---
title: Doppler, Aberration, and Appearance
module: Spacetime and the Lorentz Group
moduleNumber: 2
lessonNumber: 4
order: 204
summary: >
  Light carries a null four-momentum, and boosting it produces every optical
  effect of relativity at once. The covariant Doppler formula follows from the
  transformation of frequency, aberration from the transformation of direction,
  and the headlight effect from the resulting concentration of light forward. The
  Terrell-Penrose result shows that a fast object photographs as rotated, not
  contracted.
topics: [Spacetime and the Lorentz Group]
draft: false
sources:
  - book: Taylor & Wheeler
    ref: "Spacetime Physics, Ch. 8 — Doppler shift and aberration"
  - book: Hartle
    ref: "Gravity, §5.5 — The four-momentum of light and the Doppler shift"
---

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 $E = \hbar\omega$ and momentum $\vec{p} = (\hbar\omega/c)\,\hat{n}$.
Assembled into a four-momentum,

$$
k^\mu = \frac{\hbar\omega}{c}\,(1, \hat{n}) = \parens{\frac{E}{c}, \vec{p}},
$$

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

$$
k \cdot k = \eta_{\mu\nu}k^\mu k^\nu = \frac{\omega^2}{c^2}\parens{-1 + \abs{\hat{n}}^2} = 0.
$$

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.

$$
% caption: The photon four-momentum is a null four-vector on the light cone. A
% boost slides it along the cone, changing its time component (the frequency) and
% tilting its spatial part (the direction) while keeping the invariant square zero.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, dashed] (-2.7,-2.7) -- (2.7,2.7);
  \draw[black, dashed] (-2.7,2.7) -- (2.7,-2.7);
  \draw[->, black] (-3.0,0) -- (3.1,0) node[right] {px};
  \draw[->, black] (0,-1.0) -- (0,3.2) node[above] {E over c};
  \fill[black] (0,0) circle (1.6pt);
  % original photon momentum along cone
  \draw[->, black, very thick] (0,0) -- (1.6,1.6);
  \node[black, anchor=west] at (1.55,1.45) {source frame};
  % boosted photon momentum, higher up the cone (blueshift)
  \draw[->, acc, very thick] (0,0) -- (2.35,2.35);
  \node[acc, anchor=west] at (2.15,2.55) {observer frame};
\end{tikzpicture}
$$

## 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 $k'^0 = \gamma(k^0 - \beta k^1)$:

$$
\omega_{\text{o}} = \gamma\,\omega_{\text{s}}\,(1 - \beta\cos\theta_{\text{s}}).
$$

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
  $$
  \omega_{\text{o}} = \sqrt{\frac{1-\beta}{1+\beta}}\;\omega_{\text{s}} < \omega_{\text{s}}
  \quad (\text{redshift}).
  $$
- **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.

> **Worked example (Redshift of a receding galaxy).** A spectral line emitted at
> $\omega_{\text{s}}$ is observed from a galaxy receding at $\beta = 0.1$ along the
> line of sight. The observed frequency is
> $$
> \omega_{\text{o}} = \sqrt{\frac{1 - 0.1}{1 + 0.1}}\,\omega_{\text{s}}
> = \sqrt{\frac{0.9}{1.1}}\,\omega_{\text{s}} = 0.905\,\omega_{\text{s}}.
> $$
> The fractional wavelength shift is
> $z = \lambda_{\text{o}}/\lambda_{\text{s}} - 1 = \omega_{\text{s}}/\omega_{\text{o}} - 1 = 0.105$,
> close to but not equal to the naive $\beta = 0.1$; the difference is the
> second-order time-dilation term $\tfrac12\beta^2$.

## 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 $k'^1 = \gamma(k^1 - \beta k^0)$ and
$k'^0 = \gamma(k^0 - \beta k^1)$,

$$
\cos\theta_{\text{o}} = \frac{k'^1}{k'^0}
= \frac{\cos\theta_{\text{s}} - \beta}{1 - \beta\cos\theta_{\text{s}}}.
$$

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 $\theta_{\text{o}} \approx \theta_{\text{s}} - \beta\sin\theta_{\text{s}}$.

