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Interchange of length and duration spaces

In geometry, a spherical coordinate system specifies points by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle, which is the horizontal angle between the origin to the point of interest.

A spherical coordinate system is implicitly behind length space (space-time) (3+1), with the temporal directional ignored for scalar time. That is, every instant in duration space (3D time) is projected onto a temporal sphere centered on the origin instant. The scalar time is the radial distime of each instant.

Let the spherical coordinates of length space be (r, θ, φ) with r representing the radial distance, and θ and φ representing the zenith and azimuth angles, respectively. Let the spherical coordinates of duration space (3D time) be (t, χ, ψ) with t representing the radial distime, and χ and ψ representing the temporal zenith and azimuth angles, respectively.

To represent the fullness of length space and duration space requires six dimensions (3+3), three for length and three for duration: ((r, θ, φ); (t, χ, ψ)) or (r, θ, φ; t, χ, ψ). Then length-duration (space-time) (3+1) can be represented by the coordinates [r, θ, φ; t] and duration-length (time-space) (1+3) by the coordinates 〈r; t, χ, ψ〉.

If rectilinear coordinates are used for duration space, say (ξ, η, ζ), then the time, i.e., radial distime, t = √(ξ² + η² + ζ²). The corresponding length space concept, base, is the radial distance. If rectilinear coordinates are used for length space, say (x, y, z), then the base r = √(x² + y² + z²). For 2D applications such as mapping, polar coordinates would be used instead of spherical, in which case r = √(x² + y²) and t = √(ξ² + η²).

To convert (3+1) function to a (1+3) function requires expansion to its (3+3) function, inversion by interchange of length and duration, and then contraction to (1+3). In symbols with ↑ as expand, ↓ as contract, and ↔ as interchange: (3+1) ↑ (3+3) ↓ (1+3), or (r, θ, φ; t) ↑ (r, θ, φ; t, χ, ψ) ↓ (r; t, χ, ψ). In this way length and duration, time and base are interchanged.

In more detail, a parametric length space function is converted to a parametric time function: r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r) = [t, χ, ψ] = t.

Take for example the definition v = dr/dt. We have: v = dr/dt = [dr/dt, dθ/dt, dφ/dt] = [r(t), θ(t), φ(t)]) ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = [dt/dr, dχ/dr, dψ/dt] = dt/dr = u. The result is that length space and duration space are interchanged, with length space vectors becoming bases and times becoming duration space vectors.

Functions may be converted by differentiation, then inversion, then integration. Take for example, the function s(t) = s0 + v0t + ½at². Differentiating twice leads to s(t)´´ = a = dv/dt. Expanding to spherical coordinates, inverting, and contracting to the radial component results in t(s)´´ = du/ds = b. Integrating twice produces t(s) = t0 + u0s + ½bs², which has the same form as the original function.