Velocity-addition Formula - Derivation

Derivation

Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the x and y axes, it is convenient to use the same units for space and time, otherwise there is a unit conversion factor which converts incessantly between the units. This conversion factor is the speed of light, which turns time intervals measured in seconds to lengths measured in meters. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to 1. All velocities are then fractions of the speed of light. To translate to engineering units, replace v with v/c everywhere.

To find the relativistic transformation law, it is useful to introduce the four-velocities (V0, V1, 0) and (U0, U1, U2). The four-velocity is defined to be a four vector with relativistic length equal to 1, tangent to the space-time path of the object. It is convenient to take the x-axis to be the direction of motion of the ship, and the y-axis so that the x-y plane is the plane spanned by the motion of the ship and the fly. The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing:

\, \mathbf{v}= (V_1/V_0, 0 )
\, \mathbf{u}= (U_1/U_0, U_2/U_0)

Since the relativistic length of V is 1,

so

The Lorentz transformation matrix which boosts the rest frame to four-velocity V is then:

\begin{pmatrix} V_0 & V_1 & 0 \\ V_1 & V_0 & 0 \\ 0 & 0 & 1 \end{pmatrix}

This matrix rotates the a pure time-axis vector (1, 0, 0) to (V0, V1, 0), and all its columns are relativistically perpendicular to one another, so it defines a Lorentz transformation.

If a fly is moving with four-velocity (U0, U1, U2) in the rest frame, and it is boosted by multiplying by the matrix above, the new four-velocity is (S0, S1, S2):

\, S_0 = V_0 U_0 + V_1 U_1
\, S_1 = V_1 U_0 + V_0 U_1
\, S_2 = U_2

Dividing by the time component S0 and replacing the four-vectors for U and V by the three vectors u and v gives the relativistic composition law:

\, s_1 = { v_1 + u_1 \over 1 + v_1 u_1 }
\, s_2 = { 1 \over V_0}{u_2 \over (1 + v_1 u_1) } = \sqrt{1-v^2}\, { u_2 \over 1 + v_1 u_1 }

The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out. The failure of simultaneity means that the fly is changing simultaneity slices as the projection of u onto v. Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of 1/V0 = √(1 − v2).

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