$$
% caption: Aberration shifts apparent star positions toward the direction of
% motion. Rays arriving from the sides in the rest frame are swung forward for a
% moving observer, crowding the star field ahead and thinning it behind.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % observer at center, direction of motion to the right
  \fill[black] (0,0) circle (1.8pt);
  \draw[->, black, thick] (0,0) -- (2.6,0) node[right] {motion};
  % rest-frame ray directions (spread)
  \draw[black, thick] (0,0) -- (-1.6,1.5);
  \draw[black, thick] (0,0) -- (-2.0,0.2);
  \draw[black, thick] (0,0) -- (-1.6,-1.5);
  \node[black, anchor=east] at (-2.0,0.2) {rest frame};
  % aberrated (forward-swung) directions
  \draw[acc, very thick] (0,0) -- (1.5,1.7);
  \draw[acc, very thick] (0,0) -- (2.1,0.6);
  \draw[acc, very thick] (0,0) -- (1.5,-1.7);
  \node[acc, anchor=west] at (1.5,1.75) {moving observer};
\end{tikzpicture}
$$

## 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

$$
\cos\theta_{\text{o}} = \frac{0 - \beta}{1 - 0} = -\beta,
\qquad \theta_{\text{o}} = \arccos(-\beta) \to \frac{\pi}{2}\ \text{as}\ \beta\to0,\ \to \pi\ \text{as}\ \beta\to1.
$$

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.

$$
% caption: The headlight effect. Light a source emits isotropically at rest is
% swept into a forward cone once the source moves; the cone half-angle falls
% toward one over gamma as the speed rises, beaming the radiation ahead.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % rest source: isotropic
  \fill[black] (-3.0,0) circle (1.7pt);
  \foreach \a in {0,45,90,135,180,225,270,315}
    \draw[black] (-3.0,0) -- ++(\a:0.9);
  \node[black, anchor=north] at (-3.0,-1.05) {at rest};
  % moving source: beamed forward cone
  \fill[black] (1.3,0) circle (1.7pt);
  \fill[acc!12] (1.3,0) -- (3.4,0.75) -- (3.4,-0.75) -- cycle;
  \draw[acc, thick] (1.3,0) -- (3.4,0.75);
  \draw[acc, thick] (1.3,0) -- (3.4,-0.75);
  \foreach \a in {-18,-9,0,9,18}
    \draw[acc!70] (1.3,0) -- ++(\a:2.0);
  \draw[->, black, thick] (1.3,0) -- (1.3,-1.4) node[below] {fast};
  \node[acc, anchor=west] at (3.0,1.0) {forward beam};
\end{tikzpicture}
$$

## 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.

> **Theorem (Terrell–Penrose).** A rapidly moving object, photographed from a
> distance so that light rays reaching the camera are nearly parallel, appears not
> flattened by length contraction but rotated. A passing sphere photographs as a
> sphere of undistorted circular outline; a passing cube photographs as if turned
> through an angle set by its speed.

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.

$$
% caption: Terrell-Penrose. Length contraction shortens the moving cube along its
% motion, but differential light-travel time reveals the trailing face; the two
% combine so a photograph records an apparent rigid rotation rather than a squashed
% cube.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % static cube outline
  \draw[black, thick] (-3.4,-0.8) rectangle (-1.8,0.8);
  \node[black, anchor=north] at (-2.6,-1.0) {at rest: square face};
  % apparent rotated cube (front face + revealed side face)
  \draw[acc, very thick] (0.6,-0.8) -- (2.0,-0.8) -- (2.0,0.8) -- (0.6,0.8) -- cycle;
  \draw[acc, thick] (0.6,0.8) -- (0.05,1.2) -- (0.05,-0.4) -- (0.6,-0.8);
  \node[acc, anchor=north] at (1.3,-1.0) {moving: turned, side shown};
  \draw[->, black, thick] (2.4,0) -- (3.2,0) node[right] {motion};
\end{tikzpicture}
$$

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](/relativity/relativistic-dynamics/four-momentum-force-and-accelerated-motion)
takes the four-momentum into collisions and decays, and the general-theory modules
carry the same four-vectors onto a curved metric.

[^tw-doppler]: **Taylor & Wheeler**, _Spacetime Physics_, Ch. 8 — the relativistic Doppler shift from the transformation of the photon frequency, the transverse Doppler effect as a measurement of time dilation, and stellar aberration.

[^hartle-photon]: **Hartle**, _Gravity_, §5.5 — the null four-momentum of light, the Doppler formula $\omega_{\text{o}} = \gamma\omega_{\text{s}}(1 - \beta\cos\theta)$, and the aberration relation obtained by boosting the photon four-vector.