True where it was derived, used where it was not — part 2 of 5

“The reverberation after a sound is the room — echoes from the walls.”

A room supplies most of it and is not required for any of it. The tail computed here is in an unbounded, uniform, lossless medium with no boundary anywhere, and it is a property of two dimensions rather than of anything the wave met. A struck plate rings on for the same reason before its edges are considered.

Tested in The arrival that keeps arriving · the wave motion reading path

“The modes of a region depend on its shape, so counting them requires solving for that shape.”

Counting them does not. Three regions of exactly the same area — a square, a right triangle and an oblong four times as long as it is wide — have completely different spectra and mode counts that agree to leading order and differ only through their perimeters, with measured slopes matching −L/4π. The individual frequencies need the shape; the count needs an area and a length.

Tested in How many ways there are to vibrate · the standing waves reading path

“Air resistance reduces the range, so the best angle is still 45° and everything just falls short.”

Loss and asymmetry are separate effects and only the second moves the optimum. Integrating the trajectory at a drag force equal to 2.4 weights at launch puts the best angle at 38.1°, not 45° — and the best of five angles drawn at that drag is 32° — and the range curve leans: ten degrees under the optimum costs little and ten degrees over it costs a great deal.

Tested in The angle that drag moves · the projectile reading path

“This is parametric resonance, which is how a child pumps a swing.”

It is the same equation in a different corner of its chart. Parametric resonance drives a mode unstable by modulating at about twice its frequency; this drives a fixed point stable by modulating at many times any natural frequency. One takes energy in at resonance, the other never resonates with anything, and the boundary between them is on the same stability diagram.

Tested in Held up by a force that averages to nothing · the pendulum reading path

“The principle says nature is economical: it minimises effort, or time, or energy.”

The action is not effort, time or energy, and it is not minimised in general. It is the time integral of kinetic minus potential energy, a quantity with no independent meaning, and the content of the principle is that its first variation vanishes. Nothing in it is thrifty; the same formalism written with a plus sign describes nothing.

Tested in Least action, except that it is not least · the least action reading path

“Damp sand is stronger than dry, so a large enough damp pile can stand at any angle.”

Cohesion supplies a stress, not an angle, and the weight it must hold grows with the pile. The ratio c/ρg is a height: a sand with 200 Pa of cohesion stands vertically at a centimetre and is back within a degree of the dry angle by seventy centimetres. That is why a sandcastle works and a sand cliff does not.

Tested in The angle that does not know the size of the heap · the granular matter reading path

“Desalination is expensive because membranes are imperfect; a better membrane would make it cheap.”

A perfect membrane changes nothing about the floor. The minimum work is the free energy of mixing read backwards — 0.79 kWh per cubic metre for the first drop from seawater and 1.10 at half recovery — and a good plant already uses about three, so the whole of the remaining margin is a factor of three rather than the factor of a thousand a technological framing implies.

Tested in What it costs to take the salt out · the osmosis reading path

“Helium II is a mixture of two substances, one superfluid and one normal, in some proportion.”

There are no two kinds of atom and nothing that could be separated. The two-fluid description is a decomposition of one liquid's motion into two velocity fields, and its test is that quantities computed from it — a fountain height of 19 mm per millikelvin at 1.8 K, a counterflow speed of 0.2 m/s per watt per square centimetre — come out right, not that anybody can find the two components.

Tested in The fountain a lamp can drive · the superfluidity reading path

“A gas cloud collapses if its gravity is stronger than its pressure.”

Neither is a force until a size is chosen, and the comparison is between two times rather than two forces. Gravity collapses a region in a free-fall time that contains no radius at all; pressure can only answer on a sound-crossing time that grows with the region. So every gas is stable at small scales and unstable at large ones, and the criterion is a wavelength.

Tested in The disturbance that grows instead of travelling · the Self-gravity reading path

“Screening is the plasma's charges physically covering the test charge, like a coat.”

Nothing is stuck to anything. The electrons are moving at thermal speed and any individual one is near the charge for a moment; what is steady is the average excess density, which the Boltzmann factor gives as a small fractional enhancement over a region of size λ_D containing a hundred thousand particles. It is a statistical rearrangement, not a shell.

