Illustrated Physics
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Gravitational waves — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

    The wave that stretches one way and squeezes the other

    A gravitational wave passing through a ring of free masses lengthens one diameter while shortening the perpendicular one, then swaps. The two conservation laws that forbid anything simpler are why the effect is a part in a thousand million million million.

    part 1 · astrophysics
  2. How long an orbit has before the waves take it. The time a circular orbit has left before gravitational radiation brings it together, against its separation, for four pairs of masses. Both axes are logarithmic and every curve has the same measured slope, 4.00: the lifetime goes as the fourth power of the separation, so halving an orbit shortens its remaining life by a factor of sixteen. The Earth's orbit is 13 decades above the age of the universe and the neutron-star binary is below it, which is the whole difference between a system that is losing energy and a system that is going to merge.

    The orbit that has to shrink

    Two masses in orbit radiate gravitational waves and lose energy, so the orbit tightens, so they go faster and radiate harder. The runaway takes 10²³ years for the Earth and the Sun and eight minutes for the last thousand kilometres of a black-hole pair — and the same one-line formula gives both.

    part 2 · astrophysics
  3. Where the instrument can hear. The response of an L-shaped interferometer to the plus polarisation, over the whole sky: azimuth across, cosine of the polar angle up, so that equal areas of the picture are equal areas of the sky. The plus pattern is largest directly overhead and underfoot and along the arms, and vanishes on four lines where a wave stretches both arms equally and the interferometer has nothing to compare. The average of the square over the whole sky is 0.2333, summed over a hundred and sixty thousand directions, and the two polarisations' averages add to 0.4000 — two fifths exactly, whatever the polarisation angle. Averaged over that angle as well, each polarisation contributes a fifth, and that fifth is what turns a strain sensitivity into a range: it is why a detector's quoted reach is substantially less than what it would manage for a source overhead. There is no direction in which the instrument is deaf to both polarisations at once, and none in which it is fully sensitive either.

    What the instrument actually hears

    A gravitational-wave detector is not a ruler laid against a stretching space. It is a clock comparison, its response falls to nothing at frequencies where the wave turns over while the light is still in the arm, and there are directions in the sky where it is deaf.

    part 3 · astrophysics
  4. A few cycles, and everything about them is two numbers. The strain radiated by a remnant of 62 solar masses spinning at 0.68 of its maximum, as it settles down, with the decaying envelope of its fundamental mode drawn over it. The fundamental rings at 274 hertz and decays in 3.7 milliseconds, which is 1.0 cycles — this is not a bell and it does not sustain. Every frequency and every decay time in the sum is fixed by the mass and the spin alone; nothing about what made the remnant survives into them. What does depend on the collision is how loudly each mode is excited, and the relative amplitudes here are the rough values a merger of two comparable masses produces rather than a prediction.

    A few cycles that are only mass and spin

    After the orbit is gone there is one object left, distorted, and it settles down by radiating at frequencies that belong to it rather than to the collision. For a black hole those frequencies are fixed by the mass and the spin and by nothing else — so the first mode measured is a measurement and every mode after it is a test, and the test is that four curves in one plane pass through one point.

    part 4 · astrophysics
  5. The ring does not come back. A ring of freely floating masses at four phases of a passing gravitational wave and once after it has gone. The first four are the familiar picture: stretched one way, then the other, with the area unchanged. The fifth is the one the standard picture does not draw — the ring is permanently deformed, by 25 per cent of the largest deformation the wave itself produced, and nothing brings it back. The masses are not oscillating about a new centre; they are at rest, at new separations. Both the oscillation and the offset are exaggerated enormously: the real strain at 440 megaparsecs is 9.9e-22 and the real permanent offset is 2.5e-22, so the drawing magnifies both by about 2e+20. What is honest in the picture is the ratio between them.

