The ring that does not come back

The picture everyone knows draws the figure everyone knows: a ring of freely floating masses, a wave passing through it, the ring stretched along one diameter while it is squeezed along the perpendicular one, then the reverse, then back.

The last frame of that sequence is not right, and it is not right by an amount that is not small.

The effect is called memory, it was identified in a limited form in the 1970s and in its full nonlinear form in 1991, and it has never been observed.

Why a wave can leave something behind #

A wave that leaves nothing behind is what a wave usually is, and it is worth being clear about which property of the usual case fails here.

In the linearised theory, the strain $h$ is a small perturbation of flat spacetime that obeys a wave equation, and a solution of a wave equation driven by a source that switches off returns to zero. So the linearised theory of a burst of gravitational waves gives no memory at all. That is why the effect was not there to be found in the standard textbook treatment.

Two things escape the argument, and they have different characters.

The linear memory is what happens when the source’s own mass distribution ends up different. Consider masses that come in from infinity, interact and fly out again — a hyperbolic encounter, a supernova ejecting material, a star disrupting. The system’s quadrupole moment is not oscillating at late times; it is growing, because the pieces are separating at constant velocity. A second derivative of that growth is not zero at any finite time, and the difference in $h$ between the far past and the far future does not vanish. This part has been understood since the 1970s and is small for a binary that merges, because a merged binary has no unbound pieces.

The nonlinear memory is the larger part, and it is the reason the subject is interesting. Gravitational waves carry energy; energy gravitates, which is the same statement that makes a mirrored box of light heavier than an empty one; so the waves themselves act as a source of further gravitational field. A burst of radiation leaving a system is, from far away, indistinguishable in this respect from a cloud of massless particles flying off — an unbound flux of energy heading to infinity — and an unbound flux of energy leaves exactly the kind of permanent imprint the linear memory describes for unbound masses.

So the wave sources its own memory. That is a genuinely nonlinear statement in a theory whose waves are ordinarily treated as linear, it is not a correction of the usual kind, and its size is not small. Empty space was supposed to be the one medium that adds exactly; for electromagnetism it nearly is, and for gravity it is not, because the gravitational field carries the charge it responds to.

The size, which is computed from the flux and not chosen #

Because the memory is sourced by the radiated energy, its rate of growth is the energy flux — which is the square of the strain’s own time derivative. That fixes both the total and the shape without any separate model.

The magnitude follows from one line. The permanent offset is $4G\Delta E/c^4 r$ times an angular factor, where $\Delta E$ is the energy radiated. For an equal-mass merger $\Delta E$ is about five per cent of the total rest mass — the shortfall the remnant’s own weight records — and the peak strain of the same event is about $0.14\,GM_{\text{tot}}/c^2r$ — so both quantities are proportional to the total mass divided by the distance, and their ratio is a pure number.

That is the arithmetic worth taking away, and it removes the explanation most people reach for first. The memory is not a small correction riding on a large signal. It is a fifth of the signal, in every merger in the universe, at the orientation that shows it best. Any event loud enough to detect carries a memory that is also, by the same factor, loud enough to detect — if it could be detected at all.

Why it cannot be detected, which is about frequency and not size #

An interferometer is not a ruler. The account of what an interferometer measures establishes that it is a clock comparison whose response falls away at both ends of its band, and the low end is where the memory lives.

A permanent step is a signal whose Fourier content is concentrated at low frequency: the transform of a step falls as the inverse of the frequency, so most of its power is below anything the instrument can hear. And the low-frequency end of a ground-based detector’s band is not set by anything that can be improved cheaply. It is set by the fact that the mirrors are hung on wires in a gravitational field, so below a few hertz they swing with the ground rather than staying still, and seismic motion at those frequencies is enormous.

So the memory is a signal the instrument is deaf to by construction. What is left is the part of the step that happens quickly — the rise, which as the figure shows takes place over the last few tens of milliseconds and therefore has content at tens of hertz — and that is a small fraction of a step’s power.

