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Made into an instrument — part 2 of 3

The point at which a principle stops explaining something and starts measuring it — a critical angle sold as a refractometer, a Doppler shift weighing a planet, a null result testing a law.

$The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.$ Fluids

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

Three terms, and one distance. The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of.

Electromagnetism

The distance where a field changes its mind

An oscillating source has three fields around it, falling as the inverse cube, the inverse square and the inverse first power of distance. They are all equal at one radius, and that radius is the wavelength over 2π — a number containing nothing about the source at all. Inside it a source mostly stores energy; outside it, mostly loses it, and the two behaviours are different technologies rather than different strengths.

$What a row of sources does that one cannot. The pattern of 8 identical sources in a row, spaced 0.5 wavelengths apart, all driven in phase. Each source alone radiates the same in every direction drawn here; together they radiate almost entirely along one. The sum being performed is the sum over path differences across the row, which is the sum a diffraction grating performs over its slits — the same function with the same first null, at sin θ = 1/Nd, measured here off the curve. What has been exploited is retardation: the contributions arrive at different times, and the pattern is a map of where they arrive in step.$ Electromagnetism

When the source is not heard all at once

The dipole approximation is not a statement that a source is small. It is a statement that every part of it is heard at the same retarded time, and dropping that assumption turns one source into a sum over a source. The sum is the same one a diffraction grating performs over its slits, with the same first null and the same extra orders — so a phased array and a grating are one piece of arithmetic met twice.

The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.

Mechanics

The deflection that closes on itself

The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

Quantum

The energy that did not all arrive

A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.

A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Waves

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium.

Waves

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

An angle set by a voltage. The contact angle of water on a fluoropolymer coating with a rest angle of 115°, against the voltage across the coating, for coatings 1 and 3 µm thick with a relative permittivity of 1.93. The Lippmann–Young law cos θ = cos θ₀ + ε₀εᵣV²/2dγ makes the cosine rise as the square of the voltage, so the curve is symmetric about zero volts and steepest where the angle is near 90°. For 1 µm the law reaches complete wetting at 110 V; for 3 µm the law reaches complete wetting at 191 V. It does not get there. In most reported experiments the angle stops falling somewhere in the shaded band and then stays put however high the voltage is raised — contact-angle saturation — and why is still argued: charge trapped in the insulator, ionisation of the air at the sharp edge of the drop, and the thermodynamic stability of the contact line have all been proposed and none accounts for every case.

Fluids

The angle a voltage can set

A contact angle is treated as a fact about three materials — a solid, a liquid and the air — fixed the moment they are chosen. Put a voltage across a micrometre of insulator under a drop and the angle falls as the square of the voltage, with nothing about the materials changed. Drops can be steered across a chip with no pump, and a lens can focus with no moving part. The law that describes it is exact in its model, and real surfaces stop obeying it at an angle nobody has fully explained.

$A hemisphere outside is a cone of 16.6° inside. Rays leaving a flat face from a material of index 3.5 into air. Every direction in the hemisphere above the face is reached by a ray from inside a cone of half-angle 16.6°, arcsin(1/n), because refraction at the face fans the cone out; rays steeper than that are reflected back, drawn in the warning colour. The squeeze is exact in the way the invariant requires. The projected solid angle of the inside cone is π sin²θc, which is π/12.25, and multiplied by n² it equals π, the projected solid angle of the whole hemisphere outside — checked on the drawn geometry. So the light that does get out has not gained anything: the radiance inside is n² times higher and the directions available are n² times fewer, and the étendue passing the face is the same on both sides.$ Optics

The cone light has to find to get out

Inside a dense material the whole hemisphere of directions outside a flat face shrinks to a narrow cone, and light made inside can leave only if it happens to be travelling within it. For gallium arsenide that is two per cent of the light. Turned round, the same cone keeps light in: a slab of silicon with a rough surface holds the light it admits for fifty-one passes. Both numbers are the n² the optical invariant carries, and neither one breaks it.

A current with no voltage, and then a voltage that is a frequency. Current against mean voltage for a Josephson junction shunted by a resistance, in units of its critical current and of the critical current times the resistance. Up to the critical current the junction carries a supercurrent at exactly zero voltage, set by the difference of the two superconductors' phases. Above it a mean voltage appears, growing as R√(I² − Ic²) and approaching the ohmic line at large current; the curve is integrated from the junction equation and checked against that result at three currents. The voltage is not steady. It is a train of pulses, each the phase slipping by one turn, at a frequency of 483.6 GHz per millivolt: for a junction with Ic = 1 mA and R = 1 Ω, one unit of the voltage axis is 1 mV and 484 GHz.

Electromagnetism

The voltage that is a frequency

Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.

The month a binding energy would bend. The Moon's orbit seen with the Sun held off to the right, drawn as it would be if the Earth's binding energy fell towards the Sun more weakly than the rest of it. The Earth is bound by 4.5 × 10⁻¹⁰ of its mass-energy and the Moon by 1.9 × 10⁻¹¹, so with the Sun pulling at 5.93 mm/s² the Moon is pushed sunward relative to the Earth by 2.6 × 10⁻¹² m/s² for every unit of η. That push turns once a synodic month relative to the orbit, 29.53 days, and Hill's equations about a circular orbit — integrated from the forced solution and held on it to 4 × 10⁻¹¹ over twelve months — give a radial displacement of 8.0 m times η times the cosine of the lunar phase: outward at new moon, inward at full. The complete lunar theory, with the Sun's tide on the orbit included, gives 13.1 m. The displacement is drawn about 7 × 10⁶ times larger than it would be at η = 1.

Astrophysics

The binding energy that has to fall too

Every laboratory test of the equivalence principle compares bodies whose own gravity is a part in 10²⁵ of their mass, so none of them can ask whether gravitational binding energy falls like everything else. The Earth is bound by five parts in ten billion and the Moon by twenty times less, and if that difference fell differently the Moon's orbit would lean towards the Sun once a month — by a distance lasers have been measuring since 1969.

The electrode that walks itself negative. An electrode driven through a blocking capacitor by a symmetric waveform of amplitude 20 kTe, in an argon-like plasma whose electrons are collected 108 times as readily as its ions at saturation, starting uncharged. Potentials are relative to the plasma, in electron temperatures. On the first positive half-cycle the electrode draws a flood of electrons and nothing on the negative half-cycle can return the charge, because the ions arrive at a fixed, small rate; the electrode walks down, and within 6 cycles its mean potential is within half an electron temperature of the self-bias. That bias, found by integrating the charging, is −22.27 kTe, and the charge balance with a Bessel function gives −22.27. The floating potential of the same wall undriven is −4.68. At 3 eV, a 60 V drive makes a 67 V bias with no direct current supplied anywhere.

Astrophysics

The bias no battery supplies

Drive an electrode in a plasma through a capacitor with a perfectly symmetric waveform and it charges to a steady negative voltage almost as large as the waveform's own amplitude. No direct current flows anywhere and no battery is connected. The plasma's sheath lets electrons in far more easily than it lets them out, and every semiconductor wafer etched in the last forty years was held at a voltage made that way.

The rectangle a spin tilts. The laboratory energy of the pion from a tau into a pion and a neutrino moving at 0.8 of the speed of light, for samples of 60,000 decays whose parents have αP = +1, 0, −1 along their line of flight. Every sample fills the same interval, 313 to 2667 MeV: the edges are set by the masses and the speed, and they do not move. Inside that interval an unpolarised sample is flat and a polarised one is tilted, its height at each energy 1 + αP times the cosine of the rest-frame angle that energy corresponds to. Reading the slope back from each sample gives +0.993, −0.010, −1.006, each with a standard error near 0.006. A parent spinning along its flight throws the pion forwards in its own frame, and the boost turns forwards into more energetic, so the tilt of an energy spectrum measures a polarisation without any angle being measured at all.

Relativity

The slope a spin leaves in a spectrum

An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.

$Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.$ Relativity

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

Relativity

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

Quantum

The field an atom calls strong

A magnetic field splits sodium's two yellow lines into ten, at spacings set by Landé's factors — until the field grows past the one the electron already feels from its own motion, when the ten reorganise into the three a theory without spin predicts. Nothing about the magnet decides which pattern appears. The atom does: the same 45 tesla is a strong field for hydrogen, a middling one for sodium and a weak one for caesium.

Four stress states in one beam, and only one of them has no tension. Stress across the depth of a 300 by 600 millimetre concrete section at the middle of an eight-metre span, compression to the right, for four conditions. Under the load alone the bottom fibre is in tension at 11.1 MPa, which is four times what concrete can carry, so an ordinary reinforced beam cracks there and relies on steel to hold the crack together. With 1,500 kilonewtons of prestress 120 millimetres below the centroid and the full load applied, the section runs from 9.8 to 3.6 MPa and every fibre of it is in compression. The second case is the one that surprises: with the prestress applied and nothing whatever to oppose it, the top fibre is in tension at 1.7 MPa, because a force below the centroid bends the beam upwards. A prestressed beam is at its most vulnerable when nothing is on it, and what rescues it is its own weight: adding that alone brings the top back to 0.3 MPa of compression. Each stress block is checked by integrating it and recovering the force and the moment that produced it.

