Illustrated Physics
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What stays the same — part 2 of 3

Conservation laws, and the habit of solving a problem by refusing to look at the middle of it.
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

Ask what a boost leaves alone and the answer is 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.

 A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it. Fluids

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

 A ball dropped from 1 m with e = 0.8, bouncing for ever and stopping anyway. The height of a ball dropped from 1 m onto a floor that returns 0.8 of its impact speed, against time. The first fall takes 0.452 s. Every flight after it is 0.8 times as long as the one before, so the flight times are a geometric series and their sum is finite: the ball has made infinitely many bounces by 4.064 s, t₁(1 + e)/(1 − e), and is at rest from then on. The stepped sum of the first 60 flights plus the geometric remainder agrees with the closed form to better than a part in a billion. The dashed line marks the moment it stops; the hops just before it are too small to draw. Mechanics

The bounces that add up to a stop

A ball dropped on a hard floor bounces, and bounces again, each time a fixed fraction lower. The bounces never run out — there is always another, smaller one — and yet the ball is lying still four seconds later. The flight times form a geometric series with a finite sum, the floor's infinitely many kicks add up to exactly the ball's weight times that time, and the series is only broken where each flight becomes shorter than the impact that launched it.

 Every ray from one point, meeting again at another. Rays from a point half a radius from the centre of Maxwell's fish-eye, n = n₀/(1 + r²/R²), traced in 11 directions round the full circle. Every one is an arc of a circle, and every one passes through the point on the other side of the centre at distance R²/r from it — 2.00 radii — which is the image. The worst miss is 8.8e-4 radii, the size of the integration's own error. Rays leaving in opposite directions, rays leaving at right angles, rays that go the long way round: all arrive. The faint circles are contours of the index, which falls from n₀ at the centre to n₀/2 at one radius and n₀/5 at two. Optics

The medium that images every point

A single refracting surface can be shaped to image one point perfectly and no other. A medium whose index falls smoothly away from a centre can do better: in Maxwell's fish-eye every ray from any point, leaving in any direction, arrives at one image point, and every such path takes exactly the same time. It is a perfect instrument for all of space at once — and the image it forms is sharp everywhere and a faithful copy nowhere.

 Electrons and holes in silicon against the donors added. The concentrations of free electrons and holes in silicon at 300 K against the concentration of donor atoms added, on logarithmic axes, solved from charge neutrality and the product law n·p = nᵢ². Below the intrinsic concentration of 10¹⁰ per cubic centimetre the donors change almost nothing. Above it the electrons follow the donors one for one and the holes fall in exact proportion: a hundred times more donors, a hundred times fewer holes. At 10¹⁶ donors the electrons outnumber the holes by 10¹² to one. The dashed line is √(n·p), which does not move at all across ten decades of doping — checked to a part in a billion. Thermodynamics

The product doping cannot move

Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.

 Modes added in step: a train of pulses. The intensity of the sum of N waves of equal amplitude at equally spaced frequencies, all starting in phase, against time in units of the round trip — the inverse of the spacing — for N = 5, 10, 20. The sum is a train of pulses, one per round trip, each of peak intensity N² times one wave's and width about 1/N of the round trip. Between the pulses the waves cancel almost completely. The average intensity is N, as it must be — 5.01 for N = 5, 10.04 for N = 10, 20.19 for N = 20 — so the pulses do not create energy; they move it all into a fraction 1/N of the time, where it is N times more intense than it would be if spread evenly. Waves

The phases that turn a glow into pulses

A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

 3 beams crossing: change the phases and the pattern only slides. The intensity where 3 plane waves of one wavelength cross in a plane, their directions spread evenly round the circle, for three different sets of relative phases, each panel spanning 2.2 wavelengths. Darkest shading is brightest, in five steps of a fifth of the peak. The three patterns are the same lattice of bright spots moved sideways: for each of the altered sets there is a shift that reproduces the first pattern to 2.0 per cent of the peak. With 3 beams in two dimensions there are 2 relative phases and 2 directions to slide in, so every change of phase is a slide. Waves

The lattice no phase can bend

Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

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

The radiation a difference in falling would make

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

 The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m. Electromagnetism

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

 Where a moving charge keeps its field's energy and its momentum. Left: the density of field energy and of field momentum around a charged shell moving slowly to the right, by direction, each drawn as the mass per unit volume it contributes — energy density over c², momentum density over velocity. The energy density is the same in every direction; the momentum density goes as sin²θ, zero ahead and behind and largest broadside, because it is ε₀E × B and the magnetic field of a moving charge vanishes along its line of motion. Broadside it is twice the energy's; averaged over all directions sin²θ is 0.6667, and twice that is the 4/3. Right: the fraction of the field's energy, and of its momentum, lying within a distance r of the centre of a shell of radius a. Both are 1 − a/r, because both densities fall as the fourth power of distance: half of each lies within 2a and nine-tenths within 10a. Electromagnetism

The mass a charge's field gets wrong by a third

Give a charged sphere's field its energy U and divide by c², and the field has a mass. Move the sphere and ask the field's momentum what the mass is, and it says four-thirds of that. The discrepancy took from 1881 to the 1960s to understand, and the answer is not a better calculation of the field. It is that a charge cannot hold itself together, and whatever does has energy too.

