Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields I
Physical Science International Journal,
Page 1-20
DOI:
10.9734/psij/2021/v25i1030284
Abstract
Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE′3 can transform into a curved three-dimensionalRiemannian metric space IM′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE3 (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM3, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE′3ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE′3ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE′3. The projective ∅IE′3ab is imperceptibly embedded in ∅IE′3 and IE′3ab in IE′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM′3 only is reached.
Keywords:
- Conventional metric space
- Riemann geometry
- absolute intrinsic metric space
- absolute intrinsic Riemann geometry
How to Cite
References
Joseph OAA. Reformulating Special Relativity on a Two-world Background II. Physical Science International Journal. 2020;24(9):34-67. Available:https://doi.org/10.9734/psij/2020/ v24i830215.
Joseph OAA. Reformulating Special Relativity on a Two-world background, Physical Science International Journal; 2020. Available:https://doi.org/10.9734/psij/2020/ v24i1230229.
Joseph OAA. Re-interpretation of the two- world background of special relativity as four-world background II. Physical Science International Journal. 2021;25(2):37-57. Available:https://doi.org/10.9734/psij/2021/ v25i230243.
Einstein A. The Principle of Relativity. (London: Methuen; reprints by Dover Publications); 1923.
Spivak M. Differential Geometry, Vol. II (Boston, Massachusetts: Publish and Perish, Inc.); 1970.
Adler R, Bazin M, Schiffer M. Introduction to General Relativity, 2nd Edition. (New York: McGraw-Hill Book Co.); 1975.
Jammer M. Concepts of Space, 2nd Edition. (Cambridge Massachusetts: Havard University Press); 1953.
Giesen J, Wagner U. Shape Dimension and Intrinsic Metric from Samples of Manifolds, Discrete Comput Geom. 2004;32:245267. Available:https://doi.org/10.1007/s00454- 004-1120-8
Bennett RS. The intrinsic dimensionality of signal collections. IEEE Transactions on Information Theory. 1969;15(5):517525. Available:https://ieeexplore.ieee.org/ document/1054365
Trunk GV. Statistical estimation of the intrinsic dimensionality of a noisy signal collection. IEEE Transactions on Computers. 1976;100(2):165171. Available:https://doi.org/10.1109/TC.1976. 5009231
Pettis KW, Bailey TA, Jain AK and Dubes RC. An intrinsic dimensionality estimator from near-neighbor information. IEEE Transactions on Pattern Analysis and Machine Intelligence. 1979;1(1):2537. Available:https://doi.org/10.1109/TPAMI. 1979.4766873.
-
Abstract View: 270 times
PDF Download: 109 times