Illustrated Physics
physics, drawn
Theme

What stays the same

Conservation laws, and the habit of solving a problem by refusing to look at the middle of it.
The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix. Mechanics

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.

 A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic. Mechanics

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

 A travelling wave, caught at one instant. A sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats. The ghosted curve is the same wave a moment later. Waves

A wave is a shape that travels, and nothing else does

In a wave on water, no water goes anywhere. What moves is the shape — and separating the two motions is the whole of wave physics.

 Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing. Electromagnetism

One number for every point, and nothing at all is lost

The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.

 Pressure, counted as momentum arriving at a wall. Molecules with speeds drawn from the Maxwell–Boltzmann distribution and directions drawn uniformly in the plane of the figure. Those moving toward the wall will bounce off it, reversing the component perpendicular to it and delivering twice that momentum each — and pressure is nothing but the rate at which that momentum arrives. Because the directions here lie in a plane rather than in space, the perpendicular component carries half the energy rather than the third it carries in a real gas. Thermodynamics

Pressure is a rate of arrival, and the gas law falls out of counting

Nothing in a gas is pushing on the walls. Molecules arrive, bounce, and leave, and pressure is the momentum they deliver per second — from which the ideal gas law follows with no thermodynamics in it at all.

 The same 120 N force at 3 different arms. 3 spanners of different lengths, each with the same 120 newton force applied at its end. The torque printed under each is the force times its own moment arm, so it rises with the length while the force does not change. Mechanics

The same push, further out, and why that is a different quantity

A force is not enough to say whether something turns. What decides is where the line of the force passes, and the distance from the pivot to that line is the whole of the story.

 Released together on a 20° slope, 1.1 s later. 3 bodies of different shape, released from the same line on a 20 degree slope and drawn where each has reached after 1.1 seconds. The order is sphere, then disc, then hoop. Each spoke is turned by the distance that body has rolled divided by its radius. Mechanics

The mass, and where it sits, which is what decides the race

Release a hoop and a marble together on a slope and the marble wins, whatever they weigh and whatever their size. Neither mass nor radius survives the arithmetic; only the arrangement does.

 A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion. Electromagnetism

The field that makes the other, and only while it is changing

A magnet sitting next to a coil does nothing at all. Move it and a current flows. The law is not about the field but about its rate of change, and everything electrical since 1831 rests on that distinction.

 Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it. Relativity

Mass is a form of energy, which is not the same as a source of it

The famous equation is usually read as a promise that mass can be turned into energy. It says something stricter and stranger — that a mass is an energy already, sitting there, whether or not anything ever releases it.

 The wavelength shift against scattering angle. How much longer a scattered photon's wavelength is, against the angle it scattered through. The shift runs from nothing at 0 degrees to 4.853 picometres straight back, passing through the electron's Compton wavelength of 2.4263 picometres at 90 degrees. Nothing about the incident light or the target material appears anywhere on this axis. Quantum

A photon with a momentum, and a collision that proves it

X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.

 The energy ladder of a box. The first 6 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0, 36.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 4 to 3 releases 7.000 E₁. Quantum

No two in the same state, and why matter has volume

Nothing in the energy levels of an atom says how many electrons may occupy each one. The answer is one per state, it is not derived from any force, and it is the reason a table holds a cup up.

 Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises. Quantum

Where the quantum picture hands back the old one

A confined particle's probability density oscillates violently at every quantum number, and never stops. What makes the classical answer come back is not that the oscillations die away — it is that nothing can resolve them.

 Force multiplied, distance paid. Two pistons on one body of fluid, of areas in the ratio 16 to 1. A force of 200 N on the small one holds 3.20 kN on the large one, and pushing the small piston 16 cm raises the large one by 10.0 mm. The two products are the same number: nothing is gained except the shape of the bargain. Fluids

Force multiplied, and nothing gained

A push on a small piston becomes a much larger push on a large one, in the ratio of their areas, with no machinery in between except the liquid. What the liquid will not do is give anything away — the distances shrink by the same factor the forces grow by, and the product is untouched.

 Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it. Fluids

The weight of the water that is not there

A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.

 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. Astrophysics

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.

 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. Astrophysics

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. Astrophysics

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.

 A spacetime diagram at β = 0.6. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light. Relativity

The twin who comes back younger

If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.

 Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment. Electromagnetism

The force that does no work

A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

 The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all. Electromagnetism

The term that made light

Ampère's law contradicts itself the moment a current stops being steady, and the contradiction is visible in one picture: two surfaces on the same loop, with a current through one of them and nothing through the other. The term that repairs it turns four equations into a wave, whose speed is two constants measured in a room with the lamps off.

 What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0. Relativity

The quantity nobody argues about

Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.

 A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic. Mechanics

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

 Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy. Waves

The equation that lets a shape travel

Newton's second law applied to a piece of string a millimetre long gives T·y″ = µ·ÿ, and the derivation never once asks what the string is made of. Two things fall out immediately: the speed is √(T/µ) and belongs to the medium, and the general solution holds two arbitrary functions rather than one. The second of them is the reflection, which is why a boundary condition can be met at all.

 What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0. Relativity

The invariant that survives a boost

Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.

 Every path the body can take, at one angular momentum. The angular momentum vector, drawn in the body's own frame on the sphere its length confines it to, for a body whose principal moments are 3.068e-3, 6.817e-3 and 9.750e-3 kg m². Each closed curve is one motion, traced by integrating Euler's equations rather than by solving for the intersection of the sphere with the energy ellipsoid, so a curve closes only if the physics closes it. The low-energy curves circle the greatest-moment axis and the high-energy ones circle the least; both sets are small loops that stay near their axis, which is what stability looks like. Between them is the one curve that is not a loop at all — four arcs, drawn heavier, meeting at the intermediate axis and leaving it again. A body spun about that axis is balanced on the crossing point of paths that go somewhere else, which is the whole of why it does not stay. Mechanics

The axis that will not hold

A book spun about its long edge keeps spinning about it. Spun about the axis through its covers, it keeps spinning about that. Spun about the third axis, it flips end over end, again and again, with nothing touching it. Three numbers decide, and what matters is only their order.

 The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones. Relativity

The mass that is missing

A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

 Which happened first, asked of several observers. Two events on a spacetime diagram: one at the origin and one 3 light-seconds away and 1 second later, so that light leaving the first cannot reach the second. Through the second event runs a family of lines, each one the set of events some observer calls simultaneous with it; an observer moving at a fraction β of the speed of light has such a line of slope β on these axes. Where a line meets the vertical axis is the time that observer assigns to the second event. For a slow observer that meeting point is above the origin and the second event happens later; for a fast one it is below and the second event happens EARLIER. The changeover is at β = 0.3333, which is the time separation divided by the space separation, and it is a legal speed only because the separation is spacelike. So the order of these two events is not a property of the events. What every observer does agree on is that neither could have caused the other, because the two lie outside each other's light cones — drawn here as the diagonals — and that agreement is what causality rests on rather than on any shared notion of before. Relativity

Which came first, and who decides

Two events far apart can happen in either order, depending on who is asked, and both answers are correct. That is not a loophole in causality but the reason causality survives at all — because the pairs whose order is negotiable are exactly the pairs neither of which could have caused the other.

 Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting. Fluids

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

 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. Astrophysics

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.

 Arms in: 4.33× the rate, and 4.33× the energy. A body of 1.2 kg m² carrying two 4 kg masses on arms, spinning freely at 60 revolutions a minute with the arms out at 0.75 m, as the arms are pulled in to 0.12 m. The axis runs right to left, in the direction the arms move. Angular momentum is flat — nothing exerts a torque about the axis, and pulling inward is a force along a radius, which has no moment about the centre. The rate rises as the inverse of the moment of inertia, by a factor of 4.33 here, and the kinetic energy L²/2I rises by exactly the same factor, which is where the usual account stops and where the question starts. The fourth curve is the work done by whoever pulled the arms in, integrated from the force needed to hold each mass on its circle. It lies on the energy curve, to 1.6e-7 joules. Nothing is unaccounted for and nothing is created: the energy is bought, at full price, by pulling against the force that would otherwise fling the arms out. Mechanics

The quantity that survives a change of shape

A skater pulls her arms in and spins four times faster. Angular momentum is conserved, which is the usual explanation, and it accounts for only half of what happened — because the kinetic energy has gone up by the same factor, and something had to pay for it.

