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Lorentz from light clocks

Length and duration space are inverse perspectives on motion. Length is measured by a rigid rod at rest, whereas duration is measured by a clock that is always in motion. Duration is measured by a clock at rest relative to the time frame, whereas length is measured by a rigid rod in motion that counteracts time as it were.

This is illustrated by deriving the Lorentz factor for time dilation and length contraction from light clocks. The first derivation is in length space with scalar time and the second is in duration space with scalar stance.

light at restlight moving

The first figure above shows a light clock in length 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 h. Let one time cycle Δt = 2h/c = 2 or h = cΔt/2 = Δt/(2¢), with mean speed of light c and mean pace of light ¢.

The second figure shows the light clock as observed by someone moving with velocity v and pace u relative to the light clock. Call the path length of each half-cycle d; and call the length of the base of one half-cycle b.

From the moving observer’s perspective, the clock travels b = vΔt´ = Δt´/u, with time interval Δt´. The light beam travels 2d = cΔt´ = Δt´/¢ each cycle. If we normalize v with c and u with ¢, then v/c = ¢/u = β, c¢ = 1, b = βΔt´, 2d = Δt´, and h = Δt/2.

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

t´/2)² = (βΔt´/2)² + (Δt/2)².

Multiply both sides by 2². Gather the Δt´ coefficients to get

Δt´² (1 − β²) = Δt².

This leads to

Δt´² = Δt² / (1 − β²),

so that

Δt´ = Δt √(1/(1 − β²)) = γΔt,

which is the time dilation with the Lorentz factor γ.


Now consider the light clock in time with scalar space or stance.

light clock in timelight clock in time over space

In this case the observation in the first figure is of a light clock in time, i.e., a series of cycles. As before, it is a beam of light reflected back and forth between two mirrored surfaces. Call the half-cycle duration of the light beam between the surfaces k. Call one half-cycle Δr´ = 2k or k = Δr´/2, with the speed of light set to unity as above.

The second figure shows a light clock at rest as observed by someone moving with lenticity u relative to it. Let the round-trip duration of each cycle be e, and let the duration of the base of one triangle-shaped cycle be a.

The moving observer travels a = βΔr, with stance interval Δr. The light beam travels 2e = Δr each cycle.

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

r/2)² = (βΔr/2)² + (Δr´/2)².

Multiply both sides by 2². Gather the Δr coefficients to get

Δr²(1 − β²) = Δr´².

This leads to

Δr´ = Δr √(1 − β²) = Δr / γ,

which is the length contraction with the Lorentz factor γ.

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