= Angular size under a collinear relativistic boost
{title2=$\Theta'/\Theta=e^{-\eta}=\sqrt{(1-\beta)/(1+\beta)}$}
Let an observer move towards the centre of a small distant source with <velocity> $\beta c$, where $c$ is the <speed of light>. For source-direction <angle> $\theta$, the photon arrives with longitudinal <momentum> $-E\cos\theta/c$. A <Lorentz transformation> gives $E'=\gamma E(1+\beta\cos\theta)$ and therefore the <relativistic aberration> law
$$
\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta},\qquad
\tan\frac{\theta'}2=e^{-\eta}\tan\frac\theta2,
$$
where $\eta=\operatorname{artanh}\beta$ is the <rapidity>. For a small <angular diameter> $\Theta$, the displayed angular-size ratio follows by linearizing at $\theta=0$. Two momentarily coincident observers can consequently measure different angular sizes without any change in their observation event. If their diameter ratio is $q\ge1$, their relative speed is $c(q^2-1)/(q^2+1)$.
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