 Six observers, six energies, one loss. The kinetic energy of the same collision before and after it, as measured by observers moving at 6 different speeds. No two of them agree about how much energy there was: the totals here range from 5.50 to 25.50 in the same units. Every one of them agrees about how much was lost — the gap between the two curves is 3.413 for all of them, varying by 6.2e-15. Energy is a quantity an observer owns; a change in it is not, and that is why heat, deformation and sound can be counted at all. Mechanics

The energy that depends on the observer

A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.

 What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance. Relativity

The field nobody can transform away

A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter.

 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. Astrophysics

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 most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are. Optics

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

 Four predictions no list of answers can keep. The four measurements a three-particle GHZ state predicts with certainty, and what the best possible list of pre-agreed answers does with them. Each row is a choice of which quantity to measure on each of the three particles; the quantum value is not an average but a certainty, so a single run of any row has a determined outcome. Every list of answers assigns a value to X and to Y at each particle, which is sixty-four lists in all, and the enumeration finds that 32 of them get three of the four right and 32 get one. None gets four, and none can: multiplying the four left-hand sides together gives every X and every Y twice, so the product is +1 for any list whatever, while the product of the four quantum values is −1. The column on the right is one such list, which agrees with the first three rows and is then forced into the opposite of the fourth. That is the whole argument, and it needs no inequality, no average and no repetition: measure the first three rows on three copies, and the fourth is predicted with certainty and comes out the other way. Quantum

The disagreement that one run settles

Bell's argument is a statistical one — a correlation of 2.828 where a pre-agreed list of answers is stuck at 2, dug out of hundreds of coincidences. Add a third particle and the argument stops being about how often. Three measurements predict a fourth with certainty, every list of answers that gets the three right gets the fourth exactly backwards, and one run of the experiment is enough.

 What a collision has to spend, against what it is given. The energy available in a proton–proton collision, against the energy of one beam, on logarithmic axes. Against a stationary target the available energy is √(2mE) and the line has slope one half; head-on it is 2E and the slope is one. Bevatron at 6.2 GeV per beam reaches 3.7 GeV; SPS fixed target at 450 GeV per beam reaches 29.1 GeV; LEP at 104.5 GeV per beam reaches 209.0 GeV; Tevatron at 980 GeV per beam reaches 1960.0 GeV; LHC at 6500 GeV per beam reaches 13000.0 GeV. The gap is the whole architecture of the subject: the LHC's beams give 13000 GeV head-on and would give 110 GeV against a stationary proton, a factor of 118. Reaching the same 13000 GeV in fixed-target mode would need a beam of 90.1 million GeV. What the missing energy has gone into is not lost: it is the kinetic energy of the centre of mass, which every product has to carry away and which no experiment can use. Relativity

The collision that wastes most of the energy

The LHC's two beams carry 6,500 GeV each and 13,000 GeV are available. Fire one of those beams at a stationary block of copper instead and 110 GeV are available — the other 12,890 have gone into the motion of the wreckage and cannot be used for anything. The difference is a square root, and every accelerator built since 1970 is a consequence of it.

 The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales. Relativity

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

 Thrust against the air it is supposed to be pushing on. The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so. Mechanics

The push that needs nothing to push against

A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.

 Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives. Thermodynamics

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

 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. Astrophysics

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.

 Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another. Fluids

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

 Phase portraits of the standard map at 2 couplings. The standard map p → p + K sin θ, θ → θ + p, iterated 220 times from 12 starting points, at couplings of 0.6 and 1.3. Both coordinates run from 0 to 2π. An orbit that lies on a curve spanning the picture from left to right is an invariant circle, and nothing can cross it; an orbit that fills an area is chaotic; an orbit that circulates round a centre is trapped in a resonance island. At K = 0.6, orbits launched on p = 0 get no further than 1.55 in p, so a spanning curve is still there. At K = 1.3, orbits launched on p = 0 get no further than 10.78 in p, so a spanning curve is gone and transport is global. The point of the pair is that the change between them is not a change of character in any single orbit — chaotic orbits and regular ones coexist on both sides — but the loss of the barriers that kept the chaotic ones local. Mechanics

The last curve to go

Chaos does not arrive all at once. Order is destroyed a resonance at a time, and there is a coupling — 0.971635, known to six figures — at which the final barrier separating one part of the phase space from another gives way. Below it a chaotic orbit is still trapped; above it nothing stops it.

 Three vector potentials for one magnetic field. Three different vector potentials, drawn as arrow fields over the same square, every one of which describes the same uniform magnetic field of 1 tesla out of the page. The symmetric gauge circulates about the origin; the two Landau gauges are unidirectional and point in perpendicular directions, and neither has any circulation about anything a reader can see. Underneath each is the field recovered from its own arrows, by adding up the potential round a small square and dividing by the area enclosed — which is what the curl is. The three numbers 1.000000, 1.000000, 1.000000 agree to the last digit computed. The picture is the argument: a vector potential has no physical direction, no physical magnitude and no physical circulation of its own, because a change of gauge alters all three and alters nothing that can be measured. What survives is the curl, and the curl is the field. Electromagnetism

The potentials that are not unique

Nobody solves Maxwell's equations for the fields. They are solved for potentials instead, and the potentials are not unique — three completely different vector potentials describe the same uniform magnetic field, and one of them changes everywhere the instant a charge moves, at any distance, without anything having outrun light.

 Where 4s goes below 3d. The energies of five orbitals of a nucleus of charge 19, solved in a screened Coulomb potential, against how far out the screening extends. Nothing about the ordering is assumed: each level is found by integrating the radial equation outward and bisecting on the energy until the solution has the number of nodes that state is supposed to have, and the same solver returns hydrogen's −1/2n² to 7.3e-12 hartree when the screening is switched off. With no screening the three n = 3 levels would lie on top of one another. With screening they separate, always in the same order — s lowest, then p, then d — because a low angular momentum has no centrifugal barrier keeping it out of the core, so it spends part of its time inside the other electrons where the nuclear charge is unscreened. And at a screening length of 0.41 bohr the 4s level crosses below 3d, which is the fourth row of the periodic table: potassium and calcium put their electrons in 4s before anything goes into 3d, so they are an alkali metal and an alkaline earth rather than the first two transition metals. The model is a caricature — one screening length for every electron, no self-consistency, and no exchange — and it gets the ordering right anyway, which is the argument that the ordering is about penetration and nothing subtler. Quantum

The order the shells fill

In hydrogen every state with the same principal number has the same energy, and 4s and 3d differ by nothing. In every other atom they do not, and 4s is below 3d — which is why potassium is an alkali metal rather than the first transition metal. The difference is a small piece of probability that an s orbital has inside the innermost shell and a d orbital does not.

 The action along a family of paths. On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve. Mechanics

Least action, except that it is not least

Mechanics can be stated twice over. Once as a rule about every instant — force equals mass times acceleration — and once as a rule about the whole path at once, which says that one number computed along it is stationary. The two pick out the same trajectory, and the second name for it is wrong — past a certain duration the real path has more action than its neighbours, not less.

 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. Astrophysics

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.

 A circuit on the sphere of polarisations. The Poincaré sphere, on which every polarisation state is a point: linear states around the equator, circular at the poles, and orthogonal states at opposite ends of a diameter. The triangle is a closed circuit — linear at 0°, then linear at 23°, then left circular, and back — taken along geodesics, which is what a sequence of ideal polarisers does. Bringing a state round it returns it to exactly the state it started in, and multiplied by a phase: 0.3927 radians here, against minus half the enclosed solid angle of -0.7854 steradians. The two agree exactly, and neither calculation knows about the other — the phase is the argument of a product of three overlaps between Jones vectors, and the solid angle is spherical geometry. Nothing about the elements used, their thickness, or the wavelength enters. A phase that depends only on the shape of a path is the signature of a geometric phase, and this is the oldest known example of one. Optics

The phase that is only a shape

Take a beam of polarised light through a sequence of elements that returns it to the polarisation it started with, and it comes back with a phase it did not have before. That phase is not an optical path length — it does not depend on the thickness of anything, or on the wavelength, or on how slowly the sequence was carried out. It is minus half the area the path enclosed on the sphere of polarisation states, and nothing else.