Mechanics

A state no load could reach

The free direction a redundant structure leaves open can be driven on purpose. Tighten a tendon through a concrete beam and its whole stress state moves into the half of the range the material is good at; tighten a bolt hard and the load it carries fluctuates by a fifth of what is applied to it. Both put the structure somewhere no arrangement of external loads could.

Optics

The same cone, and a different arrival

The invariant fixes what a guide accepts and says nothing about when it arrives, and the two turn out to be nearly independent. Shaping the index so that the rays which travel furthest also travel fastest cuts the spread in arrival times by a factor of five hundred, at the cost of exactly half the light — and the acceptance cone the invariant governs is untouched throughout.

Twice the kinetic energy, and what it equals. Twice the time-averaged kinetic energy of a bound orbit, divided by its time-averaged potential energy, against the power with which that potential depends on separation. Each point is a measurement: an eccentric orbit integrated for more than a hundred radial periods, with the two averages accumulated along it, and the radius checked to vary by at least a fifth so that the orbit is not trivially circular. The line is the exponent itself, and the points miss it by at most 6.6e-4. Two cases carry everything. At n = 2, a harmonic well, the two energies are equal — which is the ordinary equipartition statement, half a kT to the kinetic term and half a kT to the potential one. At n = −1, which is gravity and the Coulomb force, twice the kinetic energy equals minus the potential energy, so the total energy of a bound system is minus its kinetic energy. Nothing about temperature entered, and nothing about equilibrium: the relation holds for one orbit averaged over time as well as for a crowd averaged over members.

Thermodynamics

Weighing what cannot be put on a scale

Summed over every coordinate of a bound system, equipartition stops being a statement about temperature and becomes a relation between two averages: twice the kinetic energy equals n times the potential energy for a potential going as the nth power. For gravity that fixes a bound system's total energy from how fast its parts move — so a Doppler shift and an angular size return a mass, and for the Coma cluster the mass they return is fifty times the mass that shines.

How large the hidden dimensions would have to be. The size extra dimensions would need, for gravity's true scale to be at a TeV rather than at 10¹⁹ GeV, against how many of them there are — a logarithmic axis of metres, for three choices of the true scale. The relation is the one that makes the arrangement work: the Planck mass observed in four dimensions is M_*^(2+n)Rⁿ, so gravity is weak because its field spreads into a volume nothing else can enter. Solved for R at a true scale of a TeV: 1 dimension needs 2.9e+13 m, 2 dimensions needs 2.4e-3 m, 3 dimensions needs 1.0e-8 m, 4 dimensions needs 2.2e-11 m, 5 dimensions needs 5.4e-13 m, 6 dimensions needs 4.5e-14 m. The two horizontal lines are where experiment has been. One extra dimension would have to be of order a hundred astronomical units, which would have wrecked the orbits of the planets and is excluded absolutely. Two would have to be of order a millimetre — which is what made the arrangement famous, because a millimetre is a distance a laboratory can test, and torsion balances have since verified the inverse-square law down to fifty-two micrometres and excluded it. Three or more sit below a nanometre, where no gravitational measurement reaches, and are untouched.

Astrophysics

The scale that may not be where it looks

Every Planck number assumes gravity is four-dimensional all the way down. If it is not — if the field spreads into dimensions compact enough to have escaped notice — the true scale where gravity becomes strong could be at a TeV, and the whole remoteness of the Planck scale would be an artefact of where the field lines go. It is the one part of the subject an experiment can address, and the experiments have addressed it.

Four outcomes, one of which happened. The correlation between the two outer particles' measurements, against the angle between their analysers, for each of the four results the middle measurement can give. Every one of the four leaves the outer pair maximally entangled — each curve reaches one and minus one — so every outcome is as good as any other, and the outer parties have a perfect Bell pair whichever it was. What differs is which correlation they have, and the four are shifted and reflected versions of each other. The flat line is their average, which is zero at every angle, checked here at seven hundred and twenty angles to twelve figures. That vanishing is the whole reason the operation cannot be used to send anything: until the middle party's two classical bits arrive by an ordinary channel, the outer parties' data is indistinguishable from noise, and no correlation appears at all. The entanglement is created instantly and is useless until a message travelling no faster than light says which of the four it is.

Quantum

A link between two that never met

Take two entangled pairs sharing no particle, measure the two inner particles jointly, and the two outer ones — which have never interacted, never been in the same place, and have no history in common — are entangled. Nothing travelled between them. What has to travel is two classical bits saying which of four results occurred, and until those arrive the outer parties see nothing at all.

Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

Waves

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

The ripple that sits on top of an absorption edge. The Cu K absorption of copper foil, 293 K through its edge and for seven hundred electronvolts above it, with the smooth atomic background it would have if the absorbing atom were alone drawn beneath it. The difference between the two is the fine structure: a modulation reaching 11 per cent, dying away as the photon energy rises, and entirely absent from a free atom. It is there because the ejected electron is a wave that the neighbouring atoms scatter back onto the atom that emitted it, so the absorption depends on whether the returning wave arrives in step with the outgoing one — which depends on the distance to the neighbour and on nothing else about the sample.

Waves

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

The pressure at which an oil becomes a glass. Viscosity against pressure for 4 liquids on a logarithmic axis, from Barus's rule with the pressure coefficient each one actually has. The rule is an exponential, so a gigapascal multiplies an ordinary oil's viscosity by a hundred million or more, and the curves cross the line conventionally taken to mark a glass — a million million pascal-seconds — at 1.37 GPa for a mineral oil, 0.98 GPa for a traction fluid. The vertical line is the peak pressure inside the loaded contact this figure is about, 1.32 gigapascals, computed from the Hertz solution for that geometry and load. Water is drawn for contrast: its pressure coefficient is thirty times smaller, it never approaches a glass over this range, and that is why it is useless as a lubricant in a rolling contact however clean it is. Each curve is drawn solid up to the glass line and dashed above it, because past that point the material is not a liquid and its viscosity is not what decides how it shears — the exponential continues, and the substance it was written for does not.

Fluids

The oil that is a glass for a quarter of a millisecond

Every other account of viscosity here varies the temperature. Pressure does something larger and in the same direction for every liquid: viscosity rises exponentially, by a factor of ten to the eight or more at the gigapascal inside a loaded gear tooth. That is not a curiosity — it is the only reason there is a film there at all. Remove the pressure dependence from the calculation and the predicted film is five nanometres, under the roughness, and the surfaces touch.

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.

Astrophysics

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.

$The rings belong to the edge, not to the size. The far-field intensity of 3 apertures of the same width, against angle in units of the diffraction limit, on a logarithmic intensity axis spanning ten decades. They differ only in how the transmission falls off toward the rim. With a hard edge the first sidelobe is 13.3 decibels down and the core is 0.89 wide. With a Hann taper the first sidelobe is 31.5 decibels down and the core is 1.44 wide. With a Blackman taper the first sidelobe is 58.1 decibels down and the core is 1.64 wide. The hard edge's rings are not a defect of the optics and are not reduced by making it larger — they are the transform of a discontinuity, and the only way to remove them is to remove the discontinuity. What it costs is the width of the core, which is the resolution.$ Optics

The rings that belong to the edge

Every account of diffraction so far asks what the size of an aperture does. The rings around a star are not about its size: they are the transform of a discontinuity, they do not shrink relative to the core when the telescope grows, and the only way to remove them is to stop the transmission falling to zero abruptly. Softening the edge buys forty-five decibels of contrast and costs eighty per cent of the resolution.

$The peak that is lost to a fraction of a wave. The height of the central peak, relative to a perfect pupil of the same size, against the root-mean-square error of the wavefront in waves, for four kinds of error — each computed from the transform and each normalised to the same rms. The dashed curve is the usual approximation, the exponential of minus the square of two pi times the error. What the figure shows is that to a good approximation it does not matter WHAT the error is, only how large it is in the mean square: four quite different shapes of wavefront give nearly the same peak. A fourteenth of a wave leaves 80 per cent of the peak, which is the conventional definition of diffraction-limited, and it corresponds to a quarter of a wave peak-to-valley for a simple defocus — which is where Rayleigh's quarter-wave rule comes from and why it is a convention laid over a computed number rather than a threshold in the physics.$ Optics

How accurate a mirror has to be

The pupil's amplitude decides the rings; its phase decides the peak. A wavefront error of a fourteenth of a wave root-mean-square leaves eighty per cent of the peak intensity, which is the whole of what 'diffraction-limited' means — a convention laid over a computed number. And the number barely depends on what the error is, only on how large: four quite different aberrations of the same magnitude give nearly the same answer.

Thermodynamics

A fridge with no work going into it

Every engine here so far turns heat into work or work into a heat flow. A machine exchanging heat with three reservoirs and doing no work at all can still move heat from cold to hot, and the ceiling on how much is the product of two Carnot expressions — an engine's efficiency times a fridge's coefficient of performance. A gas flame makes ice, and the accounting is one inequality in three entropy flows.

