Relativistic aberration

At a sightline perpendicular to Earth's instantaneous orbital velocity, a heliocentrically transverse ray acquires an apparent angular displacement towards the observer's motion. A telescope of length $L$ moves transversely by $v_{\oplus }L/c$ during the light crossing, giving the first-order result. The exact transverse Lorentz boost gives $\sin \delta =v_{\oplus }/c$. For a circular orbital radius $R$ and year $Y$, $v_{\oplus }=2\pi R/Y$. The amplitude is half the full opposite-season excursion for a suitable sightline; it is not annual parallax, which depends on stellar distance.

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.

The spinor boost $\mathrm{diag}(e^{\kappa /2},e^{-\kappa /2})$ changes the component-magnitude ratio by $e^{-\kappa }$. The associated active positive-$z$ Lorentz boost therefore changes the polar half-angle as displayed and leaves azimuth unchanged. Its null-vector time scale changes by $\cosh \kappa +\sinh \kappa \cos \theta$, which must be retained separately when normalizing a spinor to encode direction.

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 $E^{\prime }=\gamma E(1+\beta \cos \theta )$ and therefore the relativistic aberration lawwhere $\eta =\mathrm{artanh}\beta$ 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 $q\ge 1$, their relative speed is $c(q^2-1)/(q^2+1)$.

For isotropic planar photon emission in a rest frame, aberration of light maps its forward half of the angular directions into a lab cone of semi-angle $\alpha$ satisfying the displayed equation. This follows from $\cos \theta =(\cos {\theta }^{\prime }+\beta )/(1+\beta \cos {\theta }^{\prime })$ at ${\theta }^{\prime }=\pm \pi /2$. For a fast positive Lorentz boost, $\alpha \sim 1/\gamma$. The half fraction concerns photon count or directions: the relativistic Doppler effect changes the angular energy weights, so the cone need not contain half the radiant energy.