Illustrated Physics
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Where the model stops — part 1 of 3

Every picture in physics has a domain of validity, and the interesting physics usually lives at its edge.
Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place. Mechanics

The angle that throws furthest, and why nobody notices

Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.

 Refraction from n = 1 into n = 1.5. A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires. Optics

The bend at the boundary, and what it is really about

Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all.

 The field of a dipole. Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field. Electromagnetism

Field lines are a choice, not a discovery

Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation.

 The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there. Electromagnetism

The field before the lines were drawn on it

A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.

 A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light. Relativity

Two axes, one speed, and a diagram that does the arguing

Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive.

 Simultaneity at β = 0.5. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other. Relativity

Now is a choice of slicing

Two events happening at the same time is not a fact about the events. It is a fact about who is asking, and different observers slice spacetime at different angles.

 The pendulum's period against its amplitude. The exact period of a simple pendulum divided by the small-angle period, plotted against amplitude. The small-angle formula is the horizontal line at one; the exact curve leaves it slowly and then climbs without limit as the amplitude approaches a half turn. Mechanics

The period that depends on the swing, computed exactly

A pendulum's period is not independent of amplitude. The exact answer is an elliptic integral, it has no elementary form, and plotting it shows precisely where the famous approximation earns its keep and where it collapses.

 A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light. Relativity

The length that depends on when, and is not really about length

A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.

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

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

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

 The blackbody spectrum, against what classical physics predicted. Spectral exitance against wavelength for a blackbody at 3000, 4000, 5000 kelvin, in kilowatts per square metre per nanometre. Each curve peaks at the wavelength Wien's displacement law gives — 966 nm at 3000 K, 724 nm at 4000 K, 580 nm at 5000 K — and falls to nothing at short wavelengths. Quantum

The curve that would not come down

Classical physics predicted that a warm object radiates infinite power at short wavelengths. Every step of the derivation was correct, the prediction was absurd, and closing the gap required assuming that energy comes in lumps.

 A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do. Quantum

The wall that is not quite a wall

A particle without enough energy to climb a barrier sometimes appears on the other side of it. The probability falls exponentially with the barrier's width, which is why the effect is invisible at ordinary scales and why it can be turned into a microscope.

 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. Quantum

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.

 Four answers to one push. Shear stress against shear rate for four fluids. The straight line through the origin is the Newtonian definition and is the only one of the four for which the word viscosity names a number. The Bingham fluid does not move at all until the stress passes 0.4, which is why toothpaste holds a shape on a brush and why wet concrete can be stood in a heap. Fluids

The fluid that answers back

For water, stress is proportional to how fast it is sheared, and the constant of proportionality is its viscosity. For paint, blood, ketchup and cornflour in water, it is not — and once the proportionality goes, so does the idea that viscosity is a number a substance has.

 A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell. Thermodynamics

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

 Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do. Quantum

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

 Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit. Astrophysics

The bend Newton got half right

A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.

 How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical. Astrophysics

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

 The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion. Astrophysics

A charge that turns must glow

An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

 The dip that decides it. Two equally bright points seen through a circular aperture, their Airy patterns added, at 0.7, 1, 1.6 times the Rayleigh separation. Below each is the depth of the dip between the peaks, measured off the drawn sum: 0.0%, 26.5%, 91.6%. At exactly the Rayleigh separation the dip is 26.5% — the criterion is a convention about how much of a dip a detector can see, not a threshold anything crosses. Below it the two peaks merge into one and the pair is gone; above it the answer was never in doubt. Optics

How far apart two things have to be

An instrument's ability to tell two things apart is not set by the quality of its glass. It is set by the width of the hole light comes through, by a factor that is the first zero of a Bessel function — and the threshold everyone quotes is a convention laid over a computed dip of 26.5 per cent.

 Two sources 4 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths. Optics

Why two lamps never interfere

Adding amplitudes is unconditional; fringes are not. What decides is whether the phase difference holds still for longer than a detector takes to record it — and a 10 nm slice of white light holds it for 100 femtoseconds, across a path of 30 micrometres.

 The term that abolishes the inner orbits. The effective potential of the Schwarzschild geometry per unit mass, in units of c², at 4 angular momenta, against radius in Schwarzschild radii. Newton's version has a minimum — a stable circular orbit — at L̃²/GM for every angular momentum there is, however small, and it is the dashed curve at the same L̃, here with its minima at 10.13, 7.61, 6.00, 5.12 rs. General relativity adds one term, −GM L̃²/c²r³, and that minimum stops existing below a definite angular momentum. At L̃ = 4.50 GM/c the barrier and the well are still separate, at 1.83 and 8.29 rs. The two merge at L̃ = √12 GM/c, at 3 rs = 6GM/c² — the innermost stable circular orbit, where the curve drawn here has neither a maximum nor a minimum but a single inflection. Below that momentum — the curve at L̃ = 3.20 GM/c — there is no stationary point anywhere outside the horizon, so no circular orbit exists at all, at any angular momentum whatever. None of that is a property of matter; it is a statement about the geometry, and it fixes the energy of the innermost orbit at √(8/9) = 0.94281 of mc², so 5.719% of the rest mass has been radiated by anything that reached it. Every well drawn here was located on the emitted curve by golden section and checked against the closed form. Astrophysics

The orbit that cannot be made smaller

Newtonian gravity allows a stable circular path at every radius, however tight, and there is no innermost one. General relativity adds a single term to the expression that says so, and below 6GM/c² no stable circular path exists at any angular momentum whatever. Anything arriving there has radiated 5.72% of its rest mass, which is eight times what hydrogen fusion converts.

