5.4 Relativistic Doppler effect

The relativistic Doppler effect extends the classical Doppler shift into the regime where source and observer move at speeds comparable to $c$. By folding in time dilation, it predicts frequency shifts that the classical formula misses entirely, most notably a purely transverse redshift with no radial motion at all. These results are central to astrophysical spectroscopy and to experimental tests of special relativity.

Relativistic Doppler Effect

When a source of electromagnetic radiation and an observer are in relative motion, the observed frequency differs from the emitted frequency. The relativistic Doppler effect accounts for this by combining two ingredients: the geometric path-length change familiar from the classical Doppler effect, and the relativistic time dilation of the source's proper time. The result is a single, exact formula valid at all speeds $v < c$.

Classical vs. Relativistic Doppler Effect

The classical Doppler effect predicts a frequency shift based purely on the radial component of relative velocity. For sound waves in a medium, the shift depends separately on whether the source or the observer is moving, because the medium defines a preferred frame.

For light there is no medium, and no preferred frame. Special relativity demands that only the relative velocity between source and observer matters. The relativistic treatment introduces an extra factor of the Lorentz gamma, $\gamma = 1/\sqrt{1 - v^2/c^2}$, which encodes time dilation. At low speeds ($v \ll c$) the relativistic formula reduces to the classical one, so the classical result is just the leading-order approximation.

Longitudinal vs. Transverse Doppler Effect

Longitudinal (radial) case. When the motion is along the line connecting source and observer:

• Source approaching: observed frequency is higher than emitted (blueshift).
• Source receding: observed frequency is lower than emitted (redshift).

This is the direct relativistic analogue of the everyday Doppler effect for sound.

Transverse case. When the motion is exactly perpendicular to the line of sight, the classical prediction is zero shift. Relativity disagrees: the source's clock runs slow by a factor $\gamma$, so the observer measures a lower frequency. This transverse Doppler redshift is a purely relativistic effect with no classical counterpart.

For an arbitrary angle $\theta$ (measured in the observer's frame between the velocity vector and the line of sight), both effects contribute simultaneously.

Relativistic Doppler Shift Formula

For purely radial motion (source and observer moving directly toward or away from each other), the observed frequency is

$f = f_0 \sqrt{\frac{1 + \beta}{1 - \beta}}$

where $f_0$ is the emitted (proper) frequency and $\beta = v/c$. The sign convention here is that $\beta > 0$ means approach; for recession, replace $\beta \to -\beta$ (or equivalently flip the numerator and denominator).

The general formula for an arbitrary angle $\theta$ between the velocity and the direction from source to observer is

$f = \frac{f_0}{\gamma(1 - \beta \cos\theta)}$

Setting $\theta = 0$ (direct approach) or $\theta = \pi$ (direct recession) recovers the radial formula. Setting $\theta = \pi/2$ (perpendicular motion) gives

$f = \frac{f_0}{\gamma}$

which is the transverse Doppler redshift, purely due to time dilation.

Deriving the radial formula in a few steps:

1. In the source's rest frame, successive wave crests are emitted a proper time $\Delta t_0 = 1/f_0$ apart.
2. In the observer's frame, time dilation stretches this interval to $\Delta t = \gamma \Delta t_0$.
3. During that interval the source has moved a distance $v \Delta t$ toward (or away from) the observer, changing the path length by $\pm v \Delta t$.
4. The observed period is $T = \Delta t \mp v \Delta t / c = \gamma \Delta t_0 (1 \mp \beta)$.
5. Inverting gives $f = 1/T = f_0 / [\gamma(1 \mp \beta)]$. Rationalizing the denominator yields the square-root form above.

Redshift and Blueshift

Redshift ($f < f_0$, or equivalently $\lambda > \lambda_0$) occurs when the source recedes. The redshift parameter is defined as

$z = \frac{\lambda - \lambda_0}{\lambda_0} = \frac{f_0}{f} - 1$

For a radially receding source, $1 + z = \sqrt{(1+\beta)/(1-\beta)}$ with $\beta$ now representing the recession speed.

Blueshift ($f > f_0$) occurs when the source approaches.

In cosmology, the observed redshift of distant galaxies is not purely a kinematic Doppler shift; it arises from the expansion of space stretching the wavelength while the photon is in transit (cosmological redshift). The two descriptions converge for nearby sources but diverge at cosmological distances where general-relativistic effects dominate.

