Doppler, Aberration, and Appearance

Light is the one thing every inertial observer measures the same way in speed but differently in frequency and direction. All of those differences follow from a single object: the photon's null four-momentum. Boosting it once produces the relativistic Doppler shift, the aberration of starlight, the headlight beaming of a fast source, and the surprising result that a rapidly moving object looks rotated rather than flattened.

The photon four-momentum

A light wave of frequency $\omega$ traveling in the unit direction $\hat{n}$ carries energy $E=\hslash \omega$ and momentum $\vec{p}=(\hslash \omega /c)\hat{n}$. Assembled into a four-momentum,

or, stripped of the constants, the wave four-vector $k^{\mu }=(\omega /c)(1,\hat{n})$. Its invariant square is

The photon four-momentum is null: it lies on the light cone in momentum space, which is the four-vector statement that light has zero rest mass and moves at $c$ in every frame. Because $k^{\mu }$ is a four-vector, its components in a boosted frame follow from the same boost matrix that acts on any four-vector, and reading off the transformed components gives every optical effect below.

The relativistic Doppler shift

Let the source emit light of frequency ${\omega }_{\text{s}}$, and let the observer move with speed $v=\beta c$ along $x$. In the source frame the photon has $k^0={\omega }_{\text{s}}/c$ and $k^1=({\omega }_{\text{s}}/c)\cos {\theta }_{\text{s}}$, where ${\theta }_{\text{s}}$ is the angle between the light and the $x$ axis. The observer's frequency is $c$ times the boosted time component $k^{\prime 0}=\gamma (k^0-\beta k^1)$:

Three cases follow from the emission angle.

• Longitudinal, receding (${\theta }_{\text{s}}=0$, source and observer separating along the line of sight): ${\omega }_{\text{o}}=\gamma (1-\beta ){\omega }_{\text{s}}$, which simplifies through $\gamma (1-\beta )=\sqrt{(1-\beta )/(1+\beta )}$ to $${\omega }_{\text{o}}=\sqrt{\frac{1-\beta }{1+\beta }}{\omega }_{\text{s}}<{\omega }_{\text{s}}(\text{redshift}).$$
• Longitudinal, approaching (${\theta }_{\text{s}}=\pi$): the sign of $\beta$ flips and ${\omega }_{\text{o}}=\sqrt{(1+\beta )/(1-\beta )}{\omega }_{\text{s}}>{\omega }_{\text{s}}$ (blueshift).
• Transverse (${\theta }_{\text{s}}=\pi /2$, light emitted at right angles to the motion in the source frame): ${\omega }_{\text{o}}=\gamma {\omega }_{\text{s}}$.

The transverse case has no classical counterpart. A stationary Doppler analysis predicts no shift for light emitted at $90^{\circ }$, but relativity gives a pure time-dilation blueshift ${\omega }_{\text{o}}=\gamma {\omega }_{\text{s}}$ — the moving source's clock runs slow, so the observer catches its ticks at a shifted rate regardless of direction. The transverse Doppler shift is a direct measurement of time dilation, confirmed in the Ives–Stilwell experiment and in modern ion-storage-ring spectroscopy.

Aberration

The same boost that shifts the frequency tilts the direction. The observer's emission angle ${\theta }_{\text{o}}$ comes from the ratio of the boosted spatial component to the time component. Using $k^{\prime 1}=\gamma (k^1-\beta k^0)$ and $k^{\prime 0}=\gamma (k^0-\beta k^1)$,

This is the aberration of light: the angle at which a ray arrives depends on the observer's motion. A star seen at angle ${\theta }_{\text{s}}$ from the direction of motion in one frame appears at a smaller angle ${\theta }_{\text{o}}$ to an observer moving toward it — the whole sky shifts toward the direction of travel. The Earth's orbital motion produces a yearly aberration of about $20.5$ arcseconds, first measured by Bradley in 1728 and correctly given by the relativistic formula in the small-$\beta$ limit ${\theta }_{\text{o}}\approx {\theta }_{\text{s}}-\beta \sin {\theta }_{\text{s}}$.

The headlight effect

Aberration concentrates emitted light forward. Consider a source that radiates isotropically in its own frame, so half its photons go into the forward hemisphere ${\theta }_{\text{s}}<\pi /2$. In the frame where the source moves at $\beta$, the boundary ray ${\theta }_{\text{s}}=\pi /2$ maps to

Half of all the light — the entire forward hemisphere of emission — is compressed into a forward cone of half-angle ${\theta }_{\text{o}}$ that shrinks toward the line of motion as $\beta \to 1$. For an ultrarelativistic source the opening half-angle of the beam is approximately $1/\gamma$: the radiation is beamed into a narrow forward cone. This headlight effect is why synchrotron radiation from relativistic electrons arrives in sharp forward-pointing pulses and why a relativistic jet pointed at Earth appears vastly brighter than the same jet seen from the side.

Terrell–Penrose rotation

Length contraction predicts that a moving rod is measured shorter along its motion, but measuring and seeing are different operations. A photograph records photons that arrive at the lens simultaneously, which means they left different parts of the object at different times — light from the far side had farther to travel and so was emitted earlier. Combining this arrival-time spread with aberration produces the Terrell–Penrose result.

The intuition is a cancellation. Length contraction squeezes the object along its motion by $1/\gamma$; the differential light-travel time from its trailing face reveals a portion of the side that a static view would hide, and aberration bends the apparent directions. For a sphere the two effects combine so the silhouette stays exactly circular. For a cube the net visual transformation is indistinguishable from a rigid rotation by the aberration angle $\theta$ with $\cos \theta$ set by $\beta$, so the observer sees the back face swing into view as if the cube had turned rather than shortened. The contraction is real in the sense of measurement — simultaneous marking of the two ends in the observer's frame does give a shorter length — but a single camera exposure does not perform that simultaneous marking, and what it captures is a rotation.

All four effects are one four-vector transformed. Frequency is the time component of $k^{\mu }$, direction is its spatial part, and beaming and Terrell rotation are the consequences of transforming both together and then accounting for when the light was emitted. This closes the special-relativistic geometry of the module; the dynamics module takes the four-momentum into collisions and decays, and the general-theory modules carry the same four-vectors onto a curved metric.