Every essay — page 31
Mechanics
Motion, force, and the quantities that refuse to change.
The third law that falls out of a change of scale
Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.
The parabola that is the top of an ellipse
Every textbook throw is a parabola, the best angle is 45°, and the range goes as the square of the speed. All three are statements about a flat Earth with gravity pointing the same way everywhere. On the real one a throw is a piece of an orbit — an ellipse with the Earth's centre at one focus — and the parabola is what the top of that ellipse looks like when the Earth is large. Throw faster and the best angle falls, the range runs ahead of the formula, and at eight kilometres a second the best throw is horizontal and never lands.
Waves
Oscillation, and everything that turns out to be an oscillation.
The tide a lake makes by itself
On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.
The wall that stops sound by its weight
A wall between two rooms keeps out sound not by absorbing it but by being hard to shake. The sound's pressure pushes the wall back and forth, the moving wall becomes a loudspeaker on the far side, and how much gets through depends almost entirely on the wall's mass per square metre against the frequency: six decibels more for every doubling of either. Read as impedance, a wall is a mass placed in series with the air, and the same reading explains the frequency at which a stiff pane of glass nearly vanishes and why two thin walls with a gap beat one thick one.
Optics
Light, and the small number of rules it obeys.
The surface every ray is normal to
Light from a point, bent by any number of lenses and mirrors, comes out as a tangle of rays that no longer meet anywhere. Hidden in the tangle is a surface every ray crosses at a right angle: the set of points the light reaches at the same optical time. That a surface always exists is a theorem two centuries old, and it is the reason a modern wavefront sensor works — an array of tiny lenses measures only the direction each patch of light is travelling, and the measurements can be added up into a shape only because they are the slopes of one.
The dark lines a wave cannot avoid
Overlap three waves of light and there are places where they cancel exactly — not along fringes, as two waves cancel, but at isolated points, round which the phase of the light turns through a full circle. They cannot be removed by any small change, they can only be created or destroyed two at a time, and random light is full of them, about three to every square wavelength. A grating with a fork in it writes one into a beam on purpose, and the beam that comes out has a dark core and carries angular momentum in its twist.
Electromagnetism
Charge, field, and the lines drawn between them.
The field a changing field does not make
Faraday's law says a changing magnetic field comes with a curling electric field, and Maxwell's correction to Ampère's law says the reverse, and the two are almost always read as a chain of causes: each field makes the other, and that is how light carries itself across empty space. Solved for what the fields actually are, Maxwell's equations say something different. Every field at every place is an integral over the charges and currents on the observer's past light cone, and over nothing else. The changing fields are not sources; they are two descriptions of the same retarded currents, and the laws that relate them hold because both are computed from one history.
The two coils one radius apart
A single loop of wire makes a field that is strongest at its centre and falls away in every direction. Two loops on the same axis, spaced exactly one radius apart, make a field that is flat at the midpoint to a part in a thousand over a region a third of the coils' size — and flat sideways as well as along the axis, although only the axis was designed. The sideways uniformity is free because a magnetic field in empty space obeys Laplace's equation, which ties its variation in one direction to its variation in the others, term by term.
Thermodynamics
Heat, disorder, and the one law with a direction in it.
The entropy every liquid gains on boiling
Divide the heat it takes to boil a liquid by the temperature at which it boils, and for benzene, chloroform, acetone, hexane, ether and carbon tetrachloride the answer comes out within a few per cent of the same number — about ten and a half times the gas constant. Frederick Trouton noticed it in 1884. It is a statement about entropy, and once it is read that way the liquids it fails for become the interesting ones: water and the alcohols, whose hydrogen bonds hold the liquid in order; acetic acid, whose vapour is made of pairs; and helium, whose liquid is already as disordered as quantum mechanics allows.
The cluster that grows where the walkers arrive
Let particles wander in one at a time from far away, each sticking where it first touches a growing clump. The rule has nothing in it about shape, and the clump that results is a lacy, branched thing whose mass grows more slowly than its area, so it gets emptier the bigger it gets. The reason is that a random walker reaches the outermost tips long before it can find its way into a gap between branches. The same arithmetic — the probability that a walker arrives is a solution of Laplace's equation — shapes electrodeposits, lightning-like discharges in plastic, fingers of water pushed into oil, and colonies of bacteria short of food.
The heat capacity that goes up and comes down
Equipartition promises every way a system can move a fixed share of heat, and for a harmonic oscillator the promise is kept: once the temperature is high enough, it holds one k per oscillator and goes on holding it. A system with only two levels cannot keep it. It absorbs heat while its upper level is filling, then stops — the two levels equally full and nowhere higher to go — so its heat capacity rises to a peak and falls back to zero. The hump, named after Walter Schottky, is the signature of a degree of freedom with a ceiling, and the area under it counts how many states were there.
