7B (Special Relativity)

Write

(t, x)

for coordinates in the rest frame of

A_{0}

and

(t^{'}, x^{'})

for those in the rest frame of

A_{1}

x^{'} = γ_{1} (x - v_{1} t)

and

t^{'} = γ_{1} (t - v_{1} x / c^{2})

. Along the worldline of

A_{2}

x^{'} = v_{2} t^{'}

. Cancelling

γ_{1}

gives

x - v_{1} t = v_{2} (t - v_{1} x / c^{2})

v_{02} = \frac{v _{1} + v _{2}}{1 + v _{1} v _{2} / c ^{2}} .

A_{0}

and

A_{2}

A_{3}

. Its velocity in the former is

(v_{02} + v_{3}) / (1 + v_{02} v_{3} / c^{2})

. Substitution simplifies to

v_{03} = \frac{v _{1} + v _{2} + v _{3} + v _{1} v _{2} v _{3} / c ^{2}}{1 + ( v _{1} v _{2} + v _{1} v _{3} + v _{2} v _{3} ) / c ^{2}} .

Here

∣ v_{i} ∣ < c

and the formulas use signed collinear velocities. In the given positive-direction setting they are positive speeds. The limit

c \to \infty

recovers ordinary addition, providing a check on both expressions.

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