The bar a wave of gravity barely touches

The wave that stretches one way and squeezes the other mentioned Joseph Weber’s detectors in passing: aluminium cylinders, with piezoelectric crystals glued round their waists, designed to ring when a gravitational wave at the right frequency went by, and replaced in the end by interferometers. The cylinders deserve more than a mention, because they are the simplest possible gravitational-wave detector and the physics of how little they catch is clean and surprising. A bar is a lump of matter with one mode of vibration that a wave can excite. Working out how strongly the wave excites it turns into a single formula, and the formula has the speed of sound in it.

A passing wave changes the distances between free particles by a fraction $h$, the strain, which for the waves detected since 2015 is about $10^{-21}$. A bar is not made of free particles. Its atoms are held at fixed spacings by the stiffness of the metal, so the wave’s stretch has to work against that stiffness, and the bar responds by vibrating rather than by simply following the stretch. That is both the opportunity and the problem. The opportunity is resonance: a wave at the bar’s own frequency adds to the vibration cycle after cycle. The problem is that the coupling between the wave and the vibration is weak in a way that no engineering can fix.

The force a wave exerts on a bar #

In the frame of the bar’s centre, a gravitational wave acts as a tidal force. A piece of the bar a distance $x$ from the centre feels a force proportional to $x$ and to the wave’s second time derivative, $\tfrac12\rho\ddot h x$ per unit volume, pulling the two ends apart and then pushing them together as the wave oscillates. The first figure draws that force along the bar as a straight line through the middle.

The bar’s lowest mode of longitudinal vibration has the ends moving in opposite directions while the middle stays still, with a displacement along the bar shaped like $\sin(\pi x/L)$. It has the right symmetry to be driven: the force pulls the ends apart exactly when the mode wants them apart. Projecting the force onto the mode shape gives the effective drive, and the integral comes out simply. The bar’s fundamental mode responds as if it were two point masses, each with half the bar’s mass, held $4L/\pi^2$ apart by a spring tuned to the bar’s frequency, about four-tenths of its length. The third of itself a spring carries met the same reduction for a spring with mass: a distributed object vibrating in one mode behaves like a smaller lumped one, with an effective mass and an effective length set by the shape of the mode.

The frequency of that mode is the time for sound to cross the bar and back, $f_0 = v/2L$. A three-metre aluminium bar, with sound at 5.4 kilometres a second, rings near 900 hertz. That is where the bars listened, because it is where the sources thought most likely in the 1960s and 1970s — the final instants of collapsing stellar cores and merging neutron stars — were expected to put most of their power.

A burst, and the ringing it leaves #

The second figure drives the mode with a short burst of wave at its own frequency and integrates the result. While the burst lasts, the bar’s swing grows, each cycle of the wave adding to it in step; when the burst has gone, the swing stays. At a quality factor of a million, typical of a well-made aluminium bar suspended in a vacuum, the ringing would take six minutes to die away by a factor of e, so the bar holds the burst’s energy long enough for an amplifier to read it.

The swing itself is of order the strain times the bar’s length. For the burst drawn, with a strain a thousand times larger than anything detected since, it is $6\times10^{-18}$ metres, a few thousandths of the diameter of a proton. A burst of the size actually detected would leave a swing a thousand times smaller still. The whole art of the bars was in reading motions of that size against everything else that makes the bar move.

How much of a wave a bar can catch #

The energy the burst leaves in the bar can be compared with the energy the wave carried. A gravitational wave carries energy at a rate set by its strain and frequency, $c^3/16\pi G$ times the mean square of $\dot h$, per square metre. That coefficient is enormous — about $10^{34}$ in SI units — which is the sense in which spacetime is stiff: a strain of $10^{-21}$ at a kilohertz carries about a tenth of a watt per square metre, and a burst of the strengths Weber was looking for would have carried megajoules. The bar’s integrated cross-section, the ratio of the energy it absorbs to the energy per unit area per unit frequency that crossed it, works out to

$$\Sigma_0 = \frac{8}{\pi}\,\frac{GM}{c}\left(\frac{v}{c}\right)^2 .$$

The figures obtain it by integrating the driven mode and dividing by the fluence, and check the two against each other.

The formula is the argument of this essay in one line. The cross-section grows with the mass, which is why the bars weighed tonnes. It contains Newton’s constant, because the coupling of a wave of gravity to matter is gravitational. And it contains the square of $v/c$, the speed of sound in the bar over the speed of light. That last factor is about three parts in ten billion for aluminium, and it is the reason a bar catches so little. A bar responds to a strain only by deforming against its own elasticity, and the elastic forces that resist deformation are, compared with anything relativistic, very soft. A body that could transmit stress at nearly the speed of light would couple ten billion times better; no such material exists, because sound speed is bounded by how stiff matter can be, and the stiffest solids are diamond and sapphire, two or three times aluminium.

