Angular-diameter ratio under a collinear boost ($|v|/c=(r^2-1)/(r^2+1)$)

For two coincident observers with relative motion along the sightline to a small distant source, its angular diameter changes by a rapidity factor. A boost toward the source compresses its incoming angular cone: $\tan ({\vartheta }^{\prime }/2)=e^{-\eta }\tan (\vartheta /2)$ for angular radius $\vartheta$. For a small source the larger-to-smaller diameter ratio is $r=e^{|\eta |}$, giving the displayed relative speed. A finite angular size requires the half-angle formula, and arbitrary transverse relative motion cannot be inferred from one diameter ratio alone.