The bath does push back, by an unmeasurable amount. The retarding force on a perfectly absorbing body moving through isotropic radiation at 2.725 kelvin, against its speed, for three areas. Moving through a bath of radiation is not free: the radiation arriving from ahead is blue-shifted and more intense and the radiation from behind is red-shifted and weaker, so the body absorbs more momentum from the front than from the back and decelerates. A square metre at half the speed of light feels 3.7e-14 newtons. That is the reason a preferred frame exists without relativity being violated: the laws are the same in every frame and the radiation is not — it is a physical system with a state, and its state picks out the frame in which it is isotropic, exactly as a body of water does.

Relativity

The bath that pushes back

Moving through a bath of radiation is not free. The light arriving from ahead is blue-shifted and more intense and the light from behind is weaker, so a body absorbs more momentum from the front than from the back and slows down. That drag picks out the frame in which the radiation is isotropic — without violating relativity, because the laws are the same in every frame and the radiation is not.

Four states of light, each with the same area of noise. The field of a single mode of light drawn as a point in a plane whose two axes are its two quadratures — the parts of the wave in step with a reference and a quarter-cycle out of step. The distance from the centre is the amplitude and the angle is the phase. Each state is a cloud of 400 sampled measurements with its two-standard-deviation outline, in units where the vacuum's noise is one in every direction. The vacuum is a round cloud at the centre. Steady laser light is the same round cloud moved away from the centre. Light squeezed by 4 dB is an ellipse of the same area: narrower than the vacuum by a factor of 0.63 in one direction and wider by 1.58 in the other. Pointed along the direction from the centre it is quiet in amplitude; pointed across it, quiet in phase. The sampled spreads were checked against each state's widths.

Quantum

The noise pushed below the floor

A perfectly steady laser beam still flickers, by an amount set by the vacuum itself, and for most of the twentieth century that flicker was treated as the floor of any optical measurement. It is a floor only for one shape of noise. Light can be made quieter than the vacuum in one property by being made louder in another — and the price, the fragility and the use of that trade are all visible in how the noise is shaped, which is why the world's gravitational-wave detectors now run on it.

Two paths through a neutron interferometer at two heights. A neutron interferometer cut from one silicon crystal, tilted by 30 degrees about its incoming beam so that one path runs higher than the other. Left: the two paths, split at the first slab, turned at the second and recombined at the third, enclosing 10.1 square centimetres; the heights are drawn exaggerated. Taken as a rectangle of the same area, the upper path runs 15.8 mm higher for 3.2 cm. There a neutron of wavelength 1.445 Å, moving at 2738 m/s, is slower by 56.5 micrometres per second, so its wavelength is longer by 20.6 parts per thousand million. Over 3.2 cm that accumulates 28.7 radians less phase than the lower leg — 4.6 whole fringes — computed by integrating the local wavenumber along both legs and checked against 2πm²gλA sin α / h².

Quantum

The fall that leaves the mass in the phase

Every body falls the same way whatever its mass, and a neutron is no exception. But a neutron is also a wave, and the phase that wave accumulates while falling depends on the mass — as its square, at a fixed wavelength. Tilt a neutron interferometer so that one path runs a centimetre higher than the other and the neutrons swing between its two detectors, which in 1975 was the first measurement in which gravity and quantum mechanics both had to be right at once.

$The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.$ Waves

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

$Both clocks move, and their ratio does not. Two years of a clock's fractional frequency against a distant clock (upper panel), and of the ratio of two unlike clocks kept side by side (lower panel). Above, both clocks slow as the Earth nears the Sun, with an amplitude of 1.65·10⁻¹⁰, and if the redshift is universal the two curves are one curve. Below, the ratio: the flat line is what universality predicts, and the sinusoid is what a clock responding to the potential 10⁻⁶ more strongly than the other would produce — an annual term of 1.65·10⁻¹⁶, a million times smaller than the shift both clocks share and within reach of clocks that compare to parts in 10¹⁷.$ Astrophysics

The clocks that must all slow together

Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.

Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

Electromagnetism

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

A living cell and a dead one, told apart by a frequency. The real part of the dielectrophoretic factor for a living cell and a dead one, ten micrometres across, in a liquid of conductivity 0.01 S/m, against frequency. The cell is a conducting interior inside a membrane five nanometres thick. The living cell's intact membrane blocks slow fields, so at low frequency it behaves as an insulator and is pushed out of strong field; above 42 kHz the membrane is short-circuited by its own capacitance and the conducting interior is felt, so the cell is pulled in; above 140 MHz the permittivities take over and it is pushed again. The dead cell's membrane leaks, its interior has lost ions, and it changes sign only at 8.8 MHz. In 2 bands — below 42 kHz and from 8.9 MHz to 140 MHz — the same field pushes the two in opposite directions.

Electromagnetism

The force whose sign a frequency chooses

A neutral particle in a non-uniform field is pulled towards strong field if it polarises more than the liquid around it and pushed away if it polarises less. With conduction in the picture, which of the two it does is decided by how fast the field alternates — and a living cell, a conductor wrapped in an insulator five nanometres thick, changes sign twice, where a dead one changes once. The same electrodes can then send the living cells one way and the dead ones the other.

A parallel beam brought to a point on the far side. Parallel rays sent into Luneburg's sphere, whose index falls from the square root of two at the centre to one at the rim, from the left. Each beam is brought to a single point on the rim diametrically opposite the direction it came from; the worst of the 11 rays misses by 2.0e-3 radii. The sphere has no axis — every direction is the same to it — so a beam from any direction is focused as well as any other, and there is no such thing as an off-axis aberration.

Optics

The lens with no axis

Every ordinary lens has an axis, and every one of its hardest aberrations is a penalty for looking away from it. A sphere has no axis, so a sphere cannot have off-axis aberrations — but a glass ball has spherical aberration instead, because one index and one curvature are too few to focus every ray. Luneburg's sphere, whose index falls from √2 at the centre to 1 at the rim, keeps the symmetry and removes the aberration: it brings parallel light from every direction to a perfect point on its own surface.

Every colour a bound electron can meet. The light-scattering cross-section of one electron bound with a resonance at 2.105 eV (589 nm) and damped only by its own radiation, in units of the Thomson cross-section, against the frequency of the light in units of the resonance, on logarithmic axes spanning nine decades of frequency and twenty-five of cross-section. Far below the resonance it rises with slope 4.001, Rayleigh's fourth power: at a hundredth of the resonance it is 1.0 × 10⁻⁸ times the Thomson value. At the resonance it spikes to 2.5 × 10¹⁵ times the Thomson value, which is 3λ²/2π = 1.66 × 10⁻¹³ m² exactly, over a width of 2.0 × 10⁻⁸ of the resonant frequency. Above it the cross-section settles to the Thomson value, 6.652 × 10⁻²⁹ m² — 1.0002 times it at a hundred times the resonance — and stays within 10 per cent of it from 4.6 times the resonant frequency, 9.8 eV, to 29 keV, 3.5 decades, until the photon energy approaches the electron's rest energy, where the Klein–Nishina factor for a free electron takes it down to 0.431 of the Thomson value at 511 keV and 0.077 at 10 MeV.

Astrophysics

The cross-section that forgets the colour

An electron on a spring, shaken by light, scatters it — weakly and very blue below its resonance, enormously at it, and above it by exactly the same amount whatever the colour. That flat stretch is the Thomson cross-section, 6.65 × 10⁻²⁹ m², and it is why an X-ray picture of a crystal is a map of electrons and why the corona is white.

The centre of charge, and whether it moves. A pulsar of 1.46 and a white dwarf of 0.181 solar masses in a circular orbit, drawn twice about their fixed centre of mass (the cross). On the left each body carries a charge in exact proportion to its mass, as gravitational charge is under the equivalence principle: the centre of charge sits on the centre of mass and never moves, so the dipole moment is constant and nothing is radiated at dipole order. On the right the pulsar carries 60 per cent less charge per unit mass than its companion — an exaggerated mismatch, chosen so that the motion is visible at this scale — as a compact body's scalar charge would in a theory where its own binding energy couples differently: the centre of charge now circles the centre of mass at 12.6 per cent of the separation, once per orbit, and a dipole turning at the orbital frequency radiates.

Astrophysics

The radiation a difference in falling would make

Gravitational waves start at the quadrupole because the equivalence principle removes the dipole: when every body's gravitational charge is its mass, the centre of charge is the centre of mass and cannot move. If a neutron star's own gravity fell even slightly differently, a binary containing one would radiate a dipole — louder than Einstein's quadrupole in slow orbits — and a pulsar with a white dwarf in an 8.5-hour orbit has been listening for it.

