Illustrated Physics
physics, drawn
Concept

Measurement — where it appears

An interaction that yields a value, which in quantum mechanics changes the state and need not report a property the system already had. Which of those two it is is decidable: correlations that no pre-existing set of values could produce are what a Bell test measures.

Named by 38 essays across 7 fields — each of them below, with the objects they name alongside it.

A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

The answer that was not there before

Send a beam through an analyser and it splits in two. Send one half through a second analyser turned sideways, then through a third pointing the way the first did, and the property the first analyser removed has come back.

quantum · Measurement A cube photographed at four speeds. The outline a camera records of a cube passing at β = 0, β = 0.4, β = 0.7, β = 0.9, moving to the right, seen at the moment it passes. The dashed square is what the usual picture shows: the cube flattened along its motion by the Lorentz factor. It is not what the camera gets. Light from the far face left earlier than light from the near face, by the time it needed to cross the extra distance, and the cube moved in that interval — so the far face appears displaced backwards and the side of the cube comes into view. The two effects together are exactly a rotation: at β = 0.4 the cube photographs as one turned 23.6°, at β = 0.7 the cube photographs as one turned 44.4°, at β = 0.9 the cube photographs as one turned 64.2°. The contraction has not gone anywhere — a ruler laid alongside still reads 43.6 per cent of the proper length at the highest speed drawn — but it is a statement about two positions at one time, and a photograph is not that.

The contraction no photograph shows

Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed. A camera would have shown it turned. The contraction is entirely real in the measurement that defines it, and a photograph is not that measurement.

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

The delay that is not a bend

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

astrophysics · Light deflection Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

The angular momentum that is not a rotation

An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed.

quantum · Spin A quarter of the time, an object is found without being touched. The outcomes of sending one photon into a balanced interferometer, with and without an opaque object in one arm, computed from the same amplitudes. With the arm clear, every photon leaves by the bright port and the dark port receives nothing. With the object in place, 50 per cent of photons are absorbed by it, 25 per cent reach the bright port and say nothing, and 25 per cent reach the dark port — which is impossible unless something is in the arm, and which happens with the object still sitting there unabsorbed and unlit. The photon that produced that click did not go through the blocked arm, because a photon that goes through a blocked arm is absorbed. So an object has been located by light that never met it. The price is that 50 per cent of attempts destroy the thing being looked for: an efficiency of 33 per cent, which the Zeno version of the apparatus takes as close to one as one likes.

The measurement that never touched it

A balanced interferometer sends every photon to one output and none at all to the other. Put an object in one arm and the empty port starts clicking — and a click there is caused by a photon that cannot have gone near the object, because a photon that goes near it is absorbed. The object has been found by light that never met it.

quantum · Measurement The work one molecule and one bit are worth. The pressure of a gas of one molecule against its volume, at 300 K, in units of the volume it starts in. The shaded area is the work the molecule does pushing a partition out isothermally, and it is measured here by integrating the drawn curve rather than written down: expanding by 1.5× yields 0.4055 kT against ln 1.5 = 0.4055; expanding by 2× yields 0.6931 kT against ln 2 = 0.6931; expanding by 4× yields 1.3863 kT against ln 4 = 1.3863; expanding by 8× yields 2.0794 kT against ln 8 = 2.0794, agreeing to 2.0e-10. The doubling is the one that matters, because a partition inserted in the middle leaves the molecule on one side or the other, and knowing which is what lets the load be attached to the right face. That single expansion delivers kT·ln2 = 2.87 zeptojoules at 300 K. It looks like work extracted from one temperature, and it is — until the engine is asked to run again, which requires forgetting which side the molecule was on.

The bit that has to be paid for

One molecule in a box, a partition, and the knowledge of which side it went — enough, between them, to extract work from a single reservoir, which the second law forbids. The engine is real and the arithmetic is right. What closes the loophole is that the cycle does not finish until the knowledge has been thrown away, and throwing away one bit costs exactly what the expansion delivered.

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

The disagreement that one run settles

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

quantum · Entanglement Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

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

The state that cannot be copied

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

quantum · Measurement One measurement that separates unpolarised from polarised. What a rotating linear polariser passes, against its angle, for beams of the same total intensity and degrees of polarisation 0, 0.35, 0.7, 1. Every curve has the same average — a polariser passes half of any beam over a whole turn, whatever its state — and they differ only in how deeply they modulate. The depth of the modulation is the degree of polarisation, exactly: a fully polarised beam goes to zero at one angle and a beam with no preferred direction gives a flat line at a half. That is the whole measurement, and it is why "unpolarised" is a statement about a modulation depth rather than about what a wave is doing.