The usual estimate is that a single event’s memory has a signal-to-noise ratio of a few tenths at current sensitivity, against a detection threshold of several. The route to a detection is therefore to add events: the memory’s sign is predicted, so contributions from many events can be summed coherently, and the accumulated significance grows as the square root of their number. Estimates of how many are needed range from about a hundred to about two thousand depending on the detector generation assumed, which places a first detection somewhere between the end of this decade and the next instrument.

A detector that is deaf at the wrong end, and one that is not #

There is a second kind of instrument for which everything in the previous section runs backwards.

A pulsar timing array uses millisecond pulsars as clocks and the Earth as the other end of a baseline of thousands of light years. It is sensitive at nanohertz — periods of years — because that is the band in which a decade of arrival-time data has any resolution at all, and it is completely insensitive above a microhertz.

A memory signal in that band looks like this. A pair of supermassive black holes merges somewhere; the burst passes the Earth, or passes a pulsar; and the light-travel time along that line of sight is permanently changed. Since the timing measurement is of the accumulated phase of a pulsar’s pulses, a permanent change in the rate at which they arrive produces a ramp in the timing residuals that grows linearly and never stops.

That is exactly the shape a timing array is good at. A step in strain is a ramp in residuals, and a ramp accumulating over a decade against a measurement precision of a hundred nanoseconds is a far better bargain than a step in a band where the instrument’s response vanishes.

The searches have been done and the result is upper limits: no memory event has been seen, and the limits constrain how often supermassive holes of the relevant masses merge nearby. What the limits already rule out is modest; what makes them interesting is that they are a search for an effect nobody has otherwise looked for, in a band where it is the natural signal rather than the awkward one.

The general point is worth separating from the astronomy. An effect that is impossible for one instrument may be the easiest thing for another, and which it is depends on the shape of the signal rather than on its size. A step is hopeless for a device that measures oscillations and ideal for one that measures accumulated phase, and the two devices in question here differ in frequency by nine orders of magnitude.

The pattern that is the wave’s own, inverted #

There is one further obstacle, and it is the kind that is more interesting than an instrumental one.

The reason is the same one that sets the memory’s size. The memory is sourced by the energy flux, and the energy flux from a binary is concentrated along the orbital axis — a face-on binary is loud precisely because most of its radiation goes that way. Seen from along the axis, that flux is symmetric about the line of sight and produces no transverse-traceless distortion at all. Seen from the side, the same flux is maximally asymmetric, and the memory is largest.

So the binaries that are easiest to detect are the ones with the least memory to detect. A binary within ten degrees of face-on is at 99 per cent of the wave’s best amplitude and at three per cent of the memory’s — and since detection is amplitude-limited, the observed population is biased toward exactly the orientations that hide the effect.

Nothing about that is a defect of the instruments. It is a property of quadrupole radiation, it is the same geometry that makes the antenna pattern of a detector what it is, and the correct response is to fold it into the stacking rather than to hope for a fortunate event.

Seventy-five years in the equations before anybody looked #

The effect is a consequence of the field equations as Einstein wrote them in 1915, and it was not identified until 1991. That gap deserves an account.

The linear memory came first, in the 1970s, from work on the gravitational radiation of unbound systems — a hyperbolic encounter between two stars, or a supernova throwing material off. The reasoning there is not subtle once the question is asked: the source’s quadrupole moment does not return to what it was, so the field far away does not either. It was called a burst with memory, and it was expected to be small.

The nonlinear part required a change of viewpoint rather than a longer calculation. Christodoulou’s result in 1991 came out of a mathematical programme on the stability of flat spacetime, in which the asymptotic behaviour of solutions is the whole object of study rather than an afterthought — and in that setting the radiated energy flux is one of the quantities the analysis tracks by construction. The memory appears as a property of the solution at large distance, not as a correction to a wave.

What made it invisible from the usual direction is that the usual direction throws it away in the first step. Expanding around flat space and keeping terms linear in the perturbation removes the wave’s own energy from the source, because that energy is second order — and the memory is the thing that energy produces. It is not that the calculation was hard; it is that the standard approximation begins by deleting the source.