The third figure draws the cross-section against mass. Weber’s first bar, 1.4 tonnes, caught $2.3\times10^{-25}$ square metres for each hertz of bandwidth the wave covered — compared with the $6.7\times10^{-29}$ square metres with which a single electron scatters light, which the cross-section that forgets the colour found to be the smallest cross-section a charge can offer, the bar is not small. It is the energy per unit area that must be enormous. To absorb a single joule, a wave would have to bring more than $10^{24}$ joules per square metre per hertz to the bar.

Why a sharper resonance does not catch more #

It would be natural to think that a bar with a higher quality factor catches more of a wave, since it rings longer and more sharply. It does not, and the reason is the relation the width that is a lifetime established for every resonance. A resonance’s response, plotted against frequency, has a height proportional to its quality factor and a width inversely proportional to it, and the cross-section integrated across the resonance — the quantity in the formula — is the area under that curve, which does not depend on the quality factor at all. A bar with ten times the $Q$ absorbs from a narrower slice of the wave’s spectrum, ten times more strongly; for a burst whose spectrum is broad and flat across the resonance, the two effects cancel exactly.

What the quality factor does change is how long the bar remembers. The energy a burst leaves stays for $Q/\pi f_0$ seconds, and the thermal motion it has to be read against changes on the same slow timescale. That is the lever the cryogenic bars pulled: a high $Q$ does not enlarge the signal, but it slows the noise down, so a measurement short compared with the ringing time compares the bar’s state just before and just after a burst and sees only the small part of the thermal energy that could have changed in between.

A planet as a bar #

The formula says nothing about what the bar is made of except its mass and its sound speed, and the largest elastic bodies available are planets. Freeman Dyson worked out in 1969 how the solid Earth responds to a passing gravitational wave, and the Earth’s own quadrupole oscillation, the slowest way it can ring, is excited by a wave of the right frequency exactly as a bar’s mode is. The tone the whole Earth rings at met that oscillation as the free vibration set going by great earthquakes, with a period of about 54 minutes. Its mass is $6\times10^{24}$ kilograms, twenty-one orders of magnitude more than a bar’s, and the seismic wave speed is comparable with aluminium’s, so the Earth’s cross-section is vast.

It is also deafened by its own noise, which is why no one has heard a gravitational wave in it. Earthquakes, the atmosphere and the oceans keep the Earth’s free oscillations permanently excited far above any plausible gravitational signal at millihertz frequencies. The Moon is quieter, and seismometers left by the Apollo astronauts were used, much later, to set upper limits on a background of gravitational waves; proposals to place more sensitive instruments on the lunar surface, turning the whole Moon into a resonant detector for the band between the interferometers on the ground and in space, are under study.

Standing out from the bar’s own motion #

A bar at temperature $T$ is never still. Half a kT for every way of moving gives each mode of vibration an average thermal energy of $kT$, half kinetic and half potential, and the fundamental mode is no exception. A burst is detectable only if it changes the mode’s energy by something comparable. The fourth figure turns that into a strain: the burst that leaves one $kT$ in the bar.

At room temperature, which is where Weber’s bars ran, it takes a strain of $1.6\times10^{-16}$ — five orders of magnitude larger than the waves that have since been detected. Cooling helps less than it seems to, because the energy a burst deposits goes as the square of its strain, so the strain needed falls only as the square root of the temperature. Liquid helium gains a factor of eight; a dilution refrigerator at a tenth of a kelvin, another six. The cryogenic bars of the 1990s and 2000s — ALLEGRO at Louisiana State University, EXPLORER at CERN and NAUTILUS at Frascati, AURIGA near Padua — did better than their temperatures alone allow by a trick the high quality factor makes possible. The noise a high Q moves out of the way found that a resonator with a high quality factor concentrates its thermal motion into a slow random walk of amplitude and phase, so a measurement short compared with the ringing time sees only a small part of the thermal energy. With superconducting amplifiers the bars reached effective noise temperatures of a few millikelvin and strains of a few times $10^{-19}$.

Below that is the floor no cooling removes. The mode is a quantum oscillator, and its energy comes in units of $\hbar\omega_0$; a burst that deposits less than one quantum cannot be told from no burst by a measurement of energy. For a 900-hertz bar that quantum corresponds to a strain of about $2\times10^{-21}$. The noise every amplifier must add found the related limit on any amplifier that reads both quadratures of an oscillation: it must add half a quantum of noise of its own. The bars never reached it. Proposals to evade it by measuring only one quadrature — the first ideas of quantum non-demolition measurement, worked out by Vladimir Braginsky and Kip Thorne largely for the bars — were developed in the end for the interferometers, which use squeezed light to do it.