Tested in The long-range force that does not reach · the plasma oscillation reading path

“Whether radiation can blow a star apart depends on how close to it you are.”

It does not depend on distance at all. Radiation pressure goes as L/4πr²c and gravity as GM/r², so the ratio has no r in it — computed here at radii spanning four decades the ratio is identical to a part in 10¹⁵. What the comparison yields is a luminosity, and a body either is above it everywhere or below it everywhere.

Tested in The brightness a mass cannot exceed · the radiation pressure reading path

“The supernumerary arcs are interference between light from different drops.”

Every drop produces them independently. They are the interference of the two rays that a fold caustic joins — the pair that emerge from the same drop at the same angle with different impact parameters — and the calculation for a single drop reproduces their positions from that pair alone.

Tested in The fringes below the rainbow · the diffraction reading path

“A phase shift in optics is an optical path length — a thickness times a refractive index.”

Not all of it. Turning a half-wave plate changes the phase of transmitted circular light by exactly twice the angle turned, with the plate's thickness, material and the wavelength all unchanged and none of them in the answer. The measured slope is 2.000 radians per radian, and it is a property of the geometry of the path through the space of states.

Tested in The phase that is only a shape · the polarisation reading path

“Two vessels connected by a hole come to the same pressure.”

Only if the hole is larger than the mean free path. If it is smaller, what equalises is the flux each way rather than the pressure, and equilibrium sits at P₁/√T₁ = P₂/√T₂ — so a vessel four times hotter holds twice the pressure indefinitely. Every low-pressure gauge at a different temperature from its vessel reads wrong by that factor.

Tested in The gas that leaves is not the gas inside · the kinetic theory reading path

“A supersaturated vapour condenses because condensing lowers its free energy.”

It lowers the free energy of the finished droplet and raises it for every droplet on the way there. At a supersaturation of 3 the barrier is 73 kT and the critical droplet is 0.98 nm across; below that radius a droplet is more expensive than the vapour it came from and evaporates again. Thermodynamics says where the system is going and nothing about whether it can start.

Tested in The barrier a new phase has to climb · the phase change reading path

“An accelerating detector heats up because it is being shaken.”

Nothing is shaking it and no energy is being supplied by the acceleration itself. The detector's excitation is a property of how the field's ground state decomposes into modes for an accelerated observer: the inertial vacuum contains no quanta of the inertial modes and a thermal population of the accelerated ones, and the two descriptions are of the same state.

Tested in The temperature of an acceleration · the accelerated frames reading path

“The speed of light has been measured to nine decimal places, so of course it is the same in both directions.”

Every one of those measurements is a round trip — an interferometer, a time-of-flight to a mirror, a resonant cavity. The round-trip speed is c under every synchronisation convention: computed here across five of them the round-trip times agree exactly. What varies between conventions is the split of that round trip into two halves, and no measurement reads the split.

Tested in The speed that cannot be measured one way · the simultaneity reading path

“Earnshaw's theorem forbids stable levitation.”

It forbids it for a fixed charge in a static electric field in a charge-free region, which is four conditions. Diamagnetic levitation breaks the sign of the response, a Paul trap breaks the staticness, optical tweezers break it with dissipation and induced dipoles, and a maglev train breaks it with feedback. All four are real and none contradicts the theorem.

Tested in Nothing can be held still by a static field · the potential reading path

“Interference is destroyed because measuring the particle disturbs it.”

Disturbance is not the mechanism and is not needed. What suppresses the interference is the existence of a correlation between the particle and something else, whether or not anything is read out: the visibility equals the overlap of the environment's two states, and V² + D² = 1 exactly. An interaction that leaves no distinguishing record leaves the fringes untouched however violent it is.

Tested in Where the interference goes · the measurement reading path

“Noether's theorem says every symmetry gives a conservation law.”

Every *continuous* symmetry of the *action* does. A discrete symmetry gives none: reflection is a perfect symmetry of electromagnetism and there is no conserved quantity attached to it, only a selection rule. And a symmetry of the equations of motion that is not a symmetry of the action gives none either — the scaling that leaves Kepler's equation invariant changes the action by a factor and supplies nothing.

