The shift that survives at right angles

The Doppler effect for sound has two formulas, and which one applies depends on which of the source and the listener is doing the moving. That is not a subtlety anybody usually notices, because at ordinary speeds the two agree to a part in a million. At half the speed of sound they differ by a third.

Why sound has two answers #

The mechanism is worth drawing rather than deriving, because the asymmetry is visible in the geometry.

A moving listener does something else entirely. The crests in the medium are undisturbed, evenly spaced, and the listener simply runs into them faster. The wavelength is unchanged and the encounter rate goes up in proportion to the closing speed, giving $1 + \beta$.

Two different physical situations, two different answers, and an experiment can tell them apart: measure the wavelength in the medium. That is the whole content of the asymmetry — the medium is a thing, it has a frame, and motion with respect to it is detectable.

There is a third classical case, usually left out, which sharpens the point: source and listener both moving, in the same direction, at the same speed. The relative velocity is zero and so is the shift — but the wavelength in the medium is compressed all the same, and a third observer standing still measures it. Two of the three parties agree there is no Doppler effect and one of them can see the compressed wave. Nothing about that is contradictory once the medium is admitted as a physical object with its own frame, and every part of it becomes untenable when the medium is taken away.

Light has one answer #

For light there is no medium, and Michelson and Morley’s failure to find one is the experimental heart of the matter. What remains is the relative velocity of source and receiver, and the shift must depend only on that.

The relativistic formula for a source approaching directly is

$$\frac{\nu_{\text{received}}}{\nu_{\text{emitted}}} = \sqrt{\frac{1+\beta}{1-\beta}},$$

and at $\beta = 0.5$ it is $\sqrt3 = 1.732$. That it lands exactly between the two classical answers is not a coincidence: the relativistic factor is $\sqrt{(1+\beta)\cdot\frac{1}{1-\beta}}$, which is the geometric mean of $1+\beta$ and $1/(1-\beta)$, algebraically and at every speed.

The reason for the mean is instructive. Relativity does not abolish either classical mechanism; it says the two are the same situation described from two frames, and the factor that reconciles them is the square root of $\gamma^2$ split evenly between them. Half a time dilation each, so to speak.

The derivation is short enough to give in full, and it makes the structure plain. Let the source emit at frequency $\nu_0$ in its own frame and recede at $\beta$. In the receiver’s frame the source’s clock runs slow, so it emits $\nu_0/\gamma$ crests per second of receiver time. Each successive crest is emitted from further away by $\beta c$ per second, so it has that much further to travel, and the crests arrive spread by a factor $(1+\beta)$. Combining,

$$\nu = \frac{\nu_0}{\gamma(1+\beta)} = \nu_0\sqrt{\frac{1-\beta}{1+\beta}},$$

where the simplification uses $\gamma^{-1} = \sqrt{(1-\beta)(1+\beta)}$. The two factors in that expression are exactly the two mechanisms: one is time dilation, which is relativistic and depends on $\beta^2$; the other is the changing travel distance, which is classical and depends on $\beta$. Every case in this essay is those two multiplied together with the right angle in the second.

Where the extra factor comes from is a moving clock running slow by $\gamma$. A moving source therefore emits fewer crests per second of the receiver’s time than it thinks it is emitting. The classical crowding of crests is still there and unchanged; what is new is that the source’s own second is longer, so the two effects multiply rather than one replacing the other.

The term with nothing behind it classically #

Now put the source at closest approach, moving directly across the line of sight. Its distance is momentarily not changing at all, so every classical mechanism predicts no shift whatever.

The measured shift is $1/\gamma$: a redshift of 0.866 at half the speed of light, and of 0.436 at 0.9.

There is no way to read this as an arrival-rate effect, because nothing about the arrival geometry is changing at that instant. What is being observed is the source’s clock, running slow by $\gamma$, expressed as a frequency. Time dilation, measured as a colour.

That makes the transverse shift the sharpest available distinction between relativity and any theory in which the Doppler effect is about waves and motion. The longitudinal shift is a first-order effect that a classical theory can reproduce to within a correction; the transverse shift is second order and pure.

The factor doing the work is $\gamma$, and its reciprocal is the transverse shift exactly — 0.866 at half the speed of light, 0.436 at nine tenths. Everything in this essay beyond the classical crowding of crests is that one curve appearing in a different place, which is why the transverse Doppler shift is the cleanest measurement of time dilation there is: the geometry has been arranged so that nothing else survives.

There is a neat consistency check available on the transverse case that requires no new physics. Consider two observers passing each other, each looking at the other at their own moment of closest approach. Each sees the other’s clock running slow by $\gamma$, and each is right, because the two are looking at different events — the situation is symmetric in exactly the way the twin problem is not, since neither observer ever returns to compare. The transverse shift is therefore the purest available demonstration that time dilation is reciprocal, and that reciprocity is not a paradox as long as nothing brings the two clocks back together.