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

The axis that will not hold

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

 Newton's answer, Laplace's, and the measurement. The speed of sound in 4 gases at 273.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 15.5% low for air, 22.5% low for helium, 22.5% low for argon, 12.0% low for carbon dioxide, and the corrected one is right to 0.05% for every gas here. Turning it round: γ read off each pair of bars is 1.400 for air, 1.665 for helium, 1.666 for argon, 1.290 for carbon dioxide, which is 1 + 2/f with f = 5.0, 3.0, 3.0, 6.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move. Waves

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

 How large a box may be before gravity shows in it. The largest freely falling laboratory in which nothing can be detected, against how long the experiment runs, for an instrument resolving 10⁻⁹ m. Free fall removes the field and leaves the gradient: two masses released a distance apart converge across the fall and separate along it, by (GM/r³)Lt²/2, so the box that stays undetectably flat shrinks as the square of the time. At one second the limits are 1.30 mm at the Earth's surface, 5.07 mm at the Sun's surface, 8.61 nm at a white dwarf, 1.86·10⁻¹⁷ m at a neutron star, 3.88·10⁻¹⁷ m at a stellar black hole, 388 mm at a giant black hole. Two things are worth reading off it. The equivalence principle is local in time as much as in space, on the same footing and with a worse exponent — patience is more expensive than room. And the tidal parameter GM/r³ falls with mass at a horizon, so a large enough black hole is one an experimenter could fall through with a metre-sized laboratory and a very good interferometer and detect nothing at all: the curve for the giant lies above the Earth's, not below it. Astrophysics

The term free fall cannot remove

Fall freely and gravity disappears. It disappears only to the extent that the falling laboratory is small — what survives is the gradient, which pulls two released masses together across the fall and apart along it. Given an instrument, the size of the box in which nothing is detectable is computable, and that number is the whole content of the word "locally".

 Where the sky's blue goes. How many times more strongly a sphere scatters 450 nm light than 650 nm light, against its radius, with the radius on a logarithmic axis running from a couple of nanometres to twenty micrometres. On the left the ratio sits at 4.35, which is the fourth power of the wavelength ratio and the whole reason the daytime sky is blue. It does not stay there. By a radius of 245 nm the preference has halved, and by a micrometre it has essentially gone: a particle comparable with the wavelength scatters every visible colour within a few per cent of equally, which is why a cloud is white, why fog is white, why milk is white and why the exhaust of a cold diesel is white while the smoke of a cigarette — whose particles are ten times smaller — is blue. Nothing about the material changed between one end of this axis and the other; only the size did. Optics

When the particle is the size of the wave

The sky is blue because small things scatter short wavelengths far more strongly. A cloud is made of the same water and scatters every colour alike. Nothing about the material changed — only the size, and one dimensionless number crossing one.

 A peak that leaves before it should have arrived. Two pulses at the far face of a cell, both normalised to the peak the vacuum one reaches. One has crossed empty space; the other has crossed a medium with two gain lines either side of its carrier, whose group index there is -3.85 — negative, so the envelope's peak should emerge early, and it does. Measured off the two curves the advance is 616 in units where the carrier period is 2π, against 582 predicted from the group index alone; the difference is the higher-order dispersion the group index leaves out. The advance is 0.21 of the pulse's own duration, and the peak leaves the far face before the input peak has entered the near one. Nothing has outrun anything. The emergent pulse is a reshaped version of the input's leading edge, which arrived in plenty of time and already contained — for a smooth pulse — everything needed to reconstruct the rest; the medium amplifies it by 1.14× and delivers it early. Give the pulse a genuine front, a moment before which it is exactly zero, and that front travels at the speed of light in every medium there is. Waves

The speed that carries no signal

In the right medium a pulse's peak emerges from the far side before it entered the near one. The measurement is real, it has been made, and nothing has outrun light — because the peak of a smooth pulse was never carrying any information in the first place.

 The part of the curve no fluid follows. Van der Waals' isotherms in reduced units, at 5 temperatures either side of the critical one, so that nothing about any particular substance appears. Above the critical temperature the pressure falls monotonically as the volume grows, which is what a fluid does. Below it the curve develops a loop with a rising section in the middle, and that section says the pressure increases as the substance expands — a material with negative compressibility, which cannot exist, because any fluctuation would run away. What happens instead is drawn as the horizontal line: the substance separates into two phases at one pressure, and the volume moves along the line as the proportions change. Its height, 0.6470 of the critical pressure, is fixed by requiring the two areas the line cuts off to be equal, which is the condition that the two phases have the same Gibbs energy. It meets the curve at volumes 0.603 and 2.349, a ratio of 3.9, and those are the densities of the liquid and its vapour. The two turning points of the loop, at 0.72 and 1.53, bound the part that is not merely unobserved but impossible; between them and the construction the substance can be made to sit, superheated or supercooled, until something nucleates. Thermodynamics

The part of the curve no fluid follows

One equation for a real gas produces isotherms with a rising middle section, which says a substance would expand as the pressure on it grows. Nothing does that. What replaces it is a horizontal line whose height is fixed by making two areas equal, and the condition is not a convenience.