$Where to strike so the pivot feels nothing. The impulse the pivot has to supply, as a fraction of the blow, against where along the body the blow lands, measured from the pivot, for a uniform rod pivoted at one end and for a 0.84 m wooden bat held 15 cm from its knob, with the blow's position as a fraction of the length beyond the grip. For the rod the reaction is 1 − 3q/2L, positive for blows nearer the pivot, negative beyond, and zero at 0.6667 of the length. For the bat it is zero 51.9 cm from the grip, 17.1 cm from the end of the barrel. A blow there sets the body turning about the pivot as though the pivot were not needed.$ Mechanics

The point that takes the blow without a jolt

Hit a stick held at one end and the hand feels a jolt — forwards if the blow lands near the hand, backwards if it lands near the tip, and nothing at all if it lands two-thirds of the way along. That point is the centre of percussion, and it is exactly where a simple pendulum with the same period would end: two different questions about a swinging body with one answer.

The angle at which a total reflection stops being total. Reflectance of light at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond it, against the angle of incidence, for the two polarisations, from the three-layer Fresnel sum in complex arithmetic. Past the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely, as total internal reflection requires, apart from what the thin gold absorbs. p-polarised light is reflected too — except near one angle, where the reflectance falls to 0.014 at 71.52°. The dispersion relation of a surface plasmon on gold against water puts its resonance at 71.67°; the difference is the film's finite thickness and the prism on its other side pulling the resonance slightly.

Optics

The angle at which total reflection stops being total

Coat the back face of a glass prism with fifty nanometres of gold and total internal reflection still returns almost all the light — except at one angle, where the reflection of one polarisation falls nearly to zero. The light has gone into an electron wave bound to the far surface of the gold, and because that wave's angle depends on whatever touches the gold, the missing reflection has become one of the most sensitive balances in biochemistry.

Six ways into a corner, each leaving from the opposite side. A corner cube seen along its axis: its hexagonal face, divided into six sectors by the three edges where its mirror faces meet and by their extensions. A ray entering along the axis meets the three faces in an order fixed by the sector it enters: y → x → z, y → z → x, z → y → x, z → x → y, x → z → y, x → y → z. Traced reflection by reflection, every ray leaves travelling exactly back along its arrival direction, to rounding error, and it leaves from the point diametrically opposite where it entered — in the sector whose order is the reverse of its own.

Optics

The corner that sends light home

Three mirrors at right angles send light straight back the way it came, from any direction they can see, because each one reverses one component of its direction. Made from solid glass, a corner reflects by total internal reflection and needs no silver — which is why the reflectors left on the Moon are bare glass — but each of its six sectors then returns a different polarisation, and the returned spot loses three-quarters of its peak.

Why a flip of the spin takes eleven million years. Spontaneous emission rates against frequency on logarithmic axes. The two lines are the rates an electric dipole of e a₀ and a magnetic dipole of one Bohr magneton would give, both rising as the cube of the frequency; they differ by α²/4, 1.3 × 10⁻⁵, the square of the ratio of the two moments. Measured lines sit near them: Lyman α at 6.3 × 10⁸ s⁻¹, Hα and sodium's D line near 10⁸, and hydrogen's 21 cm line at 2.87 × 10⁻¹⁵ s⁻¹ — a lifetime of 11.0 million years. Its slowness is two factors multiplied: a magnetic moment rather than an electric one, and a frequency nearly a million times lower, cubed.

Quantum

The line that takes eleven million years

A hydrogen atom in its ground state can flip its electron's spin relative to the proton's and emit a photon of 21 centimetres. Left alone, one atom takes eleven million years to do it. The Galaxy is nonetheless bright at that wavelength, because there is so much hydrogen, and because the flip is so slow that the line's brightness counts atoms without caring how hot they are.

Quantum

The atom the size of a bacterium

Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

$How much slower a warm atom ticks. The fraction by which a free atom's clock runs slow on average because of its thermal motion, (3/2)kT/mc², against temperature on logarithmic axes, for five atoms used in clocks or tests of relativity: hydrogen 4.1 × 10⁻¹¹ at 300 K, aluminium ion 1.5 × 10⁻¹² at 300 K, strontium 4.8 × 10⁻¹³ at 300 K, caesium 3.1 × 10⁻¹³ at 300 K, mercury ion 2.1 × 10⁻¹³ at 300 K. Every line falls in proportion to the temperature and to one over the mass. A clock aiming at one part in 10¹⁸ must hold strontium below 0.63 mK and hydrogen below 7.3 µK — or know their temperature well enough to correct for it. Pound and Rebka's gravitational shift across 22.5 m, 2.46 × 10⁻¹⁵, is marked for scale.$ Relativity

The clock that runs slow because it is warm

Every atom in a gas is moving, and every moving clock runs slow. The first-order Doppler shifts of a hot gas cancel on average; the time dilation does not, because it has only one sign. A hydrogen maser at 40 °C runs slow by four parts in a hundred billion from this alone, a nucleus in a crystal runs slow even at absolute zero, and the first test of gravitational time dilation nearly drowned in a one-kelvin temperature difference.

Fifty-nine millivolts for every factor of ten. The voltage that holds a singly charged ion at one concentration on one side of a boundary and another concentration on the other, in equilibrium, against the ratio of the concentrations on a logarithmic axis: (kT/e) ln(c₁/c₂), from requiring the electrochemical potential to be the same on both sides. At 0 °C, 54.20 mV per decade; at 25 °C, 59.16 mV per decade; at 37 °C, 61.54 mV per decade; at 100 °C, 74.04 mV per decade. The voltage depends on the ratio and the temperature and on nothing about the ion, the solvent or the boundary: it is the entropy of dilution, expressed per unit charge.

Thermodynamics

The voltage that is a chemical potential

A battery, a nerve cell and a pH meter all turn a difference in chemical potential into a voltage, and the conversion is the same in each: fifty-nine millivolts for every factor of ten in concentration, at room temperature, whatever the ion. The voltage a battery holds as it discharges is the chemical potential of the atoms stored in its electrodes — and a battery whose voltage barely changes is one whose electrode is going through a phase transition.

The area under the spacings is the bond. The gap between each vibrational level and the next, against level number, for a Morse well and a Lennard-Jones well with the same depth and the same curvature at the bottom, Λ = 100, in units of the well depth. The sum of the gaps from the lowest level up is the energy needed to break the bond from there. For the Morse well the gaps fall in a straight line to zero, and a straight line through the first 5 of them, extended to zero, encloses 0.9473 against the true 0.9473. For the Lennard-Jones well the gaps start almost the same and fall slightly faster at first — which is what the straight line sees — then trail off in a long tail of small gaps near the top; the same extrapolation from the first 5 encloses 0.7988 against the true 0.9474 — 15.7 per cent short, because the straight line cannot see the crowded levels its data never reached.

Mechanics

The spectrum that weighs a bond

A vibrational spectrum measured near the bottom of a molecule's well gives its curvature, and the curvature says nothing about how deep the well is. Measured further up, it says everything. The levels of a real bond crowd together towards the top, the gap between neighbours falls towards zero, and the sum of all the gaps is the energy it takes to break the bond. A straight line drawn through the gaps that have been measured, and extended to zero, was for decades how bond energies were found — and for a bond with the tail of two neutral atoms it comes out a sixth short.

A strong oscillator lifts a picowatt to the quantum limit. The signal-to-noise ratio for detecting a 1 pW signal at 1.55 µm in a 1 MHz bandwidth, with a detector of 80 per cent quantum efficiency whose amplifier adds 2 pA/√Hz of noise, against the power of the local oscillator. Detected directly, the signal is 66 dB below the amplifier's noise. Heterodyned, it rises with the oscillator's power — −3.3 dB with a microwatt, 7.4 dB with a hundred — until the oscillator's own shot noise swamps the amplifier's, above about 12 µW, and then it stops rising at 8.0 dB: ηPₛ/(hνB), the number of detected signal photons per resolution time, which no receiver can beat. A stronger oscillator raises the signal and its own noise together.

Optics

The faint light a bright one makes measurable

A photodetector does not respond to a light wave's field; it responds to the square of it. Shine a strong beam onto the detector together with a weak one at a slightly different frequency, and the square contains a term that is the weak field multiplied by the strong one — oscillating at the difference frequency, carrying the weak light's phase, and as large as the strong beam cares to make it. That product lifts a picowatt out of an amplifier's noise to the limit set by counting photons, turns an optical frequency shift into a radio one, and carries with it a quantum price that decides why radio telescopes and optical telescopes are built so differently.

A temperature the circuit refuses to go below. The escape temperature — the temperature that would give the measured escape rate if escape were purely thermal — against the actual temperature of the junction, for a 10 µA, 5 pF junction at biases of 0.985 and 0.99 of its critical current. Hot, the two are equal: the phase is kicked over the barrier by thermal noise. Cooled below the crossover, 39 mK at 0.985 and 36 mK at 0.99, the escape temperature stops falling and levels off at 42 mK and 44 mK: the rate no longer depends on the temperature at all. This levelling-off, measured by Michel Devoret, John Martinis and John Clarke in 1985 with the junction's electrical environment carefully characterised, was the evidence that a macroscopic electrical variable tunnels.

