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The science
An observer moving with speed v = beta c along +x sees a star that lies at angle theta from +x in the rest frame at the aberrated angle cos(theta') = (cos(theta) + beta) / (1 + beta cos(theta)). The sky crowds towards the direction of motion. The photon frequency is multiplied by the Doppler factor D = gamma (1 + beta cos(theta)) = 1 / (gamma (1 - beta cos(theta'))) with gamma = 1 / sqrt(1 - beta^2): blueshift ahead, redshift behind.
Solid angles shrink by d(Omega')/d(Omega) = 1/D^2, so the star density ahead grows as D^2. Because the ratio I_nu / nu^3 is a Lorentz invariant, the surface brightness of a uniform glowing sky (bolometric) scales as D^4; this is the relativistic headlight effect. Half of the light of an isotropic emitter in the source frame is squeezed into a forward cone of half-angle theta' = arccos(beta) (as seen in the other frame).
Simplifications: a flat 2D slice through the observer (no rotation of the 3D sky), point stars with identical rest-frame brightness and a simple colour map for the shift, no spectral lines, and no gravitational effects. The 3D page renders the sky; this page plots the transformation. Computed exactly with the Lorentz boost, not approximated.
What to try
- Choose 0.9 c: the forward half-light cone is only about 26 degrees wide.
- Use the sweep to accelerate from 0 to 0.99 c and watch the stars flow forward.
- Compare the D^4 headlight curve with the star-count curve D^2 on the third graph.