Photons, Doppler Shift, and Aberration

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A photon has no rest frame, but every massive observer can measure its frequency and direction. Both come from decomposing a nonzero future-directed null four-momentum $k^\mu$ relative to the observer’s four-velocity $u^\mu$ .

We use $k^\mu k_\mu=0$ and absorb $\hbar=1$ , so angular frequency and photon energy have the same numerical value.

Required background. Events, Intervals, and Minkowski Geometry begins the Special Relativity sequence and supplies null geometry; Four-Vectors and Lorentz-Invariant Decomposition supplies four-vector algebra; Observers, Rest Spaces, and Measured Components supplies the observer projector. Helpful background. Calculus and Linear Algebra supplies contractions and projections; Classical and Variational Mechanics supplies relative-velocity and classical wave language.

Frequency and direction as observer measurements

The observed frequency is

\omega_{(u)}=-k_\mu u^\mu\gt0.

With

h^\mu{}_{\nu}(u)=\delta^\mu{}_{\nu}+u^\mu u_\nu,

projecting into the rest space gives

k^\mu=\omega_{(u)}(u^\mu+n^\mu_{(u)}), \qquad n^\mu_{(u)} =\frac{h^\mu{}_\nu k^\nu}{\omega_{(u)}}.

The null condition implies

u_\mu n^\mu_{(u)}=0, \qquad n^\mu_{(u)}n^{(u)}_\mu=1.

Thus $n^\mu_{(u)}$ is the unit spatial propagation direction measured by $u^\mu$ . A telescope’s line of sight to the source points oppositely to the arriving photon’s propagation direction, so observational angle conventions must say which vector is used.

The invariant Doppler ratio

If a source observer $u_{\rm s}^\mu$ emits a ray and a receiving observer $u_{\rm r}^\mu$ detects the same geometric-optics ray, then

\frac{\omega_{\rm r}}{\omega_{\rm s}} =\frac{-k_{{\rm r}\mu}u_{\rm r}^\mu} {-k_{{\rm s}\mu}u_{\rm s}^\mu}.

Here $k_{\rm s}$ and $k_{\rm r}$ are the wave covectors at the two endpoints of one consistently normalized ray. The formula is coordinate independent. In flat spacetime, an inertial ray has constant components in one global inertial chart. As a preview beyond this chapter’s flat-spacetime scope, vacuum geometric optics in curved spacetime parallel-transports $k$ along the ray, while its coordinate components and its contraction with different observers may change.

In an inertial frame $S$ , let the photon propagate at angle $\theta$ to the positive $x$ -axis:

k^\mu=\omega(1,\cos\theta,\sin\theta,0).

Let $S'$ be the rest frame of an observer moving with velocity $+\beta\hat{\mathbf x}$ in $S$ . With the passive boost convention

k'^0=\gamma(k^0-\beta k^x), \qquad k'^x=\gamma(k^x-\beta k^0),

that receiver has $u_{\rm r}^\mu=\gamma(1,\beta,0,0)$ in $S$ (and $(1,0,0,0)$ in $S'$ ), so

\omega'=-k\cdot u_{\rm r} =\gamma\omega(1-\beta\cos\theta).

For longitudinal recession, take the source to be at rest in $S$ , the photon to propagate along $+x$ , and the receiver to move along $+x$ away from the source. Then $\cos\theta=1$ and

\frac{\omega'}{\omega} =\gamma(1-\beta) =\sqrt{\frac{1-\beta}{1+\beta}}.

The low-speed limit is $1-\beta+O(\beta^2)$ .

For the approaching arrangement with the same positive receiver velocity, place the stationary source ahead of the receiver. Its photons now travel toward $-x$ , so $\theta=\pi$ and

\frac{\omega'}{\omega} =\gamma(1+\beta) =\sqrt{\frac{1+\beta}{1-\beta}}.

Alternatively, keep the original source behind and reverse the receiver velocity: use $\beta\mapsto-\beta$ while keeping $\theta=0$ . Changing the sign of the velocity and changing the photon direction are two ways to describe approach; neither should be done silently.