Tested in The conservation law a symmetry hands over · the least action reading path

“Adding supports makes a structure safer, because the load is shared more ways.”

It makes the load-sharing undetermined, which is a different thing. A four-legged table on an uneven floor puts more than half the load on two legs and can lift a third off entirely; the redistribution from a leg one part in a thousand short is computed here and does not depend on the load at all. Redundant structures are safer against a member failing and less predictable in what each member carries.

Tested in The table statics cannot settle · the Free-body reading path

“Waves add, so two sounds together are the sum of the two apart.”

Superposition is a property of linear equations, not of waves. In a nonlinear medium two tones produce their sum and difference frequencies and every combination of them, which is exactly how an acoustic parametric array makes a narrow beam of audible sound out of two ultrasonic ones. The figure of harmonic amplitudes here shows a single input tone growing five harmonics that were never present.

Tested in The front that steepens until it cannot · the wave motion reading path

“Solitons pass through each other because the equation is linear enough for superposition.”

During the overlap the sum of the two is emphatically not what is drawn — the merged profile is lower than the taller of the two constituents and is not their sum anywhere. What emerges is unchanged, which is a much stranger statement than superposition and is true of almost no nonlinear equation. The price is a phase shift, measured here at +0.55 for the fast one and −1.10 for the slow one.

Tested in The pulse two failures keep alive · the wave packets reading path

“The kinetic theory predicts viscosity proportional to the square root of the temperature.”

It predicts that for hard spheres, and real gases follow a steeper law — 0.77 for air at 300 K rather than 0.5. The excess is the attraction between molecules, which deflects a slow pair more than a fast one and so shrinks the effective cross-section as the gas warms. The exponent falls back towards a half by four thousand kelvin, which is the hard sphere being recovered when the kinetic energy is far above the well depth.

Tested in The thickness that goes both ways · the viscosity reading path

“The forces in a pack of grains can be found from the loads by statics, given enough care.”

Above the threshold the contacts are redundant, and the count of redundant contacts is measured here rather than argued: a fully packed lattice patch of 37 grains carries 19 contacts whose forces no balance of forces or moments determines. Two silos filled identically carry different force networks, and which one a given pack has depends on how it was built.

Tested in The heap that becomes a solid · the granular matter reading path

“Anything passing inside the Roche limit is torn apart.”

Only a body held together by its own gravity is. A body small enough for its material strength to beat its self-gravity ignores the limit entirely: Phobos orbits well inside Mars's and is intact, and a spacecraft or a person would be unaffected. The limit is a statement about self-gravitating bodies, and the crossover size below which strength wins is set by the material rather than by the orbit.

Tested in The distance that forgets the moon · the Self-gravity reading path

“A random walk of N steps of length λ covers a distance Nλ.”

It covers √N λ, and the difference is the whole subject. Measured over 240 independent walks here the exponent is 0.503 against an exact half, so a walk of 900 steps ends about 30 lengths from where it began rather than 900. Escaping a body therefore costs (R/λ)² steps rather than R/λ, and shrinking the free path by a thousand lengthens the journey by a million.

Tested in The light that takes a hundred thousand years to leave · the kinetic theory reading path

“The ring appears at 22° because that is the deviation an ice prism produces.”

An ice prism produces every deviation from 21.8° upwards, over a window of incidence 76° wide. What puts a ring at 21.8° is that the deviation is *stationary* there, so a wide band of orientations delivers light within a fraction of a degree of one angle while the rest is spread over forty. The histogram drawn here has a hard cliff at the minimum and a long tail beyond it, and the cliff is the ring.

Tested in The ring at twenty-two degrees · the dispersion reading path

“Speckle is noise and can be averaged away.”

It averages as the square root of the number of *independent* patterns, and independence has to be bought. The measurement here gives a slope of −0.509 against an exact −½: halving the contrast costs four patterns and dividing it by ten costs a hundred. Each independent pattern has to come from a separate polarisation, a separate wavelength or a separate moment, and every one of those was wanted for something else.

Tested in The grain that is in the light · the coherence reading path

“A single photon splits at a beam splitter and half of it goes each way.”