The light with no direction of shaking

Unpolarised light is not a state of a wave; it is the absence of one, and no description of a single wave can represent it. What can is a set of four numbers, all of them powers a detector reads — and they describe every beam there is, including the ones that are neither polarised nor not.

optics · Polarisation Which pairs have anything between them. The commutator of every pair of 4 observables of a spin-½, multiplied out in 2×2 complex arithmetic and shown by the size of AB − BA. The diagonal is exactly zero: an observable commutes with itself, which is why measuring the same thing twice gives the same answer. S², the total angular momentum, is a multiple of the identity here and so commutes with everything — its row and its column are zero, and a spin can have a definite total angular momentum and a definite component at the same time. The largest entry is 0.7071 in units of ħ². Sz: 0.000 with Sz, 0.707 with Sx, 0.707 with Sy, 0.000 with S²; Sx: 0.707 with Sz, 0.000 with Sx, 0.707 with Sy, 0.000 with S²; Sy: 0.707 with Sz, 0.707 with Sx, 0.000 with Sy, 0.000 with S²; S²: 0.000 with Sz, 0.000 with Sx, 0.000 with Sy, 0.000 with S². A zero cell is a promise that the two quantities can be sharp together; a non-zero one is an obstruction whose size sets how badly they cannot.

The questions that can be asked together

Two quantities can have definite values at once exactly when their operators commute. That is a piece of arithmetic about matrices, and everything the uncertainty principle forbids follows from it — including the fact that most of the time it forbids nothing at all.

quantum · Uncertainty The delay that stops caring how thick the wall is. The Wigner phase time — how late the transmitted packet's peak arrives, compared with a free particle covering the same distance — for an electron under a 3 eV barrier, at 0.2 nm, 0.5 nm, 1 nm thick. 0.2 nm: 0.3728 fs at half the barrier height; 0.5 nm: 0.4372 fs at half the barrier height; 1 nm: 0.4388 fs at half the barrier height. The widths span a factor of 5.0 and the times span 1.18. A thicker barrier is exponentially harder to cross and the delay in crossing it barely moves — which is either a remarkable fact about tunnelling or a sign that the phase time is not a traversal time, and the rest of the essay is about which.

How long the crossing takes

Tunnelling has a probability, and asking how long it takes turns out to be a different kind of question. The delay a transmitted packet shows stops growing once the barrier is opaque, so a thicker wall is crossed in the same time — and several defensible clocks give several different answers.

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

What two have they cannot give a third

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

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

What the instrument actually hears

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

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

Whether a charge on a table glows

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

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

The length no experiment can resolve

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

astrophysics · Planck scale The pattern the whole sky is written in. The sky as a disc — zenith at the centre, horizon at the rim, equal angles at equal distances — with the sun 30° above the horizon. Each short line is the direction the electric field vibrates in at that point, and its length and darkness are how polarised the light there is. The directions are perpendicular to the plane containing the sun, the observer and the point, which puts them tangent to circles centred on the sun. The heavy arc is the locus 90° from the sun, where the polarisation is strongest — 74 per cent here — and it is a great circle rather than a patch: a band across the sky, not a region near the horizon. This is what a polarising filter on a camera acts on, and it is why turning one darkens a band of sky and leaves the rest almost untouched, and why the effect is strongest when the sun is off to one side and absent when it is behind the photographer.

The pattern the sky is written in

Scattered sunlight is polarised, so the whole sky carries a direction of vibration at every point — arranged in circles about the sun, strongest on the great circle ninety degrees away from it, and vanishing at points that were found by looking before anyone could explain them. Bees navigate by it and a camera filter reads one band of it.

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

The area that is not allowed to shrink

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

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

The clock that measures a height

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

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

The fall that does not depend on what is falling

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

astrophysics · Equivalence principle The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

relativity · Velocity addition 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.

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.

relativity · Relativistic dynamics 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.

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.

relativity · Spacetime diagram 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.

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.

relativity · Relativistic dynamics 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.

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.

relativity · Relativistic dynamics 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.

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.

quantum · Uncertainty The modes of a wire, counted. Nyquist's argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, and the line is a one-dimensional cavity whose standing waves are spaced c/2L apart in frequency. Each mode has an electric and a magnetic energy, both quadratic, so equipartition gives it kT — the same half a kT per quadratic term that a heat capacity counts. The upper points are how many modes fall in a band of a hundred and thirty-seven megahertz, for five line lengths, from 17 on a thirteen-metre line to 8,558 on one of six kilometres. The lower points are the power each end therefore receives, as a fraction of kTΔf, and they converge on one: a longer line has proportionally more modes and takes proportionally longer to deliver them, so the length cancels. The longest line lands within 0.005 per cent and the shortest is 4.5 per cent low, because thirteen metres holds only seventeen whole modes in the band and the remainder is a real granularity rather than an error. What is left in the limit is kTΔf — a noise power with no resistance in it at all, and none of the line's properties either.

