The wave that stretches one way and squeezes the other

A gravitational wave passing through a ring of freely floating masses does something no other wave does. It does not push them along its direction of travel, and it does not push them sideways in unison. It lengthens one diameter of the ring while shortening the perpendicular one, and half a cycle later the two have exchanged roles.

Two things about that description are worth separating. The shape of the distortion is forced by a symmetry, and it is the reason the effect exists at all in a form nothing else produces. The size is forced by a coupling constant, and it is the reason it took a century to measure.

Why nothing simpler is allowed #

Start with what radiation requires. A source radiates when some property of its distribution changes with time, and the multipole expansion sorts those changes by how symmetric they are.

The monopole is forbidden by conservation of mass. A pulsating sphere would radiate a monopole wave, and to do so the total mass would have to change with time. It cannot. In electromagnetism the same rule holds — an oscillating total charge would be a monopole antenna — and it is equally forbidden, so this first exclusion is not special to gravity.

The dipole is forbidden by conservation of momentum. A dipole term requires the first moment of the mass distribution — which is the centre of mass times the total mass — to accelerate. For an isolated system it cannot: that is Newton’s first law applied to the system as a whole.

Here the two theories part company, and the reason is the sign of the charge. Electric charge comes in two signs, so the electric dipole moment $\sum q_i \mathbf{r}_i$ is not proportional to the momentum and can oscillate freely — which is exactly what every radio transmitter does. Mass has one sign, so the gravitational “dipole moment” is the momentum, and it is conserved.

What the other theory has and this one does not is a dipole. An oscillating electric dipole radiates into a two-lobed pattern — what an accelerating charge produces, and what every antenna is built around — and nothing in gravitation has that shape, because building the source would require two kinds of mass. There is no negative mass to move against positive mass, so the dipole moment of an isolated system is fixed by its momentum and cannot oscillate. The gravitational pattern starts at the next order up, with four lobes rather than two.

So the first term that survives is the quadrupole, and a quadrupole is not a moving source but a source that changes shape — which is why an orbit has to shrink and a falling stone does not radiate. A perfectly spherical star pulsating radially emits nothing whatever, however violently it pulsates. A spinning object that is exactly axisymmetric emits nothing however fast it spins. Radiation demands asymmetry, and asymmetry that changes.

Why it is so faint #

The amplitude of the wave at distance $r$ from a source is, to the accuracy that matters here,

$$h \sim \frac{G}{c^4}\,\frac{\ddot{Q}}{r},$$

with $Q$ the quadrupole moment of the source — a mass times the square of a length. Everything about the weakness of the effect is in the prefactor: $G/c^4$ is $8\times10^{-45}$ in SI units. That is the smallest coupling constant in physics by a wide margin, and it is why a phenomenon predicted in 1916 was measured in 2015.

Put a laboratory source in it. Two one-tonne masses on the ends of a two-metre bar, spun as fast as the bar’s own strength allows, produce a strain at a distance of a few metres of order $10^{-43}$. No conceivable improvement in instruments reaches that. Gravitational-wave sources are not built; they are found, and they have to involve masses of stellar size moving at appreciable fractions of the speed of light.

The same prefactor has a compensating consequence that is easy to miss. Because the coupling is so weak, the waves pass through matter essentially without absorption or scattering — nothing is opaque to them, and there is no gravitational analogue of a Faraday cage. That is the property that makes them worth detecting at all.

Strain is a fraction #

The quantity $h$ is a fractional change in length, not a length, and this changes what a detector should be.

If the response is proportional to length, then a longer baseline gives a bigger absolute displacement, and the displacement is what an instrument measures. At $h = 10^{-21}$, a metre-long bar changes by $10^{-21}$ m and a four-kilometre arm by $4\times10^{-18}$ m. Neither is a large number, and the second is four thousand times the first for no cost other than land.

This is also why the interferometric design won. An interferometer compares two perpendicular arms, and the wave’s own geometry — one arm lengthening while the other shortens — doubles the signal and cancels anything that changes both arms together. A great deal of what would otherwise be noise is common-mode, and the right-angle arrangement is chosen to exploit precisely the quadrupole pattern in the opening figure.

What a source has to look like #

The quadrupole formula, read as a design constraint on a source rather than as a prediction, says something specific: the strain is of order the source’s gravitational radius divided by the distance, times the square of how fast its parts are moving.

$$h \sim \frac{r_s}{r}\left(\frac{v}{c}\right)^2.$$

Every factor in that expression has to be large for anything to be detectable, and each of them is capped. The first is the ratio of the source’s Schwarzschild radius to its distance: for a few tens of solar masses, $r_s$ is a few tens of kilometres, so at a hundred million light-years the ratio is around $10^{-19}$. The second is the square of the speed, which cannot exceed one and reaches perhaps a third for two compact objects in their last orbits — a factor of ten down.