 Every stationary point has a way out. The potential along four lines through the most symmetric point of a symmetric arrangement of 4 equal charges at the corners of a square — the one place a trap might be expected. In the plane of the square the potential rises in every direction; out of the plane it falls. The point is stationary and it is a saddle, which is what Laplace's equation forces: the three second derivatives must sum to zero — computed here as 1.4e-5 against the individual values of order 2e+0 — so if two of them are positive the third must be negative. A particle released here rolls away along the direction that falls. No amount of ingenuity in placing the charges changes this, because the constraint is on the equation rather than on the arrangement, and it is why every real trap for a charged particle either uses time-varying fields, or a magnetic field with a velocity, or a material with a negative response. Electromagnetism

Nothing can be held still by a static field

However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.

 Three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system. Mechanics

The conservation law a symmetry hands over

Energy, momentum and angular momentum are usually presented as three separate empirical facts that happen to hold. They are one fact three times: every continuous symmetry of a system's action supplies a quantity that does not change, the correspondence is exact, and it runs both ways.

 Two ways of destroying a pulse, and their cancellation. The same starting pulse, carried forward three times by three equations that differ only in which terms are present, all shown at t = 0.097. Dotted: where it began. With the dispersive term alone the pulse spreads and sheds an oscillating tail, because its Fourier components travel at different speeds and drift out of step — the profile departs from what it was by 27 per cent of its own height. With the nonlinear term alone the tall part overtakes the shallow part and the front leans forward: the steepest gradient is 9.3 times what it started as and is on its way to vertical. With both, neither happens — the profile has moved to the right and is otherwise identical to what it was, to 2.7e-12 of its own height. The balanced run is a pseudo-spectral integration and the other two are exact, and the integration conserves the two quantities the equation conserves — mass to 2.2e-16 and the squared integral to 4.7e-15 — which is the check that the answer belongs to the equation rather than to the integrator. What the picture cannot show is why the cancellation is stable: a pulse of the wrong height for its width does not persist in the wrong shape, it sheds the excess as a dispersive tail and settles on the shape that works, which is why these objects turn up in canals and optical fibres rather than only in equations. Waves

The pulse two failures keep alive

Dispersion spreads a pulse until it is nothing. Nonlinearity steepens it until it breaks. Each on its own destroys a disturbance, and there is exactly one height for each width at which the two cancel completely — leaving a shape that travels for ever and survives being run into by another one.

 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. Astrophysics

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.

 The hole that holds exactly one particle. How likely a second particle is to be found a distance away from a first, relative to a gas with no correlation at all, for three cases that differ in nothing but the symmetry of the state under swapping the two labels. There is no interaction anywhere in this calculation: no Coulomb term, no potential, no force. Distinguishable particles give a flat line, which is what no interaction ought to give. Identical fermions dig a hole that reaches exactly zero at zero separation and fills back in over about a wavelength. Identical bosons do the opposite and pile up to twice the density. Integrating the fermion hole gives 0.9992 particles missing from around each one — exactly one, and the sum rule holds at any density, because raising the density narrows the hole in exact proportion. That is what makes the effect worth a name of its own. It is often called an exchange force and it is not a force: nothing carries momentum between the particles, and no term in the energy is proportional to a distance. It is a statement about which states exist. What follows from it is most of chemistry — the reason two atoms with filled shells repel, the reason a metal's electrons cost so much less Coulomb energy than a random arrangement would, and the reason matter takes up room. Quantum

The force with no force in it

Two identical fermions keep apart and two identical bosons crowd together, and neither is being pushed. The Hamiltonian contains no interaction at all: what produces the hole and the pile is which many-particle states exist, and the hole it digs around each electron holds exactly one particle at any density whatever.

 Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields. Electromagnetism

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

 The fuel a starship needs, which is most of the universe. The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again. Relativity

The fuel a starship needs

Tsiolkovsky's logarithm survives relativity with one substitution: what adds is the rapidity rather than the velocity. The result is that a photon rocket reaches half light speed on a mass ratio of the square root of three, and a chemical one reaches a tenth of it on a mass ratio with three thousand digits.

 Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn. Fluids

The angles a film has no choice about

A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

 Past the critical angle, all that is left of a reflection is its phase. The phase each polarisation acquires on total internal reflection at an index ratio of 1.518, against angle of incidence, together with the difference between them. Below the critical angle of 41.19° there is a transmitted beam and the reflection coefficients are real; above it the transmitted wavenumber is imaginary, the coefficients have modulus exactly one — every photon comes back — and the only thing that distinguishes one angle from another is the phase. The two polarisations acquire different phases, and their difference peaks at 46.533° at an incidence of 51.05°, which the closed form puts at the same place. That difference is a retardation: a wave plate made out of an angle, with no birefringent material anywhere in it, and — because the expression contains only the index ratio — one that barely changes with colour. Optics

The retarder with no crystal in it

A wave plate turns linear polarisation into circular by making one component travel a little further than the other, which requires a birefringent crystal cut to a thickness and works properly at one wavelength. Total internal reflection does the same job with a phase that comes from the geometry instead — and because a refractive index barely changes across the visible where a wavelength changes by a factor of two, the same block of ordinary glass is a quarter-wave plate for every colour at once.

 Three copiers, and what each of them costs. How well three copying machines reproduce a qubit, against the state's angle from the pole. The first is a linear machine built to copy the two pole states perfectly; linearity then fixes what it does everywhere else, and on an equal superposition it produces an entangled pair whose overlap with the two copies wanted is exactly 0.500. That is the no-cloning theorem written as a number rather than as an argument: no adjustment is available, because the machine's behaviour on superpositions was decided the moment its behaviour on the basis was. The second measures in a fixed basis and prepares two copies of what it found, which is perfect at the poles and averages 0.6667 over the sphere — two thirds, exactly. The third is the best machine there is, and it manages 0.8333 on every state alike: five sixths, and not one. Quantum

The state that cannot be copied

Every measurement in this collection disturbs what it measures, and the obvious way round that is to make a spare first. It cannot be done, and the reason is not a practical difficulty or a limit on how good an apparatus can be: a copier is a linear machine, so fixing what it does to two states fixes what it does to their superpositions, and what it then does is not a copy.

 The size of the quantum says what is carrying the current. What the flux quantum would be for each candidate carrier charge, in units of 10⁻¹⁵ webers, against the measured value drawn as a line. A single electron would give 4.1357, a pair 2.0678, a triple 1.3786. The measurement is 2.0678, which picks the pair and excludes the others by a factor of two — not by a few per cent, so no question of experimental accuracy arises. The whole of the argument is that the condensate's wavefunction must come back to itself round the ring, which makes the enclosed flux a multiple of h over the carrier's charge; measuring the multiple therefore measures the charge, without any charge ever being measured. That is how the pairing was established in 1961, four years after it was proposed and by an experiment that looks nothing like a measurement of a charge. Electromagnetism

The two in the flux quantum

A superconducting ring cannot hold whatever flux is applied to it. It holds a whole number of quanta and drives a current to make up the difference, and the size of that quantum is Planck's constant divided by twice the electron's charge. The factor of two was measured in 1961, four years after somebody predicted that the carriers are pairs — by an experiment in which no charge is measured at all.

 Two quantities that are never computed and never change. A two-dimensional field integrated for 110 steps using only the two equations that contain a time derivative — Faraday's and Ampère's. The two that do not, Gauss's law for the electric field and the statement that there are no magnetic charges, are never imposed and never checked during the run. Their residuals are plotted: the divergence of B stays below 2.4e-16 of the field's own size and the divergence of E below 2.4e-16, over the whole run, while the field itself moves and changes by a factor of 8.59. That is not a numerical coincidence. Taking the divergence of Faraday's law gives the divergence of a curl, which vanishes identically, so ∂(∇·B)/∂t is zero whatever the fields are doing; the same manoeuvre on Ampère's law gives ∂(∇·D)/∂t = −∇·J. So the two constraints are initial conditions, propagated for ever by the two that are laws of motion, and Maxwell's four equations are two dynamical ones and two statements about how the field was set up. Electromagnetism

The two equations that are not laws of motion

Maxwell's equations are usually presented as four laws of equal standing. Two of them contain no time derivative at all, which means they cannot be evolution equations: they are conditions on the field at one instant. What makes them consistent with the other two is that the other two preserve them exactly — and one of the two preservations holds only because charge is conserved.

 Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up. Thermodynamics

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

 Two experiments, two computations, one coefficient. The Seebeck coefficient of a resonant conductor against where its resonance sits relative to the chemical potential, together with the Peltier coefficient divided by the temperature. The first is obtained by applying a temperature difference and finding the voltage that stops the current; the second by applying a voltage at uniform temperature and taking the ratio of the heat flow to the current. Different driving, different measurement, different integral — and the two curves agree to 3.8e-7 of the sweep's own scale across the whole of where they pass through zero and change sign. That equality is Kelvin's relation Π = ST, guessed in 1854 from an argument its author knew was not sound and proved by Onsager in 1931 from microscopic reversibility. It is not a property of this conductor; it holds for every one. Thermodynamics

The second experiment that cannot disagree

Heat one end of a wire and a voltage appears across it. Pass a current through the same wire at uniform temperature and it carries heat. Those are two different experiments with two different apparatus, and the coefficient in front is the same number in both — not approximately, and not for some materials. The reason is that the equations of motion underneath look the same run backwards.

 The same source, seen coming and seen going. The observed flux of a moving source, relative to the same source at rest, against the angle between its motion and the line of sight, on a logarithmic scale, at β = 0.5, β = 0.9, β = 0.99. The curves span 8.1e+1, 1.3e+5, 1.6e+9 between the approaching and receding directions, which is ((1+β)/(1−β))⁴ exactly. The fourth power comes from the one quantity every observer agrees about: the specific intensity divided by the cube of the frequency. Three powers of the Doppler factor come from that invariance and the fourth from integrating over frequency. Two consequences are worth reading off. A source seen side-on is fainter than the same source at rest, by γ⁴ — a factor of 2.5e+3 at the fastest speed drawn. And half the light arrives inside a cone of 60.0°, 25.8°, 8.1°, which for a fast source is one over γ. Relativity

The brightness that is not the same for everyone

A moving source is not merely shifted in colour. Its light is concentrated forwards, so the same lamp is enormously brighter seen coming than seen going — by the fourth power of one number, which for a jet at ninety-nine per cent of light speed is a factor of a thousand million. The reason is that only one combination of intensity and frequency is the same for every observer, and everything else follows from it.

 The reaction that reaches zero, and where the bead lets go. The force the sphere pushes back with, in units of the bead's weight, against the angle from the top. It starts at 1.000 and falls, because the speed the bead has gained needs more centripetal force than gravity's component along the radius can supply. At 48.19° it reaches zero, and past that the surface would have to pull inward to keep the bead on it — which a surface cannot do. So the bead leaves there, and the departure angle is a statement about the sign of a constraint force rather than about a speed or a height. The multiplier is what carries that sign: solve the motion in the angle alone and the reaction is absent from every equation, so nothing in the solution knows that the constraint has stopped holding, and the bead is drawn happily circling a sphere it has already left. Mechanics

The force a coordinate cannot see

Writing a pendulum in terms of its angle is the first good move anybody learns, and it deletes the tension from every equation that follows. The string still breaks. Recovering the force that the clever choice of coordinate threw away turns out to be a computation with a sign in it, and the sign is where the bead leaves the sphere.

 Four modes of a string that is heavier in one place. The first four modes of a string whose mass per unit length rises to 5 times the light end's over a smooth bump centred 62 per cent of the way along, drawn beneath them. Nothing is symmetric any more: the shapes bunch up over the heavy region, where the local wavelength is shorter, and the amplitudes there are smaller. The frequencies are 1.70, 4.01, 6.10, 8.12, which are in the ratios 1.000, 2.361, 3.590, 4.779 rather than 1, 2, 3, 4 — this string has no harmonics and would sound like a bell rather than a violin. What has not changed is the one thing an ordering needs: the interior zeros, marked, run 0, 1, 2, 3 exactly as they do on a uniform string. That is Sturm's theorem, and the nodes here are counted on the computed shapes rather than assumed. Waves

The count that cannot be cheated

A string with a lump in it has no harmonics, no symmetry and no obvious order to its modes. It has one thing left: the nth mode crosses the axis exactly n−1 times, whatever the string is made of. Ordering by frequency and ordering by node count turn out to be the same operation, and in two dimensions the equality quietly becomes an inequality.

 Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them. Waves

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

 Where the refracted ray comes from, drawn with a compass. Wavevectors in units of the vacuum wavenumber, for light arriving at 30° from a medium of index 1.5 at a medium of index 1. Every direction available in the first medium lies on the circle of radius 1.5 and every direction available in the second on the circle of radius 1; the horizontal axis lies in the interface. The boundary cannot change the component along itself, because the two sides have to agree on the phase at every point of the interface, so the refracted wave is fixed by the vertical line at 0.7500 — and where that line cuts the smaller circle is the refracted direction, 48.59° from the normal. Nothing about least time or about wavefronts enters, and the ratio of sines is what the construction reads when the two radii are written as indices. The normal component is not conserved and is not meant to be: it goes from 1.2990 to 0.6614, which is the whole of the difference between the two rays. Optics

The law that only asks about one component

A boundary between two media cannot change the part of a wave that runs along it, because both sides have to agree on the phase at every point of the surface at every instant. That single restriction produces the refracted ray, the critical angle, the evanescent field and every order of a diffraction grating, out of one drawing made with a compass.

 Three curves that can only meet at a point. Water's phase diagram within a kelvin of its triple point, with each boundary drawn at the slope Clausius and Clapeyron give it from measured latent heats and densities: 44.4 Pa/K for boiling, 50.3 for sublimation, and -135 bar per kelvin for melting — negative, and so steep that it is drawn vertical here, because ice is the less dense phase and pressure therefore melts it. Two things follow that no amount of measurement could adjust. The three slopes are not independent: the sublimation and vaporisation curves differ in slope by 5.92 Pa/K, which is exactly what the fusion latent heat predicts, because going solid → gas directly and solid → liquid → gas must cost the same enthalpy. And the sublimation curve is the steeper of the two, so the two cross rather than touch — which is why the triple point is a crossing and not a tangency, and why ice sublimes below it and melts above it. Thermodynamics

Why the triple point is a point

Three phases of one substance coexist at one temperature and one pressure and nowhere else, and the reason is arithmetic rather than chemistry: count the numbers that describe the state, count the conditions equilibrium imposes, and subtract. The same subtraction says four phases of one substance are impossible, and it says so without knowing what the substance is.

 One factor, defined by an experiment rather than by a transformation. A observer A stays at x = 0 and flashes a light every 1 second by their own clock. B recedes at 0.6c. The flashes are the diagonal lines; where each meets B's worldline is where B receives it. B's clock reads a longer gap between arrivals than A's read between departures, by the factor k = 2.0000, and it is the same factor between every consecutive pair — measured here off the drawn meetings rather than assumed. That single number is the whole apparatus. Nobody has written down a coordinate transformation, chosen a convention for distant simultaneity, or drawn a tilted axis; the only thing used is that light travels on the diagonals and that neither observer is special, so B's flashes reach A stretched by the same k. From it: γ = (k + 1/k)/2 = 1.2500, and β = (k² − 1)/(k² + 1) = 0.6000. Relativity

Everything from an exchange of pulses

Send a flash every second and ask how often the far observer receives them. That one measured ratio generates time dilation, the composition of velocities and the twin result, with no coordinate transformation written down anywhere and no convention chosen about what "at the same time" means far away.

 The turnaround, made gentler and gentler, and the difference that does not move. A round trip to a star 4 light-years away at 0.6c, with the turnaround done at nine different accelerations from a tenth of a gravity to a thousand. The upper curve is the age difference between the twins and the lower one is how much of that difference the turnaround itself contributes. At 0.1 g the turn accounts for 51 per cent of it; at 1000 g it accounts for 0.00 per cent, and it keeps falling. The total does not follow it down: it tends to 2.67 years, which is what the instantaneous-turnaround cartoon gives. So the acceleration is not what makes the twins differ. It is what makes one twin's path the bent one, and a bent path through spacetime is shorter for the same reason a bent path on a map is longer — but the amount is in the legs, not in the corner, and the corner's contribution can be made as small as anyone likes without the difference going away. Relativity

The clock that does not feel the turn

Proper time is the integral of dt over gamma, which presumes that a clock's rate depends on its speed and on nothing else — not on its acceleration, not on how long it has been accelerating. That is an assumption about clocks rather than a theorem about spacetime, and the twin result is empty without it.

