Illustrated Physics
physics, drawn
Field

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.
Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size.

The floor that cannot be told from gravity

Seal a laboratory, take away the windows, and no experiment inside it can distinguish standing in a gravitational field from accelerating through empty space. That is not a philosophical remark — it forces light to bend, forces clocks to disagree, and has a size at which it stops being true.

 Where a clock gains, and where it loses. The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.

The clock that runs slow lower down

Two identical clocks, one on the floor and one on a shelf, do not keep the same time — and the difference is large enough that a satellite navigation system which ignored it would be useless within a morning. The derivation needs nothing but a photon and a conservation law.

 Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.

The bend Newton got half right

A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.

 How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical.

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

 What the distant observer actually receives. The frequency of a signal from a clock falling into a horizon, as received far away, against the receiver's own time. It is a straight line on a logarithmic axis, which means the fading is exponential: the e-folding time fitted to the drawn curve is 2.01 rs/c, which for a 10-solar-mass hole is 198 microseconds. Nothing hovers. The image reddens, the photons arrive at an exponentially falling rate, and within a millisecond there is nothing left to see.

Two clocks that disagree about the fall

A clock falling into a horizon crosses it in a few milliseconds by its own reckoning and never crosses it at all by a distant one. Both accounts are right, and the thing everybody remembers about the second — that the image hangs there for ever — is wrong.

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

The wave that stretches one way and squeezes the other

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

 Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Light has a pressure

Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

 The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.

A charge that turns must glow

An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

 The same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

How far a neutrino gets

A mean free path is one over the number density times the cross-section, and nothing else. Change only the cross-section — by twenty-eight powers of ten — and the same arithmetic that gives a molecule seventy nanometres in air gives a neutrino a light-year of solid lead.

 What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

The field that cannot get out

Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

 Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

The size at which a body becomes round

A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

 The two lengths every mass has. The Compton wavelength and the Schwarzschild radius of the same mass, against mass, on logarithmic axes. One falls and the other rises, so they cross exactly once — here at 1.539·10⁻⁸ kg and 2.286·10⁻³⁵ m, found by bisecting the difference rather than by writing down √(ħG/c³). The conventional Planck values are 2.176·10⁻⁸ kg and 1.616·10⁻³⁵ m; the crossing sits a factor of 1.414 away from them, which is exactly √2 and is the factor of two in the Schwarzschild radius coming through a square root. That is the whole precision this argument has, and it is worth saying, because a number written to four figures invites a reader to believe the definition is doing more work than it is. Nothing in physics is known at that length.

Where every model runs out at once

Every mass carries two lengths — one below which quantum mechanics will not let it be located, one below which gravity will not let anything escape. One falls with mass and the other rises, so they cross exactly once, at a length nothing in physics has ever probed.

 The term that abolishes the inner orbits. The effective potential of the Schwarzschild geometry per unit mass, in units of c², at 4 angular momenta, against radius in Schwarzschild radii. Newton's version has a minimum — a stable circular orbit — at L̃²/GM for every angular momentum there is, however small, and it is the dashed curve at the same L̃, here with its minima at 10.13, 7.61, 6.00, 5.12 rs. General relativity adds one term, −GM L̃²/c²r³, and that minimum stops existing below a definite angular momentum. At L̃ = 4.50 GM/c the barrier and the well are still separate, at 1.83 and 8.29 rs. The two merge at L̃ = √12 GM/c, at 3 rs = 6GM/c² — the innermost stable circular orbit, where the curve drawn here has neither a maximum nor a minimum but a single inflection. Below that momentum — the curve at L̃ = 3.20 GM/c — there is no stationary point anywhere outside the horizon, so no circular orbit exists at all, at any angular momentum whatever. None of that is a property of matter; it is a statement about the geometry, and it fixes the energy of the innermost orbit at √(8/9) = 0.94281 of mc², so 5.719% of the rest mass has been radiated by anything that reached it. Every well drawn here was located on the emitted curve by golden section and checked against the closed form.

The orbit that cannot be made smaller

Newtonian gravity allows a stable circular path at every radius, however tight, and there is no innermost one. General relativity adds a single term to the expression that says so, and below 6GM/c² no stable circular path exists at any angular momentum whatever. Anything arriving there has radiated 5.72% of its rest mass, which is eight times what hydrogen fusion converts.

 How large a box may be before gravity shows in it. The largest freely falling laboratory in which nothing can be detected, against how long the experiment runs, for an instrument resolving 10⁻⁹ m. Free fall removes the field and leaves the gradient: two masses released a distance apart converge across the fall and separate along it, by (GM/r³)Lt²/2, so the box that stays undetectably flat shrinks as the square of the time. At one second the limits are 1.30 mm at the Earth's surface, 5.07 mm at the Sun's surface, 8.61 nm at a white dwarf, 1.86·10⁻¹⁷ m at a neutron star, 3.88·10⁻¹⁷ m at a stellar black hole, 388 mm at a giant black hole. Two things are worth reading off it. The equivalence principle is local in time as much as in space, on the same footing and with a worse exponent — patience is more expensive than room. And the tidal parameter GM/r³ falls with mass at a horizon, so a large enough black hole is one an experimenter could fall through with a metre-sized laboratory and a very good interferometer and detect nothing at all: the curve for the giant lies above the Earth's, not below it.

The term free fall cannot remove

Fall freely and gravity disappears. It disappears only to the extent that the falling laboratory is small — what survives is the gradient, which pulls two released masses together across the fall and apart along it. Given an instrument, the size of the box in which nothing is detectable is computable, and that number is the whole content of the word "locally".

 The bigger the hole, the gentler the horizon. The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present.

The horizon that nothing marks

A falling body is torn apart by the difference in gravity between its ends. At the horizon of a black hole that difference goes as the inverse square of the mass, so a large enough hole can be entered intact — and nothing local happens at the crossing to say it has occurred.

 The same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

The orbit that does not come back to itself

A bounded orbit under any central force oscillates between a smallest and a largest radius for ever. That it should also return to the same point is a further demand, and only two force laws in existence meet it — the inverse square, and the linear spring.

 A radar echo past the Sun, delayed by 233 microseconds. The extra time a round-trip radar signal takes when its path passes close to the Sun, against how close, for a reflector 0.723 AU away. Grazing the Sun's limb the delay is 233 microseconds — about 70 kilometres of light travel, on a path of hundreds of millions — and it falls only as the logarithm of the impact parameter, so the effect is still tens of microseconds ten solar radii out. That slow falloff is what makes the measurement possible: the delay can be watched building and fading as the geometry changes, rather than having to be caught at one instant. The dashed curves are the delay the solar corona's plasma adds at three radio frequencies. It is the competing effect, it is larger than the gravitational one close in, and it falls as the square of the frequency while the gravitational delay does not depend on frequency at all — which is how the two are separated, and why the sharpest measurement of this was made with a spacecraft carrying three radio links instead of one.

The delay that is not a bend

The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.

 The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals.

The size the light cannot blow away

Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

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

The orbit that has to shrink

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

 The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does.

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

 A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen.

The lens with no focal length

A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

 The figure a static field is not allowed to close. A spacetime diagram of two clocks held at fixed heights 22.5 metres apart, drawn as though spacetime were flat: time upward, height to the right, light at forty-five degrees. The lower clock sends two pulses; the upper clock receives them. Because the field does not change with time, nothing about the second pulse's journey differs from the first's, so the two null lines are congruent and the four worldlines bound a parallelogram. Opposite sides of a parallelogram in flat spacetime have equal length, so the proper time between emissions must equal the proper time between receptions, and the two clocks must agree. They do not: the measured fractional difference across a tower this tall is 2.455e-15, which Pound and Rebka established in 1960 and Pound and Snider confirmed to one per cent in 1964. Every step above is either a definition, an assumption of staticity, or a theorem of flat geometry — so the measurement refutes the flatness. No field equation has been written down, and none is needed: a laboratory result twenty-two metres tall is already incompatible with a flat spacetime.

The parallelogram that will not close

Two clocks twenty-two metres apart in a lift shaft run at different rates, by two parts in a thousand million million. That measurement, on its own, is enough to prove that spacetime cannot be flat — and the proof needs no field equation, no curvature tensor and no astronomy. It needs one drawing and the fact that opposite sides of a parallelogram are the same length.

