9A (Special Relativity)

For a standard collinear Lorentz boost, put

β = v / c

and

Γ = (1 - β^{2})^{- 1/2}

. Its equations are

x^{'} = Γ (x - β c t)

and

c t^{'} = Γ (c t - β x)

ϕ = arctanh β

. Since

cosh^{2} ϕ - sinh^{2} ϕ = 1

and

cosh ϕ > 0

cosh ϕ = Γ

and

sinh ϕ = Γ β

. Therefore

x^{'} = x cosh ϕ - c t sinh ϕ, c t^{'} = - x sinh ϕ + c t cosh ϕ .

Adding and subtracting, and using

cosh ϕ \mp sinh ϕ = e^{\mp ϕ}

, gives

x^{'} + c t^{'} = e^{- ϕ} (x + c t), x^{'} - c t^{'} = e^{ϕ} (x - c t) .

A second boost multiplies these factors by

e^{\mp ϕ^{'}}

, so the combined rapidity is

ϕ + ϕ^{'}

v^{''} = c tanh (ϕ + ϕ^{'}) = \frac{v + v ^{'}}{1 + v v ^{'} / c ^{2}} .

Velocities are signed along the common axis, so this formula also covers oppositely directed boosts.

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