The four shapes a wave of gravity leaves out

The wave that stretches one way and squeezes the other drew a ring of free masses deformed by a passing gravitational wave in two patterns, plus and cross, the same shape turned through forty-five degrees. The angle came out of the graviton’s spin: a spin-2 wave returns to itself after half a turn, so its two states sit at a quarter of that. The essay took it as given that there are two states. That is general relativity’s answer, and it is worth knowing what question it answers, because a ring of masses could be deformed in more ways than two and some theories of gravity use them.

Six numbers in a tide #

Whatever a gravitational wave is, what it does to a set of free masses is a tide. Two neighbouring masses falling freely feel no gravity in their own frame — free fall removes it — but they feel the difference between the gravity at one and at the other, which no choice of falling frame can remove. The difference is linear in their separation, so it is described by a matrix that turns a separation into a relative acceleration. In any theory where gravity is the geometry of spacetime, that matrix is symmetric, and a symmetric three-by-three matrix has six independent entries.

Six entries mean six independent ways a wave can move the masses. Douglas Eardley, David Lee, Alan Lightman and two colleagues classified them in 1973 for a wave travelling along one direction, and they are the six shapes in the figure.

Two of the six act only across the direction of travel and change no area — the plus and cross of general relativity. One acts across the direction of travel and does change the area: breathing, an even swelling and shrinking of the whole ring. One acts only along the direction of travel: longitudinal, a stretching of separations parallel to the wave. And two mix the two directions, shearing the planes that contain the direction of travel: the vector modes, named for how they transform when the frame is turned about the wave’s direction. Under such a turn by an angle $\alpha$ the tensor pair rotates into itself by $2\alpha$, the vector pair by $\alpha$, and breathing and longitudinal not at all — the same rule that makes light’s handedness a spin-1 property.

Why general relativity keeps two #

The metric of spacetime has ten independent components. A wave of it ought to carry ten, or six after the tide is taken. General relativity keeps two, and the four it discards are discarded for a reason that is not specific to gravity.

Part of what is written in the metric is the choice of coordinates, and a change of coordinates changes the metric without changing anything physical. The same freedom appears in electromagnetism, where the potentials can be altered without changing a single field. In general relativity it removes four of the ten components outright. Einstein’s equations then impose four conditions on what is left that are not equations of motion at all — constraints, like Gauss’s law — and those remove four more. Two survive, and they are the plus and cross.

The discarded components were not always recognised as discardable. Einstein’s first paper on gravitational waves, in 1916, found three kinds of wave in the linearised equations. Arthur Eddington showed in 1922 that two of them were artefacts of the coordinates Einstein had chosen: they could be made to travel at any speed at all by choosing different coordinates, and so, he wrote, they travelled “with the speed of thought”. Only the third kind, the transverse wave whose two states are plus and cross, carried energy and travelled at the speed of light in every coordinate system. Einstein himself doubted for a time in the 1930s that even that one was real, and submitted a paper arguing that gravitational waves did not exist before a referee’s report, and a colleague’s patience, persuaded him that a singularity in his solution was again a feature of the coordinates. The question of which parts of a metric wave are physical and which are bookkeeping is old, and the polarisation count is its answer.

The count is the count for a massless field of spin 2. A massless field carries only the states whose spin points entirely along or entirely against its motion, two for any spin, which is why light has two polarisations as well. General relativity’s graviton is massless and is the only field gravity has. Change either of those and the count changes.

What each theory adds #

The oldest alternative, Carl Brans and Robert Dicke’s theory of 1961, adds a scalar field to the metric: a single number at every point that sets the local strength of gravity. Its waves can carry that number as well, and a ripple in the strength of gravity swells and shrinks a ring evenly — breathing. The same scalar field radiates from an orbiting pair whose members couple to it differently, at dipole order, which is how binary pulsars have bounded it. Theories that add a vector field, as Einstein–aether theory does by giving spacetime a preferred direction of time at every point, can carry vector modes as well.

The most striking change comes from giving the graviton a mass. A massive field of spin 2 is not the massless one with a small correction. A massive particle can be brought to rest, and at rest it has no direction of motion for its spin to point along, so all its spin states are on an equal footing: a spin-2 particle at rest has five. Five polarisations — tensor, vector and one scalar — for any mass at all, however small. The three extra states do not fade away as the mass is made smaller. They remain, as states, all the way down to the massless limit, and then vanish at once.

