Rapidity

a) Consider two successive Lorentz transformations of the three frames of reference K₀, K₁, K₂. K₁ moves parallel to the x axis of K₀ with velocity v, as does K₂ with respect to K₁. Given an object moving in the x direction with velocity v₂ in K₂ derive the formula for the transformation of its velocity from K₂ to K₀.

b) Now consider n+1 frames moving with the same velocity v relative to one another (see Figure 1.1). Derive the formula for a Lorentz transformation from K_n to K₀ if the velocity of the object in K_n is also v.

 Figure 1.1

Hint: You may want to use the definition of rapidity or velocity parameter, tanh ψ = β, where β = v/c.

SOLUTION

a) The velocity of the particle moving in frame K₁ with velocity v₂ in the frame K₁ is given by a standard formula:

Introducing β_i = v_i/c we may rewrite this formula in the form

(1)

Now the same formula may be written for a transformation from K₁ to K₀:

(2)

Now substituting (1) into (2), we obtain

b) If we need to make a transformation for n-frames, it is difficult to obtain a formula using the approach in (a). Instead, we use the idea of rapidity, ѱ. Indeed for one frame, we had in (1)

which is the formula for the tanh of a sum of arguments

where tanh ѱ_i = β_i. This means that the consecutive Lorentz transformations are equivalent to adding rapidities. So the velocity in the frame K₀ after n transformations (if v_n+1 = v ) will be given by

We can check that if n → ∞, then β_i → 1.