“In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified 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* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.” (*Wikipedia*)

A 3D time spherical coordinate system is implicitly behind space-time (3+1), with the two angles ignored for scalar time. That is, every instant in 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 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 3D time be (*t, β, α*) with *t* representing the radial distime, and *β* and *α* representing the temporal zenith and azimuth angles, respectively.

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

If rectilinear coordinates are used for 3D time, say (*ξ, η, ζ*), then the radial distime *t* equals √(*ξ*² + *η*² + *ζ*²). The corresponding spatial concept, *stance*, is the radial distance, which if rectilinear coordinates are used for 3D space, say (*x, y, z*), then the stance *r* equals √(*x*² + *y*² + *z*²). For 2D applications such as mapping, polar coordinates would be used instead of spherical, in which case *r* = √(*x*² + *y*²) and *t* = √(*ξ*² + *η*²).

The result is that to convert an invertible (3+1) function to a (1+3) function requires expansion to the (3+3) function, inversion, and then contraction to (1+3). In symbols, (3+1) ⇑ (3+3) ⇓ (1+3), or (*r, θ, φ*; *t*) ⇑ (*r, θ, φ*; *t, β, α*) ⇓ (*r*; *t, β, α*). In this way space and time are interchanged.

In symbols, from a invertible parametric space function inverted to a parametric time function (with ⇑ as expand, ⇓ as contract, and ↔ as invert): **r** = [*r, θ, φ*] = **r**(*t*) = [*r*(*t*), *θ*(*t*), *φ*(*t*)] ⇑ [*r´*(*t´, β´, α´*), *θ´*(*t´, β´, α´*), *φ´*(*t´, β´, α´*)] ↔ (*t´*(*r´, θ´, φ´*), *β´*(*r´, θ´, φ´*), *α´*(*r´, θ´, φ´*)) ⇓ [*t*(*r*), *β*(*r*), *α*(*r*)] = **t**(*r*) = [*t, β, α*] = **t**.

Take for example the definition **v** = d**r**/d*t*. We have: **v** = d**r**/d*t* = [d*r*/d*t*, d*θ*/d*t*, d*φ*/d*t*] = [*r*(*t*), *θ*(*t*), *φ*(*t*)]) ⇑ [*r´*(*t´, β´, α´*), *θ´*(*t´, β´, α´*), *φ´*(*t´, β´, α´*)] ↔ (*t´*(*r´, θ´, φ´*), *β´*(*r´, θ´, φ´*), *α´*(*r´, θ´, φ´*)) ⇓ [*t*(*r*), *β*(*r*), *α*(*r*)] = [d*t*/d*r*, d*β*/d*r*, d*α*/d*t*] = d**t**/d*r* = **u**. The result is that space and time are interchanged, with spatial vectors becoming radial distances and radial distimes becoming temporal vectors.

Functions that are not invertible may be inverted by differentiation, then integration. Take for example, the function **s**(*t*) = **s**_{0} + **v**_{0}*t *+ ½**a***t*². Differentiating twice leads to **s**(*t*)´´= **a** = d**v**/d*t*. Expanding, inverting, and contracting results in **t**(*s*)´´= d**u**/d*s* = **b**. Integrating twice produces **t**(*s*) = **t**_{0} + **u**_{0}*s *+ ½**b***s*², which has the same form as the original function.