    The ring that does not come back

    Every picture of a passing gravitational wave shows a ring of free masses stretched, squeezed and let go. The last frame is wrong. The ring ends a different shape — permanently, with the masses at rest at new separations — by about a fifth of the largest distortion the wave itself produced. What sources the offset is the energy the wave carried away, so the wave is remembering itself, and nobody has measured it.

    part 5 · astrophysics
  6. The strain a lumpy spinning star sends. The gravitational-wave strain at Earth from a neutron star one kiloparsec away, against the wave's frequency — twice the spin frequency — on logarithmic axes, for ellipticities of 10⁻⁹, 10⁻⁷, 10⁻⁵, 10⁻³, with the most each of three known pulsars could emit if all its observed spin-down were gravitational radiation. Crab: at most 1.4·10⁻²⁴ at 59.2 Hz, needing an ellipticity of 7.7·10⁻⁴; Vela: at most 3.3·10⁻²⁴ at 22.4 Hz, needing an ellipticity of 0.0018; J0437−4715: at most 1.6·10⁻²⁶ at 347.4 Hz, needing an ellipticity of 2·10⁻⁸. Searches have pushed the Crab's strain below about a hundredth of its limit, 1.4·10⁻²⁶. The strain grows as the square of the frequency, so a fast millisecond pulsar with a tiny mountain can send as much as a young pulsar with a large one.

    The mountain a spinning star is allowed

    A neutron star spinning thirty times a second with a bump on its surface a few centimetres high is a rotating mass quadrupole. It sends out gravitational waves at twice its spin frequency, as a steady tone lasting millions of years. None has been heard. The silence is a measurement. The Crab pulsar is slowing down, and if all the energy it loses went into gravitational waves they would have been detected many times over. Searches now say the waves carry less than a ten-thousandth of it. The mountain on the Crab is less than about ten centimetres high, on a star twenty-four kilometres across.

    part 6 · astrophysics
  7. The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.

    The wobble that radiating makes larger

    Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

    part 7 · astrophysics
  8. Six ways a wave of gravity could move a ring of free masses. Sixteen free masses on a ring (dashed) and where a passing gravitational wave of each of the six possible polarisations moves them at its crest, drawn at a strain of 0.32, some 10²⁰ times larger than a real wave's. Top row: plus and cross, the two general relativity allows, which stretch one direction across the wave while squeezing the other and change no area; and breathing, which swells the ring evenly. Bottom row, drawn in planes containing the direction of travel (arrow): longitudinal, which stretches along the travel; and the two vector modes, which shear those planes. Any metric theory of gravity permits these six and no others; which of them a theory keeps is decided by the field that carries gravity.

    The four shapes a wave of gravity leaves out

    A passing gravitational wave squeezes a ring of free masses in one of two patterns, and that is usually presented as the whole menu. It is general relativity's choice from a longer one. The tide a wave carries is a symmetric table of six numbers, so any theory in which gravity is geometry allows six shapes; a theory keeps two only if its graviton has no mass and nothing else rides along with it. Give the graviton the smallest mass imaginable and it gains three more states, and one of them changes how much the Sun bends light — by a quarter, however small the mass.

    part 8 · astrophysics
  9. How much of a wave of gravity a bar can catch. A resonant bar's cross-section for gravitational waves, integrated over its resonance, (8/π)(GM/c)(v/c)², against its mass, on logarithmic axes, for sound speeds of 3.0 km/s, 5.4 km/s, 10.0 km/s (aluminium is about 5.4). It is proportional to the mass, which is why the bars weighed tonnes, and to the square of the speed of sound over the speed of light — the bar answers a strain only by being stretched against its own stiffness, and (v/c)² is about 3 × 10⁻¹⁰. Weber's first bar caught 2.3·10⁻²⁵ m² Hz: to absorb a single joule, a burst must carry 4.4·10²⁴ J per square metre per hertz at the bar's frequency. Dots: Weber's bar, and the two cryogenic aluminium bars that ran longest, ALLEGRO in Louisiana and NAUTILUS near Rome.

    The bar a wave of gravity barely touches

    The first gravitational-wave detectors were tonnes of aluminium, hung in a vacuum and listened to for ringing. A passing wave stretches and squeezes the bar, and the bar can answer only by vibrating against its own stiffness, so the fraction of the wave it catches is set by the speed of sound over the speed of light, squared: about three parts in ten billion. That one ratio explains why the bars had to be enormous and cold, why Joseph Weber's reported detections of 1969 would have required the Galaxy to be converting dozens of suns a day into gravitational waves, and why the waves were finally found by light instead.

    part 9 · astrophysics

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