The simulated run here gives a coincidence rate 0.014 of the accidental one, seventy standard deviations below the classical floor of one. A field that divided would trigger both detectors at the accidental rate at best and more often at worst; a wave of any intensity distribution whatever is bounded below by one, which is proved here by trying two hundred of them.

Tested in The experiment a wave cannot pass · the photon reading path

“Levels of the same symmetry cannot cross, so a crossing in a spectrum means the states have different symmetry.”

That is right for a one-parameter family and wrong in general. With two parameters to vary, two levels of the same symmetry can be made degenerate at isolated points — conical intersections — and those points are where molecular photochemistry happens. The non-crossing rule is a statement about how many knobs are being turned, not about symmetry alone.

Tested in The crossing that never happens · the bands reading path

“Two electrons in a helium atom repel, so the triplet state is lower because the electrons are further apart.”

The direction is right and the reasoning is inverted. Antisymmetry of the *spatial* part is what keeps the electrons apart, and that spatial antisymmetry requires the *spin* part to be symmetric — a triplet. The spins are not doing anything; they are a label that records which spatial symmetry is available, which is why an effect with an energy of electronvolts is written as if it were magnetic and is not.

Tested in The force with no force in it · the exclusion reading path

“The field is discontinuous at the boundary because the permittivity is.”

Neither field is discontinuous in the way that suggests. The component of E along the surface is exactly continuous, because a line integral round a vanishing loop must vanish; the component of D across it is exactly continuous, because the flux out of a vanishing pillbox must equal the free charge inside, which is none. What jumps is the normal component of E, by the permittivity ratio, and the tangential component of D by the same factor.

Tested in A refraction with no wave in it · the dielectrics reading path

“Reciprocity is a consequence of the loops being identical or symmetrically placed.”

The loops here are a circle and a tilted, offset rectangle — as asymmetric an arrangement as fits on a page — and the agreement is unaffected. What the theorem needs is that the medium be linear and reciprocal, not that the geometry be symmetric; it fails in a magnetised ferrite, where the permeability tensor is not symmetric, and that failure is exactly what a circulator is built to exploit.

Tested in The coupling that is the same both ways · the induction reading path

“A superconductor screens magnetic fields better than any normal metal.”

Above about nine terahertz an ordinary copper sheet has the shorter screening length, computed here from the two expressions with no fitting. The superconductor's advantage is not that its screening length is small but that it contains no frequency at all: copper's diverges as the frequency falls and is infinite for a steady field, while the London depth is the same twenty-two nanometres at every frequency including zero.

Tested in The field that is pushed out · the superconductivity reading path

“Any system can in principle be brought to a negative temperature by adding enough energy.”

Only one with an upper bound on its energy can. A gas has no ceiling on its kinetic energy, so its entropy rises without limit and its temperature never turns over; nuclear spins in a lattice do have one, which is why the effect was first demonstrated on them in 1951 and why every subsequent realisation is a spin system, a bounded optical lattice, or a set of internal levels.

Tested in Hotter than any temperature there is · the third law reading path

“The Joule–Thomson effect is a cooling.”

It is a temperature change whose sign is set by where the gas starts. The inversion curve computed here divides the plane, and outside it every gas warms. Nitrogen's inversion temperature is 621 kelvin so air cools at room temperature; hydrogen's is 202 and helium's 43, so both warm — and a Linde machine fed room-temperature hydrogen heats it up.

Tested in The gas that cools by being let go · the phase change reading path

“Since nothing is rigid, rigid-body mechanics is wrong.”

Rigid-body mechanics is excellent wherever the transit time is short compared with the timescale of the motion, which covers essentially all of engineering. What it cannot be used for is an argument about simultaneity — the case it is invariably reached for. Bell's spaceship paradox, the pole and barn, and every 'push the rod and signal instantly' proposal are the same misuse.

Tested in Nothing is allowed to be rigid · the relativistic dynamics reading path

“A photon rocket is the ultimate drive because light carries the most momentum per unit energy.”

It carries the least. A photon carries momentum E/c, and a massive exhaust carries γmv, which for the same energy is larger at every speed below c — so a photon rocket has the highest exhaust speed and the worst momentum per joule. What makes it the limiting case is the exhaust speed, which decides the logarithm's coefficient; what makes it impractical is that converting mass to a collimated beam at any useful power is beyond anything known.