Half a kT in a piece of wire

Count the quadratic terms in a molecule's energy and equipartition gives a heat capacity. Count them on a transmission line instead and the same theorem gives a resistor's noise voltage — 4kTRΔf, with nothing in it about what the resistor is made of. A fifty-ohm input at room temperature says 0.91 nanovolts in every root hertz, and no design removes it.

thermodynamics · Equipartition 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.

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.

thermodynamics · Equipartition How large the hidden dimensions would have to be. The size extra dimensions would need, for gravity's true scale to be at a TeV rather than at 10¹⁹ GeV, against how many of them there are — a logarithmic axis of metres, for three choices of the true scale. The relation is the one that makes the arrangement work: the Planck mass observed in four dimensions is M_*^(2+n)Rⁿ, so gravity is weak because its field spreads into a volume nothing else can enter. Solved for R at a true scale of a TeV: 1 dimension needs 2.9e+13 m, 2 dimensions needs 2.4e-3 m, 3 dimensions needs 1.0e-8 m, 4 dimensions needs 2.2e-11 m, 5 dimensions needs 5.4e-13 m, 6 dimensions needs 4.5e-14 m. The two horizontal lines are where experiment has been. One extra dimension would have to be of order a hundred astronomical units, which would have wrecked the orbits of the planets and is excluded absolutely. Two would have to be of order a millimetre — which is what made the arrangement famous, because a millimetre is a distance a laboratory can test, and torsion balances have since verified the inverse-square law down to fifty-two micrometres and excluded it. Three or more sit below a nanometre, where no gravitational measurement reaches, and are untouched.

The scale that may not be where it looks

Every Planck number assumes gravity is four-dimensional all the way down. If it is not — if the field spreads into dimensions compact enough to have escaped notice — the true scale where gravity becomes strong could be at a TeV, and the whole remoteness of the Planck scale would be an artefact of where the field lines go. It is the one part of the subject an experiment can address, and the experiments have addressed it.

astrophysics · Planck scale 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.

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.

quantum · Entanglement 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.

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.

quantum · Entanglement How large now is, on this planet. By how much a synchronisation carried around a region of the rotating Earth fails to come back to itself, against the size of that region — from a metre to the whole planet, both axes logarithmic. The three horizontal lines are what three kinds of clock can resolve, and where each crosses the curve is where that clock can detect that 'now' is not a global notion: a good wristwatch at 785 thousand km, a quartz oscillator at 25 thousand km, a caesium clock at 785 km. Carried the whole way round the equator the defect is 207 nanoseconds, which is sixty metres of light travel and is the correction every satellite-navigation system applies. The effect is not small and not exotic; it is a routine engineering term, and the reason it was not an engineering term before 1955 is that nothing could measure it. A wristwatch's now is global out past the Moon; a caesium clock's reaches about the width of a large country.

How big now is

Three earlier arguments have established that a global now is a choice, that part of the choice is convention, and that for a rotating observer no consistent global choice exists at all. What survives is a size. Every observer has a local now, and how local is computable: on the rotating Earth it is 785 kilometres to the nanosecond, and a freely falling frame is inertial over the tolerance times the distance to the centre, divided by two.

relativity · Simultaneity The edge that is a straight line, not a step. The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's.

Below the gap, where there is nothing to absorb

A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about.

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

A few cycles that are only mass and spin

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

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

The ring that does not come back

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

astrophysics · Gravitational waves No beam is narrow enough and wide enough at once. Three lengths at the far end of a Stern-Gerlach magnet 10 centimetres long with a gradient of 1000 tesla a metre, against the width of the beam entering it, for an electron at 100 electronvolts. The spin splitting is a horizontal line at 2.9e-6 metres: it does not depend on the beam's width. The Lorentz blurring rises in proportion to the width, because the field a particle sees depends on where in the beam it is, and the divergence-free condition ties a gradient in one component to a gradient in another. It overtakes the splitting at 1.0e-9 metres. Narrower than that and diffraction has already spread the beam by 3.9e-3 metres, which is larger still. There is no width at which the splitting is the largest of the three, and the magnet's length and gradient cancel out of the comparison entirely — so no magnet helps.

The experiment that defines spin and cannot be done on it

A Stern–Gerlach magnet separates magnetic moments and is how spin was discovered. It cannot be made to work on a free electron, and the obstruction is not the apparatus: the field gradient that splits the beam also deflects the charge by an amount that varies across it, and the ratio of the splitting to that blurring comes out as the de Broglie wavelength over the beam width — with the magnet's length and gradient cancelling exactly.

quantum · Spin A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

relativity · Relativistic thermodynamics

Named alongside it

The objects these essays reach for when they reach for this one.

SuperpositionSpinQuantum stateEntanglementThe Lorentz transformationUncertainty principleAngular distributionBlack holeCorrelationDecayEquivalence principleGravitational waves

All concepts