Every essay — page 26
Mechanics
Motion, force, and the quantities that refuse to change.
The push that pays only for the hill
A domino can knock over one half as large again as itself, and that one a still larger one, so that thirteen steps take a fall that began with a fingernail's flick to a slab as tall as a person. Nothing is being amplified by a machine. Each domino holds energy it was given when it was stood on its end, far more than it takes to tip it, and the push that starts it pays only for the small hill in front of that store. A chain of dominoes passes along not energy but permission — which is how a spark lights a fire and a nerve carries a signal the length of an arm without fading.
The compass that finds the axis the Earth turns about
A spinning wheel held so that its axis stays level will, on a turning Earth, swing its axis round to point at true north and stay there — not magnetic north, the geographic pole. Nothing in the instrument knows where north is; it finds the one direction about which the Earth's turning stops tilting it. Built to be used at sea, such a compass has to ignore the ship's lurches, and that forces its swing to have one particular period: 84.4 minutes, the period of a pendulum as long as the Earth's radius, of a satellite skimming the ground, and of a stone dropped through a tunnel to the far side of the planet.
Waves
Oscillation, and everything that turns out to be an oscillation.
The sandbars that reflect the sea
A stack of alternating layers reflects the one colour whose half-wavelength matches the repeat. A row of sandbars on a gently sloping seabed is the same arrangement for ocean waves, and it does the same thing. Waves twice as long as the bar spacing are sent back out to sea, one small reflection per bar, all adding in step. A patch of eight bars can return most of the swell that meets it. The reflected wave, added to the incoming one, makes a standing pattern whose nodes are spaced exactly like the bars. Sand gathers in that pattern, so a field of bars can build its own next bar.
The loss that falls as the frequency rises
Every metal conductor loses more at higher frequencies, because the current crowds into a thinner skin and meets more resistance. One wave in a round metal pipe breaks the rule. The TE₀₁ mode's electric field circles the axis and touches the wall nowhere. As the frequency rises it becomes more and more nearly a plane wave running down the middle of the pipe, and it drives less and less current in the metal. Its loss falls with frequency, to about half a decibel per kilometre at 60 GHz in a pipe 60 mm across. In the 1960s it was the best way anyone knew to send a hundred thousand telephone calls across a continent. The catch was that the pipe also carried seven hundred other modes.
Optics
Light, and the small number of rules it obeys.
The mirror that lights a tenth of a micrometre
Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.
The index that falls below one
Glass bends light because light is slower in it. X-rays are not slower in glass, or in anything: every material's refractive index for them is slightly less than one, by a few parts in a million. Everything about X-ray optics follows from that small number with the wrong sign. A lens that focuses them must be thinner in the middle than at the edges, and so weak that dozens are stacked in a row. A mirror reflects them only if they arrive at a glancing angle of less than a degree — totally, from the outside, the reverse of the way a diamond traps light — which is why the mirrors of an X-ray telescope are long, nested barrels rather than dishes.
Electromagnetism
Charge, field, and the lines drawn between them.
The attraction that weakens when light is too slow
Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.
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.
The handedness a mirror cannot keep
Maxwell's equations in empty space do not change if the electric field is turned partly into the magnetic field and the magnetic partly into the electric. A symmetry that exact must hand over a conserved quantity, and this one hands over the handedness of light — how many more photons spin one way than the other. Every mirror breaks it: circularly polarised light comes back from a mirror with its handedness reversed. Only a material that responds to the magnetic field exactly as strongly as to the electric one keeps it, and such a material, it turns out, sends nothing straight back.
The force between pieces of a current
Ampère spent six years finding the force one short piece of current exerts on another, and the law he found is not the one used today. The modern law, which follows from the Biot–Savart field, disagrees with his for almost every pair of pieces — his obeys Newton's third law and the modern one does not. Both give exactly the same force between any two complete circuits, which is the only thing that has ever been measured. The force between two pieces of current is a quantity with two correct values, because a piece of current on its own does not exist.
The sheet resistance that forgets the shape
To measure how well a thin film conducts, the obvious method is to cut it into a neat strip and pass a current along it. In 1958 Leo van der Pauw, an engineer at Philips, showed that the strip is unnecessary. Put four small contacts anywhere on the edge of a film of any shape, take two readings, and one equation returns the film's resistance per square — the shape and the spacing of the contacts drop out entirely. The proof is a single fact about logarithms and the maps that preserve angles, and it fails for exactly one kind of film: one with a hole in it.
Relativity
Space and time, drawn on the same axes.