 The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter.

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

 The self-force matters at 6.27 × 10⁻²⁴ s, and nowhere a charge has ever been. The ratio of the radiation reaction to the applied force, which is τ divided by the time the force takes to change, on a logarithmic axis. τ = μ₀q²/6πmc = 6.266e-24 s for an electron, and light crosses 1.879 femtometres in that time — two thirds of the classical electron radius. a 3 GHz accelerating cavity: 1.9e-14; a 500 nm optical field: 3.8e-9; an electron orbiting a proton, ground state: 4.1e-8; an X-ray at 0.1 nm: 1.9e-5; over a classical electron radius: 6.7e-1. The largest of them, "over a classical electron radius", is still 1.5e+0 times too slow. So the correction is never large for any force anybody can apply, and the only regime where it would be is one in which the charge's own structure has already made the whole description meaningless.

The force a charge exerts on itself

Larmor's formula says how much an accelerating charge radiates and says nothing about who pays. Conservation says the charge does, so there is a force on it — and the equation that force produces has a free particle accelerating for ever with nothing pushing it, or else beginning to move before it is pushed. Both solutions are absurd, and the interval over which they are absurd is smaller than the electron the equation was written for.

 One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round.

The frequency below which nothing gets in

Free charges give a medium a permittivity that is negative, and a negative permittivity is not an absorbing medium — it is one in which no wave exists at all. Below that frequency the reflection is total, exactly rather than nearly, because there is no transmitted wave and nothing to absorb. The same expression puts the number at 9 MHz for the ionosphere and 3.8 PHz for aluminium.

 Cold matter, and the mass above which nothing holds it up. The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it.

The mass no cold matter can hold up

A white dwarf gets smaller as it gets heavier, which no ordinary object does. Follow that curve upward and the radius reaches zero at 1.46 solar masses — because once the electrons are relativistic the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball does not depend on its radius at all.

 A click that arrives sorted by pitch. Arrival time against frequency for a broadband pulse travelling 40 megametres along a magnetic field line through a plasma of 1e+8 electrons per cubic metre in a field of 300 nanotesla — the magnetosphere, roughly. The gyrofrequency is 8.4 kilohertz and the plasma frequency 90 kilohertz, so this is the regime where a right-hand circularly polarised wave travels happily below both of them. It does not travel at one speed. The group velocity is 2c√ω(ω_c − ω)^(3/2)/(ω_p ω_c), which is zero at zero frequency and zero again at the gyrofrequency and peaks in between — at 2.10 kilohertz here, which is a quarter of the gyrofrequency, found by searching the drawn function rather than quoted. Everything either side of that peak is slower, so the curve has a nose: 3.21 s at 500 Hz, 2.20 s at 2000 Hz, 3.59 s at 5000 Hz. The branch below the nose is the classic whistler — a lightning stroke in one hemisphere reaching a receiver in the other as a note gliding downward over about a second, which is what gave the phenomenon its name — and the branch above it arrives as a rising tone at the same time. Both are observed, and a recording that shows the two joined at the nose is how the gyrofrequency along the path is read off directly. Read the other way it is an instrument: the product of the delay and the square root of the frequency is 79 s·Hz^½ here, nearly constant across the low end of the band, and it measures the electron content of a path through the magnetosphere that nothing else could reach.

The whistle that arrives sorted

A lightning stroke in one hemisphere reaches a receiver in the other as a note gliding downward over about a second. Nothing dispersed the sound; there was no sound. A radio pulse travelled forty megametres along a magnetic field line through a plasma whose group velocity depends on frequency, and arrived with its frequencies separated by up to three seconds.

 Which wavelengths grow instead of oscillating. The dispersion relation of a self-gravitating isothermal gas, ω² = c²k² − 4πGρ, with each axis measured in the scale the gas sets for itself. Short wavelengths oscillate: pressure wins, and the disturbance is a sound wave. Long wavelengths do not: ω² is negative, so the disturbance grows exponentially instead of travelling, and the region collapses. The changeover is at λ_J = c√(π/Gρ), and it happens because pressure support acts on a sound-crossing time that grows with the region while gravity's collapse time does not depend on the size at all. Four densities spanning 6 decades are drawn and they lie exactly on top of one another, to 6e-16, because the criterion has no scale of its own: it is the same curve for a diffuse cloud and for a protostellar core, with different numbers written on the axes. In this gas at 10 K those numbers are 2.12 pc and 28.72 solar masses at 10² cm⁻³, 0.21 pc and 2.87 solar masses at 10⁴ cm⁻³, 4372 AU and 0.29 solar masses at 10⁶ cm⁻³, 437 AU and 0.03 solar masses at 10⁸ cm⁻³.

The disturbance that grows instead of travelling

A sound wave in a gas oscillates because pressure restores what the disturbance displaced. Add the gravity the gas exerts on itself and the restoring force acquires a competitor that does not weaken with size — so above one wavelength the sum changes sign, the frequency becomes imaginary, and the disturbance stops travelling and starts growing. It is the same wave equation with one term subtracted.

 The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis.

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

 The brightest anything of a given mass can be. The Eddington luminosity against mass, with the main sequence drawn beside it. Radiation pushes outward on the electrons and gravity pulls inward on the protons, and both go as one over the distance squared — so the radius cancels out of the comparison entirely, checked here at three radii spanning four decades and coming out identical to 1e-20. What is left is a luminosity: L = 4πGMc/κ, which is 1.47 × 10³¹ watts per solar mass, or 3.8·10⁴ solar luminosities. Above it, radiation drives the outer layers away faster than gravity can hold them. The Sun is at 2.6e-5 of its own limit and in no danger; a star of 10 solar masses is at 1.2e-2; and a star of 100 is at 0.83, which is why the two lines converge at the top of the chart and why the most massive stars known are a few hundred solar masses rather than a few thousand. They do not fail to form for lack of gas; they blow away the gas that would have made them heavier, and once formed they shed mass continuously in a radiation-driven wind. The limit is the same expression for an accreting black hole, where it caps not the brightness but the rate at which mass can be taken on.

The brightness a mass cannot exceed

Light pushes outward on the electrons of a star and gravity pulls inward on its protons, and both forces fall off as one over the distance squared. The distance therefore cancels, and what is left is a limit on brightness rather than on size — 3.8 × 10⁴ times the Sun's luminosity for every solar mass, above which a body drives its own outer layers away.

 The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all.

The distance that forgets the moon

A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

 Nine hundred steps and hardly anywhere. On the left, a walk of 900 steps of unit length in uniformly random directions, which is what a photon does inside a star: it goes a mean free path, scatters, and starts again in a direction that has forgotten the last one. After 900 steps it is 7.5 lengths from where it began, against 900 if it had gone straight. On the right, the root-mean-square distance over 240 independent walks against the number of steps, both logarithmic: a straight line of slope 0.4920 against an exact one half. The square root is the whole of the result and it is brutal. Escaping a body of radius R takes not R/λ steps but (R/λ)² of them, so a mean free path a thousand times smaller costs a million times as long. That is the difference between a photon leaving the Sun's core and a neutrino doing it: one takes a hundred thousand years and the other takes two and a third seconds, through the same material, and the only thing that differs is λ. What the picture cannot show is the sense in which the escaping energy is not the photon that started: it is absorbed and re-emitted countless times, at falling temperature, so what leaves is a gamma ray's worth of energy arriving as a great many visible photons.

The light that takes a hundred thousand years to leave

A neutrino made in the Sun's core is at the surface in two and a third seconds. A photon made beside it takes something like a hundred thousand years, through the same material, over the same seven hundred thousand kilometres — and the whole of the difference is one length, entering the answer squared.

 Five rays, and the one that decides which side everything falls on. Light traced past a non-rotating mass at five impact parameters, by integrating the exact null geodesic rather than the weak-field formula. The shaded disc is the horizon and the dashed circle is the photon sphere at three masses. Rays passing closer than 5.196 masses are captured — 2 of the five here — and rays passing further escape, however far they are bent on the way. The two nearest the critical value behave in the most striking way: one wraps 1.06 times round before leaving and the other wraps round before falling in, and between them lies a ray that would circle for ever on the unstable orbit. Far away the same integration reproduces the familiar weak-field deflection: at sixty masses it gives 0.07021 radians against 4M/b = 0.06667, so the strong field and the weak one are one calculation.