A quarter of the bending, at any mass #

Whether the extra states vanish from the physics as well as from the count is what Hendrik van Dam, Martinus Veltman and Valentin Zakharov asked in 1970, and the answer was no.

The scalar state of a massive graviton couples to matter through the trace of its energy and stress, which for a slow body is its mass. It therefore adds to the pull a planet feels from the Sun. Light has a stress tensor with zero trace, so the scalar state does not touch light at all. Match the theory to the planets — fix the total pull so that the orbits come out right — and the part of the pull that comes from the tensor states has to be smaller than in general relativity to leave room for the scalar. Light feels only that smaller part.

The bookkeeping is the same as in the parametrised description of the solar-system tests, where light’s deflection is $(1+\gamma)/2$ times general relativity’s and $\gamma$ measures how much space curvature accompanies a given Newtonian pull. General relativity has $\gamma = 1$. The linear massive theory, once its pull on the planets is matched, behaves as though $\gamma$ were one half, and the deflection is $(1 + \tfrac12)/2 = \tfrac34$ of the observed value.

The result is that light is bent by exactly three-quarters of general relativity’s angle, for any graviton mass, however small. It is the same structure as the factor of two Newton missed: light’s deflection measures something about gravity that planetary orbits do not, and here the something is whether the graviton has a mass. The radio-tracking of the Cassini spacecraft in 2002, as its signals passed close to the Sun, measured the ratio to be one to within a hundred-thousandth. The linear theory of a massive graviton fails by twenty thousand of the measurement’s standard deviations.

That was regarded as the end of massive gravity for a while. Arkady Vainshtein pointed out in 1972 that the linear theory cannot be trusted near a mass as large as the Sun, because the scalar state’s self-interactions become large there, and that they could suppress its effect inside a radius that, for the Sun, extends far beyond the planets. Inside that radius the theory would look like general relativity and outside it like massive gravity. Making a consistent theory that does this, without an extra state that has negative energy, took until 2010, and whether such theories describe the universe is open. What the light-bending measurement did was turn a question about the number of polarisations into a question about how the theory behaves near the Sun.

There is also a direct bound. A massive graviton’s waves would travel at a speed that depends on frequency, lower frequencies slower, and the chirps of merging black holes would arrive distorted. None has been. The current limit from the catalogue of detected mergers is a mass below $1.3 \times 10^{-23}$ electronvolts, a Compton wavelength longer than about three parsecs.

What a detector hears of each #

An interferometer does not see a ring. It measures one number, the difference in length between its two arms, and each polarisation contributes to that number with a weight that depends on where the wave comes from and how it is oriented. The instrument’s response to plus and cross has a pattern over the sky with zeros along certain lines. The other four have patterns of their own.

A wave from directly overhead moves the masses in the plane of the arms only through its tensor part. Breathing swells both arms equally and leaves their difference unchanged; longitudinal and vector modes move the masses along the vertical, where there are no arms. From the side the other modes appear, with their own patterns. Averaged over the sky, the squared response of an L-shaped detector is 0.400 for the tensor pair, 0.400 for the vector pair and 0.067 for breathing — a breathing wave would be harder to see than a tensor wave of the same strain by a factor of six in power.

The two scalar modes are worse than faint. At every angle the longitudinal response is exactly the breathing response with the sign reversed. Their effects on an interferometer are not merely similar but identical, so no number of interferometers, arranged in any way, can tell a breathing wave from a longitudinal one. What can be measured is the scalar content as a whole.

One wave, three detectors #

The general way to measure polarisation content is to look at a single wave with several detectors at different orientations. Each records one combination, weighted by its own response pattern. With enough detectors and a known source direction the combinations can be solved for the amplitudes of each kind.

The two LIGO detectors in Washington and Louisiana were built with their arms as nearly parallel as the curvature of the Earth allows, so that a wave seen in one would be seen in the other with the same weights. That doubles the confidence that a signal is real and halves the information about its shape: they measure almost the same combination. On 14 August 2017 a black-hole merger was recorded by both LIGO detectors and by Virgo, near Pisa, whose arms point in quite different directions. It was the first wave seen with three independent orientations.