That is a recurring shape and it is worth naming. A systematic approximation deletes whole effects rather than shrinking them, and which effects it deletes is decided at the first step and not visible afterwards. The same thing happens to a fluid’s steady streaming, which is second order in an acoustic amplitude and is therefore absent from a first-order theory that is otherwise perfect — and there too the effect turned out to be the one that transported anything.

A model waveform, a leading-order size, and one polarisation #

The waveform here is a model. It is a Newtonian chirp joined to a ringdown at the peak, and no part of it is a numerical relativity template. What is computed from it is the shape of the energy flux, which is what the memory integrates, and the shape is right where the timing is not: a real merger’s flux peaks a little later and falls a little differently, and the resulting memory differs from this one by tens of per cent.

The expression for the size is the leading order. The angular function used here is the leading-order one for a binary, and it omits contributions from the higher radiative multipoles and the spin. Numerically computed memory from a full simulation differs from it by a comparable amount, which is fine for an argument about a factor of five and not fine for a template.

There is also memory in the second polarisation, and there is not. The memory appears in the plus polarisation for a non-precessing binary and vanishes in the cross polarisation, which is a statement about the source’s symmetry rather than a theorem; a precessing binary has memory in both, and the amount depends on the geometry in a way no simple angular function captures.

And the linear memory has been left almost entirely out. For a binary that merges it is small, because there are no unbound masses. For a supernova, a hyperbolic encounter or a neutrino burst it can be the whole effect — and the radiation of an accelerated body is the argument gravitational radiation inherits from electromagnetism — and a core-collapse supernova’s neutrinos carry away a hundred times the energy its gravitational waves do — so the largest memory signal in the local universe may come from neutrinos rather than from waves at all.

Twenty orders of magnitude of exaggeration, and an instrument’s response #

They cannot show the effect at its true size. The ring figure exaggerates both the oscillation and the offset by twenty orders of magnitude, and it says so on the canvas; at the real scale a four-kilometre arm changes length permanently by about four times ten to the minus nineteen metres, which is a thousandth of a proton’s width. What the picture is honest about is the ratio of the permanent part to the oscillating part, and that is the only quantity in it worth reading.

Nor can they show what a detector would actually record. An interferometer’s output is not the strain but the strain filtered through the instrument’s response, and a step passed through a response that vanishes at zero frequency comes out as a bump that decays back to nothing. The permanent offset is not merely hard to see; in the recorded data it is not permanent, because the instrument cannot represent a permanent anything.

And they cannot show that the masses have changed state rather than position. A ring caught mid-wave is distorted and its masses are moving; a ring after the wave has passed is distorted and its masses are at rest. Those are different situations and a still frame draws them identically. The distinction is the whole content of the word memory: what is left is not a motion but a configuration.

Still open: whether the memory is one corner of something much larger #

Since the 2010s the memory has been tied to two apparently unrelated results — theorems about the emission of very low-energy gravitons in any collision, and an infinite family of symmetries of spacetime at the place light ends up — as three faces of one structure. Each of the three can be derived from either of the others.

What that structure means is not settled. The symmetries are far larger than the symmetries of flat spacetime, they were found in 1962 and treated for decades as an embarrassment of the formalism, and they have since been argued to constrain what happens to information falling into a black hole. Whether any of that survives as physics, or whether it is a reformulation of results already known, is an active question — and a measurement of the memory would be the only experimental contact any of it has.

That is an unusual position for an effect to be in. It is a clean prediction of general relativity, computable to a few tens of per cent, of a size that is a fixed fraction of a signal detected routinely — and it is also the single observable quantity attached to a structure that several people believe is the beginning of a different way of formulating the theory.

The habit worth carrying away is about what a linear theory cannot be asked. When a wave carries energy and the energy gravitates, the wave is a source, and any statement derived by treating it as a solution alone will be missing exactly the part that persists. The linearised theory of gravitational waves gets the oscillation right and the offset exactly zero, and the offset is a fifth of the oscillation — which is a useful measure of how far “linear to a part in a thousand million million million” is from linear.