Quantum

The circuit that goes through a wall

Tunnelling is usually told about electrons and alpha particles — single, small, quantum things passing through barriers they cannot climb. In 1985 it was seen in an electrical circuit: the phase difference across a Josephson junction, a coordinate shared by all the electrons in two pieces of superconducting metal, trapped in a well and escaping from it through the barrier rather than over it. The evidence was a temperature below which the circuit's escape rate stopped caring how cold it was. The same circuit showed discrete energy levels, and its descendants are the qubits of superconducting quantum computers.

The self-bias set by a phase alone. The dc self-bias against the phase between the fundamental and its second harmonic, each of amplitude A, for three chambers: symmetry parameter ε = 0.5, 1, 2, where ε = 1 is a chamber with equal sheaths on both sides. In the symmetric chamber the bias runs from −0.438 A at 0° through zero at 45° to 0.438 at 90°, and back — −43.8 V to 43.8 V at A = 100 V — staying within 0.011 A of a straight line between 0° and 90°. An asymmetric chamber adds a fixed offset of the kind unequal electrodes produce and leaves the phase control in place, so the geometry and the waveform add rather than compete.

Astrophysics

The bias a waveform can choose

A plasma between two identical electrodes, driven by a sine wave, has no reason to prefer either side, and it does not: no direct voltage appears. Add the second harmonic and it does, although the chamber is still symmetric, no direct current flows and the power spectrum does not depend on the phase between the two. The phase alone sets a bias of up to 0.44 of each harmonic's amplitude, of either sign, and moves the energy of the ions from one electrode to the other while their number stays put.

The wobble slows as the body yields. The period of the Earth's free wobble against how far the body yields to its own rotation, measured as the Love number k over its value for a fluid Earth, kₛ = 0.941, computed from the Earth's flattening. A rigid Earth, at zero, wobbles in 303.6 days, Euler's period. The observed 433 days requires k/kₛ = 0.299, an effective k of 0.281: the Earth yields about 30 per cent as much as a fluid would. The period grows without limit as the body approaches a fluid, because a fluid body's bulge re-forms round any new spin axis and leaves nothing to push the axis back.

Mechanics

The wobble that is slow because the Earth gives

Euler worked out in 1765 that a spinning, flattened Earth whose axis is knocked slightly off its figure should wobble, the pole wandering in a small circle once every ten months. Astronomers looked for a ten-month period for a century and found nothing. In 1891 Seth Chandler found a wobble of fourteen months instead. The four extra months are not an error in Euler's arithmetic. They are the Earth yielding: its bulge partly re-forms round the displaced axis, weakening the torque that drives the wobble, and the lengthening measures how soft the planet is.

Optics

The full moon that is too bright

Half a moon shows half the lit disc, so it ought to be about half as bright as a full one — a little less, allowing for the angle. It is a tenth as bright. In the last few days before full the Moon brightens far faster than its growing area can explain, and the extra light is not reflected any more strongly. It comes from shadows disappearing. The lunar soil is a loose heap of grains, and when the Sun is directly behind the observer every grain hides its own shadow.

Quantum

The spectrum that is a thermometer at every height

A satellite looking down at the Earth in the infrared sees, at each wavelength, the glow of whatever layer of air that wavelength escapes from. In the wings of carbon dioxide's great absorption band the air is transparent and the satellite sees the ground; towards the band's centre the gas grows opaque and each wavelength escapes from higher up. A single spectrum is therefore a stack of thermometers at different heights, and weather forecasts are built on reading it. What cannot be read from it is anything much thinner than the ten to twenty kilometres of air each of those thermometers averages over.

The orbits of an electron in a travelling wave. The possible motions of an electron in a plasma wave whose phase moves at γₚ = 10 and whose field is 0.2 of the cold wave-breaking field, drawn as its energy against its position in the wave over two wavelengths — the level curves of H = γ − βₚ u − ε cos ψ, which the electron's motion conserves. Inside the separatrix (thick) an electron is trapped: it slides back and forth through the wave, gaining energy while it outruns the wave and losing it while the wave outruns it. The separatrix reaches γ = 99 at its top, near 4γₚ²ε = 80, starting from electrons moving at the wave's own speed. Outside it, electrons that are too slow are overtaken by the wave and those that are fast enough outrun it; neither gains energy on average.

Relativity

The wave an electron rides to a gigaelectronvolt

A laser pulse fired into a thin plasma pushes its electrons aside and leaves them oscillating behind it, a wave of charge travelling at nearly the speed of light. The field in that wave is a thousand times stronger than any metal accelerator can hold, and an electron caught in it at the right moment is carried along like a surfer, gaining a gigaelectronvolt in a few centimetres. Then it outruns the wave. The distance it can ride, and the energy it can gain, are both set by how nearly the wave keeps pace with it — which is to say, by relativity.

The same thermal motion, spread three ways. How the thermal jiggling of a spring-mounted mass is distributed over frequency, for quality factors 3, 30, 300, with frequency in units of the resonance and the spectrum in units that make the area under each curve its mean-square displacement. All three curves enclose exactly the same area, kT/k — 1.000, 1.000, 1.000 of it in the units drawn — because equipartition fixes the energy in the spring and not how fast it changes. What the damping chooses is where that motion goes: a high Q concentrates it in a peak 100 times taller than the low-Q one, and leaves the rest of the spectrum 100 times quieter at 10 times the resonance frequency.

Thermodynamics

The noise a high Q moves out of the way

Anything held by a spring at a temperature jiggles, and equipartition says exactly how much: the mean square of its displacement is kT divided by the spring's stiffness, whatever the spring is made of and however it is damped. That looks like a floor no measurement can go below. It is a floor on the total, and the total can be moved. An oscillator with little damping puts almost all of its thermal motion into a narrow peak at its own frequency and leaves every other frequency quiet — which is why the mirrors of gravitational-wave detectors hang from fibres of glass that ring for minutes.

Echoes from a layer that thins to nothing. Synthetic seismic echoes from a layer between two identical rocks — the top reflecting +R, the base −R — for thicknesses of 0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 75, 100 m, sounded with a 30 Hz pulse in rock of 3000 m/s, whose wavelength in the layer is 100 m. Time runs downward; each trace is one thickness. Thick, the layer gives two separate echoes, one from each face, and its thickness can be read from the time between them. As it thins the two echoes overlap and merge. Near 20 m, about a fifth of a wavelength, they interfere constructively and the combined echo is strongest, 1.45 times one face's echo. Thinner still, the combined echo keeps a fixed shape — the pulse's derivative — and simply weakens in proportion to the thickness; nothing in its timing changes.

Waves

The layer too thin to see

A seismic survey sends a pulse of sound into the ground and listens to the echoes from every boundary where the rock changes. A thick layer returns two echoes, one from its top and one from its base, and the time between them gives its thickness. A thin one returns one echo, and below about a quarter of a wavelength the timing says nothing more: every thin layer looks the same, just weaker as it thins. It has not become invisible. It has become unmeasurable in the one way that was being used, and its thickness has moved into the echo's strength and its spectrum.

$Colours that leave a grating on different roads. A pair of parallel diffraction gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for three wavelengths, 760, 800 and 840 nm. The first grating sends each colour off at its own angle — 24.3° for 760 nm, 27.4° for 800 nm, 30.5° for 840 nm — and the second turns them all back parallel to the input, side by side. Between the gratings the red travels the longest road: for gratings 10 mm apart the slant paths are 10.97 mm, 11.26 mm, 11.61 mm. Measured to a common wavefront after the second grating, the longer road still wins, so red arrives last: the geometry has given the pulse the opposite ordering to the one glass gives it.$ Optics

The dispersion made of angles

Glass delays blue light more than red, so a short pulse that passes through glass comes out stretched with its red end first. At the wavelengths where most short-pulse lasers work, no transparent glass does the opposite. Two diffraction gratings do it with no material at all. The first spreads the colours in angle, the second turns them back parallel, and in between the red has travelled the longer road. A spread in angle, however it is made, always adds this kind of dispersion, of the sign glass cannot give. It is how every high-power ultrafast laser squeezes its pulse back after amplifying it.

A sheet of light a tenth of a micrometre thick. The intensity of the evanescent field in water beyond a glass coverslip (index 1.518 against 1.33), against distance from the glass, for 488 nm light totally reflected at 62°, 65°, 70°, 75°, s-polarised, relative to the light arriving in the glass. The critical angle is 61.2°. At 62° the field at the surface is 3.79 times the incident intensity and falls by e in 234 nm; at 65° the field at the surface is 3.07 times the incident intensity and falls by e in 110 nm; at 70° the field at the surface is 2.01 times the incident intensity and falls by e in 75 nm; at 75° the field at the surface is 1.15 times the incident intensity and falls by e in 63 nm. Ordinary illumination (dashed) lights the whole depth of a cell equally; totally reflected light lights only the layer against the glass, which is where a cell's membrane and the proteins in it are.

Optics

The mirror that lights a tenth of a micrometre

Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.