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

The point that takes the blow without a jolt

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

 The thickness fixes the area, and the design only moves it. How much a metal-backed resistive sheet 7.5 mm from the metal absorbs, measured as ln(1/|Γ|), against wavelength, for sheets of 150, 250, 377, 800 Ω per square. Rozanov's bound says the area under any such curve, over all wavelengths, can be at most 2π² times the thickness: 147.9 mm here. The areas computed are 147.9 mm for 150 Ω, 147.9 mm for 250 Ω, 147.9 mm for 377 Ω, 69.7 mm for 800 Ω. Every sheet up to the impedance of free space meets the bound exactly and only moves its area between wavelengths; the more resistive sheet wastes part of it. Waves

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

 A smooth swing that sharpens into pulses. The shape of a string whose far end moves back and forth by 1% of its length at 2 times its fundamental frequency, started swinging in its fundamental, after 0, 10, 20, 30, 40 round trips of a wave. The smooth arch gathers into two ever-narrower pulses running in opposite directions, drawn at the moments they cross in the middle, with their height unchanged and their sides ever steeper; each reaches the moving end just as it moves inwards fastest. The energy has become 1.00, 1.09, 3.15, 10.9, 38.3 times the starting value. Every shape is computed exactly from the moving end, with no approximation to the wall's motion. Waves

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

 Why the meander has to move west. A line of fluid columns at 45° pushed into a meander 3,000 km long and 300 km in amplitude, seen from above with north up and east to the right. Columns pushed north find the planet spinning faster beneath them and turn clockwise; columns pushed south turn anticlockwise. Each turning column drives the fluid north on one side of it and south on the other, and the arrows are that flow, from integrating the columns' spin along the line: it peaks at 2.32 m/s, a quarter of a wavelength west of each crest, so the columns just west of a crest are carried north and the crest itself is rebuilt further west. The dashed meander is the same pattern 2 days later, moved 638 km west at β/k² = 3.69 m/s. Nothing in the argument depends on which way the meander was first pushed. Mechanics

The wave that can only travel west

A column of air or water on a spinning planet keeps one quantity as it moves: its total spin divided by its depth. Carry it north, where the planet's own spin about the local vertical is larger, and it must turn the other way to compensate. That single bookkeeping rule makes a wave that exists nowhere else — one whose crests can only ever move west — and the same rule holds the jet stream's meanders still over the mountains, makes the ocean take decades to notice a change in the wind, and pins the Gulf Stream to the western side of the Atlantic.

 The stiffness that falls below four-thirds. The first adiabatic exponent Γ₁ — how steeply pressure rises when a gas is squeezed without heat escaping — of hot oxygen with its radiation and its equilibrium electron–positron pairs, against temperature, at densities of 10³, 10⁴, 10⁵, 10⁶ g/cm³. At 10³ g/cm³ its least value is 1.222, near 1.2 billion kelvin; at 10⁴ g/cm³ its least value is 1.235, near 1.3 billion kelvin; at 10⁵ g/cm³ its least value is 1.285, near 1.8 billion kelvin; at 10⁶ g/cm³ its least value is 1.339, near 3.1 billion kelvin. A star can hold itself up only if its core's Γ₁ averages above 4/3, the value for a gas of light alone; a core that dips below it has no stable size, and the pair production that pulls it below is strongest where the gas is hot and thin — the cores of the most massive stars. Relativity

The heat that turns into matter

Thermodynamics counts particles and relativity lets them be made. Heat a gas to a billion kelvin and its light carries enough energy to create electrons and positrons, which the gas then holds in equilibrium like any other ingredient. The heat spent making them raises neither the temperature nor the pressure, so the gas grows soft — softer, in the cores of the most massive stars, than a star can survive. Those stars collapse, ignite their oxygen all at once and blow themselves entirely apart; and the same pairs, vanishing a few seconds after the Big Bang, left the universe's light hotter than its neutrinos by a factor that is the cube root of eleven over four.