Relativistic Doppler Effect and Time Dilation

Time dilation is not an add-on correction; it is built into the derivation at step 2 above. A moving source's oscillation period is longer in the observer's frame by the factor $\gamma$. This is why the transverse effect exists at all: even with no change in path length, the dilated period lowers the observed frequency.

A common point of confusion: the transverse Doppler effect always produces a redshift, never a blueshift, because $\gamma > 1$ for any nonzero speed. This makes it a clean, direct test of relativistic time dilation.

Experimental Verification

• Ives–Stilwell experiment (1938). Measured the frequency of light emitted by fast-moving hydrogen ions (canal rays) both forward and backward. The average of the two shifted frequencies differed from $f_0$ by exactly the amount predicted by the transverse Doppler formula, confirming the $\gamma$ factor. This was one of the earliest direct tests of relativistic time dilation.
• Modern Ives–Stilwell tests. Saturation spectroscopy on lithium ions at storage rings (e.g., the TSR at Heidelberg) has verified the relativistic formula to parts in $10^{9}$.
• Mössbauer rotor experiments. A gamma-ray source mounted on a spinning rotor experiences a transverse Doppler shift relative to a stationary absorber. The measured shift matches $\Delta f / f = -v^2/(2c^2)$ (the leading-order expansion of $1/\gamma$) with high precision.
• Pound–Rebka experiment (1959). Strictly a test of gravitational redshift (general relativity), not the kinematic Doppler effect, though the two are sometimes discussed together. It confirmed that photons climbing out of a gravitational well lose energy as predicted.

Relativistic Beaming

When a source emits isotropically in its own rest frame but moves at relativistic speed relative to the observer, the radiation pattern in the observer's frame is compressed into a forward cone of half-angle $\sim 1/\gamma$. This happens because of two combined effects:

• Aberration of light. Photons emitted at large angles in the source frame are swept forward in the observer's frame.
• Doppler boosting. Photons emitted toward the observer are blueshifted and arrive at a higher rate; those emitted away are redshifted and arrive less frequently.

The net result is that the apparent luminosity of an approaching relativistic source is amplified by a factor that scales as $\delta^{3+\alpha}$ (for a continuous jet) or $\delta^{2+\alpha}$ (for a discrete blob), where $\delta = 1/[\gamma(1-\beta\cos\theta)]$ is the Doppler factor and $\alpha$ is the spectral index. This is directly observed in the jets of active galactic nuclei (AGN) and gamma-ray bursts.

Superluminal Motion

Some radio-bright AGN jets appear to move across the sky faster than $c$. This is a projection effect, not a violation of relativity.

How it works:

1. A blob in a jet moves at speed $v$ (with $v < c$) at a small angle $\theta$ to the line of sight.

2. Between two emission events separated by time $\Delta t$ in the observer's frame, the blob moves a transverse distance $v \Delta t \sin\theta$.

3. The second signal has a shorter travel path to the observer by $v \Delta t \cos\theta$, so it arrives sooner by $v \Delta t \cos\theta / c$.

4. The apparent transverse speed is $v_{\text{app}} = v \sin\theta / (1 - \beta \cos\theta)$.

5. For $\beta$ close to 1 and $\theta$ small, $v_{\text{app}}$ can greatly exceed $c$. The maximum apparent speed is $v_{\text{app,max}} = \beta\gamma c$, occurring at $\cos\theta = \beta$.

Superluminal motion is routinely observed with VLBI (very long baseline interferometry) in quasar jets, and its measurement provides direct constraints on the jet Lorentz factor and viewing angle.

Applications in Astronomy and Cosmology

• Hubble's law and cosmic expansion. Spectroscopic redshifts of galaxies, interpreted through the Doppler/cosmological redshift framework, yield recession velocities that scale with distance. This underpins the measurement of the Hubble constant $H_0$.
• Galaxy dynamics. Doppler shifts of spectral lines across a galaxy's disk map its rotation curve, revealing the distribution of visible and dark matter.
• Relativistic jets. Doppler boosting and superluminal motion diagnostics constrain the bulk Lorentz factors and orientations of jets from AGN and microquasars.
• Redshift surveys. Large-scale surveys (SDSS, DESI) use millions of galaxy redshifts to map the three-dimensional structure of the universe and constrain dark energy parameters.
• High-redshift universe. Observations of objects at $z > 6$ probe the epoch of reionization and the formation of the first galaxies, relying entirely on accurate redshift measurements.