Tested in The fuel a starship needs · the Mass-energy reading path

“Dissipation destroys structure, so a damped system ends up somewhere simple.”

This map contracts area by exactly 0.3 at every point, without exception, so a region of any size shrinks to nothing under iteration — and what it shrinks onto is more intricate than what it started as, not less. The contraction is what makes the layers thin enough to be distinguished at every magnification; a volume-preserving system has no attractor to be fractal. Dissipation is a necessary ingredient of a strange attractor rather than an obstacle to one.

Tested in The fold that has to be there · the chaos reading path

“The cycloid is the isochronous curve because of a special property of the cycloid.”

It is the isochronous curve because it is the curve whose height above the lowest point is exactly proportional to the square of the arc length from it: y = s²/8a, with no approximation. That condition is a differential equation, its solution is a cycloid, and the physics never mentions rolling wheels at all. The same condition read as a statement about restoring force says the force along the curve is proportional to the displacement along it, which is the definition of simple harmonic motion.

Tested in The curve that does not ask where it started · the pendulum reading path

“The edge wave is a convenient fiction — nothing is actually radiating from the rim.”

It is a decomposition rather than a claim about atoms, and it makes quantitative predictions no fiction could. Its amplitude falls as the distance to the power −1.0000, measured on the computed field over a decade, which is what a line source and nothing else does; its magnitude is the same at equal distances into the light and into the shadow; and its strength on the boundary is exactly half the incident amplitude, fixed by nothing but the requirement that the total field be continuous where the geometrical one is not.

Tested in The wave that comes from the rim · the huygens reading path

“Surface waves do most of the damage in an earthquake because they carry most of the energy.”

They do not necessarily carry most of it at the source. They arrive largest because they spread over a circle rather than a sphere, so their amplitude falls as the inverse square root of distance where a body wave's falls as the inverse first power — a ratio that grows as the square root of the distance and reaches 31.6 at a thousand kilometres. The dominance is geometry acquired on the way, not a property of the source, and it is why a distant earthquake is felt as a long roll and a nearby one as a jolt.

Tested in The wave a surface is enough to hold · the guided waves reading path

“Foam geometry is complicated, so its angles vary with the liquid and the bubble size.”

Two angles are available and no others, and neither contains anything about the liquid. Three films meeting along a line make 120.00 degrees with one another because three equal coplanar vectors sum to zero only at that angle; four such lines meeting at a point make 109.47 degrees because four equal vectors in space sum to zero only at the tetrahedral angle. Both are solved here from the balance rather than quoted, and a junction with any other number of films rearranges within milliseconds because the tensions there cannot balance at all.

Tested in The angles a film has no choice about · the surface tension reading path

“The Weissenberg number is a Reynolds number for elastic fluids.”

A Reynolds number compares inertia with viscosity and is a ratio of two terms in the same equation. The Weissenberg number compares two stresses the fluid itself produces — the elastic against the viscous — and is a shear rate multiplied by the fluid's own relaxation time, so it can be large in a flow whose Reynolds number is a millionth. Rod climbing happens in creeping flow, where inertia is negligible everywhere and the only reason anything moves outward at all is the small centrifugal term the elasticity has to beat.

Tested in The liquid that climbs the rod · the rheology reading path

“Strong-field light bending is a different theory from the 1.75-arcsecond deflection measured at the Sun.”

It is the same equation integrated further in. The null geodesic solved here — d²u/dφ² + u = 3Mu² — gives capture below 5.196 masses and, run at an impact parameter of sixty masses, returns a deflection of 0.0666 radians against 4M/b = 0.0667. One integration covers both, and the weak-field formula is what its first term gives; there is no boundary between the regimes, only a term that stops being small.

Tested in The circle light cannot leave · the horizons reading path

“Landau damping produces entropy, since a wave has become particle motion.”