 The same top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° from the vertical. Released from rest the axis falls, and the fall is what generates the sideways motion: the path comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Launched at exactly the steady rate the path is a circle and the nutation is absent. Launched slower it waves; launched faster it loops: at 0× the steady rate the path comes to cusps, at 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin. Mechanics

The top that nods before it settles

A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping. Mechanics

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

 Two principles, two classes of path, two things left free. On the left, five curves from the same launch point to the same target: the true trajectory of a particle of energy 0.7 in a uniform field, and four deformations of it that share both ends. On the right, four quantities computed along that family and plotted as departures from their values on the true path. Maupertuis' abbreviated action ∫p·ds, computed at fixed energy, is stationary — flat at the centre. Hamilton's action ∫L dt, computed at fixed duration, is stationary too. The other two are not: the time a fixed-energy path takes changes at first order in the deformation, and so does the energy a fixed-duration path carries. That is the whole difference between the two principles. Each holds one of those quantities fixed and lets the other vary, and neither can hold both. Mechanics

The principle that fixes the energy instead of the clock

There are two principles of least action, they compare different sets of paths, and they are not the same statement. One holds the duration fixed and lets the energy vary; the other holds the energy fixed and lets the duration vary — and written that way, mechanics turns into optics with a refractive index.

 The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which. Electromagnetism

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

 A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way. Electromagnetism

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

 A cone of 14.3°, from two indices and nothing else. On the left, the acceptance cone of a step-index fibre with a core index of 1.4677 and a cladding of 1.4624. A ray entering steeper than the cone reaches the wall inside the critical angle and is refracted out at the first bounce; one inside it is trapped. The sine of the half-angle is √(n₁² − n₂²) = 0.125, which is 7.2° in air. On the right, that number against the fractional index difference between core and cladding. Nothing about the core's diameter appears: a fibre a hundred times thicker accepts exactly the same cone, and takes a hundred times the area's worth of light through it. Optics

The cone a fibre will accept

A fibre takes light from a cone whose half-angle depends on two refractive indices and nothing else — not on how thick it is, not on how long, not on what is shining into it. That single number, squared and multiplied by the core's area, is all the light it will ever carry.

 Why only a sideways scattered wave takes anything away. The transmitted amplitude behind a thin scatterer, drawn as a phasor: the incident wave of unit length along the axis, plus a forward-scattered wave of length 0.12 at 0°, 60°, 90°, 150°. What a detector reads is the square of the total length. A scattered wave along the incident one lengthens or shortens the sum in proportion to itself; one at right angles changes the length only in second order, because a small perpendicular addition to a long vector barely alters its length. So a scatterer that removes energy from the beam at first order must scatter forward with a component perpendicular to the incident wave, and the size of that component is the whole extinction — which is the optical theorem. Optics

Everything a scatterer removes, from one direction

How much light a particle takes out of a beam — by scattering it anywhere at all, and by absorbing it — is fixed entirely by what it does in the forward direction, where its scattered wave cannot be told apart from the incident one. The mechanism is interference, and it also gives the refractive index.

 The cross term, and the fact that it averages to nothing. The intensity of two waves of amplitude 1 and 0.7 added together, against the phase difference between them, in turns. A detector reads the square of the summed amplitude, which is the sum of the two intensities plus a cross term that swings between plus and minus twice the product. At no phase difference the reading is 2.89 and at half a turn it is 0.09; the flat line is what the two would give with no interference, 1.49, and it is exactly the average of the curve over a whole turn. Interference redistributes and does not create — which answers the question of where the energy goes at a dark fringe by saying that it never left. Waves

What adding does to the energy

Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.

 Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them. Thermodynamics

The gradient that drives the other thing

A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

 A bundle that swings and never spreads. A Gaussian of the ground state's own width, released at rest from x = 3 in a harmonic well and propagated on a grid by split-step Fourier, drawn at 0 of a period, 0.25 of a period, 0.5 of a period. The packet slides from side to side and its shape does not change: over 2.2 full periods the width moves by 6.6e-5 per cent, and its centre tracks x₀cos t to 8.6e-6. Every other initial width breathes. This one is the displaced ground state, and it is the closest a quantum state comes to being a classical oscillator — a definite thing at a definite place, moving on the classical trajectory, staying the size it was. Quantum

The state that swings like a pendulum

Most quantum states of an oscillator look nothing like a swinging weight. One family does: it follows the classical trajectory exactly, never spreads, and sits at the uncertainty minimum for ever — and it is the state a laser and a driven circuit actually produce.

 Two pairs, and only one of them may cheat. The CHSH value a party shares with a second, against the value the same party shares with a third. Every quantum state lies inside a quarter circle whose radius is Tsirelson's bound, 2.8284, because the sum of the two squared values cannot exceed eight. The classical limit is 2 on each axis, and the square that would hold both violations sticks out of the circle everywhere except at its corner: the best both can manage at once is exactly 1.999998, which is the classical value and no violation at all. So a party maximally entangled with one other is correlated with everybody else exactly as a classical object would be. Nothing about the measurement or the apparatus was assumed; this follows from the state alone. Quantum

What two have they cannot give a third

Entanglement will not be shared. A pair that violates a Bell inequality is correlated with everything else exactly as a classical object would be, and the trade is exact enough to be drawn: two CHSH values must fit inside a circle of radius 2√2.

 Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more. Quantum

Why an atom is the size it is

A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

 Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500. Relativity

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

 The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting. Relativity

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

 Two things that change and one that does not. How a boost treats a piece of charged matter: the charge density rises by the Lorentz factor because the same charges occupy a contracted length, and the length falls by the same factor. At β = 0.6 the density is 1.250 times what it was and the length is 0.800 times, and their product is one to fourteen decimals across the whole range drawn. So the total charge is the same number in every frame, and it is the only quantity in the transformation that is. Charge density is the time component of a four-vector and transforms like an energy; charge itself is a scalar, and nothing about the observer changes it. Relativity

The one quantity a boost leaves alone

Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.

 Every detour costs time. 4 routes between the same two events, 10 seconds apart in the frame drawn, each swinging out and back 1 time on the way. The proper time each carries is the integral of the square root of one minus the speed squared, computed by Simpson's rule along each curve: the straight route, 10.0000 s; wandering 1 light-seconds, 9.7485 s; wandering 2 light-seconds, 8.9245 s; wandering 3 light-seconds, 7.0935 s. The straight one carries the most, and every other one carries less — checked, on each drawn route. That is the opposite of what a length behaves like on paper, where the straight line is the shortest, and the whole difference is the minus sign in front of the space term. Relativity

The longest way round is the shortest clock

Of all the routes between two events, the one with no acceleration in it carries the most time on its own clock. That is the opposite of the Euclidean statement about straight lines, it comes entirely from one minus sign, and in a gravitational field it is why a thrown ball follows the path it does.

 The wedge that proves it. A wedge of water 4 mm on its vertical side, at a depth of 3 m, with the pressure on each of its three faces as an unknown. The two force balances decide them. Horizontally, the sloping face's push has a component that must exactly cancel the vertical face's, and since the sloping face is longer by exactly the factor its slope reduces the component by, the two pressures are equal — the geometry cancels, at every angle, for every size. Vertically the same cancellation happens except for the wedge's own weight, which needs the bottom face to carry 0.0667 per cent more. That excess falls in proportion to the size of the wedge, so at a point it is nothing and the three pressures are one number. Pressure being the same in every direction is the conclusion of that argument, not an assumption in it. Fluids

The push that has no direction

That the pressure at a point in a still fluid is the same whichever way the surface faces is not a definition. It is a theorem, and its proof is an argument about how two kinds of force scale with size — which is also the exact statement of when it stops being true.

 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. Astrophysics

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.

 What comes back after one turn, and what needs two. A spin-½ pointing along z and rotated about the x axis through 720°, with the rotation integrated step by step rather than evaluated from a formula. The direction of the spin — the quantity a Stern–Gerlach magnet, a compass or any other instrument reports — is back where it started after 360°, exactly as the orientation of any other object would be. The state is not: its overlap with the state it began in has reached −1 there, and returns to +1 only after 720°. At 360° the overlap is -1.000 and ⟨σz⟩ is 1.000; At 720° the overlap is 1.000 and ⟨σz⟩ is 1.000. Both curves come off one integration of dψ/dθ = −(i/2)σx ψ whose norm is checked before anything is drawn, so the factor of two between their rates is a property of the propagation rather than of two separate formulae that were chosen to differ. Quantum

The turn that has to be made twice

Turn a spin-½ through a full circle and it does not come back. The direction it points in does, and every measurement on it does, but the state itself has changed sign — and a second full turn is needed before anything is where it started. The sign is invisible on one spin and measurable the moment a superposition has one branch turned and the other not.