 The bigger the hole, the gentler the horizon. The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present. Astrophysics

The horizon that nothing marks

A falling body is torn apart by the difference in gravity between its ends. At the horizon of a black hole that difference goes as the inverse square of the mass, so a large enough hole can be entered intact — and nothing local happens at the crossing to say it has occurred.

 The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term. Optics

The reflection that happens where the glass is not

Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed.

 A disc that generates 0.785 V with no changing flux anywhere. The force per unit charge on a carrier in a conducting disc of radius 100 mm spinning at 3000 revolutions a minute in an axial field of 0.5 tesla, against how far out the carrier is. Each carrier is moving through the field at ωr, so v × B pushes it radially with a force per unit charge of ωrB — zero at the axle and largest at the rim. The area under the line is the potential difference between the axle and the rim, which is ½BωR² = 0.7854 volts. Nothing in this calculation mentions a loop, a flux or a rate of change, and the flux through the circuit formed by the disc, the contacts and the external wire does not change at all while the machine runs. Electromagnetism

The rule that is two laws wearing one coat

The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.

 A siphon's pressure, and the 10.09 m it cannot pass. The absolute pressure of the liquid along a siphon, from the upper surface, over the crown, and down to an outlet 1.2 m below the surface, for 4 crown heights. The profile is hydrostatic and depends on nothing but height: the tube's shape, its length and its bore do not appear. At the crown the liquid is below atmospheric pressure by ρg times the lift, and the whole question of how high a siphon can reach is whether that number stays above the liquid's vapour pressure — 2.34 kPa for water at 20 °C, which puts the ceiling at 10.09 m. a 2 m crown sits at 81.7 kPa and holds, a 6 m crown sits at 42.5 kPa and holds, a 9.5 m crown sits at 8.1 kPa and holds, a 11.5 m crown sits at -11.5 kPa and is below the vapour pressure, so it boils. The ceiling is a property of the liquid, not of the mechanism: a degassed liquid that can be pulled into tension has no such limit, and siphons in a vacuum. Fluids

The height a siphon cannot pass

A siphon will not lift water more than about ten metres, and the usual explanation for the limit is also given as the explanation for the mechanism. It cannot be both. A siphon runs in a vacuum, with degassed water, over a crown no atmosphere could support.

 Stationary, and not always shortest. Three mirrors through the same point, with the same source and the same detector, and the length of the reflected path against where on the mirror the ray strikes. The actual ray is the one at the centre in all three, because the tangent there is horizontal and the two angles are equal whatever the curvature. Its path is a minimum at 0.6× the ellipse's curvature, every path equal at 1× the ellipse's curvature, a maximum at 1.6× the ellipse's curvature. On the most curved mirror every path the light did not take is shorter than the one it did, so the principle cannot be stated as least time and survive; it has to be stated as stationary time, and the difference is not a technicality. Optics

The path that takes the longest time

Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.

 Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years. Thermodynamics

The point at which the two become one

Heat a sealed tube of carbon dioxide and the meniscus inside it does not boil away — it fades, the two densities converging until there is nothing to separate. Twenty millikelvin before that happens the fluid turns milky, and the exponent describing the last approach is a number van der Waals got wrong and could not have got right.

 The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does. Astrophysics

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

 The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain. Electromagnetism

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

 What a coordinate is worth, by the shape of its energy. The mean energy stored in one coordinate, in units of kT, against the power with which that coordinate enters the energy. The curve is 1/n and the points are the same average obtained by integrating the Boltzmann weight numerically, agreeing to 3.3e-7 per cent at worst. an energy going as |x|^1 holds 1.0000 kT; an energy going as |x|^1.5 holds 0.6667 kT; an energy going as |x|^2 holds 0.5000 kT; an energy going as |x|^3 holds 0.3333 kT; an energy going as |x|^4 holds 0.2500 kT; an energy going as |x|^6 holds 0.1667 kT. The familiar half a kT is the case n = 2 and nothing more general than that: a coordinate whose energy is linear in it — the momentum of an ultrarelativistic particle, or a field with no restoring force but a constant tension — carries a whole kT, and one confined by a very steep wall carries almost nothing. Equipartition is a theorem about quadratic terms, and calling it a theorem about degrees of freedom is the substitution that makes it fail. Thermodynamics

The share that is not half a kT

Equipartition is quoted as half a kT for every degree of freedom, and it is nothing of the kind. It is half a kT for every *quadratic* term. A coordinate whose energy is linear in it carries a whole kT, and a gas hot enough that its particles' energy is pc rather than p²/2m therefore holds twice what the counting says — which drops its ratio of specific heats to four thirds and puts a star on the edge of being able to hold itself up.

