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Lorentz factor from light clocks

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 Δt = 2h/c or h = cΔt/2, with speed of light c, which is the maximum speed.

The second figure shows frame 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 2d = cΔt´ each cycle. From the moving observer’s perspective , the clock travels b = vΔt´, with time interval Δt´. If we normalize v with c, then v/c = β and b = βcΔt´.

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

(cΔt´/2)² = (βcΔt´/2)² + (cΔt/2)².

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

Δt´² (1 − β²) = Δt².

This leads to

Δt´² = Δt² / (1 − β²),

so that

Δt´ = Δt /√(1 − β²) = γΔt,

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 Δr = 2g/k or g = kΔr/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 2e = kΔr´ each cycle. From the moving observer’s perspective, the clock travels in time a = ℓΔ, with placepoint interval Δr´. If we normalize ℓ with k as k/ℓ = v/c = β (since k is a minimum), then a = ℓΔr´ = βkΔ, and g = kΔr/2.

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

(kΔ/2)² = (βkΔ/2)² + (kΔr/2)².

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

Δ²(1 − β²) = Δr².

This leads to

Δr´ = Δr/√(1 − β²) = γΔr,

which is placepoint interval dilation with the Lorentz factor γ.

First version April 4, 2019.

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