 What the plane between them carries. The stress transmitted across the plane halfway between two charges of 10 nC held 1 cm apart, against distance from the axis in units of the half-separation. For two like charges the field on that plane lies entirely in it — checked here rather than assumed — so the plane sees only the pressure across the lines, the stress is negative everywhere, and the two halves are pushed apart. For opposite charges the field on the plane is entirely perpendicular to it, the plane sees only the tension along the lines, the stress is positive, and the halves are pulled together. Faraday's two words for a field line, tension along it and pressure across it, are exactly these two curves; the whole content of the tensor is that they are the same quantity, ε₀E²/2, wearing two signs. The force is what is left after integrating either curve over the plane, and both integrals come to the same magnitude — the Coulomb force — which is the next figure. Electromagnetism

The force read off a surface that touches nothing

Draw any closed surface through empty space, measure the field on it, and the sum of one expression over that surface is the total force on everything inside — whatever the contents are, and without knowing anything about them. The expression is Maxwell's stress tensor, and it turns Faraday's guess about tension along a field line into an exact statement.

 Three charges, three orbits, one drift. Three particles released at rest in crossed fields — 1000 V/m across 0.1 T — with their paths integrated by a scheme that rotates the velocity rather than adding to it, so the magnetic part changes no speeds. The three loops have wildly different sizes and periods: the the electron turns at 2799.25 MHz, the proton turns at 1.52 MHz, the α particle turns at 0.76 MHz. Their guiding centres all creep along the same line at the same rate, measured here from the orbits at -1.000e+4, -1.000e+4 and -1.000e+4 m/s against −E/B = -1.000e+4 m/s, a spread of -0.00 per cent. Neither the charge nor the mass nor the sign appears in the answer. A plasma in crossed fields therefore moves bodily and carries no current from this drift at all, which is the opposite of what an intuition built on ions being heavier than electrons expects. Electromagnetism

The drift that does not care what the charge is

A charge in a uniform magnetic field goes round in a circle and arrives nowhere. Add anything at all — an electric field, gravity, a gradient in the magnetic field itself — and the circle's centre creeps sideways, at right angles to both. One of those drifts is the same for every particle regardless of charge, sign or mass; the others are not, and the difference decides what a plasma does.

 The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off. Mechanics

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

 The energy that goes and comes back. A chain of 32 masses with springs a few per cent nonlinear, started with all its energy in its longest mode, with the energy of the first five modes followed against time in units of that mode's own period. The first mode gives up most of what it has — down to 9 per cent by 104 periods — and the energy appears in the second, third and fourth. Then it comes back: at 154 periods the first mode holds 98 per cent of the total again. Equipartition would put an equal share in every one of the thirty-two modes and leave it there. What happens instead is that a handful of modes trade with each other and return almost exactly to where they began, and go on doing so. The total energy is checked against its starting value throughout and holds to 9.2e-5, so nothing here is the integrator losing track of what it was given. Thermodynamics

The energy that refuses to be shared

Put all the energy of a chain of masses into its longest mode and add a few per cent of nonlinearity, and equipartition says it should spread out among all thirty-two modes and stay there. It does not. It leaks into three or four neighbours and then comes back — almost exactly — and goes on doing so, and the calculation that found this was expected to be a demonstration that it would not happen.

 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. Astrophysics

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.

 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. Astrophysics

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.

 The one number the boost leaves alone. The mass of a lambda into a proton and a pion, reconstructed from the two products' laboratory energies and momenta alone, against how fast the parent was moving — for five different rest-frame emission angles. Every curve is the same horizontal line. The lab energies vary by more than a factor of ten across this range and the angles between the products vary from almost 180° to a few degrees; the combination E² − p² of the pair does not vary at all, to 3.2e-15, which the figure requires before drawing anything. The light curves are the energies of the two products, on the same axis and to a different scale, drawn to show how much is moving while the invariant does not. This is the whole method of particle physics. A parent that lives for 10⁻²³ seconds is never detected; what is detected is two tracks, and their invariant mass is computed and histogrammed over millions of events. A parent that exists shows up as a peak at its own mass, at the same place whatever the beam energy, which is what makes the peak believable. Relativity

The cone a decay cannot leave

A particle at rest breaks into two and they go opposite ways. Set the parent moving and the whole pattern folds forward — into a cone with a hard edge, beyond which nothing is emitted at any rest-frame angle at all. The energy spectrum that comes out is exactly rectangular, and the one number the boost leaves alone is how the parent is identified at all.

 One boost, three constants, three pictures. The axes of a frame moving at 0.5 in units where the constant is one, drawn for the three signs the constant can have. The faint cross is the original frame's axes; the two heavy lines are the moving frame's, obtained by boosting them rather than by tilting them by hand. With a positive constant the two axes close in on one another symmetrically, and the line they are closing on is the invariant speed. With a zero constant only the time axis tilts and the space axis stays where it was, which is absolute simultaneity — every frame agrees which events are at the same time. With a negative constant the pair rotates rigidly, like a pair of axes turned in a plane. Nothing about light has been used to draw any of them. The three are the whole of what homogeneity, isotropy, the group property and the relativity principle permit, and choosing between them is a measurement rather than a postulate. Relativity

The transformation that never mentions light

Assume space and time are homogeneous, that space is isotropic, that two changes of frame compose into a third, and that the relativity principle holds. Those four leave exactly one free constant — and three possible worlds, one of them Galileo's and one of them Einstein's. Light appears nowhere in the derivation; it enters only when the constant has to be measured.

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

The deflection that closes on itself

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

 Two intersections, and the rule that picks one. The phase-matching construction for light arriving at 40° from a medium of index 1 into one of index -1. The circles are each medium's own relation between wavevector and frequency; the vertical line is the tangential wavenumber, which the boundary conserves. The line crosses the second circle twice, and the construction alone does not say which point is the answer — the rule that does is that energy must travel away from the interface. For a positive index the energy runs along the wavevector and the upper point is taken. For a negative one the energy runs against it, so the lower point is taken, the wavevector points back toward the boundary, and the ray leaves at -40.0° — on the same side of the normal as it arrived. Nothing in the drawing has changed except which intersection is circled. Optics

The ray on the wrong side of the normal

The phase-matching construction draws a circle and a line, and the line crosses the circle twice. Every earlier construction silently took the upper intersection. Which one is physical is decided by where the energy goes rather than by where the wavevector points, and in a medium whose group velocity opposes its phase velocity the answer is the other one — so the refracted ray leaves on the same side of the normal it arrived on, a flat slab focuses, and a lens can beat the diffraction limit until loss stops it.

 Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore. Quantum

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

 Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change. Relativity

Charge and current are one thing

The rung below asks what a boost leaves alone and answers charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

 Six numbers, one object. The electromagnetic field tensor written out as the four-by-four array it is, for a field with E = (0.4, 1.2, 0) and cB = (0, 0, 0.7), and again after a boost of rapidity 0.9 along the first axis. The array is antisymmetric — checked entry by entry — so of its sixteen slots only six are independent, and those six are the three components of E and the three of cB. The boost does not act on E and B separately; it acts on the array, mixing the top row into the lower block, which is what 'the electric field in one frame is partly magnetic in another' means written down. Its two scalars, E·B = 0.000 and E² − c²B² = 1.110, are unchanged, and they are the only two an antisymmetric rank-two tensor has. Relativity

Six numbers, one object

Three components of E and three of B mix into each other under a boost and never into anything else. Six numbers that transform among themselves are the independent entries of a four-by-four antisymmetric array, and writing them that way is not notation — it turns Maxwell's four equations into two, makes the two invariants the only two there could be, and shows that "electric" is a choice of axes rather than a kind of field.

 The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers. Thermodynamics

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

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

The cone light has to find to get out

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

 A quasi-probability that goes negative. The Wigner function of the oscillator's state at a quarter of its revival time, with 4 quanta on average: two copies of the packet, at x = ±2.83, in equal superposition. It is computed from the wavefunction by the Wigner transform on a lattice, and it integrates to one. The two copies are the two positive blobs. Between them lies a pattern of stripes with no classical counterpart, running from 0.289 down to −0.289, where the shaded warm regions and their outlines mark negative values. A negative probability is not a probability, so this distribution cannot describe a cloud of classical particles. The stripes are also taller than the blobs, so the interference carries more structure than the copies themselves: the highest stripe reaches 0.289 and the centre of a blob 0.159. Quantum

The probability that goes below zero

Classical mechanics describes an uncertain state as a cloud of points in the plane of position and momentum. Quantum mechanics has an exact counterpart, the Wigner function, whose shadows are the true position and momentum distributions — and which goes negative. It goes negative for a single photon, for every superposition of two packets, and for every pure state that is not a Gaussian. Where it is negative no classical cloud can imitate the state, and losing energy to the surroundings erases the negative regions first.

