iSoul In the beginning is reality.

Dual Galilean transformation

The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to

x = ∫ v dt = x0 + vt

So for two observers at constant velocity in relation to each other we have

x′ = x + vt

with their time coordinates unchanged: t′ = t if their origins coincide.

The dual Galilean transformation is based on the definition of lenticity: w = dt/dx, which for constant lenticity leads to

t = ∫ w dx = t0 + wx

So for two observers at constant lenticity in relation to each other we have

t′ = t + wx

with their length coordinates unchanged: x′ = x if their origins coincide.

These transformations reinforce the proposition that time is not necessarily the independent variable, and so is best understood as measured by a stopwatch rather than a clock.

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