The circle light cannot leave

A black hole has three radii and they are all sometimes called its size. The horizon is where nothing can come back from; the photon sphere, half again as far out, is where light can circle and cannot keep circling; and the shadow a distant observer sees is larger than either, because the ray that just escapes was bent on its way out.

 A wave that dies with nothing to rub against. The electric field of a plasma wave at kλ = 0.5, against time in plasma periods, on a logarithmic scale, obtained by integrating the collisionless kinetic equation as an initial-value problem. There are no collisions in the equation, no viscosity and no resistance; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field nevertheless falls exponentially, at 0.1534 per plasma period, against the published root of the kinetic dispersion relation at this wavenumber, 0.1534, and Landau's asymptotic formula's 0.1514. Meanwhile the free energy of the perturbation — the weighted norm of the distribution plus the field energy, which the equation conserves exactly — moves by 2.4e-10. So nothing has been dissipated: every joule the field loses is still in the distribution, and the accounting closes to a part in ten thousand million. The energy has gone into the particles' ordered motion, and the information about the wave is wound into structure at finer and finer scales in velocity.

The wave that dies with nothing to rub against

Every damping in this collection so far removes energy from a wave and puts it somewhere warmer. This one removes it and produces no heat at all: there are no collisions in the equation, the entropy is unchanged, the whole thing runs backwards perfectly, and the wave still dies exponentially. What it dies into is structure in velocity too fine for a field to see.

 The one place in a plasma where the charges do not balance. The potential and the two densities through a sheath, in Debye lengths from the sheath edge, for a plasma of hydrogen ions. The potential is measured downward in units of the electron temperature and reaches 2.84 at the wall — the value at which the two fluxes balance. The layer is 15.1 Debye lengths thick — for 3 eV electrons at 1e+16 per cubic metre, a Debye length of 129 µm and a sheath of 1.9 mm. Inside it the electron density falls as the Boltzmann factor while the ion density falls only as the ions speed up, so the two separate: at the wall the ion density exceeds the electron density by 85 per cent of itself. That is the whole of the difference between a sheath and the plasma it borders — quasineutrality is not an assumption that can be made here, and Poisson's equation has to be solved instead of replaced.

The wall a plasma builds against itself

A plasma is quasineutral everywhere except where it touches something. Electrons are faster than ions by the square root of the mass ratio, so any surface is struck by far more of them, charges negative, and goes on charging until the two arrivals balance — leaving a layer a few Debye lengths thick in which the charges do not cancel at all.

 A force that points across the beam, not along it. The intensity of a beam focused to a waist of 0.5 µm, and the force it exerts on a 60 nm polystyrene sphere in water, across the beam. The force is not a pressure and does not point along the light: it is the pull on an induced dipole sitting in a non-uniform field, proportional to the gradient of the intensity rather than to the intensity, and it points up the gradient — towards the bright axis from either side. It vanishes exactly on the axis, which is what makes the axis a trap rather than a place. The well is 47 times kT deep at 100 milliwatts, which is why the particle stays: thermal motion explores it and does not escape it. The stiffness near the bottom is 3.070 piconewtons per micrometre, and a particle in a spring that stiff wanders 36 nanometres from centre.

The light that pulls rather than pushes

Radiation pressure is momentum arriving, and it points along the beam. A gradient of intensity does something else entirely — it polarises a particle and then pulls the induced dipole up the gradient — so a focused beam holds a particle at its waist against the push, and the force that does it is not a pressure at all.

 The arrow the orbit cannot turn. A Kepler orbit of eccentricity 0.6, integrated for two revolutions, with the Laplace–Runge– Lenz vector constructed from the position and velocity at five points along it. Every one of the five is the same arrow: its length varies by 3.7e-11 over the whole run and its direction by 2.1e-10 radians. It points at the perihelion and its length is 0.600000, which is the orbit's eccentricity measured independently from the closest and furthest radii as 0.600000. Energy and angular momentum fix the size and shape of an orbit and say nothing about which way it points; this vector is the missing statement, and only an inverse square has one.

The arrow that says which way the orbit points

Energy and angular momentum fix the size and shape of an orbit and say nothing about its orientation. The inverse-square force has a third conserved quantity that supplies it — a vector pointing at the perihelion whose length is the eccentricity — and no other force law does.

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

What the instrument actually hears

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

 Colder the bigger it is. The Hawking temperature against mass, on logarithmic axes, with the microwave background drawn across it. The slope is minus one exactly, so a heavier hole is colder — a negative heat capacity, which is the fact everything else here follows from. The two lines cross at 4.50e+22 kg, about a hundredth of the Moon's mass. Anything heavier than that is colder than the sky it sits in and absorbs more than it emits, so it grows rather than evaporates. A stellar-mass hole is at 6.2e-8 kelvin and will not begin to lose mass until the background has cooled below that, which takes something like 10¹² years. Evaporation is not something happening now to any hole anybody has observed.

The hole that outlives everything and then does not

A black hole radiates at a temperature that rises as it shrinks, so losing energy makes it lose faster. The whole history follows from that one sign: a life proportional to the cube of the mass, nearly nothing happening for almost all of it, and an end that arrives in a second.

 The power at stake, which is none. The power an electron radiates by Larmor's formula, against its acceleration, with 5 cases marked. a charge on a table: 9.8e+0 m/s², 5.49e-52 W; a laboratory centrifuge: 1.0e+6 m/s², 5.71e-42 W; a proton at the LHC: 1.9e+16 m/s², 2.06e-21 W; an electron in a linac: 2.0e+19 m/s², 2.28e-15 W; an electron in a hydrogen atom: 9.0e+22 m/s², 4.62e-8 W. The slope is two, measured on the drawn line. A charge held at one gravity radiates 5.49e-52 watts, which over the whole age of the universe comes to 2.39e-34 joules — far less than one photon of any kind. So the question of whether it radiates is not an experimental question about a charge on a table, and never has been. Every number here is a straight line on logarithmic axes with an exponent the figure measures.

Whether a charge on a table glows

The equivalence principle says a charge at rest in a gravitational field is a charge accelerating in empty space, and an accelerating charge radiates. Nothing is supplying the energy. The argument has run for eighty years, and its resolution is that radiation is not something a single observer can define.

 Two terms, and the distance neither can beat. The smallest distance a probe of a given momentum can resolve, as the sum of two terms. The falling one is the uncertainty relation: more momentum, shorter wavelength, finer resolution. The rising one is gravity: the probe's own energy curves the region it is probing, and past a point it makes a horizon larger than the thing being looked at. With the gravitational term at 1× the Planck area, the least resolvable distance is 2.286e-35 m — each located by scanning the drawn curve over four hundred thousand momenta rather than by substituting into a formula. There is no momentum at which the resolution is better than that, so the ordinary procedure for measuring a distance has a floor, and the floor is the Planck length up to a factor of order one.

The length no experiment can resolve

Measuring a small distance needs a short wavelength, a short wavelength needs a large energy, and a large energy in a small region makes a horizon. Past a point, pushing harder makes the probe bigger — and the distance where that turns round is the Planck length.

 The quantity that went down, and the one that went up. The books for GW150914: two black holes of 36 and 29 solar masses merging into one of 62, with a final spin of 0.67. 3.0 solar masses left as gravitational waves, so the mass fell by 4.6 per cent. The total horizon area rose, from 107417 to 168330 in units of the Sun's gravitational radius squared — an increase of 57 per cent. The two progenitors are taken as non-spinning, which is the assumption that makes the test hardest to pass: a spinning hole of the same mass has a smaller horizon, so any spin they actually had would only widen the gap. Mass is the quantity that behaves the way energy usually does and it is not the one with a direction. Area is, and it is the reason the area has been read as an entropy ever since.

The area that is not allowed to shrink

Two black holes merge and the result weighs less than the sum, because three solar masses left as gravitational waves. The horizon area went up by more than half. Mass is the quantity that behaves like energy and it is not the one with a direction; area is, and the theorem saying so has been tested against a real merger.