 The figure a static field is not allowed to close. A spacetime diagram of two clocks held at fixed heights 22.5 metres apart, drawn as though spacetime were flat: time upward, height to the right, light at forty-five degrees. The lower clock sends two pulses; the upper clock receives them. Because the field does not change with time, nothing about the second pulse's journey differs from the first's, so the two null lines are congruent and the four worldlines bound a parallelogram. Opposite sides of a parallelogram in flat spacetime have equal length, so the proper time between emissions must equal the proper time between receptions, and the two clocks must agree. They do not: the measured fractional difference across a tower this tall is 2.455e-15, which Pound and Rebka established in 1960 and Pound and Snider confirmed to one per cent in 1964. Every step above is either a definition, an assumption of staticity, or a theorem of flat geometry — so the measurement refutes the flatness. No field equation has been written down, and none is needed: a laboratory result twenty-two metres tall is already incompatible with a flat spacetime. Astrophysics

The parallelogram that will not close

Two clocks twenty-two metres apart in a lift shaft run at different rates, by two parts in a thousand million million. That measurement, on its own, is enough to prove that spacetime cannot be flat — and the proof needs no field equation, no curvature tensor and no astronomy. It needs one drawing and the fact that opposite sides of a parallelogram are the same length.

 A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration. Quantum

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

 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. Quantum

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.

 N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here. Optics

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

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

The staircase that never reaches the floor

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

 The self-force matters at 6.27 × 10⁻²⁴ s, and nowhere a charge has ever been. The ratio of the radiation reaction to the applied force, which is τ divided by the time the force takes to change, on a logarithmic axis. τ = μ₀q²/6πmc = 6.266e-24 s for an electron, and light crosses 1.879 femtometres in that time — two thirds of the classical electron radius. a 3 GHz accelerating cavity: 1.9e-14; a 500 nm optical field: 3.8e-9; an electron orbiting a proton, ground state: 4.1e-8; an X-ray at 0.1 nm: 1.9e-5; over a classical electron radius: 6.7e-1. The largest of them, "over a classical electron radius", is still 1.5e+0 times too slow. So the correction is never large for any force anybody can apply, and the only regime where it would be is one in which the charge's own structure has already made the whole description meaningless. Astrophysics

The force a charge exerts on itself

Larmor's formula says how much an accelerating charge radiates and says nothing about who pays. Conservation says the charge does, so there is a force on it — and the equation that force produces has a free particle accelerating for ever with nothing pushing it, or else beginning to move before it is pushed. Both solutions are absurd, and the interval over which they are absurd is smaller than the electron the equation was written for.

 Rough contact grows as W^1.000, and one smooth bump as W^0.667. Real contact area against load, both logarithmic, three ways. Plastic flow of the asperities gives A = W/H, a slope of exactly one, which is the account Bowden and Tabor's argument uses. A single elastic sphere gives Hertz's answer, A ∝ W^(2/3), and a friction coefficient falling as W^(−1/3) — Amontons' law would be false. A rough elastic surface, with many asperities spread exponentially in height and integrated here rather than quoted, gives a slope of 1.0000, because pressing harder mostly recruits new contacts rather than enlarging existing ones. The dotted curve repeats that integration with a Gaussian spread of heights and gives 0.9695 — the upper tail of a Gaussian is nearly exponential, so the answer is nearly and not exactly linear. Amontons' law does not need plasticity. It needs roughness, it is exact for one distribution and good to three per cent for another, and it survives whichever way the individual junctions deform. Mechanics

The grip that is not a coefficient

A friction coefficient is independent of load and of area, and the reason is usually given as plastic flow of the asperities. It is not — a rough elastic surface gives the same law, integrated here to a slope of 1.0000, because pressing harder recruits new contacts rather than enlarging old ones. What breaks the law is a contact that is smooth, or a material that dissipates in its bulk — where μ passes one and stops being a coefficient at all.

 Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve. Mechanics

The error that doubles on a schedule

Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

 A pore that lifts 100 m has to be 0.1 µm or finer. Capillary rise against pore radius, both logarithmic, for a liquid of surface tension 72.8 mN/m at a contact angle of 20°. The relation is a straight line of slope −1 — halve the pore and double the rise — and the two horizontal marks are the height in question, 100 m, and the 10.3 m that one atmosphere supports. 0.01 µm lifts 1394.7 m; 0.1 µm lifts 139.5 m; 1 µm lifts 13.9 m; 5 µm lifts 2.8 m; 20 µm lifts 69.7 cm; 50 µm lifts 27.9 cm. The conducting vessels of a tree are tens of microns across and lift under a metre; the pores in the membranes between them are tens of nanometres and would lift kilometres. Those are the same expression at two scales, and only one of them is a pipe. Fluids

The column that is pulled, not pushed

A capillary fine enough to lift a hundred metres is far too fine to carry any flow, and one wide enough to carry the flow lifts under a metre. Neither is how the water gets up a tree. The column is under tension — an absolute pressure of −0.88 MPa at the top, which a gas cannot have — held together by cohesion and prevented from tearing by pores a few tens of nanometres across.