 A map that keeps every cone and bends every worldline. Left: a grid of inertial worldlines, drawn solid, lines of simultaneity, faint, and light lines at 45°, dashed, in one space dimension. Right: the same grid after a map that stretches the light-cone coordinates u = t − x and v = t + x by two different increasing functions, u + 0.45·tanh(1.5u) and sinh(0.45v)/0.45. Light lines go to light lines, still at 45° to within a part in a billion, and the causal order of every one of 4000 sampled pairs of events is unchanged: whatever could influence what still can, and nothing new can. But the straight worldlines are bent — the one through x = 1 by 0.19 across the window — and the lines of simultaneity are no longer straight. In one space dimension the light cones cannot tell this picture from the inertial one. Relativity

What the light cones alone can decide

Keep nothing of spacetime but its light cones — which events could influence which — and ask how much geometry survives. With one dimension of space, almost none: any pair of increasing stretches of the two families of light lines preserves every cone and bends every straight worldline. With two or more, almost all of it: the only maps that keep every cone are Lorentz transformations, shifts and a uniform stretch, and nothing about straightness has to be assumed.

 All of flat spacetime in a diamond. The whole of flat spacetime with one space dimension, squeezed into a finite diamond by applying arctan separately to the two light-cone coordinates u = t − x and v = t + x. Solid curves are the worldlines of observers at rest at x = −3, −2, −1, 0, 1, 2, 3; faint curves are the instants t equal to the same values; the dashed lines are the two light rays through the origin, still at 45°, as every light line is — checked to a part in a billion — and the causal order of 4000 sampled pairs is unchanged. Infinity is not one place. Every worldline at rest runs from the bottom corner i⁻ to the top corner i⁺; every instant runs between the side corners i⁰; and light rays begin on the lower edges ℐ⁻ and end on the upper edges ℐ⁺. Each of these limits is checked at ten million units out. Relativity

The five places infinity turns out to be

Flat spacetime goes on for ever in every direction, and it can still be drawn whole on a page. Squeeze each family of light rays with a function that keeps their order and the infinite plane becomes a diamond with every light cone still at 45°. The price is distance, which the picture no longer shows. What it shows instead is that infinity is not one place: observers slower than light all end at a single point, instants end at another, and light ends along a whole edge of its own.

 Two bodies an engine draws together. Two equal bodies of 4.186 kJ/K — a kilogram of water each — one at 90.0 °C and one at 10.0 °C, against the heat drawn from the hot one. The solid curves are the best possible engine running between them, a reversible one, which leaves the product of the two temperatures unchanged and brings both to the geometric mean, 320.7 K (47.5 °C). It draws 177.8 kJ from the hot body and delivers 20.8 kJ of work, C(√T₁ − √T₂)², checked against the heat balance. The dashed lines are the same bodies simply touching: they meet at the arithmetic mean, 323.1 K, having exchanged 167.4 kJ and delivered nothing. The 2.5 K between the two endpoints is the work, left behind as heat. Thermodynamics

The work left in two buckets of water

Carnot's ceiling assumes reservoirs so large that taking heat from one and giving it to the other changes neither temperature. Two buckets of water are not reservoirs. Run the best possible engine between a hot one and a cold one and both temperatures move, the efficiency available shrinks as they do, and the engine stops when they meet — at the geometric mean of the starting temperatures, not the ordinary one. The work it delivered is exactly the difference between those two meeting points, and it is far less than the starting temperatures promise.

 Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends. Waves

The end that knows how the middle was cut

A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

 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. Astrophysics

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 potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic. Electromagnetism

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

 Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength. Optics

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

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

The slope a spin leaves in a spectrum

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

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

The plane in which three bodies are flat

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

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

The rocket that leaves its fuel at home

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

 The floor a state's survival cannot go below. The probability that a quantum state is still found in its initial state, against time measured as its energy spread times time over ħ, for four states with the same spread. The shaded region under cos²(ΔE t/ħ) is forbidden by Mandelstam and Tamm's theorem, and every curve — each a sum of phases over the state's energies — stays out of it. Two equally weighted levels run along its edge and reach an orthogonal state at exactly π/2, the fastest any state with this spread can. The same two levels driven off resonance, with the same spread, never get further than a survival of 0.500. Three equally spaced levels become orthogonal only at 1.7101, 1.0887 times the limit, and a coherent state never does, bottoming out at 0.0183. A spread of energy is permission to change, not an obligation. Quantum

The fastest a state can stop being itself

Time has no operator, so the energy–time relation cannot be the commutator inequality it resembles. What stands in its place is sharper: a state whose energy is spread by ΔE cannot become a different, orthogonal state in less than πħ/2ΔE, and cannot do it faster than its mean energy above the ground state allows either. Two equally weighted levels reach both limits exactly. Nothing else does.

 One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel. Waves

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

 How close a network gets to a load that stores charge. The fraction of a wave's amplitude reflected from a resistance shunted by a capacitance, with RCωc = 2, against frequency in units of the band edge ωc, for the load alone and for networks of inductors and capacitors whose values, with an ideal transformer at the source, were optimised numerically to keep the reflection low across the band. The bare load holds it to 0.707 across the band; the 1-element network holds it to 0.392 across the band; the 2-element network holds it to 0.320 across the band; the 3-element network holds it to 0.289 across the band. The dashed line is Bode and Fano's floor, exp(−π/RCωc) = 0.208, which no network of any size can go beneath over the whole band; each added element brings the design closer to it, and each buys its flatter band with a reflection that climbs to total just beyond the band edge. Waves

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

 Two energies with opposite slopes, and the width between them. The energy per unit area of a domain wall in iron, against the width the wall is assumed to have, split into the two terms that decide it. The exchange term falls as one over the width, because a reversal spread over more atoms turns through a smaller angle between each neighbouring pair. The anisotropy term rises in proportion to the width, because every atom inside the wall points away from an easy direction and pays for it. The sum has a least value, found here by scanning four hundred thousand widths: 92.9 nanometres and 4.46 millijoules per square metre. Neither constant is a dimension of the sample, so neither is the width: it is the first length in magnetism that belongs to the material. The assumed profile turns at a uniform rate, which is not the cheapest way to turn, so this width is 41 per cent above the conventional πδ of the exact profile, and this energy 11 per cent above the exact 4√(AK). A trial function can only ever overestimate, and eleven per cent is how much this one costs. Electromagnetism

The first length that belongs to the substance

Every length in magnetism so far has been a length of the sample — a demagnetising factor is a shape, an avalanche cutoff is a sample's own restoring field. A domain wall's width is not. It is √(A/K), made of two material constants and nothing else, and across seven ordinary magnets it runs from two and a half nanometres to nine hundred.

 Four tolerances, four elastic answers, one collapse load. The force in each of a table's four legs against the load on it, for four different manufacturing errors, with the legs made of a material that yields at 25 kilonewtons. While everything is elastic the short diagonal pair takes more than its share by a fixed amount that depends on the error and not at all on the load, so the load at which the first leg reaches its capacity runs from 25 to 100 kilonewtons — a spread of more than a factor of three. Past that point the yielded legs hold a constant force and the others take the rest, and the difference the tolerance made is erased. Every one of the four cases collapses at 100 kilonewtons, which is four times one leg's capacity, checked here to a part in a million across the four. The quantity nobody could compute and the quantity that decides whether the structure stands are not the same quantity. Mechanics

The one number the tolerances cannot touch

A redundant structure's load sharing depends on stiffnesses and manufacturing errors that nobody knows. Its collapse load does not depend on either. Once members yield they hold a known force instead of a force proportional to a displacement, the compatibility equations that needed the unknowns drop out, and the load at which the structure becomes a mechanism follows from a work balance with no stiffness in it at all.

