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 *x _{1}* space coordinate (with no movement in the

*x*and

_{2}*x*directions) and absolute time

_{3}*t*. That changes to constant

*lenticity*(inverse of velocity) ℓ in the direction of the

*t*

_{1}*time coordinate*(with no movement in the

*t*and

_{2}*t*directions) and absolute space

_{3}*r*, the travel length. The dual Galilean transformations are then

*t _{1}´ = t_{1} − ℓr*

*t*

_{2}´ = t_{2}*t*

_{3}´ = t_{3}*r´ = r*

What does this mean? It means that there is a constant movement measured by lenticity so that as the travel length increases, the duration changes from *t* to *t _{1}´* such that

*t − t*, which is a constant ratio of duration over length multiplied times the travel length.

_{1}´ = ℓrThe Lorentz transformation is analogous to this with the absolute *time speed* of light, c, replaced by the absolute *space speed* of light, *k*, which is the inverse of *c*.

*t _{1}´ = γ (t_{1} − ℓr)*

*t*

_{2}´ = t_{2}>

*t*

_{3}´ = t_{3}*r´ = γ (r − t*

_{1}ℓ/k²)in which

*γ² = 1* / (*1 − ℓ²/k²*).

What does this mean? It means that time (duration) appears dilated and length 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.