Space and time are inverse perspectives on motion. Space is three dimensions of length. Time is three dimensions of duration. Space is measured by a rigid rod at rest, whereas time is measured by a clock that is always in motion relative to itself.

This is illustrated by deriving the Lorentz factor for time dilation and length contraction from light clocks. The first derivation is in space with a time parameter and the second is in time with a space parameter (placepoint).

The first figure above shows frame *S* with a light clock in space as a beam of light reflected back and forth between two mirrored surfaces. Call the height between the surfaces that the light beam travels distance *h*. Let one time cycle Δ*t* = 2*h*/*c* or *h* = *c*Δ*t*/2, with speed of light* c*, which is the maximum speed.

The second figure shows frame *S´* with the same light clock as observed by someone moving with velocity *v* relative to *S*. Call the length of each half-cycle *d*, and call the length of the base of one cycle in space *b*.

The light beam travels 2*d* = *c*Δ*t*´ each cycle. From the moving observer’s perspective *S´*, the clock travels *b* = *v*Δ*t*´, with time interval Δ*t*´. If we normalize *v* with *c*, then *v*/*c* = *β* and *b* = *βc*Δ*t*´.

From the Euclidean metric for space, (*d*)² = (*b*/2)² + (*h*)². Substitution gives

(*c*Δ*t*´/2)² = (*βc*Δ*t*´/2)² + (*c*Δ*t*/2)².

Multiply both sides by (2/*c*)². Gather the Δ*t*´ coefficients to get

Δ*t*´² (1 − *β*²) = Δ*t*².

This leads to

Δ*t*´² = Δ*t*² / (1 − *β*²),

so that

Δ*t*´ = Δ*t* /√(1 − *β*²) = *γ*Δ*t*,

which is time dilation with the Lorentz factor *γ*.

Now consider the light clock *in time* with a space parameter (placepoint).

The first figure above shows frame *S* with a light clock in time with a space parameter (placepoint). As before, it is a beam of light reflected back and forth between two mirrored surfaces. Call the time that the light beam travels between the surfaces the duration *g*. Let one placepoint cycle Δ*r* = 2g/*k* or *g* = *k*Δ*r*/2, with pace of light *k*, which is the minimum pace.

The second figure shows the light clock at rest as observed by someone moving with lenticity ℓ relative to it. Call the duration of each half-cycle *e*, and call the duration of the base of one triangle-shaped cycle in time *a*.

The light beam travels 2*e* = *k*Δ*r*´ each cycle. From the moving observer’s perspective, the clock travels in time *a* = ℓΔ*r´*, with placepoint interval Δ*r*´. If we normalize ℓ with *k* as *k*/ℓ = *v*/*c* = *β* (since *k* is a minimum), then *a* = ℓΔ*r*´ = *βk*Δ*r´*, and *g* = *k*Δ*r*/2.

From the Euclidean metric in time, (*e*)² = (*a*/2)² + (*g*)². Substitution gives

(*k*Δ*r´*/2)² = (*βk*Δ*r´*/2)² + (*k*Δ*r*/2)².

Multiply both sides by (2/*k*)². Gather the Δ*r´* coefficients to get

Δ*r´*²(1 − *β*²) = Δ*r*².

This leads to

Δ*r*´ = Δ*r*/√(1 − *β*²) = *γ*Δ*r,*

which is placepoint interval dilation with the Lorentz factor *γ*.

*First version April 4, 2019.*