 The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for. Optics

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

 Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K. Optics

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

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

Weighing what cannot be put on a scale

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

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

A link between two that never met

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

 How much a block knows about the rest. The entanglement between a block of a one-dimensional chain and everything outside it, against how long the block is, for three states of the same number of particles. The straight line is a randomly chosen state, whose entanglement is the block's length times the logarithm of two — a volume law, and what almost every state in Hilbert space does. The flat curve is the ground state of a chain with a gap: it saturates, varying by less than a twentieth of a per cent from a block of eight to one of forty, because the boundary of a one-dimensional block is two points however long the block is. The middle curve is the ground state of a gapless chain, which grows as the logarithm of the size with a coefficient measured here as 0.333 against the third that conformal field theory gives. Both ground states are enormously less entangled than a random state, and that is not a detail about chains: it is why a ground state can be written down at all. Quantum

The corner of Hilbert space that is ever visited

Monogamy between three parties says how much of a correlation a pair may hold. Read across a boundary in a many-body system it says something much stronger: the entanglement between a region and the rest scales with the boundary rather than the volume, for the ground state of anything with local interactions. That is why such a state can be written down at all — and why almost every state in Hilbert space is one that nothing ever prepares.

 How far the vacuum is from being a nonlinear medium. The size of the vacuum's departure from linearity, as a fraction, against the electric field it is subjected to — thirteen decades of field and twenty-eight of correction, both logarithmic. The scale is the Schwinger field, computed here from the electron's mass and the fundamental constants as 1.32e+18 volts per metre: the field at which a pair gains its own rest energy over a Compton wavelength, and therefore the field at which the vacuum stops being a passive backdrop. The marks are the strongest fields that exist. a laboratory magnet, 10 T is 2.3e-9 of it; a hydrogen atom's own field is 3.9e-7 of it; a 10²² W/cm² laser focus is 2.1e-4 of it; the Schwinger field is 1.0e+0 of it; a magnetar, 10¹¹ T is 2.3e+1 of it. So a laboratory is twenty-eight decades from making the effect large, and a magnetar's field is above the critical one — which is why the only places the vacuum's nonlinearity has been seen are the two where the fields are not human: the ultraperipheral collision of two heavy nuclei, and the surface of a neutron star. Waves

The one medium that was supposed to add exactly

Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

 14 decades of fluid, and a floor none of them reaches. The ratio of viscosity to entropy density for 8 substances, in units of the proposed lower bound, on a logarithmic axis. The quantity is a viscosity divided by how much entropy a cubic metre of the substance holds, and it has the dimensions of Planck's constant over Boltzmann's — so a bound on it is a statement with no material properties in it at all. The span here is a factor of 1.6e+14, from pitch at 20 °C at the top to the quark–gluon plasma at the bottom, which sits a factor of 1.6 above the floor. The values marked as computed are worked out from a viscosity and a tabulated entropy; the rest are quoted from the compilations, because a minimum along an isobar is an inference from many measurements rather than a single one. Fluids

Whether a fluid can be made arbitrarily thin

Viscosity has no units anybody would call fundamental, and nothing obviously stops it being as small as you like. Divide it by the entropy in a cubic metre and the units become Planck's constant over Boltzmann's — and a conjecture from 2005 says that ratio has a floor. Across fourteen decades of ordinary fluid nothing has been measured below it, the closest thing to it is the hottest matter ever made, and a kinetic-theory argument reaches the same number from the opposite direction.

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

A few cycles that are only mass and spin

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

 A potential that is lower every time round. The magnetic scalar potential along a path circling a wire carrying 10 amps, against the angle turned through, for 2 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 10 amps, and a second circuit lowers it by 10 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in. Electromagnetism

A potential that does not come back to itself

Where no current flows, the magnetic field has no circulation round any small loop, so it is the gradient of something and a magnetic problem becomes an electrostatic one. The catch is not that the potential fails to exist. It is that walking once round a wire lowers it by the current, and walking round again lowers it by the current again — so Ampère's law survives the translation as a statement about what the path encircles rather than about where it went.

 The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing. Mechanics

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

 A wall pulled away fast leaves the energy behind. The energy of a ball bouncing in a box whose wall is moved, against the length of the box, for 3 wall speeds — each a fraction of the ball's own starting speed — with the adiabatic prediction drawn dashed. Every collision is solved for exactly rather than stepped, so nothing here assumes the wall is slow. Moved slowly, the wall takes energy from the ball at the adiabatic rate, and the energy falls as the inverse square of the length. Moved as fast as the ball is moving, it takes almost nothing: the ball cannot catch a wall retreating faster than it travels, so the collisions stop and the energy stops falling. That is the difference between a gas pushing a piston and a gas expanding into a vacuum, and it is drawn here for one particle. At the end of the range the slowest wall leaves the energy at 0.058 of its starting value and the fastest at 1.000, against an adiabatic 0.065. Mechanics

The wall that moves while the ball is in flight

Energy is conserved because the rules do not depend on the time. Move the walls of a box and the rules do depend on the time, so what is inside gains or loses without limit — and how much depends entirely on how fast. Moved slowly, a wall takes energy at exactly the rate the adiabatic law says; moved faster than the ball travels, it takes none at all, and the same box is a piston or a vacuum according to a speed.

 The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all. Quantum

Two states where the counting says three

A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

 A fridge with no work going into it, and its ceiling. How much heat a three-reservoir machine can lift out of a cold space per unit of heat supplied to drive it, against the temperature of the driving heat, for 3 cold temperatures and an ambient of 300 kelvin. No work enters or leaves: the machine takes heat in at the top, takes heat in at the bottom, and rejects the sum at ambient. That it can do anything at all is the surprise — the second law allows heat to be moved up a gradient provided a larger flow is moved down one, and the accounting is a single inequality in the three entropy flows. The ceiling is the product of two familiar expressions, and at 450 kelvin driving a 253-kelvin space it is 1.79. The marks are what real machines achieve, which is a fifth to a third of it — absorption refrigeration is not efficient and is chosen when the heat is free and the silence and the absence of moving parts are worth something. Thermodynamics

A fridge with no work going into it

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

 What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all. Relativity

The count that no observer can disagree about

A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

 The action at one instant, drawn as a map. Trajectories leaving one point at the same moment, at launch speeds 0.6, 1 and sixteen directions each, in a uniform field pulling downward, drawn up to time 1. Behind them, dashed, are the level curves of the action at that instant, regarded as a function of where a trajectory ends. They are circles, and their common centre is neither the launch point nor anywhere the particles have reached: it is 0.500 above the launch point, while the whole swarm has fallen by the same 0.500. Every arriving velocity points straight out from that centre, so every trajectory crosses the level curves at right angles, and the arriving momentum equals the gradient of the action to one part in ten thousand. The action integrated along each path agrees with the map's value at its end. Mechanics

The action that knows where every path ends

The action is usually a number attached to one path. Treat it instead as a function of where the true path ends, and a single function of position and time holds every trajectory at once — its slope is the momentum, its rate of change is the energy, and its level curves are wavefronts, drawn about a point that sits above the source while everything falls.

 Every reaction's free energy has its lowest point inside. An ideal reaction A ⇌ B at 298 K. Across: how far it has gone, from pure A towards pure B. Up: the Gibbs energy of the mixture per mole, relative to pure A. Left, for standard reaction Gibbs energies ΔG° of −4, 0, +4 kJ/mol: the dashed straight lines are what the energy would be if A and B did not mix, and the solid curves add the entropy of mixing them. For ΔG° = −4 kJ/mol the lowest point is at 83.4 per cent B; for ΔG° = 0 kJ/mol the lowest point is at 50.0 per cent B; for ΔG° = +4 kJ/mol the lowest point is at 16.6 per cent B. Right, magnified near pure A, a reaction with ΔG° = +10 kJ/mol, whose straight line climbs from the start and which looks as if it should not proceed at all: its curve first falls, to a minimum of −43 J/mol at 1.74 per cent B, because the mixing term falls infinitely steeply away from a pure end. Each minimum was found by search and sits where the ratio of B to A equals exp(−ΔG°/RT). Thermodynamics

The reaction that cannot go all the way

Chemistry speaks of reactions that go to completion and reactions that do not happen, and at equilibrium there are neither. The reason is a logarithm. The free energy of a half-finished reaction contains the entropy of mixing, whose slope is infinite at both pure ends, so every reaction's lowest point lies strictly inside — and the slope of that free energy, the chemical potential, is to particles what temperature is to heat.

 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. Astrophysics

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. Astrophysics

The clocks that must all slow together

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

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