A delay the Planck scale would write into a burst. The arrival delay, relative to a low-energy photon emitted at the same moment, of photons of each energy from a gamma-ray burst at redshift 0.903, if light slowed in proportion to its energy over an energy scale, on logarithmic axes, for that scale at the Planck energy and at ten times and a tenth of it. At the Planck energy a 31 GeV photon would arrive 1.16 s late. The 31 GeV photon from GRB 090510 arrived 0.83 s after the burst began (marked), so a delay of more than that is excluded, and with it any scale below 1.4 times the Planck energy on this simple reading. Fermi's own analysis, allowing for when in the burst the photon could have been emitted, set the limit at 1.2 times the Planck energy: a speed that depends on energy at first order has been excluded at the scale where it was expected.

Astrophysics

The photons that raced for seven billion years

Nothing on a laboratory bench can reach the Planck scale, but a gamma-ray burst halfway across the universe can amplify a Planck-scale effect until it is a second long. If spacetime has structure at the Planck length, the simplest guess is that light's speed depends slightly on its energy. Then a burst's high-energy photons would arrive measurably later than its low-energy ones, after a race lasting billions of years. In 2009 a 31 GeV photon from a burst 7.5 thousand million light-years away arrived within 0.83 s of the burst's start. That excludes the simplest version of the idea at the Planck energy itself.

Astrophysics

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.

Two pulls that should be equal and opposite. Two bodies of equal passive mass, one of an ordinary material and one of a material whose active mass is 30 per cent larger than its passive mass (the difference exaggerated to be seen). Each pulls the other with its active mass and is pulled on its passive mass. The pull the unusual body exerts is therefore larger than the pull it receives, the two arrows are not equal and opposite, and the pair, with nothing outside it, accelerates towards the ordinary body. Newton's third law and the conservation of momentum both require the ratio of active to passive mass to be the same for every material, and that is a separate claim from the one torsion balances test, that passive and inertial mass agree.

Astrophysics

The pull that has to equal the pull back

Every test of the equivalence principle drops two things and asks whether gravity pulls them equally. None of them asks whether they pull equally. A body's mass enters gravity twice — once as what is pulled and once as what does the pulling — and if the two ratios differed from one material to another, Newton's third law would fail and a lopsided body could push itself through space. The first test of this used a plastic cylinder floating in a liquid. The best uses the Moon, whose light crust is thicker on the side that never faces the Earth.

Three shapes of film, one sheet resistance. Three thin films of the same material, of sheet resistance one ohm per square, with four small contacts on each edge labelled A to D in order. For each, the two readings van der Pauw's method takes — the voltage between D and C per unit current from A to B, and between A and D per unit current from B to C — computed by solving for the current flow on a fine grid. For a square, contacts at the corners: 0.221 and 0.221 Ω, giving 1.000 Ω per square; a disc, contacts unevenly spaced: 0.089 and 0.450 Ω, giving 0.999 Ω per square; an irregular lamina: 0.209 and 0.232 Ω, giving 0.999 Ω per square. The readings change with the shape and with where the contacts are; the sheet resistance recovered from them by exp(−πR₁/Rₛ) + exp(−πR₂/Rₛ) = 1 does not.

Electromagnetism

The sheet resistance that forgets the shape

To measure how well a thin film conducts, the obvious method is to cut it into a neat strip and pass a current along it. In 1958 Leo van der Pauw, an engineer at Philips, showed that the strip is unnecessary. Put four small contacts anywhere on the edge of a film of any shape, take two readings, and one equation returns the film's resistance per square — the shape and the spacing of the contacts drop out entirely. The proof is a single fact about logarithms and the maps that preserve angles, and it fails for exactly one kind of film: one with a hole in it.

A gyroscope swinging into the meridian. The direction of a gyrocompass's axis, in degrees east of true north, against time in hours, released 20° east of north at latitude 50°. The Earth's rotation tilts the horizon under the spinning axis, a pendulous weight turns the tilt into a torque, and the torque precesses the axis towards the meridian. Undamped (dashed) it swings through north and back with a period of 84.4 minutes, the compass having been built to that period. With damping it settles: after 8 hours it points within 0.03° of north. Nothing in the instrument knows where north is. It finds the direction about which the Earth turns, because only when its axis lies in the meridian does the turning horizon stop tilting it.

Mechanics

The compass that finds the axis the Earth turns about

A spinning wheel held so that its axis stays level will, on a turning Earth, swing its axis round to point at true north and stay there — not magnetic north, the geographic pole. Nothing in the instrument knows where north is; it finds the one direction about which the Earth's turning stops tilting it. Built to be used at sea, such a compass has to ignore the ship's lurches, and that forces its swing to have one particular period: 84.4 minutes, the period of a pendulum as long as the Earth's radius, of a satellite skimming the ground, and of a stone dropped through a tunnel to the far side of the planet.

Optics

The beams that add only if they differ

Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment.

Electromagnetism

The field a spinning superconductor makes of itself

Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

$Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories.$ Electromagnetism

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

$Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.$ Quantum

The grating made of light

A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

The drift is the area under the height. Height against time for the two balls: the throw (dashed curve) rising to 100 m in 4.52 s and falling back in as long again, and the drop (solid) falling from 100 m. For vertical motion the east–west velocity is always −2Ω cos λ times the height above the starting point, so the east–west drift is −2Ω cos λ times the area between the path and the starting height. The throw's area (lower shading) is (4/3)·h·t = 602 m s, above its start, so it drifts west; the drop's (upper shading) is h·t/3 = 151 m s, below its start, so it drifts east. The ratio of the two areas is four.

Mechanics

The ball thrown up that lands to the west

Drop a ball from a tower a hundred metres high and, because the Earth turns, it lands a centimetre and a half east of the point directly below. Throw a ball straight up so that it just reaches the same height, and it lands six centimetres west of where it was thrown — four times as far, the other way. The asymmetry has a one-line explanation that turns a problem in rotating frames into a problem about areas: for anything moving straight up or down, the sideways drift is proportional to the area under its graph of height against time. A dropped ball spends its flight below where it started, a thrown one above, and the thrown one's area is four times as large.

$The orbit an observer would reconstruct. The star's velocity along the line of sight, measured from the Doppler shift of its spectrum, as a fraction of its true orbital speed, plotted against the time the light arrives, in orbital periods, if light left the star at c plus the star's velocity. With light at c (dotted) the curve is the sine wave of a circular orbit. At 0.6 times the critical distance (solid) the same circular orbit arrives as a lopsided curve — a slow rise and a steep fall — that an astronomer would fit with a highly eccentric orbit that is not there. At twice the critical distance (dashed) the curve doubles back on itself: at some moments the spectrum would show two or three velocities at once, as if the star were in several places. Willem de Sitter pointed out in 1913 that spectroscopic binaries show nothing of the kind.$ Relativity

The binary star that would be seen twice

If light thrown forward by a moving source travelled faster, the way a ball thrown from a moving train does, then light from a star swinging towards the Earth in a binary orbit would catch up with light it had sent out earlier. From far enough away the two would arrive together, and the star would be seen in two places at once. Willem de Sitter pointed out in 1913 that nothing of the kind is seen, and that this settles whether the speed of light depends on its source. The argument had a loophole that took half a century to notice — interstellar gas re-emits light at its own speed — and closing it needed X-ray stars.

Relativity

The spin that runs ahead of its own motion

A charged particle circling in a magnetic field has two directions that turn: the direction it is moving, and the direction its spin points. If the particle's magnetism were exactly what Dirac's equation first gave — a g-factor of exactly two — the two would turn together at every speed, the spin locked to the motion as a compass needle is locked to a car. The muon's g is not quite two, and its spin creeps ahead by a fixed rate that is the same at a crawl and at 99.94 per cent of the speed of light. That constancy is a delicate cancellation between a magnetic field made stronger by motion and a rotation that exists only because the particle is being turned, and it is what made the most precise measurements of the muon possible — at one magic speed, where the electric fields that steer the particles stop affecting the spin at all.

$A scan into a layered sample, with a broad source and a narrower one. The envelope of the interference signal against depth in a sample with five reflecting boundaries (marked), at 60, 72, 150, 158 and 260 µm, scanned with a 50 nm source (solid) and a 10 nm source (dashed), both centred at 840 nm. The broad source, with a 6.2 µm resolution, separates every boundary, including the two that are only 8 µm apart. The narrow one, with 31 µm, merges the pairs into single blurred echoes. Every boundary in a real eye or skin sample is a small change of refractive index that reflects a tiny fraction of the light; the scan is a map of where those changes are.$ Optics

The depth a broad spectrum can see

Interference is usually spoiled by a broad spectrum: many colours drift out of step, and fringes wash out a few wavelengths from where two paths are equal. Turned round, that failure is a ruler. Light whose colours stay in step only when two paths match to within a few micrometres can report the depth of every reflecting layer inside something translucent — a retina, a skin, a coat of paint — to that precision, without cutting it open. Optical coherence tomography reads depth off the width of a spectrum, and it has become one of the most used imaging methods in medicine; its resolution is set by how many colours the light contains, and the finest instruments use sources whose spectra span hundreds of nanometres.