Measuring something quadratically small #

Ives and Stilwell measured the transverse shift in 1938, and their method is worth knowing because the problem it solves recurs everywhere: how to isolate a second-order effect that is swamped by a first-order one.

Aiming a spectrograph at right angles to a beam is hopeless. The first-order shift goes as $\cos\theta$, so an alignment error of one degree admits a first-order contamination of $\beta \times 0.017$, which at their beam speed of about 0.005$c$ is comparable with the entire $\beta^2/2$ they were looking for.

So they did not observe transversely at all. They looked along a beam of hydrogen ions and observed both the light emitted forward and the light emitted backward — the latter via a mirror — giving two lines shifted by $+\beta$ and $-\beta$ to first order. The first-order shifts are equal and opposite, so the average of the two line positions cancels them exactly, and what survives is the second-order term, common to both.

The measured displacement of the midpoint matched $\beta^2/2$ and, by extension, the time dilation factor. The trick — arrange for the large unwanted effect to appear with both signs and average it away — is the same one Pound and Rebka used two decades later, and the same one behind every differential measurement in physics.

The reciprocal property, and what it is good for #

One structural feature of the relativistic factor separates it from both classical ones and is worth stating on its own: approaching and receding at the same speed give factors that are exact reciprocals.

$$k(\beta)\,k(-\beta) = \sqrt{\frac{1+\beta}{1-\beta}}\cdot\sqrt{\frac{1-\beta}{1+\beta}} = 1.$$

Neither classical formula does this. A moving source gives $1/(1-\beta)$ and $1/(1+\beta)$, whose product is $1/(1-\beta^2)$; a moving listener gives $(1+\beta)(1-\beta) = 1-\beta^2$. Both differ from one at second order, and the difference is exactly the time dilation the classical account is missing.

The reciprocal property is what makes the Doppler factor a natural way to label a boost. Combining two boosts multiplies their factors — a chain of relative velocities becomes a product rather than the awkward addition formula that velocities themselves obey — and the quantity whose logarithm adds under composition is the rapidity. The Doppler factor is $e^{\varphi}$, and the reason speeds do not add and rapidities do is visible here as the statement that shifts multiply.

It is also what makes the twin count work so cleanly. The outbound and inbound legs have reciprocal factors, so the number of pulses received over a round trip can be summed with no algebra at all — and that argument, which contains no frames, is the one to reach for whenever a relativistic problem starts producing contradictions.

The angle, and the ambiguity in “transverse” #

There is a genuine trap in the transverse case, and it is not a subtlety about precision — it is about what the word means.

The general formula involves the angle between the source’s velocity and the line of sight, and which frame’s angle is being used changes the answer. Take the angle as measured in the receiver’s frame at the moment of reception, and the shift at ninety degrees is $1/\gamma$, a redshift. Take the angle as measured in the source’s frame at the moment of emission, and the shift at ninety degrees is $\gamma$ — a blueshift.

Both are correct statements about different experiments, because the two conditions pick out different events: light received at right angles was emitted before the source got there, and light emitted at right angles arrives from an angle that is not a right angle. The discrepancy is aberration, and it is the same phenomenon that sweeps an accelerating charge’s radiation forward into a cone.

Any quoted transverse shift that does not say which convention it is using is incomplete, and the two answers differ by $\gamma^2$, which is not a small discrepancy.

Where the shift became a tool #

Two applications are worth naming because they use the effect in opposite directions.

Slowing atoms with light. An atom absorbs strongly only at its own resonant frequency, and a moving atom sees an incoming beam Doppler shifted. Tune a laser slightly below the resonance and only atoms moving toward it are shifted into resonance — so only those atoms absorb, and each absorption delivers a photon’s momentum against their motion. The result is a force that depends on velocity, which is to say a friction, and it will cool a gas to microkelvin temperatures. The whole technique rests on the shift being large enough to distinguish one velocity class from another, which for optical frequencies means a few metres per second.

Reading a velocity off a spectrum. The inverse operation — infer the speed from the shift — is the routine use, and everything from a police radar to a Doppler weather profile is doing it. What relativity contributes is not the technique but the correction: at $\beta = 10^{-6}$ the second-order term is $5\times10^{-13}$, which is far below any radar’s resolution and far above the stability of an optical clock. So the relativistic term matters for the instrument that is sensitive enough and not for the one that is fast enough, which is the usual division.

The two together make a general point about this collection’s habit of asking where a model stops. The classical formula is not wrong for a radar gun; it is exact to twelve decimal places there, and the reason to know the relativistic one is that the same apparatus, made a million times more sensitive, is a test of relativity rather than a speedometer.

The same experiment at a third of the speed of light #

Ives and Stilwell worked at about half a per cent of the speed of light, where the effect they were after is a part in eighty thousand. The experiment has since been redone in storage rings, at speeds two orders higher, and the modern design is the reciprocal property of the previous section turned into an apparatus.