The river that sound cannot swim up
Sound travels at a fixed speed through the air or water carrying it, so a flowing medium adds its own speed to the sound's: faster downstream, slower upstream. That is the plainest velocity addition there is. Where the flow speeds up past the speed of sound, upstream-going sound is carried backwards, and nothing made beyond that point can be heard before it. The point is a horizon. Sound cones tip over there exactly as light cones do at a black hole, and the rate at which the flow speeds up plays the part of the surface gravity. A kitchen sink under a running tap makes the reverse kind of horizon, a white hole for ripples.
The contraction the forces work out for themselves
Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.
The heat that arrives before it could
Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.
Quantum
Where the continuous picture runs out, and what replaces it.
The barrier the metal cannot choose
Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.
The angle the uncertainty principle cannot be written for
Position and momentum trade exactly: squeeze one and the other spreads, with a product never below ħ/2. Angle and angular momentum look like the same pair and are not. A rotor in a state of definite angular momentum has zero spread in angular momentum and a finite spread in angle, π/√3, and their product is zero. Nothing is wrong with the rotor. The relation was written with a commutator that does not hold, because an angle lives on a circle and a line has no ends to wrap round. The relation that does hold measures the angle's spread by how far its average position on the circle falls short of the rim.
The value a gentle measurement returns
Couple a pointer to a spin so weakly that the pointer's shift is far smaller than its own blur, and a single run learns almost nothing about the spin. Now keep only the runs in which the spin is later found in a chosen, nearly opposite state. The kept pointers move as though the spin's value were ten, or a hundred, though it can only be plus or minus one. That weak value was predicted in 1988 and has been measured many times. It is not a hidden value of the spin, and it does not break the uncertainty principle: it is paid for exactly in discarded runs. What it does give is a way to watch a quantum system without disturbing it.
Fluids
Matter that will not hold a shape, and the forces that act in it anyway.
The paste that forgets it was stirred
A jar of clay slurry left on a shelf sets like a jelly; shaken, it pours. That is thixotropy, and the tempting description — its viscosity falls when it is stirred — misses what makes it strange. A thixotropic paste has no viscosity at a given rate of stirring until it has been told how long it has been stirring, and under a steady push it cannot settle on a slow flow at all. Push slightly less than a critical stress and it creeps to a halt; push slightly more and it runs. Between the two there is nothing, and a paste made to flow slowly splits into a layer that flows and a layer that does not.
The viscosity that belongs to the tube
Measure the viscosity of blood in a wide glass tube and it is about three times that of the plasma it is made from. Measure it in a tube a tenth of a millimetre across and it is lower; in one a hundredth of a millimetre across, lower still, barely more than plasma's. Then, below about six micrometres, it shoots up. The blood has not changed. The red cells push themselves away from the wall and leave a sleeve of plain plasma a few micrometres thick for the rest to slide in, and in a narrow tube that sleeve is a large part of the flow. A viscosity that depends on the pipe is no longer a property of the fluid alone.
Astrophysics
Gravity read as geometry, and the laws carried where no laboratory can follow.
The photons that raced for seven billion years
Nothing on a laboratory bench can reach the Planck scale, but a gamma-ray burst halfway across the universe can amplify a Planck-scale effect until it is a second long. If spacetime has structure at the Planck length, the simplest guess is that light's speed depends slightly on its energy. Then a burst's high-energy photons would arrive measurably later than its low-energy ones, after a race lasting billions of years. In 2009 a 31 GeV photon from a burst 7.5 thousand million light-years away arrived within 0.83 s of the burst's start. That excludes the simplest version of the idea at the Planck energy itself.
The mountain a spinning star is allowed
A neutron star spinning thirty times a second with a bump on its surface a few centimetres high is a rotating mass quadrupole. It sends out gravitational waves at twice its spin frequency, as a steady tone lasting millions of years. None has been heard. The silence is a measurement. The Crab pulsar is slowing down, and if all the energy it loses went into gravitational waves they would have been detected many times over. Searches now say the waves carry less than a ten-thousandth of it. The mountain on the Crab is less than about ten centimetres high, on a star twenty-four kilometres across.
The radius a mass adds that no circumference shows
Walk round the Earth's equator and divide by 2π, and the number is the Earth's radius. Dig to the centre and measure the same radius with a tape, and the tape comes out about two millimetres longer. Nothing is wrong with either measurement. A mass does not only slow the clocks near it; it stretches the rulers too, and the stretch is exactly as large as the slowing. Clocks cannot see it. Light can, and half of every bend and every delay that light suffers near the Sun is this.
The pull that has to equal the pull back
Every test of the equivalence principle drops two things and asks whether gravity pulls them equally. None of them asks whether they pull equally. A body's mass enters gravity twice — once as what is pulled and once as what does the pulling — and if the two ratios differed from one material to another, Newton's third law would fail and a lopsided body could push itself through space. The first test of this used a plastic cylinder floating in a liquid. The best uses the Moon, whose light crust is thicker on the side that never faces the Earth.
The pressure that weighs down what it holds up
In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.
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