Aberration

Boosting the spatial momentum along $x$ and dividing by the transformed frequency gives

\cos\theta' =\frac{\cos\theta-\beta}{1-\beta\cos\theta}, \qquad \sin\theta' =\frac{\sin\theta}{\gamma(1-\beta\cos\theta)}.

These relations preserve $\cos^2\theta'+\sin^2\theta'=1$ . They give the directional aberration that underlies relativistic beaming; an intensity-beaming statement additionally needs the solid-angle Jacobian and the transformation of spectral intensity. For an observer boosted along $+x$ relative to an angular distribution in $S$ , apparent source positions concentrate toward $+x$ ; because $\theta'$ labels photon propagation, the corresponding arriving propagation vectors concentrate toward $-x$ . Keeping those opposite directions distinct prevents a sign reversal in aberration statements.

Combining the formulas gives a useful rapidity form for a boost along the ray:

\omega'=e^{-\chi}\omega, \qquad \beta=\tanh\chi.

The exponential appears because collinear rapidities add.

Worked example: ninety degrees in one frame

There are two inequivalent transverse arrangements. First suppose the photon is transverse in $S$ , so $\theta=\pi/2$ . Then

\omega'=\gamma\omega, \qquad \cos\theta'=-\beta.

This is a blueshift. By contrast, suppose the photon is transverse in the receiver frame, so $\theta'=\pi/2$ . The aberration formula then gives $\cos\theta=\beta$ , and hence

\frac{\omega'}{\omega} =\gamma(1-\beta^2) =\frac{1}{\gamma}.

This is the standard transverse Doppler redshift. Because aberration changes the angle, “transverse in $S$ ” and “transverse in $S'$ ” are different experimental arrangements. An angle must always be tagged with its observer.

Wave crests and phase

For a plane wave with phase

\Phi=k_\mu x^\mu,

an observer samples

\frac{d\Phi}{d\tau}=k_\mu u^\mu=-\omega_{(u)}.

Frequency is therefore the rate at which phase changes along the observer worldline, up to the stated phase-sign convention. This interpretation extends beyond a particle picture and links directly to geometric optics.

Limits and common mistakes

A null vector cannot be normalized to $-1$ , and a photon has no proper time or comoving rest frame.
The endpoint contraction always defines the local measured frequency. The single constant-boost formulas above assume inertial motion in one flat chart; accelerated emitters or receivers can use an instantaneous velocity at each endpoint, while the timing and spectrum of an extended signal depend on their histories.
A frequency ratio is not automatically caused only by relative velocity. Gravity, cosmological expansion, media, and source dynamics can also change it.
Confusing propagation direction with direction toward the source reverses apparent-angle statements.
A plane-wave four-vector and its Lorentz transformation are exact in flat vacuum spacetime. Using a single local ray for a varying wave field additionally requires its wavelength to be short compared with the background and envelope variation scales. Outside that regime, wave packets and full field propagation are needed.

The paired plane-wave transformation of frequency and propagation direction was derived in Einstein 1905, §7, printed pp. 910–912. That historical source supports the special-relativistic Doppler and aberration formulas, not the modern observer contraction or the curved-spacetime preview.

Exercises

Small-speed aberration

Expand the aberration formula through first order in $\beta$ and show that, for $0\lt\theta\lt\pi$ ,

\theta'=\theta+\beta\sin\theta+O(\beta^2).

Solution

Expanding the cosine formula gives

\cos\theta' =\cos\theta-\beta\sin^2\theta+O(\beta^2).

Write $\theta'=\theta+\delta\theta$ and expand $\cos(\theta+\delta\theta)=\cos\theta-\delta\theta\sin\theta+O(\delta\theta^2)$ . Comparing terms gives $\delta\theta=\beta\sin\theta+O(\beta^2)$ . The result refers to the propagation direction; the apparent direction toward the source is opposite.

Two transverse conditions

For the boost convention used above, recover both frequency ratios when (a) $\theta=\pi/2$ and (b) $\theta'=\pi/2$ . Explain why the signs of the shifts differ.