 The thickness a paste can hold on a slope. The greatest thickness a yield-stress fluid can rest at without flowing, against the angle of the surface it is resting on, on a logarithmic vertical axis, for four materials. A layer of thickness h puts a shear stress ρgh sin α on its own base; it stays put while that is below the yield stress and flows when it is not, so the critical thickness is τ_y divided by ρg sin α and it depends on nothing else — not on the viscosity, not on how long it is left, not on how it was put there. On a vertical wall the numbers are 1.4 mm of ketchup, 9.2 mm of mayonnaise, 15.7 mm of toothpaste, 15.1 mm of basaltic lava, which is why toothpaste stays on a brush and ketchup does not stay on a plate held up. Read the other way it is a measurement: a lava flow that came to rest 3 centimetres thick on a 30° slope had a yield stress of about 400 pascals, and that is how the rheology of a flow nobody was standing next to is recovered from its shape a thousand years later. The model stops where the layer is thin enough for surface tension to matter and where the material's yield stress depends on how long it has been left alone, which for most of these it does. Fluids

The paste that holds up its own hill

Toothpaste stands on a brush and ketchup does not stand on a plate, and the difference is a single number with the units of a pressure. Below it a material does not flow slowly — it does not flow. That threshold turns a rheological property into a length, and the length is why a lava flow's thickness says what the lava was made of.

 How much of the answer a finite wire gives back. The field beside a straight segment of wire, divided by what the infinite-wire formula would give, against the length of the segment in units of the distance to the field point, on a logarithmic horizontal axis. The field is integrated element by element along the segment rather than evaluated from a closed form. A wire ten times as long as the distance already gives 92.8 per cent of the infinite answer, and one as long as the distance gives 45 — which is the practical content, and the reason the infinite-wire result is used for laboratory wires without apology. The second column is the part that matters for the law rather than for the number. The circulation of this field round a circle of radius d is not μ₀I; it falls short by exactly the fraction the segment fails to subtend. What makes up the difference is the displacement current of the charge piling up at the segment's two ends, and the two terms, both integrated here, sum to μ₀I to within 2.2e-8 per cent at every length. So Ampère's law is not approximately true for an open circuit and exactly true for a closed one — it is exactly true always, and it is a computation only when symmetry supplies the direction and the magnitude along the loop. Symmetry is doing the work; the law is doing the bookkeeping. Electromagnetism

The law that is always true and rarely useful

Ampère's law holds for every loop and every current. Apply it to a wire of finite length and it gives an answer that is wrong by half — until the displacement current of the charge piling up at the wire's two ends is put back in, whereupon the two terms sum to μ₀I exactly at every length. The law was never approximate. It was never a computation either.

 How far a 1-gravity ship gets, against its own clock. The distance covered by a ship accelerating steadily at 1 gravity for half the trip and braking for the other half, against the time on its own clock, on a logarithmic vertical axis. The curve is a cosine hyperbolic and therefore an exponential once the ship is relativistic, which it is after about a year: Proxima Centauri in 3.5 shipboard years, Sirius in 4.6 shipboard years, the Pleiades in 11.9 shipboard years, the galactic centre in 19.8 shipboard years, Andromeda in 28.6 shipboard years. Nothing about this violates anything. The speed never reaches c — it is the hyperbolic tangent of the rapidity and after 3.5 years it is 0.9495 at turnover — and every one of those journeys takes slightly more than the distance in years as measured from home. What is growing exponentially is not the speed but the length contraction, and the traveller's honest description of the trip is that the distance shrank. The rapidity is what accumulates steadily: it grows by one unit every 0.969 years of shipboard time, for ever, with no ceiling anywhere in the arithmetic. That is the whole reason the numbers come out survivable, and the reason the fuel does not. Relativity

The ship that never arrives at c

Accelerate at one gravity and never stop. The speed creeps toward light and never reaches it, and meanwhile the galactic centre is twenty shipboard years away and Andromeda twenty-nine. What makes the journey survivable is that rapidity has no ceiling; what makes it impossible is that the fuel goes as the exponential of the same quantity.

 One sharp kick, heard 6 pulse-lengths away. The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 12.1 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three. Waves

The arrival that keeps arriving

A clap heard across a field arrives and stops. The same clap in two dimensions arrives at the same instant and then goes on arriving for ever, fading as one over the time — and the difference is not absorption, or echo, or scattering. It is the number of dimensions, and sharp arrival happens in three of them and in no even number at all.

