Along the particle's worldline,

x = u_{x} t, y = u_{y} t .

The boost from

S

S^{'}

gives

x^{'} = γ (x - V t), y^{'} = y, t^{'} = γ (t - \frac{V x}{c ^{2}}) .

Therefore

\frac{d t ^{'}}{d t} = γ (1 - \frac{V u _{x}}{c ^{2}}),

and division of the transformed coordinate velocities by this factor gives the relativistic velocity-addition formula

u_{x}^{'} = \frac{u _{x} - V}{1 - V u _{x} / c ^{2}}, u_{y}^{'} = \frac{u _{y}}{γ ( 1 - V u _{x} / c ^{2} )}, u_{z}^{'} = 0 .

ii

Solution

To first order in

V / c

γ = 1 + O (V^{2} / c^{2})

and

\frac{1}{1 - V u _{x} / c ^{2}} = 1 + \frac{V u _{x}}{c ^{2}} + O (V^{2} / c^{2}) .

Thus

u_{x}^{'} = u_{x} - V + \frac{V u _{x}^{2}}{c ^{2}} + O (V^{2} / c^{2}), u_{y}^{'} = u_{y} + \frac{V u _{x} u _{y}}{c ^{2}} + O (V^{2} / c^{2}) .

Since

V = (V, 0, 0)

and

V \cdot u = V u_{x}

, these components combine into

u^{'} = u - V + \frac{V \cdot u}{c ^{2}} u + O (V^{2} / c^{2}) .

u = c (cos θ, sin θ, 0) .

Part (ii) gives the first-order changes

δ u_{x} = - V sin^{2} θ, δ u_{y} = V sin θ cos θ .

For a vector of fixed leading-order magnitude

c

, its small change of angle is

δ θ = \frac{u _{x} δ u _{y} - u _{y} δ u _{x}}{c ^{2}} .

θ^{'} - θ = \frac{V}{c} sin θ

to leading order.

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