 The height a clock can see. The fractional difference in rate between two clocks against how far apart in height they are, on logarithmic axes — a straight line of slope one, since the shift is gh/c² and comes to 1.09e-16 per metre near the ground. a caesium fountain, good to 1e-16, resolves 91.6 cm; an optical lattice clock, good to 1e-18, resolves 0.9 cm; the best clocks built, good to 8e-19, resolves 0.7 cm. The caesium fountains that define the second reach about a metre. The optical clocks that will replace them reach a centimetre, and the best of them a few millimetres. That is the whole of why this has stopped being a test of relativity and become a way of measuring the ground. A shift once so small that it took a Mössbauer experiment in a tower to see at all is now large enough to be a nuisance: two clocks in the same building disagree, and the disagreement has to be corrected for before either can be used to keep time.

The clock that measures a height

A clock a metre higher runs faster by a part in ten thousand million million million. That was once so small it took a tower and a Mössbauer source to see; the best clocks now resolve a centimetre of height, at any distance, without a line of sight. What began as a test of general relativity has become a surveying instrument that measures the quantity surveying actually wants.

 Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive.

The fall that does not depend on what is falling

Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards.

 One magnitude, and a direction that turns the wrong way. The force per unit area a magnetic field of 0.01 tesla exerts on a surface, drawn for surface normals at 0°, 30°, 45°, 60°, 90° to the field. The short grey arrows are the normals; the long coloured ones are the tractions. Every traction has the same length, 40 pascals, because the magnitude of T·n does not depend on the orientation of n at all — and the direction is the normal reflected in the field rather than carried round with it, so as the surface turns one way the force turns the other. A surface cutting across the field is pushed (the magnetic pressure); a surface cutting along it is pulled (the magnetic tension); and at 45° the traction lies in the surface, which is a shear and is neither.

The same force whichever way the surface faces

A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.

 Three speeds, drawn against direction. The phase speed of the three magnetohydrodynamic waves against the angle between the wavevector and the field, drawn as a polar diagram with the field horizontal and the sound speed 0.6 times the Alfvén speed. Along the field the two magnetosonic branches take the values of the sound speed and the Alfvén speed themselves and swap which is which as the ratio crosses one; across the field the slow branch vanishes entirely and the fast branch runs at the quadrature sum. The shear branch is the figure-of-eight, v_A cos θ, and it is the only one of the three whose speed contains no thermodynamic quantity — no pressure, no temperature, no sound speed. It does not know what the gas is made of.

The wave that does not know what the gas is made of

Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.

 Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

 A hill that is always ahead of the atom. The light-shift potentials of the two ground sublevels of a spin-½ atom in two counter-propagating beams with crossed linear polarisations, over one and a half wavelengths, in units of the well depth. The polarisation of the light turns from σ+ to linear to σ− every quarter wavelength, and the two sublevels see sinusoidal potentials a quarter wavelength out of step. Optical pumping transfers the atom from one sublevel to the other fastest exactly where its own potential is highest — the fastest pumping and the hilltop coincide, which the figure checks — and that is the bottom of the other potential. The heavy line is an atom that starts with 3.3 well depths of kinetic energy: each time it reaches a hilltop it is pumped down by exactly one well depth, climbs the next hill, and is pumped down again, 3 times, until it no longer has the energy to reach a hilltop and is left oscillating in one well. The energy is carried off by the pumping photon, which leaves bluer than the light that drove it by the depth of the well.

The limit that belonged to a simpler atom

The theory of laser cooling predicted a floor, and the first careful measurement came in six times below it. Nothing was wrong with the measurement or the arithmetic. The floor belonged to an atom with one ground state, and real atoms have several — which lets the light build a hill in front of every atom, move it to the bottom before it can roll back, and repeat the trick until the atom is a few microkelvin from rest.

 Radiated while the push holds, paid for when it stops. The power a charge radiates, the power the radiation reaction force takes from its motion, and the rate of change of the Schott term mτ a·v, through a push that rises over the first 20 per cent of its duration, holds steady, and falls away over the last 20, in units of mτa₀² where a₀ is the steady acceleration. Radiated power is a², the reaction force's take is −ȧv, and the Schott rate is found by differencing a·v along the trajectory; at every instant the first equals the sum of the other two. While the push is steady the reaction force is exactly zero and the charge still radiates at the full rate, all of it drawn from the Schott term. When the push stops, ȧ is large and negative while the charge is moving fast, and the reaction force takes 0.775 units, more than the 0.750 radiated over the whole push. The difference is what it handed back while the push was starting: then the charge is still slow, the reaction force points along the rising acceleration, and it does 0.025 units of work on the charge instead of taking any. The totals agree to a part in a hundred thousand.

The bill that arrives when the pushing stops

A charge accelerating steadily radiates at the full Larmor rate while the radiation reaction force on it is exactly zero, so for as long as the push holds, nothing about the charge's motion pays a single watt. The energy is lent by the field that travels with the charge, the loan is called the Schott term, and it is repaid the moment the acceleration changes.

 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.

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.

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 one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

The estimate that misses by a hundred and twenty

Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

 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.

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.

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

A few cycles that are only mass and spin

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

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

The ring that does not come back

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

 The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

 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.

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.

 Four bodies sent past one mass at 30 masses, at four speeds. Paths traced from the exact geodesic of a non-rotating mass, every one aimed at the same impact parameter of 30 masses and differing only in speed. The slower the body, the harder it is turned: 0.4c by 32.9°, 0.6c by 16.6°, 0.8c by 11.1°, light by 8.5°. The weak-field rule (2GM/bv²)(1 + v²/c²) gives 27.7°, 14.4°, 9.8°, 7.6° — close for the fast bodies and increasingly short for the slow ones, which pass near enough to feel the field's strong part. The angles are the true ones: near a compact mass nothing has to be exaggerated.

The half of the bend a slow body never feels

Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.

 A rosette: the orbit of a body whose inertia is its energy. An orbit of eccentricity 0.45 under inertia from energy, Newton's pull, integrated for 7 revolutions from the closest approach of a body with k/Lc = 0.35. The dashed curve is the Newtonian ellipse from the same starting position and momentum, which closes on itself. The integrated orbit does not: its closest approach moves forward by 24.3° each revolution, against the exact 24.3° of 2π(1/Γ − 1) with Γ = √(1 − k²/L²c²). The dots are the successive closest approaches.

The orbit special relativity cannot close

An inverse-square orbit closes on itself because of a conserved vector that nothing else has. Give the orbiting body an inertia that grows with its speed, as special relativity requires, and the vector turns — the orbit becomes a rosette, advancing by a sixth of Mercury's famous 43 arcseconds a century. A different theory that respects special relativity just as well turns the rosette backwards by the same amount. The 43 is not special relativity plus a correction; it is a measurement of what gravity pulls on.

 One turn of a circling charge, as it arrives. The radiation field reaching a distant observer in the plane of a charge's circular orbit, over one full turn, for speeds of 0.2c, 0.6c, 0.9c, each scaled to its own peak and plotted against arrival time as a fraction of the orbital period. At 0.2c the flash arrives 20.9 per cent of a turn wide at half height, having been emitted over 25.4 per cent; at 0.6c the flash arrives 6.0 per cent of a turn wide at half height, having been emitted over 14.2 per cent; at 0.9c the flash arrives 0.7 per cent of a turn wide at half height, having been emitted over 6.3 per cent. The slowest charge sends something close to a sine wave, with a trough 0.44 of its peak; the fastest sends a spike whose peak, divided by its speed, is 100 times a slow charge's, and a trough of 0.0028 of that peak. The field integrates to zero over every turn, as a radiation field must.

The flash a circling charge sends once a turn

A slow electron going round a magnet broadcasts one radio frequency. The same electron at nearly the speed of light, on a circle of the same kind, sends X-rays. Nothing about the mechanism changes between the two. The charge chases its own light, and the chase compresses one power of γ worth of emission into three.

 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.

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.

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.