 The height a 1 mK difference lifts helium. The head of liquid that a temperature difference of 1 millikelvin can support across a superleak — a plug fine enough that the normal fluid cannot pass and the superfluid can — against the temperature it is done at. Only the normal component carries entropy, so warming one side makes the superfluid flow toward the warm side until the pressure difference balances, and equilibrium is at ΔP = ρSΔT. The height that supports is SΔT/g, in which the density cancels exactly; computed both ways here the two agree to machine precision. The numbers are the striking part: 2.0 mm at 1.2 K, 4.6 mm at 1.4 K, 9.2 mm at 1.6 K, 19.4 mm at 1.8 K, 46.9 mm at 2 K, from a temperature step a thousand times smaller than anything a hand could feel. Aim a light at the warm side and the liquid does not merely rise but jets out of the tube, which is the fountain effect Allen and Jones found in 1938 and the most direct demonstration that helium II is two fluids rather than one. The entropies used are measured values; everything else on this chart is computed from them. Fluids

The fountain a lamp can drive

Below two degrees above absolute zero, liquid helium behaves as though it were two fluids occupying the same space — one carrying all the entropy and all the viscosity, the other carrying neither. It is not a metaphor and not a mixture. Shine a light on one side of a fine plug and the liquid jets out of the tube, because a temperature difference of a thousandth of a degree is a pressure of a hundred pascals.

 Which wavelengths grow instead of oscillating. The dispersion relation of a self-gravitating isothermal gas, ω² = c²k² − 4πGρ, with each axis measured in the scale the gas sets for itself. Short wavelengths oscillate: pressure wins, and the disturbance is a sound wave. Long wavelengths do not: ω² is negative, so the disturbance grows exponentially instead of travelling, and the region collapses. The changeover is at λ_J = c√(π/Gρ), and it happens because pressure support acts on a sound-crossing time that grows with the region while gravity's collapse time does not depend on the size at all. Four densities spanning 6 decades are drawn and they lie exactly on top of one another, to 6e-16, because the criterion has no scale of its own: it is the same curve for a diffuse cloud and for a protostellar core, with different numbers written on the axes. In this gas at 10 K those numbers are 2.12 pc and 28.72 solar masses at 10² cm⁻³, 0.21 pc and 2.87 solar masses at 10⁴ cm⁻³, 4372 AU and 0.29 solar masses at 10⁶ cm⁻³, 437 AU and 0.03 solar masses at 10⁸ cm⁻³. Astrophysics

The disturbance that grows instead of travelling

A sound wave in a gas oscillates because pressure restores what the disturbance displaced. Add the gravity the gas exerts on itself and the restoring force acquires a competitor that does not weaken with size — so above one wavelength the sum changes sign, the frequency becomes imaginary, and the disturbance stops travelling and starts growing. It is the same wave equation with one term subtracted.

 The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis. Astrophysics

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

 The brightest anything of a given mass can be. The Eddington luminosity against mass, with the main sequence drawn beside it. Radiation pushes outward on the electrons and gravity pulls inward on the protons, and both go as one over the distance squared — so the radius cancels out of the comparison entirely, checked here at three radii spanning four decades and coming out identical to 1e-20. What is left is a luminosity: L = 4πGMc/κ, which is 1.47 × 10³¹ watts per solar mass, or 3.8·10⁴ solar luminosities. Above it, radiation drives the outer layers away faster than gravity can hold them. The Sun is at 2.6e-5 of its own limit and in no danger; a star of 10 solar masses is at 1.2e-2; and a star of 100 is at 0.83, which is why the two lines converge at the top of the chart and why the most massive stars known are a few hundred solar masses rather than a few thousand. They do not fail to form for lack of gas; they blow away the gas that would have made them heavier, and once formed they shed mass continuously in a radiation-driven wind. The limit is the same expression for an accreting black hole, where it caps not the brightness but the rate at which mass can be taken on. Astrophysics

The brightness a mass cannot exceed

Light pushes outward on the electrons of a star and gravity pulls inward on its protons, and both forces fall off as one over the distance squared. The distance therefore cancels, and what is left is a limit on brightness rather than on size — 3.8 × 10⁴ times the Sun's luminosity for every solar mass, above which a body drives its own outer layers away.

 The temperature of an acceleration. The Unruh temperature against proper acceleration, both axes logarithmic. An observer accelerating through empty space finds it is not empty: the state that an inertial observer calls the vacuum, an accelerated one finds populated, with a thermal spectrum at T = ħa/2πck — which is 4.06e-21 kelvin for every metre per second squared. The line is straight because the relation is exactly proportional, and the numbers on it are what make the effect so hard to see: one gravity gives 3.98·10⁻²⁰ K, a centrifuge at 10⁵ g gives 3.97·10⁻¹⁵ K, an electron in a strong laser gives 40.6 K, the surface of a solar-mass hole gives 6.17·10⁻⁸ K. Reaching one kelvin requires 2.5·10²⁰ m/s², which is 2.5·10¹⁹ gravities and beyond anything that can be sustained. What makes the effect worth taking seriously despite that is not its size but its structure: it says the number of particles present is not a property of the field alone but of the observer as well, and the same expression with a black hole's surface gravity in place of the acceleration is the Hawking temperature exactly, checked here to a part in 10¹². Relativity

The temperature of an acceleration

Empty space is empty for an observer who is not accelerating. For one who is, the same state of the same field is a thermal bath at a temperature proportional to the acceleration — and the constant of proportionality is 4 × 10⁻²¹ kelvin for every metre per second squared, which is why nobody has felt it. What the effect changes is not what can be measured but what a particle is.

