which emerges at the first stage of evolution of metric spacetime and intrinsic metric spacetime in long-range metric force fields, isolated in the first three parts of this paper, endures for no moment before transforming into a curved two-dimensional relative proper intrinsic metric spacetime with pseudo-orthogonal curvilinear intrinsic dimensions, (\(\varnothing \rho^{\prime}\) and \(\varnothing c_s \varnothing t^{\prime}\)), on the vertical intrinsic metric spacetime hyperplane, on the larger spacetime and intrinsic spacetime of combined positive (or our) universe and the negative universe. It therefore possesses intrinsic Lorentzian metric tensor at every point. It projects an underlying flat relativistic intrinsic metric spacetime \(\left(\varnothing \rho, \varnothing c_s \varnothing t\right)\), which is made manifested outwardly in a flat four-dimensional relativistic metric spacetime \(\left(\mathbb{E}^3, c_s t\right)\), at the second (and final) stage of evolution of metric spacetime and intrinsic metric spacetime in long-range metric force fields. The conclusion that the four-dimensional metric spacetime is everywhere flat in every Iong-range metric force field is reached. The curved 'two-dimensional' absolute intrinsic metric spacetime (\(\varnothing \hat{\rho}, \varnothing \hat{c}_s \varnothing \hat{t}\)) with absolute intrinsic sub-Riemannian metric tensor \(\varnothing \hat{g}_{i k}\), which evolves at the first stage is brought forward to the second stage. The basic aspects of the theory of relativity on the flat relativistic metric spacetime, intrinsic theory of relativity on the underlying flat relativistic intrinsic metric spacetime and absolute intrinsic metric theory on the curved absolute intrinsic metric spacetime, associated with the presence of a metric force field in spacetime and intrinsic metric force field in intrinsic spacetime, are developed in terms of certain derived geometrical parameters, referred to as relative proper static flow speed, relative proper intrinsic static flow speed and absolute intrinsic static flow speed respectively. Particularization to the gravitational field will be a straight forward process, while using the results of this paper as template.