 Tadpoles and a horseshoe, drawn in the turning frame. Orbits of small bodies near the circle of a planet's orbit, integrated in the frame turning with it, with the distance from that circle exaggerated 12 times so the shapes can be seen. With Jupiter's mass ratio, 0.000954, a body started 10° and 25° ahead of L4 at rest in the turning frame circles round L4 rather than falling off the hill: it swings between 51° and 70°, and between 42° and 85° from the planet — a tadpole. With a mass ratio of 10⁻⁴, a body started opposite the planet sweeps round through L4, L3 and L5 and back, never coming within 24° of the planet — a horseshoe, the kind of orbit Saturn's moons Janus and Epimetheus follow relative to each other, exchanging places every four years. In every case the Jacobi constant, the one quantity conserved in the turning frame, is held by the integration to better than a part in a million.

The hilltop that holds the Trojans

Sixty degrees ahead of Jupiter and sixty behind it, thousands of asteroids share the planet's orbit, held at the corners of the equilateral triangles that Jupiter and the Sun make. Seen from the frame that turns with Jupiter, those corners are the tops of two hills in the effective potential — the worst places to try to keep anything still. They hold asteroids anyway, because in a turning frame a body sliding off a hill is swung sideways by the Coriolis force into an orbit round the top. The trick works only if the planet is light enough: above a mass ratio of about one in twenty-six, the hilltops let go.

 The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

 Five images of one source, and the curves where they are made. The sky around a lens with a core of 0.10 and an external shear of 0.15, in Einstein radii, with its two critical curves: an outer one near the Einstein ring and a small inner one round the core. A source at (0.06, 0.04) in the source plane — inset, with the caustics the critical curves map to — lies inside the diamond-shaped caustic and has 5 images. Their magnifications are +4.99, −3.76, +3.42, −2.63, +0.01 — a minus sign for an image turned over — and the parities sum to +1. Four sit near the outer curve and carry nearly all of the 14.8-fold total; the fifth, at the centre, is magnified by only 0.013. A source further out, at (0.55, 0.1), has 3.

The fifth image a galaxy hides

A single star bends light into two images of whatever lies behind it. A galaxy, whose mass is spread out, has to make an odd number — one, three or five — because the map from the sky to the source is smooth and has nowhere to put an even count. The famous cross of four quasar images is therefore five images, with the fifth thousands of times too faint to see, sitting at the centre of the galaxy that made it. Images are born and destroyed only in pairs, and only on the curves where the map folds.

 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.

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.

 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.

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.

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

The mountain a spinning star is allowed

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

 The slice through a mass that a ruler has to climb. A plane through the centre of a uniform body whose compactness 2GM/Rc² is 0.34 — about a neutron star's — drawn as a curved surface on which distances along the surface are the distances rulers measure. The body's edge is a circle whose circumference is 2πR; a ruler laid from the centre to that edge along the surface measures 1.068 R, 6.8 per cent more. Outside the body the surface is Flamm's paraboloid; inside it is a cap of a sphere. The flat line beneath is what Euclid would give. The height of the surface means nothing; only lengths along it are measured. For the Earth the same excess is 1.5 mm in 6,371 km and the surface would be indistinguishable from the flat line.

The radius a mass adds that no circumference shows

Walk round the Earth's equator and divide by 2π, and the number is the Earth's radius. Dig to the centre and measure the same radius with a tape, and the tape comes out about two millimetres longer. Nothing is wrong with either measurement. A mass does not only slow the clocks near it; it stretches the rulers too, and the stretch is exactly as large as the slowing. Clocks cannot see it. Light can, and half of every bend and every delay that light suffers near the Sun is this.

 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.

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.

 The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up.

The pressure that weighs down what it holds up

In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

 A target that grows as the traveller slows. The capture cross-section of a non-rotating black hole for particles arriving with speed v, in units of the horizon's own cross-sectional area π(2GM/c²)², against v/c, both on logarithmic axes (solid). For light it is 27/4 = 6.75 times the horizon's area. It grows as the particle slows, following 4c²/v² (dashed) — 404 at a tenth of light speed and 7.43·10⁶ at the 220 km/s of the Galaxy's dark matter. The dotted curve is a Newtonian ball of radius 8GM/c² — four horizon radii — capturing by ordinary gravitational focusing, and for slow particles the hole and the ball cannot be told apart.

The target that grows as the traveller slows

Light aimed at a black hole is swallowed if it passes within about five of the hole's gravitational radii, a disc six and three-quarter times the horizon's own. A particle arriving slowly is swallowed from much further out, and the disc grows as the inverse square of its speed: dark matter drifting at 220 kilometres a second sees a target seven million times the horizon. The reason is that slow matter carries almost no angular momentum for a given aim, and angular momentum is the only thing that keeps anything out. For slow traffic the hole behaves exactly like an ordinary ball four times its own size.

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

The wobble that radiating makes larger

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

 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.

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.

 Rays at a frequency the ionosphere cannot stop head-on. Ray paths at 14 MHz launched at angles from 10° to 75° from the vertical into a Chapman layer peaking at 300 km with a critical frequency of 8 MHz, over a flat Earth. A wave sent straight up at 14 MHz passes through: it is above the layer's plasma frequency everywhere. A ray at φ₀ from the vertical turns back where the plasma frequency equals f cos φ₀ — the secant law — so only rays more oblique than arccos(8/14) = 55.2° return. Those that do land beyond about 1133 km; nothing comes down closer than that: the skip zone, a ring of silence round a transmitter that is audible much further away. Vertical scale exaggerated about six times.

The frequency the sky returns only at a slant

The ionosphere turns back any radio wave below its plasma frequency, about eight megahertz on a summer afternoon, and lets anything above it out into space — if the wave is sent straight up. Send a fourteen-megahertz wave up at a slant and it comes back down, a thousand kilometres away. Only the part of the wave's motion across the layer has to be stopped, and that part is the frequency times the cosine of the angle. The rule turns one frequency into a range of them, sets the highest frequency that can reach any given distance, and leaves a ring of silence round every short-wave transmitter, inside which it cannot be heard at all.

 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.

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.

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.

 The redshift of a white dwarf weighs it. The gravitational redshift at the surface of a cold white dwarf, expressed as the velocity that would give the same Doppler shift, against the star's mass; computed from the radius each mass is forced to have. The Sun's surface redshift is 636 m/s. A white dwarf of 0.6 solar masses shifts its light by 30.1 km/s, one of 1.0 by 77.6, of 1.2 by 128 and of 1.35 by 223. Because the radius is not free, the curve is single-valued and steepening: a measured shift returns one mass. The cold model puts Sirius B, at 1.018 solar masses, at 80.8 km/s; its measured mass and radius together give 80.0.

The redshift that weighs a dead star

Light leaving the surface of an ordinary star is reddened by gravity, but the shift alone says little, because a star's radius is free and the shift depends on mass over radius. A white dwarf has no free radius. Degenerate electrons fix its size from its mass, so one measured line shift returns one mass — and the gauge grows sharper the closer the star is to collapse. The first measurement of it, in 1925, was hailed as a confirmation of general relativity and was wrong by a factor of four, in agreement with a forecast that was wrong by the same factor.

 A line carved into the X-rays of a magnetised star. A model X-ray spectrum of the kind fitted to accreting pulsars, drawn as energy per logarithmic interval on a logarithmic scale: a hard continuum that bends over above about 25 keV, with the photons near the cyclotron energy removed by resonant scattering. Dashed, the same continuum without the field. The fundamental is placed at 38 keV, where lines of this kind are seen; thermal motion of the scattering electrons along the field at kT = 8 keV broadens it to a width of about 4.8 keV. The second harmonic falls at 73.0 keV, a little below twice the fundamental. Read through ħeB/m with the relativistic level spacing and the redshift of a 1.4 solar-mass, 12 km star, z = 0.24, the fundamental says the field where the line forms is 4.24 × 10⁸ T. The depths and widths here are illustrative, not fitted to a star.

The circling that comes in quanta

An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

 The ring where an axisymmetric field vanishes. Field lines of the slowest-decaying dipole field of a conducting sphere, drawn in a plane through its axis (the axis is the left edge, the sphere's surface the arc). Each line is a contour of the flux function, which is zero on the axis and at infinity and positive between, so it must have a maximum. Here the maximum is on the equator at 0.873 of the radius: there the lines shrink to a point, which in three dimensions is a ring round the axis on which the meridional field is exactly zero. 7 of the 11 drawn lines leave the sphere and close outside; the other 4 close inside, round the ring. The toroidal current that makes the field is not zero on the ring: it is 89 per cent of its largest value.

