Illustrated Physics
physics, drawn
Concept

Strain — where it appears

The fractional deformation of a material, being a change in length divided by a length or a change in shape measured as an angle. It is dimensionless, which is why the same number describes a stretched rubber band and the passage of a gravitational wave through a four-kilometre arm.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

astrophysics · Gravitational waves Solid and liquid are answers about a duration. The relaxation time of seven materials, on a logarithmic axis spanning 39 decades, against the length of one observation. A material behaves as a solid when its relaxation time is longer than the observation and as a liquid when it is shorter, so the vertical line is what decides which — and it is a property of the observer. At 1 s, 4 of these are solids. Move the line six decades to the right and pitch joins the liquids; move it far enough left and water is a glass, which is not a figure of speech but what a picosecond pulse measures.

The liquid that remembers

Pitch shatters like glass under a hammer and flows through a funnel over a decade. Neither behaviour is the true one. What decides which a substance shows is not the substance but the length of the observation, and the ratio between the two has a name and a number.

fluids · Rheology 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.

astrophysics · Gravitational waves 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.

astrophysics · Gravitational waves Two fine gratings that make a coarse one. Top: two gratings of dark bars laid over each other, one with a period of 10 units and the other of 11, both with bars half a period wide. Bottom: the average transmission of the pair, smoothed over a few periods. Where the bars of the two gratings coincide, light passes through the gaps between them; where the bars of one fill the gaps of the other, nothing passes. The two drift in and out of step once every 110 units — the moiré period, p₁p₂/|p₁ − p₂| — producing broad dark and light bands 11 times coarser than either grating. Neither grating contains a period of 110; their product does.

The pattern two gratings make that neither contains

Lay one fine net curtain over another and broad, swirling bands appear that are in neither curtain. Two sets of lines a tenth of a millimetre apart, differing by one part in a hundred, make bands a centimetre apart. This is moiré: the beat of two gratings in space, as two notes a little out of tune beat in time. It magnifies whatever makes the gratings differ — a twist, a stretch, a shift — by the ratio of the bands to the lines, so it is used to measure strain in bridges and to align the layers of microchips. And two sheets of carbon, twisted by just over one degree, make a moiré on which electrons stop moving and the material superconducts.

waves · Superposition 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.

astrophysics · Gravitational waves

Named alongside it

The objects these essays reach for when they reach for this one.

Gravitational wavesConservation lawsDetectorMeasurementPolarisationAliasingAngular distributionAntenna patternAsymptoticsBeatsCreepCross-section

All concepts