User:Vish-aero/Sandbox

Temporal Dimension

Mapping the fourth dimension

As pointed out by Coxeter, it is foolishness to regard the fourth dimension as time when the 4-space is Euclidean; the fourth dimension is time only when the 4-space is a Minkowski 4-space. On the other hand there is a connection between Euclidean 4-space and Minkowski 4-space. When the fourth dimension x₄ in the Euclidean 4-space is appropriately related to the fourth dimension t in the Minkowski 4-space, the two four-dimensional systems map one to the other. Because of the existence of this mapping, the visual representations shown above can be applied to space-time physics. The temporal dimension in the Euclidean 4-space is called geometric time x₄ and is related to conventional time t in the Minkowski 4-space by x₄ = ict where c is the speed of light and $i={\sqrt {-}}1$.

Cross Sections

Since the fourth dimension can be regarded as geometric time, it becomes useful to look at 3-dimensional cross-sections of four-dimensional objects as an alternative to looking at three-dimensional shadows of four-dimensional objects, as was done in sections 2.2.1 and 2.2.2. When the 3-dimensional cross-sections are perpendicular to geometric time, they represent the spatial shape of the 4-dimensional object at a particular time. The evolution in geometric time of the three-dimensional cross-sections produces a 3-dimensional shape of a body as we would see it over time.

Bodies

To better understand the shapes being visualized, we start with the two-dimensional problem, then the three-dimensional problem, followed by the four-dimensional problem. The cross-sections of a square produce an evolving one-dimensional body, the cross-sections of a cube produce an evolving two-dimensional body, and the cross-sections of a tesseract produce an evolving three-dimensional body. It is particularly interesting to see how the appearances of these bodies evolve depending on the orientations of the temporal dimension relative to the orientations of the square, cube, and tesseract. The two figures below show the set-ups for the evolving one-dimensional body (cross-section) and the evolving two-dimensional body (cross-section). Each body evolves in the direction of the unit vector {\b n} along the direction of geometric time. The bottom table shows movies of the evolving cross-sections. The square is aligned with the (1, 0) and (0, 1) directions; the cube with the (1, 0, 0), (0, 1, 0), and (0, 0 ,1) directions and the tesseract with the (1,0,0,0), (0,1,0,0), (0, 0, 1, 0), and (0, 0, 0, 1) directions. The cross-sections move in the directions of geometric time indicated in the figures. As shown, the directions of geometric time considered are near 45⁰ to an axis, and then slightly skewed from a third axis are considered.

	2D	3D	4D
Near 450	$\mathbf {n} ={\begin{bmatrix}0.809\\-0.588\\\end{bmatrix}}$ $\mathbf {n} ={\begin{bmatrix}0.588\\0.809\\\end{bmatrix}}$	$\mathbf {n} ={\begin{bmatrix}0.2938\\0.4045\\0.8660\end{bmatrix}}$	$\mathbf {n} ={\begin{bmatrix}-0.2629\\-0.3514\\-0.1537\\0.9009\end{bmatrix}}$ $\mathbf {n} ={\begin{bmatrix}0.0002\\0.3529\\0.6112\\0.7085\end{bmatrix}}$
Shallow angle	$\mathbf {n} ={\begin{bmatrix}0.0747\\-0.997\\\end{bmatrix}}$ $\mathbf {n} ={\begin{bmatrix}0.997\\0.0747\\\end{bmatrix}}$	$\mathbf {n} ={\begin{bmatrix}0.0522\\0.0039\\0.9986\end{bmatrix}}$	$\mathbf {n} ={\begin{bmatrix}0\\0\\0.0031\\0.9999\end{bmatrix}}$ $\mathbf {n} ={\begin{bmatrix}0\\0.0004\\0.7057\\0.7085\end{bmatrix}}$