The field no symmetric flow can keep

The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.

 How well a distance can be known, by how long it is. The smallest uncertainty in a distance l, on logarithmic axes, by three rules. Solid: the clock argument, 1.89 (ℓₚ² l)^(1/3), which grows as the cube root of the distance. Dotted: the Planck length, the floor usually quoted, which does not grow at all. Dashed: the guess that Planck-scale jitters add like a random walk, √(ℓₚ l). a metre: 1.2·10⁻²³ m by the clock argument, 4·10⁻¹⁸ m as a random walk; the Earth to the Sun: 6.4·10⁻²⁰ m by the clock argument, 1.6·10⁻¹² m as a random walk; the observable universe: 9.2·10⁻¹⁵ m by the clock argument, 8.4·10⁻⁵ m as a random walk. By the clock argument the size of the observable universe is uncertain by about the size of an atomic nucleus; as a random walk, by about a tenth of a millimetre. All three rules meet at the Planck length.

The clock that cannot be both light and good

The Planck length is usually quoted as the smallest distance that means anything, as though every longer distance could be known to that precision. Time a distance with light against a clock and the floor turns out to rise with the distance. A light clock wanders while the light is away; a heavy clock wanders less but is closer to being a black hole; and the best compromise leaves a metre uncertain by 750 thousand million Planck lengths and the size of the observable universe by the size of an atomic nucleus. The same compromise, counted up, says a region can hold only as many distinguishable places as its surface has Planck areas.

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

The four shapes a wave of gravity leaves out

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

 Rays that leave the far side of a compact star. Light rays leaving the surface of a star of radius 2.9 Schwarzschild radii — a neutron star of 1.4 solar masses and about 12 km — and travelling to an observer far off to the right, traced along the exact paths of Schwarzschild's geometry, shown above and below the line of sight. The last ray to escape leaves the surface tangentially from a point 121° round from the point facing the observer, so the bright arc — everything the observer sees — covers 76 per cent of the surface rather than the half that straight rays would show. The dashed circle is the horizon a star of the same mass would have if it were compressed inside it.

The star whose far side can be seen

Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.

 Every steady flow of gas onto a point mass. The Mach number of steady, spherical, isothermal flow round a point mass against distance from it, both on logarithmic axes, distance in units of the sonic radius, where the inflow reaches the speed of sound. Every curve is a contour of Bondi's conserved integral, which ties the Mach number to the distance. The faint curves are flows that never pass the speed of sound, or that exist only close in or only far out. Two curves cross at the saddle (1, 1): the solid one is accretion, starting from rest far away, reaching the speed of sound exactly at the sonic radius and falling supersonically within it; the dashed one is its time-reverse, a wind. Only the crossing flow joins a gas at rest to a hole that swallows it, and it fixes the rate: λ = exp(3/2)/4 = 1.120 in Ṁ = 4πλ(GM)²ρ∞/c³.

The gas a hole catches from far away

A particle drifting past a black hole is captured only if its aim is almost perfect; miss by a little and it swings round and leaves. Gas is different. Streams passing on opposite sides collide, lose their sideways motion as heat, and fall in, so a hole in warm gas draws from the whole sphere inside which its gravity beats the gas's thermal motion. Bondi worked out the flow in 1952: one steady solution passes smoothly from rest to free fall, it crosses the speed of sound at a single radius, and the rate it fixes grows as the square of the hole's mass — fast enough, left alone, to run away in a finite time.

 An orbit, and the circle its velocities make. Left: an orbit of eccentricity 0.6 round a central mass (dot), integrated step by step, with the velocity drawn as an arrow at twelve points equally spaced in time. Right: the same twelve velocities drawn from a single point, the origin of velocity. Their tips lie on a circle — the hodograph — of radius GM/h, whose centre is displaced from the origin by e times that radius, here 0.6: the integrated velocities stay on it to within one part in ten thousand. The fastest velocity, at the closest approach, is 1.60 times GM/h and the slowest, at the furthest point, 0.40 times. The tips are bunched near the slow end, because the body spends most of its time far away.

The velocity that always lies on a circle

A planet on an elliptical orbit speeds up and slows down, its velocity swinging round as it goes. Draw all its velocities from a single point and their tips fall exactly on a circle — not an ellipse, a circle — offset from the point by the orbit's eccentricity. Hamilton noticed this in 1846; it holds for every orbit under an inverse-square force and for no other force; and it turns the ellipse into something that can be proved with a ruler and compass, as Feynman did in a lecture that was lost for thirty years.

 Two ways to spin, against angular momentum. The spin Ω²/πGρ of the Maclaurin spheroids (blue) and the Jacobi ellipsoids (red) against their angular momentum, in units of √(GM³R) with R the radius of a sphere of the same volume. The two families meet at the bifurcation, angular momentum 0.304. Below it only spheroids exist. Above it a body of given angular momentum could be either, and the two differ: the ellipsoid spins more slowly, its spin falling as its angular momentum grows, because it gets longer and its moment of inertia climbs faster than its angular momentum. Every Maclaurin spheroid past the bifurcation is a shape a slightly viscous body will leave.

The spinning fluid that becomes a cigar

A ball of fluid held together by its own gravity flattens when it spins, as the Earth bulges at the equator. Spin it faster and it flattens further, into a lens — but only up to a point, and long before that point something unexpected happens. In 1834 Jacobi found that a fast-spinning fluid can also take a shape that is not symmetric about its axis at all: an ellipsoid with three different axes, stretched along its equator, tumbling end over end. Past a definite spin the stretched shape has less energy than the flattened one, and a body that can shed energy will choose it.

 Every colour at its own angle, from one charge. The wavelength radiated by a charge passing over a grating, as a multiple of the grating's period d, against the angle from the charge's direction, for speeds of 0.3, 0.776, 0.99 of light's, in the first order: λ = d(1/β − cos θ). The shaded band is visible light for a grating of period 1.67 micrometres, the one Smith and Purcell used in 1953. At 0.776c — 300 keV electrons, their speed — visible light comes out between 0° and 30° from the beam, blue-green straight ahead and red at the wider angles; straight ahead the wavelength is 0.29d, 482 nm, and straight back 2.29d. A slow charge at 0.3c radiates only at wavelengths longer than 2.33d. A fast one at 0.99c sends 1.0 per cent of a period straight ahead, the d/2γ² of an undulator.

The rainbow a passing electron leaves on a grating

An electron moving steadily in a straight line through empty space does not radiate, at any speed. Send it skimming over a ruled metal grating, without touching it, and it lights up — every colour at its own angle, blue-green ahead and red further out, from an electron that is neither turning nor slowing. The grating hands the electron's field the one thing it lacked: a wavenumber that lets part of it escape. It is the same mechanism as the blue glow in a reactor pool, with a ruled surface standing in for the slow light in the water.

 A sound that survives only in a hot-electron plasma. The damping of an ion-acoustic wave at kλD = 0.3, as the fraction of its amplitude lost per radian of oscillation, −γ/ω, against the ratio of electron to ion temperature, for hydrogen and argon plasmas, on logarithmic axes: from the kinetic dispersion relation (solid) and from the weak-damping formula (dashed), which is valid only where the damping is small. With equal temperatures the wave loses 0.45 of its amplitude per radian in hydrogen — it is gone within a cycle. At Tₑ/Tᵢ = 10 the loss is 0.031; at 100 it is 0.0126, set by the electrons alone. Argon's floor is lower, 0.0020, because its ions are heavier and its electrons a smaller fraction of a sound wave's speed.

The sound a plasma carries on its electrons' heat

Sound in a gas needs two things: something heavy to be pushed, and a pressure to push it. In an ordinary gas the same molecules supply both. In a plasma they can be split — the ions carry the inertia and the electrons, a thousand times lighter and often a hundred times hotter, carry the pressure — and the result is a sound that travels at a speed set by one particle's temperature and the other's mass. It travels only while the two temperatures stay far apart. Warm the ions to match the electrons and the sound dies within a cycle, with nothing to rub against.