The boiling curve, which turns back. Heat flux from a horizontal surface into water boiling at one atmosphere, on a logarithmic axis, against how far the surface is above the boiling point, also logarithmic, from standard correlations. At a few degrees heat leaves by convection; from about 5 K bubbles nucleate and the flux climbs as the cube of the superheat, to the critical heat flux of 1.11 MW/m² at 20 K. Beyond it the flux FALLS as the surface gets hotter — the transition regime, drawn dashed as the interpolation it is — down to 19 kW/m² at 67 K, the Leidenfrost point, where a continuous film of vapour insulates the surface. Past that, film boiling and radiation let the flux rise again, slowly: 310 kW/m² at 1000 K above boiling. Between the crisis and the Leidenfrost point, a surface 3 times hotter passes 58 times less heat.

Thermodynamics

The boiling curve that turns back

Heat a surface under water and more heat flows from it the hotter it gets — up to a point. Past about twenty degrees above boiling the bubbles crowd together, merge, and begin to blanket the surface with vapour, and from then on a hotter surface passes less heat, not more, until by a few hundred degrees above boiling it is wrapped in an insulating film and passes fifty times less than it did at its best. Shigeo Nukiyama found the turn in 1934 by melting wires. It is why a water drop skates for a minute on a very hot pan but vanishes in two seconds on a merely hot one, and why every boiler, steam generator and water-cooled reactor is designed around a number called the critical heat flux.

The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper.

Waves

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm.

Quantum

The bounce that comes in fixed heights

A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

Two factors of γ between a magnet and an X-ray. An electron of 6 GeV (γ = 11742) crossing an undulator, a row of magnets alternating every λᵤ = 3 cm. Top: the laboratory, where the electron wiggles once per period. Middle: the electron's own frame, where the row is contracted by γ to 2.55 μm and rushes past at nearly the speed of light; to the electron it is a wave of that wavelength, and the electron radiates at it, in the infrared. Bottom: that light, emitted forward and seen back in the laboratory, is Doppler-shortened by a further factor of about 2γ, to λᵤ/2γ² = 1.088 Å; the wiggle's own slowing of the electron adds a factor (1 + K²/2), giving 1.632 Å for a deflection parameter K = 1 — hard X-rays from centimetre magnets. The drawing is schematic: the three periods differ by factors of about twelve thousand and a hundred million, and cannot be drawn to one scale.

Relativity

The magnet period that comes out as an X-ray

A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

The cone a charge leaves behind in water. A charged particle crossing water at 0.99 of the speed of light in vacuum, which is 1.317 times the speed of light in water, drawn at one instant. From each point it passed, its field set off a spherical wavelet travelling at c/n; the wavelets from earlier points have grown larger. Because the particle outruns them, they all touch one cone with its tip at the particle, at an angle θ = arccos(1/nβ) = 40.6° to the path — the light of Cherenkov radiation, which travels out perpendicular to the cone. No point on the path is special and the particle never accelerates: the radiation is a property of the field moving faster than the medium can carry it, the electromagnetic counterpart of a sonic boom or a boat's wake.

Astrophysics

Why the glow of a fast charge is blue

A charge moving in a straight line at constant speed does not radiate — in a vacuum. In water, if it moves faster than light does in water, it glows, and the glow is blue. The cone of the light is the familiar half of the story. The other half is a count: Frank and Tamm's formula says each centimetre of water yields about two hundred visible photons, in numbers that rise as the inverse square of the wavelength, which is why a reactor pool shines blue and why the light keeps getting stronger into the ultraviolet until water stops being transparent. The charge never accelerates. What radiates is the water.

Optics

The stress that polarised light can see

Squeeze a disc of clear plastic and look at it between two crossed polarising filters, and it fills with dark lines and coloured bands that crowd towards the points where it is pressed. Each line is a contour of stress. Glass, plastic and most transparent solids become doubly refracting when they are loaded, in proportion to the difference between their largest and smallest stresses, and polarised light turns that difference into a pattern that can be counted. David Brewster found it in 1816, the same Brewster whose angle polarises reflected light, and for a century the way to learn the stresses in a bridge or a dam was to cut its shape from plastic, load it and photograph the fringes.

$How bright a nebula looks through a 200 mm telescope. The surface brightness of the image of an extended object — a nebula, a galaxy, the Moon — through a lossless 200 mm telescope, as a fraction of its brightness to the naked eye, against magnification on a logarithmic axis, for eye pupils of 7 and 5 mm. It is never above one. For a 7 mm pupil it is one up to a magnification of 28.6, where the exit pupil equals the eye's; below that magnification the exit pupil is wider than the eye's and part of the collected light falls on the iris; above it the image dims as the square of the exit pupil: 0.327 at 50×, 0.082 at 100×, 0.020 at 200×. The telescope makes the nebula larger, not brighter; at the magnification where its exit pupil matches the eye's it makes it as large as it can while keeping it as bright as it is.$ Optics

The nebula no telescope can brighten

A telescope eight metres across gathers a million times more light than the eye, and a star seen through it is a million times brighter. The Orion nebula seen through it is not brighter at all. It is bigger, and spread over a bigger patch of the retina, and the two exactly cancel: no instrument made of lenses and mirrors can show an extended object brighter than the naked eye sees it. What a telescope does is decided by the small disc of light behind its eyepiece, the exit pupil, and by whether that disc is larger or smaller than the pupil of the eye looking through it.

Light whose crests wind round its axis. Cross-sections of three beams carrying orbital angular momentum, with phase exp(iℓφ) round the axis, for ℓ = 1, 2 and 3, at one instant: solid lines where the field is at a crest, dashed where it is at a trough. Along the beam the crests form ℓ interleaved helices, and at a fixed plane the pattern turns about the axis once every ℓ optical periods. Looked at from a frame turning about the axis at a rate Ω, every crest arrives ℓΩ/2π times a second more or less often: the field's frequency is shifted by ℓΩ. That is a Doppler shift with no approach and no recession — the angular counterpart of k·v, with the winding number ℓ standing in for the wavenumber and the rotation rate for the speed. Each photon of such a beam carries ℓħ of orbital angular momentum about the axis.

Waves

The Doppler shift with nothing coming closer

Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.

Light from a charge that crosses into glass at a steady speed. The transition radiation emitted as an ultrarelativistic charge crosses from vacuum into a dense medium, per unit solid angle, against the angle to the path in units of 1/γ, for photons at three frequencies: one tenth, three tenths and one times γωₚ, where ωₚ is the medium's plasma frequency. The scale is αħγ²/π² per unit frequency. Nothing is emitted straight ahead; the light comes out in a narrow hollow cone with its brightest ring at γθ = 0.98, 0.88, 0.68 for the three. At a tenth of γωₚ the peak is about ten times higher than at γωₚ. The charge never accelerates: it is the field's sudden change of shape at the surface, from the flattened disc it carries in vacuum to the one the medium's electrons allow, that is radiated.

Astrophysics

The light a charge makes by changing medium

A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.

$Which way the atoms leave, against the bias. A toy version of the 2023 ALPHA-g measurement: the fraction of trapped antihydrogen atoms that leave through the bottom mirror as both mirrors are lowered over twenty seconds, against the magnetic bias, in units of the gravitational tilt, for three assumed accelerations. Each atom is given an energy between 0.02 and 0.5 K and a random starting phase, bounces between the mirrors, and leaves through the first one it reaches that has dropped below its energy. With no bias, 100 per cent leave downward if antihydrogen falls normally, 49 per cent if it does not fall and 0 per cent if it falls up. Each curve crosses one half exactly where the bias cancels the assumed gravity, and the crossing point is what the experiment measures. The curves are not steps — each falls from nine-tenths to one-tenth over about 0.6 g of bias — because the mirrors keep falling during the few milliseconds an atom takes to cross the trap: an atom that happens to arrive at the higher mirror just after it has dropped below its energy leaves that way. ALPHA-g's measured balance gave an acceleration of 0.75 g, with an uncertainty of about 0.2 g, ruling out falling upwards.$ Astrophysics

The antiatom that had to be weighed anyway

Every test of the equivalence principle for three centuries compared two lumps of ordinary matter. Whether antimatter falls down was settled on paper in the 1950s, by an engine that would run for ever if it fell up and by the electrostatic energy of nuclei, part of which is carried by fleeting positrons. It was not measured until 2023, because on a charged antiparticle gravity is ten million times weaker than a field of one volt per metre. The measurement needed neutral antihydrogen, held at half a kelvin in a magnetic bottle a quarter of a metre tall and let out over twenty seconds. It falls down, at 0.75 g with an uncertainty of a fifth of a g.