 What is lost is exactly what is recorded. Fringe visibility against the distinguishability of the record left in the environment, for six couplings between the interferometer and a marker. The points lie on the quarter circle V² + D² = 1, computed here to a part in 10¹² — the visibility from the output probabilities with the marker traced out, and the distinguishability from the overlap of the marker's two states, with nothing shared between the two calculations. The relation is the quantitative form of complementarity, and it is stronger than the usual statement: interference is not lost because something was disturbed, and not lost only when a measurement is made. It is lost exactly to the extent that the environment could in principle say which way the particle went, whether or not anybody looks at the environment. That is why decoherence is a matter of correlation rather than of disturbance: the coherence has not been destroyed but relocated, into a correlation between the particle and something else. Quantum

Where the interference goes

A superposition does not stop being a superposition when something interacts with it. What happens is that the coherence moves — out of the system and into a correlation between the system and its surroundings — and the interference disappears from any measurement made on the system alone. For a dust grain in air the move takes 10⁻²⁸ seconds, which is why nothing large has ever been seen in two places.

 A sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain. Waves

The front that steepens until it cannot

In a linear medium every wave keeps its shape, because every part of it travels at the same speed. Let the speed depend on the height by even a little and the crest overtakes the trough, the front leans forward, and after a time that can be written down the wave demands two values at one place — which is where the description ends and a shock begins.

 The far end that has not been told yet. A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving. Relativity

Nothing is allowed to be rigid

A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.

 The set an orbit that never repeats settles onto. 24,000 successive positions of one orbit of the map x' = 1 − 1.4x² + y, y' = 0.3x, after five hundred steps of transient have been discarded. Nearby points separate at e^0.4188 per step, so the orbit is unpredictable in the way a doubling error measures; and every one of the 24,000 points lies inside a box 2.558 by 0.767, a diagonal of 2.670, so it is going nowhere. Those two statements are not compatible with a smooth stretching: something has to bring the separated points back, and the bringing back is the visible fold at the left-hand end. The curve is not a curve. Every strand of it is a bundle of strands at any magnification, which is what an area contraction of 0.3 per step leaves behind when the stretching along the other direction is e^0.419. The 24,000 points paint 7,352 distinct marks at the resolution this is drawn at, which is itself a measurement of how little of the plane the set occupies. Mechanics

The fold that has to be there

Two trajectories that separate exponentially, in a region they can never leave, are being asked to do two incompatible things. The resolution is that the motion is folded back on itself over and over, and the object that survives infinitely many foldings is neither a curve nor a patch of surface — it has a dimension between the two, and the number can be measured two entirely different ways.

 The same liquid climbs a thin rod and is thrown off a thick one. The height of the free surface against distance from the axis, for rods of 10.0, 25.0, 60.0 millimetres radius turning at one revolution a second in the same polymer solution, with the Newtonian answer dashed. Two effects compete and they fall off at different rates: a hoop tension along the curved streamlines pulls the fluid inward and lifts it, falling as the fourth power of the distance, while the centrifugal term pushes it outward and lowers it, falling as the second. Near a thin rod the fourth power wins and the liquid climbs — 2.21 millimetres at the thinnest. Beyond a radius of 34.64 millimetres it does not, and the same fluid at the same speed is thrown outward exactly as water would be. The changeover is a property of the fluid and the rod together, so a demonstration that works on a glass stirring rod fails on a spoon handle. Fluids

The liquid that climbs the rod

Stir water with a rod and it is flung outward, leaving a dip at the centre. Stir a polymer solution and it climbs the rod instead. Nothing about the viscosity can produce that, however large it is made — what produces it is a second stress that appears in shear and has no Newtonian counterpart at all, growing as the square of the rate where the familiar one grows in proportion.

 A wave that dies with nothing to rub against. The electric field of a plasma wave at kλ = 0.5, against time in plasma periods, on a logarithmic scale, obtained by integrating the collisionless kinetic equation as an initial-value problem. There are no collisions in the equation, no viscosity and no resistance; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field nevertheless falls exponentially, at 0.1534 per plasma period, against the published root of the kinetic dispersion relation at this wavenumber, 0.1534, and Landau's asymptotic formula's 0.1514. Meanwhile the free energy of the perturbation — the weighted norm of the distribution plus the field energy, which the equation conserves exactly — moves by 2.4e-10. So nothing has been dissipated: every joule the field loses is still in the distribution, and the accounting closes to a part in ten thousand million. The energy has gone into the particles' ordered motion, and the information about the wave is wound into structure at finer and finer scales in velocity. Astrophysics

The wave that dies with nothing to rub against

Every damping in this collection so far removes energy from a wave and puts it somewhere warmer. This one removes it and produces no heat at all: there are no collisions in the equation, the entropy is unchanged, the whole thing runs backwards perfectly, and the wave still dies exponentially. What it dies into is structure in velocity too fine for a field to see.