 The shift a member galaxy shows against the centre. The predicted gravitational shift of a cluster member's light relative to the central galaxy's, in km/s, against the member's projected distance from the centre, averaged along the line of sight with the members assumed to follow the mass out to twice R200, for clusters of 1 × 10¹⁴ M☉, 3 × 10¹⁴ M☉, 1 × 10¹⁵ M☉. Every member is blueshifted relative to the centre, because its light climbs out of a shallower part of the well. For the heaviest the shift is −13.2 km/s at half a megaparsec and −21.6 at two; for the lightest, whose members end at 1.9 Mpc, −4.4 and −5.7 at one and a half. Projection blunts the profile: members seen near the centre include many that lie far in front or behind it.

The light that climbs out of a cluster of galaxies

A cluster of galaxies is a well a few megaparsecs wide, and its galaxies sit at different depths in it. Light from the one at the bottom climbs furthest, so every other member should look slightly blue beside it — by ten kilometres a second, against galaxies that rush about at a thousand. No single cluster can show it. Thousands stacked together can, but only once a second effect of exactly the same size, the slowing of the galaxies' own moving clocks, has been taken out — and it cannot be separated by choosing heavier clusters, because both are set by the same depth.

 A peak that keeps growing while the light stays on. The momentum distribution of 20,000 atoms in a one-dimensional dark-state scheme after 100, 1,000 and 10,000 scattering times, as a density per unit of ħk, in bins a hundredth of ħk wide. There is no steady state. The peak at zero grows taller and narrower for as long as the light is on — its height reaches 29 per ħk by the last time — and the share of atoms within a tenth of a photon's momentum climbs from 55 to 87 to 96 per cent. The atoms are not pushed into the peak; they fall into it at random and the light stops touching them.

The atoms the light stops touching

Every photon an atom scatters kicks it by the same small amount in a random direction, so a gas that keeps scattering light cannot be colder than one kick. The way below is not a stronger friction but a weaker touch: arrange for the light to stop scattering off atoms that are nearly at rest. Then the coldest atoms are not cooled, only left alone, for times that have no average — and the gas never settles into a temperature at all, but piles into a peak that narrows for as long as the light stays on.

 The circle of electrons a wave can talk to. Electron velocities along and across the magnetic field, in units of c. Shaded: a horseshoe of 10 keV electrons (speed 0.195c), the shape a beam takes when it is driven down into a strengthening field, with the electrons moving up along the field missing. Curves: the electrons a wave travelling across the field can exchange energy with, for wave frequencies of 0.995, 0.980, 0.960 times the cyclotron frequency. Because a faster electron is heavier and circles more slowly, at Ω/γ, each resonance is a semicircle of radius c√(1 − ω²/Ω²): 0.100, 0.199, 0.280c. A wave at 0.981Ω resonates with the horseshoe's middle.

The radio that only heavier electrons can make

Above every aurora the Earth broadcasts a gigawatt of radio that no one on the ground can hear. It comes out just below the frequency at which electrons circle the field there, and it is a maser — amplified by the electrons themselves. The surprise is what makes the amplifier work. Without relativity it could not: summed over every electron, a wave at the circling frequency is always absorbed. The few per cent by which a fast electron is heavier, and circles more slowly, is not a correction to the mechanism. It is the mechanism.

 Two beams that each stay straight, and wind each other into whirlpools. Phase space of a one-dimensional particle simulation: every dot is an electron, plotted at its position along a periodic box one fastest-growing wavelength long (10.26 v₀/ωₚ) and its velocity, in units of the beam speed. One beam starts at +v₀, the other at −v₀, with a 0.0001 ripple in position. At t = 18/ωₚthe beams are visibly rippled; by 26 the ripple has grown into overturning, and by 40 the two beams have wound round each other into a single vortex, filamented into strands. When growth first stops, at ωₚt = 25.2, the wave's field is 0.56 m v₀ωₚ/e; electrons trapped in it bounce at 0.59 ωₚ, faster than the 0.354 ωₚ at which it was growing, and it traps any electron within ±1.92 v₀ of the wave's speed — zero, by symmetry — so both beams are inside its separatrix; total energy was conserved to 0.31%.

The two beams that unravel each other

A cold beam of electrons, alone, is perfectly stable: disturb it and it rings at its plasma frequency and carries the ringing downstream. Send a second beam through it the other way and the pair is not stable at all. A ripple of the right length grows by a factor of e in less than half a plasma period, takes a third of the beams' drift energy into an electric field, and then winds both beams into a single whirlpool in which their drift is gone. Neither beam can do it. The growth exists only in the pair, and a beam a millionth as dense as the plasma it crosses is enough to start it.

 A disc that makes its own field once it turns fast enough. A self-exciting disc dynamo started from rest, its motor's torque set so that, with no current, the disc would run at 0.8, 1.5, 2.5 times the critical speed R/M. Dashed: the disc's speed, in units of the critical speed. Solid: the current in its coil, in units of √(G/M), starting from a stray seed of a millionth, of either sign. Time is in units of the circuit's own L/R. At 0.8 the disc settles at its free-running speed, below critical, and the seed dies. Above it the disc spins up past the critical speed and nothing happens for a while — the current is growing exponentially from a millionth — until it reaches a tenth, at t ≈ 29 for 1.5 and 11 for 2.5 and becomes large enough to brake the disc. Speed and current then ring and settle at exactly the critical speed and a current of ±√(g − 1): 0.71 and 1.22, of the sign the seed happened to have.

The dynamo that reverses with nothing pushing it

A copper disc spinning in a magnetic field drives a current from its axle to its rim. Wire that current through a coil wound round the disc and the coil supplies the field the disc needs — and above one definite speed, a resistance divided by an inductance, the arrangement makes its own field out of any stray one, with nothing else magnetic anywhere. It cannot choose which way the field points; the stray field chooses. Couple two such discs, each feeding the other's coil, and with the motors' torque held perfectly steady the field flips direction again and again at intervals that follow no pattern. The simplest dynamo that works is also the simplest one that reverses.

 A cloud that grows wider instead of denser. The density of rubidium atoms in a magneto-optical trap against distance from its centre, for 10⁵, 10⁶, 10⁷, 10⁸ atoms, computed from the trap's spring (200 Hz), the atoms' temperature (100 μK) and the repulsion of rescattered light, with σᵣ/σₗ = 1.30. The smallest cloud is the trap's Gaussian, 78 μm in radius, with a peak of 0.91 × 10¹⁰ per cm³. Larger clouds stop getting denser: each fills a sphere at the same 2.5 × 10¹⁰ atoms per cm³ (dashed), out to 0.21, 0.46, 0.99 mm, with a soft edge about a Debye length, 45 μm, wide. More atoms make a bigger ball, not a denser one.

The light that makes a cold cloud repel itself

Load more atoms into a magneto-optical trap and the cloud does not get denser. It gets bigger, at a density that stays fixed whatever the number. The cause is the trap's own light. A photon one atom scatters can be absorbed by another, which is pushed away, and the push falls off as the inverse square of the distance, like the force between two charges. A cloud of laser-cooled atoms is therefore a plasma of neutral particles. Each behaves as if it carried a ten-thousandth of an electron's charge, and the trap fills with them as a harmonic well fills with charge: at one density, out to whatever radius the number requires.

 The power a cavity holds depends on where its mirror is. The power circulating in a cavity of finesse 10,000, a metre long and lit at 1064 nm, as a fraction of its value on resonance, against the detuning of the laser from the cavity's resonance in units of the cavity's half-linewidth (7.5 kHz). Lengthening the cavity lowers its resonance and moves every point of the curve to the right: 38 half-linewidths for each nanometre of mirror travel. On the blue side (laser above resonance, right) that motion loses power, so the radiation force on the mirror falls as the mirror moves out — a restoring spring, steepest at δ/γ = 0.58 (dashed tangent). On the red side the same motion gains power and pushes harder: a spring of the opposite sign.