Circulate lithium ions at a third of the speed of light. Point one laser along the beam and another against it, and tune each until the ions absorb from it — the first is tuned to the ions’ resonance seen with a blueshift, the second with a redshift. If the relativistic factor is correct, those two laboratory frequencies satisfy

$$\nu_a \nu_p = \nu_0^2,$$

because the two Doppler factors are exact reciprocals and their product is one. The speed of the ions has cancelled out entirely.

That is what makes the measurement so sharp. Nothing has to be known about how fast the beam is going, which is the quantity hardest to control and easiest to get wrong; what is compared is a product of two measured optical frequencies against the square of a rest frequency measured separately. Any classical or partly-classical alternative predicts a product differing from $\nu_0^2$ at second order in $\beta$, which at a third of the speed of light is a departure of several per cent rather than the parts per million Ives and Stilwell could reach for.

The measurements agree with the relativistic prediction to a few parts in a hundred million. They are now the tightest laboratory bound on the time-dilation term, and the design is a good example of a general habit: when the awkward quantity appears in two places with opposite exponents, arrange for the experiment to measure the product.

The redshift that is not one of these #

One shift that is routinely quoted with this essay’s formula does not belong to it, and the confusion is worth heading off because it produces a specific wrong conclusion.

The redshift of a distant galaxy is not a Doppler shift. In an expanding universe the wavelength of a travelling wave is stretched along with the space it crosses, so what is measured is the ratio of the universe’s scale factor now to its scale factor when the light set out — a ratio of two epochs, containing no velocity. Writing $1+z$ and solving the relativistic Doppler formula for $\beta$ gives a number, and that number is not the speed of anything.

Doing so anyway produces the standard error: since the formula cannot return a $\beta$ above one whatever $z$ is, a galaxy at $z = 8$ is reported as receding at 0.98 of the speed of light, and the whole affair looks reassuringly bounded. It is not. Galaxies whose present distance exceeds the Hubble radius are receding faster than light in the only sense the word can be given here, and they are perfectly visible — because the light they emitted long ago was in a region that was not, at the time, receding that fast.

Nothing about that violates the ceiling this collection keeps invoking. Special relativity bounds the relative speed of two things passing each other at the same place; the recession of a distant galaxy is not a measurement of that kind, and the rate at which a distance between two far-apart points grows is not constrained by it. What has to be given up is the habit of turning every redshift into a velocity — and the tell that the habit is being applied wrongly is that the answer approaches a limit it has no reason to respect.

What it costs, and where the model stops #

The formula is for a source and receiver in uniform relative motion. Acceleration during the light’s flight is not covered, and neither is a source whose velocity changes appreciably while the signal is in transit.

The source is assumed to be a source of a definite frequency. A real emitter has a line of finite width, from its own thermal motion among other things, and the second-order shift being sought here is often smaller than that width. Ives and Stilwell’s method works because it measures the centroid of a broad line rather than its edge, and the centroid can be located to a small fraction of the width.

Frequency, not colour. The ratio applies to frequency; the perceived colour of a shifted source involves the whole spectrum moving through the eye’s response, and a source shifted enough to look blue may be one whose ultraviolet has arrived rather than one whose blue has intensified.

Gravity is not included. A gravitational shift is a separate contribution with its own $gh/c^2$, and any real measurement between two places at different heights carries both.

The one-way shift is not directly measurable without a synchronisation convention. Comparing an emitted frequency with a received one requires knowing what the distant source emitted, which requires a convention about simultaneity. Every clean experiment — including Ives and Stilwell’s — is really a two-way or differential measurement, and this is the same structural point that makes a one-way measurement of the speed of light impossible in principle.

Two mechanisms, one formula, and which is which #

The tidiest way to hold all of this is to notice that every case in the essay is the product of exactly two factors, and to keep track of which is doing what.

The first factor is classical and geometric: the distance to the source is changing, so successive crests have different distances to cover, and the arrival rate changes in proportion to $1 \pm \beta$. It depends on the direction of motion and vanishes at right angles.

The second is relativistic and scalar: the source’s clock runs slow by $\gamma$, so it emits fewer crests per second of the receiver’s time. It does not depend on direction at all, and it is what survives when the first factor vanishes.

Sound has only the first, which is why it needs two formulas — the geometry differs depending on which body moves through the medium. Light has both, and because the second is direction-independent the two combine into one expression that depends only on relative velocity. Every result on this page is a statement about which of those two factors is being looked at.

The ladder from here #

Later rungs on this anchor: the general angular formula, with both conventions written out and the aberration between them; the relativistic beaming of a moving source’s output, which is the same transformation applied to intensity; the Doppler factor as the quantity that makes the twin count elementary; laser cooling, in which the shift is used to make absorption depend on velocity and a gas is slowed by light; and the Mössbauer version, where a shift of a part in $10^{15}$ is produced by moving a source at micrometres per second.

The neighbouring ladders are the classical Doppler effect, whose two formulas this rung replaces with one, and time dilation, which the transverse term measures directly.