$Three cylinders that focus electrons. An einzel lens: three coaxial tubes of radius R, the outer two at the potential the electrons arrived with and the centre one, 2R long, at 4 times it. Electrons enter from the left parallel to the axis at five heights, from 0.1R to 0.7R, and are traced through the field without approximation. In each gap the equipotentials bulge between the tubes, and an electron crossing them is refracted as at a curved glass surface: pushed towards the axis where the potential is low and away from it where the potential is high. It is slower where it is pushed inward, so it spends longer there, and each gap converges on balance; the electron leaves with the speed it came in with. The rays cross the axis at 4.00R, 3.92R, 3.78R, 3.56R, 3.25R beyond the lens centre: the outer ones are bent too much and cross first, which is spherical aberration.$ Electromagnetism

The voltage that bends an electron like glass

An electron that has fallen through a potential difference V moves at a speed proportional to √V. Crossing a surface of constant potential, it keeps the part of its momentum that runs along the surface and changes the part across it, and the two rules together are Snell's law with √V in place of the refractive index. Every arrangement of charged electrodes is therefore a piece of glass for electrons. Three metal tubes make a lens that focuses whichever way its middle tube is charged — and, by a theorem proved in 1936, no lens built from round electrodes can be free of spherical aberration. That theorem kept electron microscopes some fifty times coarser than their wavelength for sixty years.

One number decides the efficiency. The best efficiency a link of two resonant circuits can reach, with its load chosen to maximise it, against the figure of merit U = k√(Q₁Q₂) on a logarithmic scale: the coupling coefficient times the geometric mean of the two coils' quality factors. The curve is η = U²/(1 + √(1 + U²))², and nothing else about the coils, the frequency or the distance enters: 0.25 per cent at U = 0.1, 17 per cent at U = 1, 52 per cent at U = 3, 82 per cent at U = 10, 94 per cent at U = 30. A weak coupling is rescued by a high Q: k = 0.003 with Q = 1,000 on each side gives U = 3 and 54 per cent, where the same coils used without tuning capacitors, as a plain transformer, deliver under one per cent, with the source driving mostly reactive current that does nothing.

Electromagnetism

The coupling a resonance rescues

Two coils a hand's breadth apart share a few thousandths of their magnetic flux, and as a transformer they deliver almost nothing. Tune each with a capacitor to the same frequency and the same pair can carry most of the power across. Resonance does not strengthen the coupling, which is fixed by the geometry and falls as the cube of the distance. It changes how many times the energy goes round before it is lost. The efficiency then depends on one number, the coupling times the quality factor, and that one number explains a toothbrush on its stand, a phone on its pad, a car parked over a plate in the road and a light bulb lit two metres away.

$A gas pressed against the wall of a spinning tube. The density of uranium hexafluoride gas in a rotor spinning at a steady rate, relative to its density on the axis, on a logarithmic scale, against the distance from the axis as a fraction of the radius, for wall speeds of 300, 450 and 600 m/s at 300 K. In the turning frame the gas sits in the centrifugal potential −½mω²r² and settles into the same exponential that thins the atmosphere with height: n ∝ exp(mω²r²/2kT). At 600 m/s the density at the wall is e^25.4 — about 10¹¹ — times that on the axis; at 300 m/s, e^6.4. Almost all the gas lies in a thin layer against the wall, and the middle of the rotor is close to vacuum.$ Mechanics

The spin that sorts by a difference of mass

In a spinning tube a gas settles against the wall the way the atmosphere settles towards the ground, with an exponential whose scale is set by the rotor's speed rather than by gravity. Heavier molecules settle more steeply, and the ratio of two isotopes changes from axis to wall by a factor that depends on the difference of their masses, not on the ratio. That one change from a ratio to a difference is why a spinning rotor separates uranium-235 from uranium-238 more than fifty times as strongly per stage as the porous barriers that first did it — and why the history of the machine is a history of how fast a material can turn before it flies apart.

Optics

The surface that cancels whatever skims it

Glass reflects four per cent of light falling straight on it, and water at radio frequencies two-thirds of the wave's amplitude. Tilt either towards grazing and the difference disappears: every surface, whatever it is made of, reflects a wave that only skims it completely, and turns it upside down. A wave travelling along a surface therefore meets its own reflection reversed and cancels on the surface itself. That one limit explains a dark fringe Humphrey Lloyd saw with a single mirror in 1834, why a mobile signal over flat ground fades with the fourth power of distance rather than the second, why a radar above the sea is blind at the horizon, and how a cliff in Sydney became the first radio interferometer.

Three planes that meet in one line. A thin lens of focal length 50, drawn as a vertical line, and an object plane tilted at 45° to the axis, crossing it 150 units in front of the lens. Four points on the object plane are traced to their images through the lens's centre and along the axis-parallel ray. The images lie on a straight line too — a thin lens maps planes to planes — but one tilted far less steeply, and when the object plane, the lens plane and the image plane are extended they meet in a single line, here 150 units below the axis. That is the Scheimpflug rule. A sensor laid along the tilted image plane records the whole tilted object plane sharp, which no sensor square to the axis can do.

Optics

The plane of focus a tilted lens tilts

An ordinary camera focuses on a plane square to its axis, so a road receding from it, a table top or a field of flowers can be sharp only over the depth an aperture allows. Tilt the lens, or the sensor, and the plane of sharp focus tilts too, by a rule a surveying officer patented in 1904: the plane of the subject, the plane of the lens and the plane of the image always meet in a single line. Turn the focus and the plane of focus does not slide back and forth; it swings about a fixed hinge. The same geometry lets an eye camera see the whole front of the eye in one exposure and a lidar keep a laser beam kilometres long in focus.

Position written as frequency. Four tubes of water side by side in a 1.5 T magnet, their protons all precessing at 63.87 MHz, until a gradient of 10 mT/m is switched on along the row. Then the field, and so the precession frequency, rises steadily across the row, by 425.8 Hz for every millimetre. The signal the receiving coil picks up is a sum of waves at different frequencies, and its spectrum is a picture: the amount of signal at each frequency is the number of protons at the corresponding position. The axis below is position and the axis above it frequency, and they are the same axis. Nothing has been focused. The radio wave at 64 MHz has a wavelength of 4.7 m in air, and the tubes are centimetres apart.

Quantum

The image made of frequencies

A hydrogen nucleus in a magnetic field precesses at a frequency proportional to the field — 63.87 megahertz in a 1.5 tesla scanner. Make the field rise steadily across the body and every position precesses at its own frequency, so the radio signal the body gives off is a chord whose spectrum is a map of where the protons are. A magnetic resonance image is never focused. It is computed, as a Fourier transform of a signal recorded one number at a time, and its resolution, a millimetre, is set by how strong the gradient is and how long the signal is listened to — not by the wavelength of the radio waves, which is several metres.

$The oldest age a trace of modern carbon allows. The age a radiocarbon measurement would report against the sample's true age, when a fraction of the carbon in it is modern contamination — from groundwater, roots, handling or the chemistry of preparation — of 5, 1 and 0.1 per cent; dashed, no contamination. Young samples are barely affected. Old ones are ruled by the contamination, because their own carbon-14 has dwindled to less than it: with 1 per cent, a sample of any age beyond about 60,000 years reads 38.1 thousand years, and 5 per cent caps it at 24.8 thousand. Even 0.1 per cent, a microgram of modern carbon in a milligram sample, caps it at 57 thousand. The limit of radiocarbon dating, usually quoted as about fifty thousand years, is a limit on cleanliness, not on the half-life.$ Quantum

The clock read by counting atoms

Radiocarbon dating reads a clock that ticks slowly: a gram of living carbon holds sixty thousand million carbon-14 atoms, and only thirteen and a half of them decay each minute. Counting those decays to date a milligram of bone would take years. Counting the atoms themselves, one by one in an accelerator, takes an hour, because in any measurement shorter than the clock's mean life almost every atom is still there, undecayed and countable. And the method's famous limit of about fifty thousand years turns out not to be set by the half-life at all. It is set by how much modern carbon a sample has picked up, because a trace of contamination outweighs whatever carbon-14 a very old sample has left.

Relativity

The neutrino energy an angle fixes

A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.

An atom's line and a nucleus's, on one scale. Three energies, in electronvolts on a logarithmic scale, for an atomic transition, sodium's yellow line, and a nuclear one, the 14.4 keV gamma ray of iron-57: the amount by which emission and absorption by free atoms miss each other because of recoil, E²/Mc²; the natural width of the line, ħ over the excited state's lifetime; and the Doppler width from thermal motion at room temperature. For sodium the recoil miss is 2.1·10⁻¹⁰ eV, 197 times smaller than the natural width, so an atom absorbs its neighbour's light. For iron-57 it is 0.00392 eV, 839,015 times its natural width of 4.67·10⁻⁹ eV. The recoil grows as the square of the transition energy and the width does not, so a gamma ray emitted by a free nucleus misses an identical nucleus by nearly a million line widths.

Relativity

The gamma ray its own kind cannot absorb

An atom can absorb the light another atom of its kind emits; that is how a sodium flame casts a shadow in sodium light. A nucleus cannot. When it emits a gamma ray it loses the photon's energy as mass, and the photon's momentum kicks it backward, taking a small share of the energy with it; to absorb the same gamma ray, an identical nucleus would need that share added rather than removed. For iron-57's 14.4 keV line the shortfall is nearly a million times the line's own width. In 1958 Rudolf Mössbauer found that a nucleus held in a crystal can hand its recoil to the whole crystal and lose almost nothing — and the resulting line is so sharp that it could weigh the fall of light down a twenty-two metre tower.

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