February 19, 2021
Proper Velocity, Coordinate Velocity, Relative Motion
Since the time of Galileo, the same definition of velocity has been generally accepted. The definition (usually called Coordinate Velocity) is now shown to be secondary to a more fundamental measure of relative motion (usually called Proper Velocity). The method is that the transformation equations for Special Relativity are derived without introducing velocity or assuming a definition of velocity. For Inertial Frames K and K′, the transformation equations are: x′ = ax – bct, ct′ = act – bx, x = ax′ + bct′, ct = act′ + bx′ , a² = b² + 1; c is the velocity of light; a and b are constants. Four velocity measures are found. Coordinate Velocity, v, = x/ct or − x′/ct′ = b ∕ a = b/√(b² + 1). Proper Velocity, w, = x/ct′ or − x′/ct = b. Thus, v = w/√(w² + 1). The Transformation Equations for Proper Velocity replace b with w. They are x′ = ax – wct, ct′ = act – wx, x = ax′ + wct′, ct = act′ + wx′ , a² = w² + 1. The algebra shows v as a function of w, making w, Proper Velocity, the primary value. At slow velocities, Proper Velocity is an exact fit in the Galilean transformation.
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