The Vlasov equation is exactly reversible and conserves every functional of the distribution, entropy included. What the damping does is wind the perturbation into finer and finer oscillations in velocity — ten sign changes at ten plasma periods, twenty-four at twenty-four, forty at forty — and the field, being an integral over velocity, stops being able to see it. Entropy appears only when a collision rate, however small, smooths that structure away, and until then the loss is recoverable: reversing the velocities brings the wave back.

Tested in The wave that dies with nothing to rub against · the plasma oscillation reading path

“A zone plate is a poor lens because of its chromatic aberration.”

Its focal length varies by 62 per cent across the visible where a fused-silica singlet's varies by 2.7 — a hundredfold worse, and of the opposite sign, since the plate's focus moves *in* as the wavelength grows. That is disqualifying for white light and irrelevant for the application it is actually used in: at X-ray wavelengths there is no glass to make a lens out of, the source is monochromatic, and a set of rings that absorbs is the only focusing element available.

Tested in The lens that is a set of rings · the diffraction reading path

“A rhomb is achromatic because glass has low dispersion.”

It is achromatic because its retardation depends on the index rather than on a thickness measured in wavelengths, which is a difference of kind and not of degree. Across the visible the rhomb's retardation moves by 4.05 degrees and a quartz quarter-wave plate's by 62 — a factor of fifteen — and the reason is that the plate's retardation is proportional to one over the wavelength, which changes by a factor of two across the band, while the rhomb's depends on an index that changes by one per cent.

Tested in The retarder with no crystal in it · the polarisation reading path

“A scanning tunnelling microscope resolves single atoms because its tip is atomically sharp.”

The tip is not made sharp and cannot be. What supplies the resolution is that the current falls by 0.94 decades for every angstrom, so an atom of the tip standing 2.5 angstroms further back than the apex carries 1/230 of the apex's current: for the three tip shapes computed here the apex carries 99.52, 96.29 and 81.92 per cent of the total, the last of them being a tip whose neighbours are only 0.9 angstroms behind. A tip is prepared by crashing it into the surface until some protrusion sticks out furthest, and the exponential does the rest.

Tested in The last atom does all the seeing · the tunnelling reading path

“No-cloning means an unknown quantum state carries no usable information.”

It carries information that can be extracted once, and the best universal copier reaches a fidelity of five sixths on every state. The best universal copier reaches a fidelity of five sixths on every state — better than the two thirds a measure-and-prepare machine averages, and short of one — so an approximate copy is available and is quantitatively limited. Quantum key distribution rests on precisely that number: an eavesdropper who copies gets 0.8333 rather than 1, and the shortfall shows up as errors in the legitimate channel at a rate that can be measured.

Tested in The state that cannot be copied · the measurement reading path

“Flux quantisation is a small quantum correction to a classical effect.”

It is a quantum condition holding over centimetres, and its consequences are macroscopic and periodic. The transition temperature of a thin-walled cylinder oscillates with the flux threading it, with a period of exactly one quantum and a cusp at every half — a shape that no smooth field effect produces, and a periodicity in a steady applied field that nothing classical does at all. The size of the effect is a few millikelvin and the period is what carries the physics.

Tested in The two in the flux quantum · the superconductivity reading path

“Maxwell's equations are four laws that all have to be solved.”

Two of them contain no time derivative, so there is nothing in them to solve forward. Integrating only the two that do — Faraday's and Ampère's — for a hundred and ten steps on a two-dimensional grid leaves the other two satisfied to 2.4 × 10⁻¹⁶ of the field's own size, while the field itself changes by a factor of 8.6. Neither divergence is computed during the run, neither is imposed, and neither drifts, because taking the divergence of a curl gives zero identically.

Tested in The two equations that are not laws of motion · the maxwell equations reading path

“Radiation pressure is negligible, so the adiabatic index of radiation is a curiosity.”

It is negligible in a room and it is not in a star. The index computed here by compressing a box at constant entropy is 1.333333, which is exactly the value at which a self-gravitating sphere stops being stable against collapse — so a star whose support comes mostly from radiation rather than from gas is marginally stable, and how close to four thirds its effective index sits decides whether it can exist at all. The index is not a curiosity; it is the reason there is an upper limit on stellar mass.

Tested in The gas that nobody counted · the blackbody reading path

“Kelvin proved the relation between them in 1854.”