Multiply and $h \approx 10^{-21}$, which is the number the opening figure is drawn at. It is worth seeing that this estimate uses nothing but the two scalings and produces the right order of magnitude, because it explains why the number is what it is rather than merely asserting it: the strain is small because the source is far away in units of its own gravitational radius, and there is no way to be closer to something that violent.

It also fixes the frequency. Two masses orbiting near their own gravitational radius orbit at a frequency of order $c/r_s$, which for tens of solar masses lands in the audio band — hundreds of hertz. That is a coincidence of scale rather than a deep fact, and it is the reason a merger can be played through a loudspeaker.

Forty-five degrees, not ninety #

The two polarisations are drawn in the two rows of the hero figure, and their relative orientation is the sharpest structural claim on this page.

The general rule is that a field of spin $s$ has polarisation states separated by $90°/s$, and returns to itself under a rotation of $360°/s$. Light is spin 1 and returns after a full turn. A gravitational wave is spin 2 and returns after half a turn, which is what the figure shows: the plus pattern rotated by $180°$ is unchanged.

That is not a decorative fact. It is a direct, observable consequence of the field’s tensor character — of gravity being described by a metric rather than by a vector potential — and it can be measured by comparing what several detectors at different orientations record from the same event.

It really is a wave #

Everything above concerns the shape of the disturbance. The remaining question is whether it propagates, and the linearised theory answers it in the ordinary way: the metric perturbation satisfies a wave equation, so it travels at $c$, is transverse, and carries energy and momentum away from its source.

A shape that travels, drawn twice a moment apart, is what the metric perturbation does — and the only unusual thing about it is what is oscillating. Not a displacement of anything material, but the coefficients that convert coordinate differences into measured distances. So the wave has no medium, and unlike sound it does not need one: what it propagates through is the thing that was going to measure it.

There is one respect in which the analogy with an ordinary wave has to be watched. What oscillates is the metric — the rule for turning coordinate differences into measured lengths — so the wave is not a disturbance in space in the way a sound wave is a disturbance in air. Coordinates can be chosen in which the free masses of the opening figure never move at all and every change is in the rule; or in which the rule is unchanged and the masses move. The measurable quantities, such as the light travel time round a closed circuit, come out the same in both, and any statement that survives only one of those choices is a statement about the bookkeeping.

The energy claim is the one with a history. For decades it was unclear whether gravitational waves carried energy at all or were an artefact of a badly chosen coordinate system, since a coordinate wave can be constructed in flat spacetime with no physics in it. Feynman’s sticky-bead argument settled the physics in 1957: if a passing wave moves two beads along a rod against friction, it does work, and work requires energy. The formal version took another decade.

The measurement that fixed the speed #

The claim that the disturbance travels at $c$ was made above on the strength of the linearised equations, which is a theoretical argument. It has since been measured, to a precision that is worth stating because almost nothing else in gravitation is known this well.

On 17 August 2017 two detectors recorded the inspiral of a pair of neutron stars in a galaxy about forty megaparsecs away — a hundred and thirty million light-years. Gamma-ray telescopes recorded a short burst from the same direction 1.7 seconds later.

The two signals had been travelling side by side for a hundred and thirty million years, which is about four thousand million million seconds, and they arrived within two of each other. The fractional difference in their speeds is therefore under a part in $10^{15}$, and allowing generously for the possibility that the gamma rays were emitted some seconds after the merger rather than at it, the published bound is between $-3\times10^{-15}$ and $+7\times10^{-16}$.

That single number closed a large part of a research field overnight. A great many attempts to modify gravity — theories with extra fields introduced to account for the accelerating expansion of the universe — predict that the tensor modes propagate at a speed slightly different from light’s. Slightly, in most of them, meant parts in a hundred or a thousand. A bound at a part in $10^{15}$ is not a constraint on such theories; it is a refutation, and dozens of them were abandoned within weeks.

The event settled two other things worth naming. A graviton with mass would be dispersive — lower frequencies travelling slower — so the arrival of a signal sweeping upward in frequency with no measurable dispersion bounds that mass, and the bound is now around $10^{-23}$ electronvolts. And because the source was located precisely enough for optical telescopes to find it, the merger’s afterglow could be watched for weeks, and its spectrum showed the signature of freshly made heavy elements — which is the first direct evidence of where the periodic table’s upper half is manufactured.

A detector made of pulsars #

Everything above concerns waves in the audio band, from sources of a few tens of solar masses. The same quadrupole physics operates at frequencies nine orders of magnitude lower, and the instrument used there is not built.

A pair of supermassive black holes, left over from the merger of two galaxies, orbits with a period of years rather than milliseconds. The waves it emits have periods of years and wavelengths of light-years, and no interferometer of any conceivable size responds to them.

What does respond is a pulsar. A millisecond pulsar is a rotating neutron star whose pulses arrive with a regularity that rivals an atomic clock — timing residuals of a hundred nanoseconds accumulated over a decade. A gravitational wave passing between such a pulsar and the Earth changes the light travel time along that path, so the pulses arrive slightly early or slightly late, by a few hundred nanoseconds, over years. The detector’s arm is the distance to the pulsar, which is thousands of light-years.

