iSoul In the beginning is reality

# Dual Galilei and Lorentz transformations

I keep going over this because it has been so overlooked for 100 years. The Galilei transformation (GT) is based on three space dimensions and one time dimension (3S+1T). Once it is realized that time is just as dimensional as space, there is a dual Galilei transformation based on one space dimension and three time dimensions (1S+3T). The one dimensional time or space are actually “uni-dimensional” since they combine all their dimensions of time or space together.

In standard configurations all coordinates are zero at one point and only one coordinate in space and one in time are non-zero. That is, in the moving plane r1 = vt, which is the same as saying r1 = vt1 or t1 = r1/v since t = t1 in this case. The GT are then:

3S+1T: t1´ = t1, r1´ = r1 – vt1, r2´ = r2, r3´ = r3.

1S+3T: r1´ = r1, t1´ = t1 – r1/v, t2´ = t2, t3´ = t3.

There are various derivations of the Lorentz transformation (LT). Take Rindler’s from his 2006 book Relativity. The key step is this (p.44):

Next, suppose x´ = γx + Fy + Gz + Ht + J. By the choice of coordinates, x = vt must imply x´ = 0, so γv + H, F, G, J all vanish and x´ = γ(x – vt).

Let’s add the subscripts as above:

Next, suppose r1´ = γr1 + Fr2 + Gr3 + Ht1 + J. By the choice of coordinates, r1 = vt1 must imply r1´ = 0, so γv + H, F, G, J all vanish and r1´ = γ(r1 – vt1).

We can also proceed in terms of t1´ instead of r1´:

Next, suppose t1´ = γt1 + Ft2 + Gt3 + Hr1 + J. By the choice of coordinates, t1 = r1/v must imply t1´ = 0, so γ/v + H, F, G, J all vanish and t1´ = γ(t1 – r1/v).

That leads, as we’ve shown several times (e.g., here) to a dual LT, not differentiated by dimension but by speed:

|v| < c: r1´ = γ (r1 – vt1), r2´ = r2, r3´ = r3, t1´ = γ (t1 – r1v/c²), t2´ = t2, t3´ = t3, with γ = √(1 / (1 – v²/c²));

|v| > c: t1´ = γ (t1 – r1/v), t2´ = t2, t3´ = t3, r1´ = γ (r1 – t1c²/v), r2´ = r2, r3´ = r3, with γ = √(1 / (1 – c²/v²)).

Otherwise, |v| = c: r1´ = r1, r2´ = r2, r3´ = r3, t1´ = t1, t2´ = t2, t3´ = t3, with γ = 1.

In summary, there are dual transformation for both GT and LT, and the total number of dimensions is six. In GT the transformations are distinguished by collapsing the six dimensions to four in either of two ways. In LT the transformations are distinguished by the relationship between the speed of the object (or the frame) and the standard speed.