The mirror held on a spring of light

Light bouncing between two mirrors pushes them apart, and how hard it pushes depends on how long the gap is, because the gap decides how much light the cavity holds. A mirror free to move therefore sits on a spring made of light: with a kilowatt stored in a metre-long cavity it is four-fifths as stiff as a steel wire a millimetre thick. The spring has a flaw no steel spring has. Stored light takes time to rise and fall, so the push always arrives a little late, and a late spring either winds the mirror up or calms it, depending only on which side of the cavity's resonance the laser sits.

 A lens that cannot focus a wave longer than itself. How much a point mass brightens a wave passing behind it — the factor on the wave's power, on a logarithmic scale — against the dimensionless frequency w = 8πGMf/c³, for a source exactly behind the lens and 0.1, 0.3, 1.0 Einstein radii off it, from the exact diffraction integral. Geometric optics puts the magnification at 10.0, 3.4 and 1.34 for the three offsets and at infinity on the axis, whatever the frequency. The wave disagrees: at w = 0.01 every curve is within 1.6 per cent of 1 — a wave a hundred times longer than the lens passes as if it were not there — and on the axis the power factor is πw/(1 − e^(−πw)): 3.28 at w = 1, 31.4 at w = 10, finite at every frequency. The off-axis curves rise to their geometric values only above w ≈ 1 and then oscillate about them, as the two images begin to interfere.

The lens a long wave walks round

Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.

 The strong coupling, run from the proton to the Planck mass. The inverse of the strong interaction's coupling, 1/αₛ, against energy on a logarithmic axis, run at one loop from its measured value at the Z mass (1/8.47, dot), with its slope changing at the charm, bottom and top masses as each quark joins in. On this plot the running is a set of straight lines: the coupling weakens by a fixed amount for each factor of ten in energy, about 2.57 per decade above the top mass. At the Planck mass, 1.22 × 10¹⁹ GeV, 1/αₛ is 52.5 — a coupling of 0.0191. Run down instead, it reaches zero — the coupling runs off to infinity — at 143 MeV, the three-flavour scale of the strong interaction at this order. Nineteen decades of energy are crossed by a line that falls from 52 to 0.

The nineteen decades a slow coupling buys

A proton is ten million million million times lighter than the Planck mass, and that ratio, squared, is why gravity between two protons is 10³⁶ times weaker than the electric force between them. It looks like a number demanding an explanation, and it has one. Almost none of the proton's mass comes from the masses of its quarks; it comes from the strong force, whose strength changes with energy by a logarithm. A coupling of about 1/50 at the Planck mass, weakening by a fixed step for every factor of ten in energy, takes nineteen factors of ten to become strong — and the proton's mass is where it does.

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

The bar a wave of gravity barely touches

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

 The same grains as a crystal and as a liquid. 160 grains repelling one another by a screened Coulomb force, held together by a bowl-shaped confinement as a levitated layer is, and jostled by gas collisions at two temperatures; each picture is the end of a simulation of their motion. Grains with exactly six close neighbours are coloured. Left: strongly coupled — thermal energy about a thousandth of the interaction at the spacing — the grains settle into a hexagonal lattice, 79 per cent of those away from the edge with six neighbours. Right: the same grains with their thermal energy raised seventy-fivefold, the lattice melted, 9 per cent. In the laboratory the grains are a hundred micrometres apart and big enough to film with an ordinary camera, so the melting of a crystal can be watched particle by particle.

The dust a plasma freezes into a crystal

Sprinkle microscopic plastic spheres into a glowing discharge and each one charges itself to tens of thousands of electrons, because the plasma's light, fast electrons reach it far more often than its heavy ions. The charge holds the grains up against gravity in the sheath above the electrode, and it makes their mutual repulsion so strong compared with their jiggling that, at room temperature, they freeze into a hexagonal crystal with a lattice spacing a camera can resolve. A plasma, the hottest and most disordered state of matter, is where the most strongly ordered one was found to form most easily.

 The deeper the line, the closer to gravity. The shift of a solar line in sunlight seen from the Earth, against the line's depth, in the two-stream model with the convective velocities dying away above the photosphere over 150 km and a deeper line taken to form higher (600 km × depth²). Weak lines, formed low among the fastest flows, show 411 m/s — about two-thirds of the gravitational value; the deepest lines, whose cores form in the upper photosphere where the granules' motion has nearly stopped, show 627 m/s and are still climbing towards the 633 m/s that gravity alone gives, drawn as the dashed line. That is the measurement made with integrated sunlight reflected from the Moon and calibrated with a laser frequency comb: plot the shift against line depth, and read the gravitational redshift off where the strongest lines are heading. Its published value, 638 ± 6 m/s, agrees with general relativity.

The redshift the Sun's boiling hides

Light leaving the Sun climbs out of a gravitational well and should arrive redder by 633 metres a second in velocity terms — a prediction Einstein made in 1907, before he had general relativity. It took half a century to confirm, and the reason was not the size of the effect but the Sun's surface, which boils. Bright hot gas rising in granules and fainter cool gas sinking between them blueshift every line by a few hundred metres a second, by an amount that depends on the line and on where on the disc it is seen. The gravitational shift was found by treating that nuisance as a second measurement: deeper lines and lines near the limb both converge on the same number.

 The bend that depends on when the planet was there. The difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals.

The bend a moving planet gives

Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.

 The product a fusion plasma has to beat. The density, temperature and confinement time multiplied together at which a deuterium–tritium plasma heats itself, against temperature. Multiplying by T turns the previous curve into a pressure times a time, which is what a magnet can actually hold, since the field limits the pressure nT and not the density alone. The curve has a clear minimum of 3.0 × 10²¹ keV s/m³ near 14 keV, and that is why magnetic machines are designed to run hot but not hotter. ITER's design point, a density of 10²⁰ per cubic metre at 8.9 keV and 3.7 s, gives 3.3 × 10²¹: a little under this line, because ITER is meant to be driven by external heating with a tenfold gain and not to ignite outright.

The product a fusion plasma has to beat

A star makes less power per litre than a compost heap, and it has burned for four and a half thousand million years. A fusion plasma makes thousands of times more and has to be held for seconds, and it has taken seventy years to get close. Both are obeying one inequality, written down in a classified report in 1955, in which density and time appear only as their product.

 A drift that twist turns into a circle. An ion's guiding centre at 10 keV in a torus the size of ITER, seen in the plane across the bore, with a vertical drift of 609 m/s from the field's curvature and gradient. With no twist it drifts straight up and reaches the wall in about 2.8 ms, ions one way and electrons the other. With the field lines twisted so that they circle the bore once for every three trips round the torus, the same drift is carried round by the motion along the line, up on the outside and back down on the inside, and the integrated path closes into a circle offset by only 1.3 cm. The offset is the drift speed over the rate of circulation. The twist does not stop the drift; it makes the drift average to nothing, which is all a bottle needs.

The twist a bottle of plasma has to have

A straight magnetic bottle leaks out of its ends, so the obvious repair is to bend it round into a ring. A ring of field with no twist in it loses its plasma in about three millionths of a second. Adding a twist repairs that by making a drift average to nothing, and adding too much makes the whole column kink like an overwound rubber band. A tokamak is built in the narrow band between those two failures.

 The one number a burn depends on. The fraction of a deuterium–tritium fuel ball that burns before it flies apart, against its density times its radius. The fuel is held by nothing: a sound wave crosses it in a time R/c, the burn rate goes as the density, and so the fraction burned goes as the product ρR, flattening as the fuel runs out, f = ρR/(ρR + 7 g/cm²). A disassembly-time estimate at 30 keV gives 7.6 g/cm² for the same constant, close to the fitted value. The radius and the density do not appear separately. A third of the fuel burns at ρR = 3 g/cm², and that value does not depend on how large the ball is; what sets the size of an inertial-fusion capsule is what has to be done to reach it.

The fuel held together only by its inertia

On 5 December 2022 a laser put 2.05 megajoules into a gold can the size of a pencil rubber, and a speck of fuel inside it released 3.15. Nothing held that fuel together for the tenth of a nanosecond it burned. It needed to be crushed to a thousand times the density of solid, and it was pushed there by something light cannot do by itself: light's own pressure is hundreds of times too feeble, so a capsule is a rocket that fires inward.

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