What Weber’s events would have cost #

In 1969 Weber reported coincident excitations of bars at Maryland and at Argonne, a thousand kilometres apart, at a rate of about one a day, and later claimed that they came preferentially from the direction of the Galactic centre. The fifth figure puts his claim through the cross-section. An event large enough to stand out in a room-temperature bar had to deposit roughly $kT$, and to deposit that, the wave had to bring some eighteen thousand joules per square metre per hertz to the bar, about twenty million joules per square metre over the bandwidth of a millisecond burst. If its source was at the centre of the Galaxy and radiated equally in every direction, it emitted the rest energy of about eighty suns in each event — at one a day, tens of thousands of solar masses a year.

The Galaxy cannot afford that. Its whole stellar mass is about a hundred billion suns, and at that rate it would have radiated itself away in a few million years. The argument was made at the time, by Dennis Sciama and others, in versions that allowed the waves to be beamed into the plane of the Galaxy or concentrated near the bars’ frequency, which reduced the bill by large factors and never far enough to be credible. Other groups built bars and found nothing, and by the mid-1970s the community had concluded that the events were not gravitational waves. Weber never accepted the conclusion. He is nonetheless remembered as the founder of the field, because the instruments, the coincidence technique and the very idea that the waves could be looked for were his.

Why light won #

The interferometers that detected the first waves in 2015 escape the bar’s limitation in a specific way. A bar measures a strain through the elastic deformation of a solid, and pays $(v/c)^2$ for doing so. An interferometer’s mirrors are free masses, hung on pendulums whose own resonance is far below the frequencies being measured, so they move with the wave as free particles would; and the distance between them is measured with light, which travels at $c$. There is no factor of $(v/c)^2$ to pay, and the arms can be made kilometres long, so the change in distance to be read is $10^{-18}$ metres rather than $10^{-21}$. What the instrument actually hears took apart how that reading is made and where it has blind spots of its own.

The interferometers are also broadband. A bar responds strongly only within a small fraction of a hertz of its resonance, extended by clever readout to tens of hertz and, in the best of the bars, about a hundred, and the waves from merging black holes and neutron stars are chirps that sweep through hundreds of hertz in a fraction of a second. The orbit that has to shrink found why: the frequency of the waves rises as the orbit decays, faster and faster. A chirp spends only milliseconds crossing a bar’s band, less time than the bar needs to ring up, and the bar sees a fraction of it.

Where the cross-section formula holds #

The formula is for a thin cylinder lying broadside to the wave with the wave’s polarisation along its axis — the most favourable orientation. Any other orientation reduces it, by the fourth power of the sine of the angle between the bar and the direction the wave comes from, and by the cosine of twice the polarisation angle. It treats the bar as a single mode; higher modes couple too, at their own frequencies, but more weakly. It assumes the wave’s wavelength, about 300 kilometres at a kilohertz, is far longer than the bar, which it is by five orders of magnitude.

Spheres were proposed as the successors to bars because a sphere has five quadrupole modes at nearly the same frequency, one for each independent pattern of tidal stretch, so it hears waves from every direction and can tell their polarisation. Two small spherical detectors, MiniGRAIL in Leiden and Mario Schenberg in São Paulo, were built and cooled in the 2000s. They run into the same factor of $(v/c)^2$ as bars, and none approached the interferometers’ sensitivity.

A ringing drawn without its noise #

The figures draw the mode’s response to a single burst in a noiseless bar; the ringing a real bar holds is that response added to a thermal motion whose amplitude, at room temperature, is a hundred times larger, and the detection is a statistical decision made on the output of an amplifier, not a glance at a trace. Nor do they show how the bar’s vibration is converted into a signal: Weber’s piezoelectric crystals, the resonant transducers and superconducting quantum interference devices of the cryogenic bars, each with its own noise and its own coupling losses. And the energy budget of the fifth figure assumes a source radiating equally in all directions with a spectrum about a kilohertz wide; every serious attempt to rescue Weber’s events altered those assumptions, and the figure’s numbers would move with them.

Still open: whether a resonant detector can still find something an interferometer misses #

Bars and spheres are not entirely retired as ideas. A resonant detector listens at a single frequency for a long time, and that suits sources that emit continuously at a known frequency, or high-frequency waves above the interferometers’ bands, where several speculative sources — from the early universe, or from hypothetical light dark-matter particles clustered round black holes — have been proposed. Small quartz and sapphire resonators cooled to millikelvin temperatures are being run at megahertz frequencies for exactly that reason, and two unexplained events seen in one in 2021 are still being argued over. Whether any gravitational wave at megahertz frequencies exists, and whether a resonant detector of any size can reach the strains needed to see it, is not known.

The habit worth carrying away is to find the soft link in a chain of coupling. A bar catches a gravitational wave with a cross-section $(8/\pi)(GM/c)(v/c)^2$, about $2\times10^{-25}$ square metre-hertz for Weber’s 1.4 tonnes, because it can answer a strain only by deforming against stiffness that transmits stress at the speed of sound; so a burst large enough to stand out in a room-temperature bar would have cost a source at the Galactic centre about eighty solar masses, and the waves were found instead by free mirrors read with light.