One pulsar proves nothing: a timing residual could be the pulsar’s own wobble, an error in the Earth’s ephemeris, or dispersion in the interstellar medium. What distinguishes a gravitational wave is that it affects every pulsar, in a pattern fixed by geometry.

And the pattern is the one this essay derived. Because the wave is spin-2, with its polarisations at forty-five degrees rather than ninety, the correlation between the residuals of two pulsars depends on the angle between them on the sky in a specific quadrupolar way: strongly positive for pulsars close together, negative for pairs about ninety degrees apart, and slightly positive again for pairs on opposite sides of the sky. That curve was worked out by Hellings and Downs in 1983, it has no free parameters, and it is different for a scalar or a vector wave.

Four collaborations announced in 2023 that they had found that correlation, at moderate significance, across arrays of dozens of pulsars timed for fifteen years or more. The measurement is a direct test of the polarisation structure drawn in the figures above, made with an instrument whose components are natural, whose baseline is a kiloparsec, and whose signal takes a decade to accumulate.

What it costs, and where the model stops #

All of this is linearised. The wave has been treated as a small perturbation on a flat background, which is excellent at any distance from a source and fails completely at the source itself, where the amplitudes are not small and the equations are not linear. The waveform emitted during the final merger of two compact objects cannot be obtained this way and is computed numerically.

The quadrupole formula is the leading term of an expansion in $v/c$. For a source whose parts move at a tenth of the speed of light the corrections are at the per-cent level; for one at a third they are not corrections. Practical waveform models carry the expansion to high order and then hand over to numerical relativity.

“Space stretches” is a phrase with a trap in it. The masses in the figures are free, and their separation genuinely changes. A solid ruler is not free — it is held together by electromagnetic forces with a length of their own — so it resists the stretching almost entirely, and the strain in a solid object is smaller than $h$ by a large factor. Which is why a detector measures a light travel time against an atomic clock rather than laying a ruler along an arm.

The obvious objection about the ruler has a second layer. The first answer — that a solid ruler is held together and resists — is correct. But an interferometer does not use a solid ruler; it uses light, and light’s wavelength is stretched by the passing wave along with everything else, so it is fair to ask why the fringes move at all.

The resolution is that the interferometer is not comparing an arm against a wavelength. It is comparing the time light takes to make the round trip in one arm against the time in the other, and the clock that times them is an atomic transition whose frequency is set by atomic structure and is not affected. Light already in flight when the wave arrives does have its wavelength stretched; the point is that the round trip takes a different number of clock ticks in the two arms, and that difference is what the fringe pattern reports. Working the whole calculation in a gauge where the mirrors do not move and the light travel time changes gives the same answer as working it in a gauge where the mirrors move — which is the test that the effect is not a coordinate artefact.

A single detector cannot locate a source. It measures one projection of the wave, so direction comes from timing between widely separated instruments — which is why they are built in threes and fours rather than singly, and why a network’s sky localisation is a geometry problem rather than a pointing one.

From bars to interferometers #

The instrumental history is worth a paragraph because the first attempt was a genuinely different idea.

Joseph Weber’s detectors from the 1960s were aluminium cylinders with piezoelectric sensors, designed to ring when a wave of the right frequency passed. A bar is a resonant instrument: its response is sharply peaked at its own frequency, and everything away from that frequency is rejected.

A driven resonance is what a bar detector is. High $Q$ buys amplitude at the resonant frequency and pays for it in bandwidth, so a bar is a narrow-band instrument — excellent if the signal’s frequency is known and blind otherwise. A merging pair sweeps upward in frequency through the audio band in under a second, which is the worst possible signal for an instrument shaped like that, and is why the bars were replaced by interferometers rather than improved.

Weber announced detections in 1969 that nobody could reproduce, and the episode did lasting damage to the field’s reputation. What replaced the bars was broadband by design: an interferometer responds across a wide range of frequencies, which is what a signal sweeping upward through hundreds of hertz in a fraction of a second requires. The lesson is a general one about instrument design — a resonant detector is the right choice only when the frequency is known in advance, and it was not.

The ladder from here #

Later rungs on this anchor: the quadrupole formula derived rather than quoted, and the energy it says is carried; the inspiral, in which the energy loss makes an orbit shrink and the frequency sweep upward; the interferometer as an instrument, and the quantum limits that decide its noise floor; the polarisation content as a test of the field’s tensor character, where alternative theories predict extra modes; and memory, the permanent displacement a passing wave leaves behind, which is the strangest prediction on the list.

The neighbouring ladders are radiation from an accelerating charge, which supplies the multipole machinery and the one term gravity is not allowed, and travelling waves, whose ordinary treatment of a shape moving at a fixed speed is exactly what the linearised theory reduces to.