Relativistic Doppler shift and aberration

Let the source emit a wavefront every T₀ = 1/f₀ by its own clock while it recedes from the observer at speed v, and do the bookkeeping in the observer's frame. First, time dilation: the source's clock runs slow, so wavefronts are emitted every γT₀ of observer time. Second, light-travel time: between two emissions the source moves vγT₀ farther away, so the second wavefront has that much more path and arrives a further vγT₀/c behind. The arrival interval is T = γT₀(1 + β), so f = f₀/[γ(1 + β)]. Now tidy the algebra: 1/γ = √(1 − β²) = √((1 − β)(1 + β)), and dividing by (1 + β) leaves f = f₀√((1 − β)/(1 + β)). For approach the path shrinks instead, the sign in the bracket flips, and f = f₀√((1 + β)/(1 − β)). At β = 0.60, γ = 1.25 and the factors are 1/(1.25 × 1.60) = 0.50 for recession and 1/(1.25 × 0.40) = 2.00 for approach. Wavelength scales the other way, λ = c/f, so a receding source at 0.60c stretches every wavelength by exactly 2.

For sound, a source approaching a fixed listener gives f₀/(1 − β), while a listener approaching a fixed source gives f₀(1 + β), with β the speed over the speed of sound: different numbers, because the air decides who is moving. Light in vacuum has no medium, and the principle of relativity forbids any experiment that tells 'source moving' from 'observer moving'. The relativistic factor √((1 + β)/(1 − β)) is the geometric mean of the two classical answers, √[(1 + β) × 1/(1 − β)], and it is the one answer both frames can share. Expand it: 1 + β + β²/2 + β³/2 + …. The classical forms are 1 + β and 1 + β + β² + …, so all three agree at first order and the relativistic one splits the difference at second order. At β = 0.10 they predict 1.1000, 1.1111 and 1.1055. Ives and Stilwell (1938) isolated that β²/2 term with hydrogen ions at β ≈ 0.005, viewing the beam head-on and in a mirror at once: the mean of the blue- and red-shifted lines sits at γλ₀, so the first-order shifts cancel and only the time-dilation remainder survives.

Let the source move along a direction making angle θ, in the observer's frame, with the light's path to the observer: θ = 0 is head-on approach and 180° is direct recession. Between two emissions γT₀ apart the source moves vγT₀, and the component of that displacement along the line of sight, vγT₀ cos θ, is the amount by which the second wavefront's path is shorter — the observer is far enough away that the two paths are parallel. The arrival interval is therefore γT₀(1 − β cos θ), and f = f₀/[γ(1 − β cos θ)]. Two factors do two jobs: the 1/γ is time dilation and never changes sign; the bracket is the light-travel term, blue for cos θ > 0 and red for cos θ < 0. They cancel where 1 − β cos θ₀ = 1/γ, that is cos θ₀ = (1 − 1/γ)/β. At β = 0.80, γ = 5/3 and cos θ₀ = 0.40/0.80 = 0.50, so θ₀ = 60°: a source still moving partly towards you shows no shift, and every angle beyond 60° is redshifted. Dimensions check — β and cos θ are pure numbers, so f carries the unit of f₀ — and the limit β → 0 returns f = f₀ at every angle.

Set θ = 90° in the observer's frame. The light was emitted from the point of closest approach, the light-travel term is exactly 1, and f = f₀/γ. Nothing about the path is changing; the shift is pure time dilation, and classical physics predicts none. Be careful which frame owns the right angle. Written with the emission angle θ′ in the source's frame the same physics reads f = f₀γ(1 + β cos θ′), and at θ′ = 90° the observer sees a blueshift of γ, because that light arrives from ahead of the source, at cos θ = β. Both statements are correct; they describe different photons. The cleanest measurement uses a source emitting in two opposite directions at once, such as the twin jets of the microquasar SS 433 at β ≈ 0.26. The approaching and receding jets show wavelengths λ₀γ(1 − β cos θ) and λ₀γ(1 + β cos θ), whose mean is γλ₀ whatever θ is, so the midpoint of the two lines sits redward of the rest wavelength by z̄ = γ − 1 ≈ 0.036 — the transverse effect read straight off a spectrum.

In the source's frame a photon leaves at angle θ′ to the x-axis with velocity components (c cos θ′, c sin θ′). The source frame moves at +v along x relative to the observer, so the velocity transformation of the previous topic gives uₓ = (c cos θ′ + v)/(1 + β cos θ′), and because the speed is still c, cos θ = (cos θ′ + β)/(1 + β cos θ′). A photon emitted sideways in the source frame, θ′ = 90°, is seen at cos θ = β: thrown forward. Half of an isotropic source's photons therefore arrive inside a cone of half-angle θ½ with sin θ½ = √(1 − β²) = 1/γ — 36.9° at β = 0.80, and 5.7° for a jet knot with γ = 10. This beaming, compounded by the blueshift of each photon and the bunching of their arrival times, is why an approaching jet outshines its receding twin; nothing about what the jet emits has changed. The effect exists at any speed. Earth's orbital speed of 29.8 km s⁻¹ gives β = 9.94 × 10⁻⁵, an angular displacement of β rad = 20.5″, and every star traces an ellipse of that semi-major axis through the year — Bradley's aberration of 1729, the first direct evidence that Earth moves. As β → 0 the formula returns θ = θ′, as it must.

A star's radial velocity comes from a line whose rest wavelength λ₀ is known. Write z = λ/λ₀ − 1; for β ≪ 1, z ≈ β and vᵣ ≈ cz, with a fractional error of β/2, of order 10⁻⁴ or less at typical stellar speeds of tens of km s⁻¹. A Hα line at 656.28 nm observed at 656.42 nm gives z = 2.13 × 10⁻⁴ and vᵣ = 64 km s⁻¹, receding — the number whose periodic wobble reveals binary companions and planets. When β is not small the inverse formula β = ((1 + z)² − 1)/((1 + z)² + 1) is compulsory: z = 0.5 means β = 0.385, not 0.5, and z = 2 means β = 0.80, not a speed above c. Then the boundaries. The derivation assumed vacuum, where the wave speed is c in both frames; in a medium the phase speed differs and the formula changes. It assumed inertial source and observer, so only the instantaneous velocity at emission enters for anything accelerating. And it does not describe cosmological redshift: a galaxy at z = 2 is not moving through space at 0.80c; its light has been stretched by expanding space, and turning that z into a distance belongs to cosmology, not to this formula.