 A current that falls by a decade for every ångström. Tunnelling current against the width of a vacuum gap, on a logarithmic scale, for three work functions covering the range of clean metal surfaces. The curves are straight because the transmission is an exponential in the gap, and their slopes are 0.833, 0.944, 1.044 decades per ångström — fitted to the drawn curves and agreeing with 2κ/ln 10 to a part in a million. At 4.5 electronvolts a change of ten picometres, a tenth of an atomic radius, changes the current by 24 per cent. That is the sensitivity a scanning tunnelling microscope lives on, and it is why the instrument measures height by holding the current fixed and recording what the piezo had to do: the current is far too steep a function of height to be read as one. Quantum

The last atom does all the seeing

A tunnelling rate falls by a factor of eight for every tenth of a nanometre of extra barrier, which is normally quoted as the reason nothing ever tunnels anywhere. Read the other way it is a microscope: the second-nearest atom of a blunt metal tip carries a two-hundredth of the current the nearest one does, so a tip nobody sharpened resolves a single atom, and the resolution comes from an exponential rather than from any piece of engineering.

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

The state that cannot be copied

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

 A stiffness that does not go to zero — it falls off a cliff. The renormalised stiffness of a two-dimensional superfluid against temperature, obtained by integrating the flow of the bare stiffness and the vortex fugacity, with the bare stiffness drawn for comparison. Below the transition the vortices are bound in pairs, screen one another and leave a finite stiffness; above it they unbind and there is none. What is remarkable is the value at which it disappears: 0.6430, against 2/π = 0.6366. That number does not depend on the material, on the bare fugacity, on the core energy, or on anything else — every two-dimensional superfluid loses its stiffness at exactly two over π times its own transition temperature, and measurements on helium films of widely different thicknesses fall on that line. A transition at which a quantity jumps to zero from a value nobody can adjust is unlike anything the usual classification of transitions describes. Thermodynamics

The transition with nothing to order

Every transition in this collection so far has an order parameter — a quantity that is zero on one side and not on the other. In two dimensions a continuous symmetry cannot break at any temperature above zero, so there is nothing for such a quantity to be, and by the usual reckoning there can be no transition. There is one anyway, and what changes at it is whether vortices are bound in pairs.

 A circumference that is more than 2π times the radius. The ratio of a rotating disc's measured circumference to 2π times its measured radius, against the speed of the rim, together with the rate of a clock carried on the rim. Rulers laid round the rim lie along their own direction of motion and are contracted; rulers laid along a radius lie across it and are not. So the circumference takes more of them than a stationary observer counts and the radius takes the same number, and the ratio is γ: 1.091 at β = 0.4, 1.400 at β = 0.7, 2.294 at β = 0.9. The geometry a rotating observer measures is therefore not Euclidean, and it is not Euclidean by an amount that depends on where on the disc the measurement is made. That is the observation Einstein said set him on the road to describing gravity with curved geometry: here is an accelerated frame, and here is a geometry in it that no choice of Cartesian coordinates can flatten. The rim's clock runs slow by the same factor, so a rotating frame has neither a common time nor a flat space. Relativity

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

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

The force a coordinate cannot see

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

 The boundary of everywhere a given speed can reach. Trajectories at one speed and five launch angles, with the curve that bounds all of them. The boundary is not one of the trajectories and is not the 45° launch: it is the envelope of the whole family, the locus of points where two neighbouring launches cross. Distances are in units of v²/g, so the greatest range is one and the greatest height a half, and the envelope is the parabola y = ½ − x²/2 joining them. Two things about it are worth having. Its focus is the launch point exactly — every point on it is as far from the gun as from the line y = v²/g — so the safety parabola is a conic with the same focus as the trajectories themselves. And a boundary made of crossings of neighbouring members of a family is a caustic: the same construction that makes the rainbow's edge bright, drawn here with cannon shells instead of light rays. Mechanics

Everywhere a throw can reach

Fix the speed and let the angle be anything. The trajectories fill a region, and the region has an edge — a curve that is not one of the trajectories, that touches each of them exactly once, and that turns out to be the same kind of object as the bright rim of a rainbow.

 The only part of the field that can pull. A dielectric slab part-way into a parallel-plate capacitor. Everywhere except at the slab's edge the field is perpendicular to the plates and therefore perpendicular to the direction the slab can move, so it exerts no force along that direction at all: inside the parallel-plate model, which is uniform between the plates and zero outside them, nothing pulls the slab anywhere. The force lives in the bowed lines drawn at the edge, where the field leaks past the end of the dielectric and acquires a component along the plates. Those lines are what the model throws away as a small correction near the boundary, and they are the entire mechanism. The energy method sidesteps the drawing altogether: differentiate the total energy with respect to the insertion and the answer is 1.328e-3 newtons, inward, without ever asking where on the slab the force is applied. Electromagnetism

The force that lives where the model is not

A slab of glass held at the mouth of a charged capacitor is pulled in. Inside the parallel-plate model there is no force at all — the field is perpendicular to the slab's motion everywhere — and the same model's energy nevertheless gives the pull exactly right. The mechanism is entirely in the part of the field the model throws away.

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