 The melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK. Thermodynamics

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

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

The hilltop that holds the Trojans

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

 Three ships, three local times, one clock. The fair-weather field at sea level, in the same model, as recorded by ships at longitudes 120° W, 0°, 120° E, each plotting it against its own local solar time. An ordinary daily effect — sunshine, heating, turbulence — would peak at the same local time everywhere. This one peaks at 11.4, 19.4, 3.4 hours local time in turn, because every ship is reading the same global potential, which follows the world's storms and peaks at one moment in Universal Time. Seeing the maxima line up in Greenwich time rather than local time, over clean ocean air far from any storm, was how the global circuit was demonstrated: the field in fair weather is the far end of a circuit whose generators are thunderstorms thousands of kilometres away. Electromagnetism

The field that keeps Greenwich time

Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.

 The path the tip of the field traces. The tip of the total field over one cycle, for four arrangements of coils, with each coil's axis as a faint line and the field at one moment as an arrow. One coil: forward-turning part 0.50, backward-turning part 0.50; Two coils, 90° and a quarter cycle apart: forward-turning part 1.00, backward-turning part 0.00; Three balanced coils: forward-turning part 1.50, backward-turning part 0.00; Three coils, the third at 70 per cent current: forward-turning part 1.35, backward-turning part 0.15. One coil traces a line: equal forward and backward parts, a field that pulses and does not turn. Two coils at right angles a quarter cycle apart trace a circle of radius 1, three balanced coils a circle of 1.5, and a weakened third coil an ellipse, the sum of a large forward circle and a small backward one. Waves

The field that turns with nothing turning

A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

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

The noise a high Q moves out of the way

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

 A clock that takes the drive's frequency over a whole range. The frequency a self-sustained oscillator actually runs at, relative to the frequency of a weak drive, against how far its own natural frequency is from the drive's, both in units of the locking strength K. Without the drive it would lie on the dashed diagonal, keeping its own frequency. With it, every natural frequency within ±K of the drive's is pulled all the way in: the oscillator runs at exactly the drive's frequency, a flat plateau where it has abandoned its own. Outside the plateau it runs free again but not at its own frequency; it is pulled towards the drive, its offset shrinking from the detuning to the square root of (detuning squared minus K squared) — 0.66 K instead of 1.2 K just outside, 2.83 K instead of 3 K further out. Waves

The clock that is pulled into step

In 1665, ill in bed, Christiaan Huygens noticed that two of his pendulum clocks hanging from the same beam were swinging in exact opposition, and that however he disturbed them they returned to it within half an hour. A clock is not a passive resonator waiting to be pushed; it keeps its own time, by its own mechanism. What a weak push can do to it is not change how strongly it swings but pull its phase — and within a range of frequencies set by the push's strength, pull it all the way into step. The same arithmetic locks lasers, pacemaker cells, power stations and the body's daily clock.

 The gas that has to stop expanding. The thermal expansion coefficient at constant pressure times the temperature, αT, against temperature over the Fermi temperature, on logarithmic axes, for an ideal classical gas and an ideal gas of spin-half fermions at the same density. For the classical gas αT is exactly one at every temperature, so α = 1/T grows without limit as the gas is cooled, which the third law forbids: by a Maxwell relation α is minus the pressure derivative of the entropy per unit volume, and that must vanish at absolute zero. The Fermi gas agrees with the classical one when hot and departs from it near the Fermi temperature; at a tenth of it αT is 0.048, and at a hundredth 0.002, falling as π²/2 times the square of the temperature over the Fermi temperature. Its expansion vanishes as the law requires. Thermodynamics

The two laws absolute zero will not allow

The ideal-gas law and Curie's law are the first two equations of state most people meet. The third law of thermodynamics says both must be wrong at low temperature, and it says so without any model of what replaces them. A Maxwell relation turns every derivative of the entropy into something measurable with a ruler or a magnetometer: the thermal expansion, the slope of the magnetisation with temperature. If the entropy stops depending on pressure and field at absolute zero, those must vanish. The classical gas expands as 1/T and Curie's magnetisation climbs as 1/T, and both slopes grow without limit instead.

 A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is. Mechanics

The molecule that lives outside its own well

The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

 Four places where three charges' fields cancel. Field lines in the plane of three equal positive point charges at the corners of an equilateral triangle, with every point where the field is zero marked. There are four: one at the centre, and three more on the lines from the centre to the middle of each side, 0.164 of a side from the centre. Four is (n − 1)² for three charges, the largest number Maxwell conjectured any three charges could make. Each zero is a saddle: along some directions the field pushes a test charge back and along others away, so none of them can hold anything still. Electromagnetism

The zeros a handful of charges can make

Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.

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