iSoul In the beginning is reality

Duals for Galilean and Lorentz transformations

In Newtonian mechanics inertial frames of reference are related by a Galilean transformation and time is absolute. In the special theory of relativity (STR) inertial frames of reference are related by a Lorentz transformation and the speed of light is absolute. By taking account of the three dimensions of time with a single dimension of space, we may derive a dual mechanics to each of these: (1) a Newtonian dual with a single dimension of absolute space and three dimensions of relative time, and (2) an STR dual with a single dimension of relative space and three dimensions of relative time.

In the usual exposition there is a reference frame S’ moving with constant velocity v in the direction of the x1 space coordinate (with no movement in the x2 and x3 directions) and absolute time t. That changes to constant argosity (inverse of velocity) w in the direction of the t1 time coordinate (with no movement in the t2 and t3 directions) and absolute space r, the distance traveled. The dual Galilean transformations are then

t1‘ = t1 – wr

t2‘ = t2

t3‘ = t3

r’ = r

What does this mean? It means that there is a constant movement measured by argosity so that as the distance increases, the duration changes from t to t1 such that t – t1‘ = wr, which is a constant ratio of duration over distance multiplied times the distance traveled.

The Lorentz transformation is analogous to this with the absolute time speed of light, c, replaced by the absolute space speed of light, which is the inverse of c, or 1/c.

t1‘ = γ (t1 – wr)

t2‘ = t2

t3‘ = t3

r’ = γ (r – t1wc²)

in which

γ = 1 / sqrt(1 – w²c²)

What does this mean? It means that time (duration) appears dilated and length (distance) appears contracted, which is the same as the standard Lorentz transformation (known as a Lorentz boost). The laws of physics remain the same despite changing to a space reference from a time reference.

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