iSoul In the beginning is reality

# From time to space and back

One question is how to translate from time rates to space rates and vice versa. Consider scalar space and scalar time, and designate the spatial position, s, initial spatial position, s0, temporal position, t, initial temporal position, t0,, velocity, v, initial velocity, v0, acceleration a (assumed constant over time), legerity, u, initial legerity, u0, and expedience, b (assumed constant over space). Then the linear equations of motion are as follows (see also the equations of motion at the top menu or here):

s = s0 + vt; v = v0 + at; s = s0 + v0t + ½at²; = v0² + 2a(s s0);

t = t0 + us; u = u0 + bs; t = t0 + u0s + ½bs²; u² = u0² + 2b(t t0)

From these we have the following derivatives:

ds/dt = v = 1/u; dv/dt = a; dt/ds = u = 1/v; du/ds = b;

du/dv = –1/v² = –u² and dv/du = –1/u² = –v².

If we are given the acceleration, a, what is the expedience, b? This may be determined as follows:

b = du/ds = dv/dt * dt/ds * du/dv = a * (1/v) * (–1/v²) = –a/v³ = –a/(v0 + at)³.

If v0 = 0, then b(t) = –1/(a²t³).

So the expedience is a function of scalar time.

Similarly, if we are given the expedience, b, what is the acceleration, a? This may be determined as follows:

a = dv/dt = du/ds * ds/dt * dv/du = b * (1/u) * (–1/u²) = –b/u³ = –b/(u0 + bs)³.

If u0 = 0, then a(s) = –1/(b²s³).

So the acceleration is a function of scalar space.