Fred measures (x, y, z, t) ⇒ Jane measures (x′, y′, z′, t′) x, y, z, t and relative speed v along x axis x′, y′, z′, t′ and relative speed − v along x axis

\begin{aligned}x′ & = x − vt \\ y′ & = y \\ z′ & = z \\ t′ & = t \end{aligned}

\begin{aligned}x′ & = γ(x − vt) \\ y′ & = y \\ z′ & = z \\ t′ & = γ(t − \frac{vx}{c^{2}}) \end{aligned}

t′ = γ(t − vx) v′ = v − (−w) = v + w

TWEET  (x_{1}, t_{1})
meet dog  (x_{2}, t_{2})
TWEET  x′_{1} = x_{1} − (−w)t_{1} , t′_{1} = t_{1}
meet dog  x′_{2} = x_{2} − (−w)t_{2} , t′_{2} = t_{2}

v′ = \frac{(x′_{2} − x′_{1})}{(t′_{2} − t′_{1})}

= \frac{(x_{2} − (−w)t_{2}) − (x_{1} − (−w)t_{1})}{(t_{2} − t_{1})}

= \frac{(x_{2} − x_{1})}{(t_{2} − t_{1})} + \frac{w(t_{2} − t_{1})}{(t_{2} − t_{1})}

= v + w Galilean version (if w < 0)

TWEET  x′_{1} = γ(x_{1} − (−w)t_{1})
t′_{1} = γ (t_{1} − \frac{(−w)x_{1}}{c^{2}} )
meet dog  x′_{2} = γ(x_{2} − (−w)t_{2})
t′_{2} = γ (t_{2} − \frac{(−w)x_{2}}{c^{2}} )
v′ = \frac{(x′_{2} − x′_{1})}{(t′_{2} − t′_{1})}

= \frac{γ(x_{2} − (−w)t_{2}) − γ(x_{1} − (−w)t_{1})}{γ (t_{2} − \frac{(−w)x_{2}}{c^{2}} ) − γ (t_{1} − \frac{(−w)x_{1}}{c^{2}} )}

= \frac{x_{2} + wt_{2} − x_{1} − wt_{1}}{t_{2} + \frac{wx_{2}}{c^{2}} − t_{1} − \frac{wx_{1}}{c^{2}}}

= \frac{(x_{2} − x_{1}) + w(t_{2} − t_{1})}{(t_{2} − t_{1}) + \frac{w(x_{2}−x_{1})}{c^{2}}}

v′ = \frac{v + w}{1 + \frac{vw}{c^{2}}} Lorentz version (if w < 0) v′ = \frac{v − w}{1 − \frac{vw}{c^{2}}} Lorentz version (general formula)

spark 1  x_{1} = −10  t_{1} = 0
spark 2  x_{2} = 10  t_{2} = 0
spark 1  x′_{1} =  t′_{1} =
spark 2  x′_{2} =  t′_{2} =