The metal that feels colder than the wood
A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.
Relativity
Space and time, drawn on the same axes.
The rod that is level for only one observer
A metre rod slides at high speed over a metre-wide hole in a thin plate. In the plate's frame the rod is contracted and drops through. In the rod's frame the hole is contracted, the rod is far too long for it, and it drops through anyway — because the instruction 'every point starts to fall now' names a different now for the rod. Its front end starts first, the start runs back along it, and the rod tips into the hole like a spear. Whether a moving thing is level is a question about which events are simultaneous, and so it has an answer only for an observer.
The rocket in which light travels on circles
Inside a rocket that accelerates for ever at a constant rate, a beam of light sent across the cabin sags towards the floor, as Einstein argued it must if acceleration and gravity are indistinguishable. Followed further than any cabin, the sag turns out to be exact geometry: every ray inside the rocket is a perfect semicircle centred on the horizon below, and the rays obey the rules of a non-Euclidean plane in which the angles of a triangle add to less than two right angles. The spacetime in the rocket is flat; the curvature belongs to the clocks.
The spot that outruns light and carries nothing
Flick a laser pointer across the face of the Moon and the spot it makes crosses the Moon faster than light. Nothing is wrong with relativity: the spot is not a thing, and no part of it travels from one side to the other. Each lit point is lit by its own light, sent from the hand. What makes the spot worth studying is how strange such a non-thing is allowed to be. It can be born in the middle of the screen as a pair running apart; its speed can be made infinite, or reversed, merely by watching it from a moving frame; and the composition of velocities that caps everything real has nothing to say about it.
The wheel whose top cannot go twice as fast
The top of a rolling wheel moves at twice the speed of its axle, and the bottom is momentarily at rest — two facts every cyclist's spokes show, blurred at the top and sharp at the bottom. Roll the wheel at a large fraction of the speed of light and the second fact survives exactly while the first cannot: the top moves at 2v/(1 + v²/c²), never as fast as light. Caught at one instant of the ground's time the spokes are bent, and over one turn the wheel advances not the circumference measured from its axle but γ times it — the rim's own length, laid down on the ground by a contact point that does not move.
Quantum
Where the continuous picture runs out, and what replaces it.
The qubit that carries two bits
A single quantum two-state system, sent from one person to another, can carry at most one bit of message — a theorem, not an engineering limit. If the two already share half each of an entangled pair, the sender can do one of four things to her half alone, send just that half, and the receiver can tell which of the four she did. Two bits arrive on one qubit. Nothing is free: the shared pair had to be delivered, and counted honestly the rate is still a bit per qubit. What entanglement buys is the freedom to send half the message before the message exists, and it also buys privacy, since the qubit in transit carries nothing at all on its own.
The wait that sets the precision
To measure an atom's frequency, the obvious method is to drive it and look for the response, and the sharpness of the answer is set by how long the drive lasts. Norman Ramsey found in 1949 that most of the drive is wasted. Two short pulses with a long wait between them do better than one long pulse, because during the wait the atom keeps its own time undisturbed, and any difference from the drive's time piles up as an angle the second pulse can read. Every primary clock in the world runs on that idea, and its precision is set by two numbers: how long the atoms can be left alone, and how many of them there are.
The hydrogen atom inside a crystal
Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.
The atom with no nucleus
An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.
Fluids
Matter that will not hold a shape, and the forces that act in it anyway.
The rod that tumbles for ever in a stream
Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.
The drop that runs to the narrow end
Put a drop of water into a glass tube that narrows towards one end and it moves, on its own, towards the narrow end — horizontally, with nothing pushing it. Francis Hauksbee reported it to the Royal Society in 1712, and the explanation is two curved surfaces pulling with different strengths: each meniscus lowers the pressure of the water behind it by an amount that grows as the tube narrows, so the end in the narrower part sucks harder. A drop of mercury, which does not wet glass, goes the other way. Which way a drop goes depends on the cosine of one angle, and, for a short drop, on its length.
Astrophysics
Gravity read as geometry, and the laws carried where no laboratory can follow.
The star whose far side can be seen
Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.
The gas a hole catches from far away
A particle drifting past a black hole is captured only if its aim is almost perfect; miss by a little and it swings round and leaves. Gas is different. Streams passing on opposite sides collide, lose their sideways motion as heat, and fall in, so a hole in warm gas draws from the whole sphere inside which its gravity beats the gas's thermal motion. Bondi worked out the flow in 1952: one steady solution passes smoothly from rest to free fall, it crosses the speed of sound at a single radius, and the rate it fixes grows as the square of the hole's mass — fast enough, left alone, to run away in a finite time.
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