The method is to ask, for each hypothesis, whether a single wave arriving from a single direction could produce the three recorded signals with the amplitudes and timings they had. A pure tensor wave from a given direction predicts a definite ratio between the strains in the three detectors, set by their tensor patterns; a pure scalar wave from the same direction predicts a different ratio, set by their scalar patterns; and the direction itself is fixed, more or less, by the arrival times, a few milliseconds apart. The two LIGO detectors alone could have fitted almost any hypothesis by moving the source somewhere else on the sky. With a third detector at a different orientation the fits stop being interchangeable, and some hypotheses fit far worse than others.

The odds favoured a purely tensor wave over a purely vector one by more than two hundred to one and over a purely scalar one by more than a thousand to one. A week later the merger of two neutron stars, seen in gravitational waves and in γ-rays from the same event 1.7 seconds apart after a trip of 130 million years, showed that the waves travel at the speed of light to about one part in $10^{15}$. That single coincidence eliminated a large class of scalar–tensor and vector–tensor theories in which the extra fields change the speed of the tensor waves.

Where the comparison stops #

The tests so far compare pure hypotheses: all tensor against all scalar. A theory in which gravity is mostly general relativity with a small admixture of breathing — which is what a scalar–tensor theory consistent with the solar system would predict — needs a measurement of mixtures, and with three detectors and a source direction to be fitted as well, there are not enough independent combinations to fit six amplitudes. With KAGRA in Japan operating and a detector planned in India, five orientations will be available, and mixed-polarisation tests become possible event by event. A wave from a continuous source, such as a spinning neutron star, would be better still, because the Earth’s rotation sweeps the detectors’ patterns across it every day and samples many combinations.

The argument for six is itself restricted. It holds for theories in which matter falls along the geodesics of a metric, so that the effect of a wave is a tide. Theories in which gravity is not geometry, or in which different kinds of matter respond to different fields, can do other things, and those are constrained by the universality of free fall rather than by polarisation counts.

What the pictures cannot show #

The rings are drawn at a strain some $10^{20}$ times larger than any real wave, as every picture of a gravitational wave must be, and at that size the deformations would not be the linear ones drawn. They are snapshots at a crest; at the next trough each shape reverses. The longitudinal and vector rings are drawn in planes containing the direction of travel, so that their motion can be seen, while the tensor and breathing rings are drawn across it; a single ring, in a single plane, would show only some of the six.

The bending figure draws the linear theory of a massive graviton with its discontinuity sharp. That is exactly what the linear theory predicts and exactly what nobody believes about the full theory near the Sun, where non-linear effects may smooth it away. The figure shows why massive gravity needed rescuing, not whether it has been rescued.

The domain of the six-polarisation classification is weak plane waves in metric theories, far from their sources. Near a merger, or in strong fields, the separation into polarisations is not clean, and the theory-by-theory counts assume the extra fields are light enough to be radiated at all.

Still open: how small a scalar part can be #

Every observation so far is consistent with a gravitational wave that is purely tensor, travelling at the speed of light, carried by a graviton with no measurable mass. What is not known is how small an admixture of the other shapes the data can exclude — whether a breathing component at a per cent of the tensor amplitude, or a tenth of a per cent, would have been noticed. Networks of five detectors, and space-based detectors whose arms sweep round the Sun over a year, are expected to bound the scalar and vector content directly, and pulsar-timing arrays, which see the low-frequency background as correlated jitter between pulsars, are sensitive to breathing and longitudinal modes in a quite different way. A detection of any extra polarisation would be the first observation of a gravitational field beyond the metric.

The two familiar patterns are a selection, not the menu. The tide a metric wave carries has six independent entries, so a ring of free masses can be moved in six shapes; general relativity’s massless spin-2 graviton keeps two, an extra scalar field adds breathing, and a graviton with any mass whatever carries five — with a scalar state that, in the linear theory, leaves light bent by three-quarters of the observed angle however small the mass. The first wave seen by three detectors at three orientations favoured the two by two hundred to one over vector shapes and a thousand to one over scalar ones.