Spacetime-Odyssey - A-08b

Spacetime Odyssey
(Οδύσσεια του Χωρόχρονου)

Ευκλείδειος Χώρος & Χρόνος
(Πίνακες)

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Εδώ βρίσκουμε τα περιεχόμενα της σειράς: Spacetime-Odyssey

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Ακολουθούν Ανακεφαλαιωτικοί Πίνακες

Evolutional Correspondence between Spacetime & Geometry
α/α	Physical Space (Position vector)	Geometry (Transformation Symmetry)
Euclidean Spaces
1.	one-dimensional (1D) Euclidean Space  r⃗  =[x]	1D-Space Rotation R(θ)=[θ]
2.	two-dimensional (2D) Euclidean Space  r⃗  =[xy] &  r⃗  =[xy]	2D-Space Rotation R(θ)=[0+θ−θ0]
3.	three-dimensional (3D) Euclidean Space  r⃗  =⎡⎣xyz⎤⎦ &  r⃗ =[xyz]	3D-Space Rotation R(θ)=⎡⎣⎢0+θz−θy−θz0+θz+θy−θx0⎤⎦⎥
Minkowskian Times
1.	one-dimensional (1D) Minkowski Time rt=[t]	1D-Time Rotation R(ϕ)=[0]
2.	two-dimensional (2D) Minkowski Time  r⃗ t=[0t] &  r⃗ t=[0−t]	2D-Time Rotation R(ϕ)=[0+ϕ−ϕ0]
3.	three-dimensional (3D) Minkowski Time  r⃗ t=⎡⎣00t⎤⎦ &  r⃗ t=[00−t]	3D-Time Rotation R(ϕ)=⎡⎣⎢00−ϕy00+ϕz+ϕy−ϕz0⎤⎦⎥

Evolutional Correspondence between Geometry & Mechanics
α/α	Geometry (Transformation Symmetry)	Mechanics (Invariant Quantities)
Euclidean Spaces
1.	1D-Space Rotation R(θ)=[0]	1D-Angular momentum L=[0]
2.	2D-Space Rotation R(θ)=[0+θ−θ0]	2D-Angular momentum L=[0+Lz−Lz0]
3.	3D-Space Rotation R(θ)=⎡⎣⎢0+θz−θy−θz0+θz+θy−θx0⎤⎦⎥	3D-Angular momentum  L⃗  =⎡⎣⎢0+Lz−Ly−Lz0+Lx+Ly−Lx0⎤⎦⎥
Minkowskian Times
1.	1D-Time Rotation R(ϕ)=[0]	1D-mass-Center position C=[0]
2.	2D-Time Rotation R(ϕ)=[0+ϕ−ϕ0]	2D-mass-Center position C=[0+Cz−Cz0]
3.	3D-Time Rotation R(ϕ)=⎡⎣⎢00−ϕy00+ϕz+ϕy−ϕz0⎤⎦⎥	3D-mass-Center position  C⃗  =⎡⎣⎢00−Cy00+Cz+Cy−Cz0⎤⎦⎥

Evolutional Correspondence between Geometry & Electrodynamics
α/α	Geometry (Transformation Symmetry)	Electrodynamics (Invariant Quantities)
Euclidean Spaces
1.	1D-Space Rotation R(θ)=[0]	1D-Magnetic Strength B=[0]
2.	2D-Space Rotation R(θ)=[0+θ−θ0]	2D-Magnetic Strength B=[0+Bz−Bz0]
3.	3D-Space Rotation R(θ)=⎡⎣⎢0+θz−θy−θz0+θz+θy−θx0⎤⎦⎥	1D-Magnetic Strength  B⃗  =⎡⎣⎢0+Bz−By−Bz0+Bx+By−Bx0⎤⎦⎥
Minkowskian Times
1.	1D-Time Rotation R(ϕ)=[0]	1D-Electric Strength E=[0]
2.	2D-Time Rotation R(ϕ)=[0+ϕ−ϕ0]	2D-Electric Strength E=[0+Ez−Ez0]
3.	3D-Time Rotation R(ϕ)=⎡⎣⎢00−ϕy00+ϕz+ϕy−ϕz0⎤⎦⎥	3D-Electric Strength  E⃗  =⎡⎣⎢00−Ey00+Ez+Ey−Ez0⎤⎦⎥

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Εδώ βρίσκουμε τα περιεχόμενα της σειράς: Spacetime-Odyssey

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