He derived it and did not believe he had proved it, and he was right to be uneasy. The argument treats the thermoelectric parts of the process as reversible while the ordinary heat conduction going on alongside them is not, and there is no justification within thermodynamics for separating the two. The relation was verified experimentally for seventy-seven years before Onsager supplied a reason for it, and the reason turned out to come from the microscopic equations rather than from thermodynamics at all.

Tested in The second experiment that cannot disagree · the diffusion reading path

“Below the transition the system is ordered and above it disordered, as in any other transition.”

Below the transition the correlations decay as a power law — slowly, but to zero — so there is no long-range order there either. What the low-temperature phase has is *quasi*-long-range order, with an exponent that varies continuously with temperature from zero up to a quarter at the transition. The renormalisation flow shows why: the whole low-temperature phase is a line of fixed points rather than a single one, so every temperature below the transition is critical and none of them is ordered.

Tested in The transition with nothing to order · the phase change reading path

“The illusion can be made arbitrarily large by choosing the geometry.”

Only by making the source fast. The maximum over all angles is γβ, which is a function of the speed alone, so an apparent speed of ten requires γ of at least ten and a true speed within half a per cent of light's. And below β = 1/√2 there is no angle at all that produces an apparent speed above one, so the effect has a threshold rather than being available to any source seen from the right direction.

Tested in The motion that measures faster than light · the Velocity-addition reading path

“Beaming is an astrophysical effect about jets.”

It is a kinematic fact about any moving source, and its most precisely measured instance involves no jet at all. The microwave background's dipole is the same Doppler factor applied to a thermal spectrum: a speed of 369 kilometres a second gives a sky hotter ahead and colder behind by 3.35 millikelvin, against a measured 3.362. That is a hundred times larger than every other feature of the microwave sky, and it is the only part of it that is about the observer rather than about the universe.

Tested in The brightness that is not the same for everyone · the doppler reading path

“The rotating disc paradox is about whether the circumference or the radius contracts.”

Both answers are known and neither is in doubt: the circumference contracts because it lies along the motion and the radius does not because it lies across it. The difficulty was never which contracts but what to make of the result — a ratio of circumference to radius that exceeds two pi, in a frame where nothing has changed but the state of rotation. That is a statement about the geometry the rotating observer measures, and it cannot be removed by any choice of coordinates.

Tested in The disc that cannot be spun · the length contraction reading path

“Higher modes have more nodes because the wavelength is shorter, so the nodes are evenly spaced.”

The count is right and the spacing is not. On a string whose density rises fivefold over part of its length the nodes bunch where the medium is heavy and spread where it is light, because the local wavelength follows the local speed — and the fourth mode still has exactly three of them. The count survives an arbitrary medium; the spacing does not survive any departure from uniformity at all.

Tested in The count that cannot be cheated · the standing waves reading path

“Make the pulse shorter to measure range better; the velocity measurement is unaffected.”

A pulse of duration T has a bandwidth of about 1/T, and a frequency shift smaller than that bandwidth cannot be resolved. Halving the pulse halves the range uncertainty and doubles the smallest measurable speed, exactly. It is the same trade as any other between a duration and a spectral width, and no waveform escapes it — the sophisticated ones spread the product over a longer transmission rather than beating it.

Tested in The shift a mirror gives twice · the doppler reading path

“An optical isolator proves that light can be made to travel one way, so reciprocity is only a rule of thumb.”

An isolator contains a magnet, and the magnetic field is what breaks the time-reversal symmetry the theorem rests on. Without it the same device is exactly reciprocal. The other physical exception is a moving medium, which is why an ultrasonic flowmeter can read a difference of 91 nanoseconds between an upstream and a downstream pulse over ten centimetres of water — and why neither exception can be built out of shape alone.

Tested in Swap the ends and nothing changes · the wave motion reading path

“Photons arrive independently, so the arrival times at two detectors cannot be correlated.”

They are correlated, and the excess is exactly |γ|². For thermal light the correlation is a statement about a classical fluctuating wave — the beat between the many independent emitters producing intensity that wanders — and the photons are simply reporting where the wave was large. The independence that fails is the independence of the *classical intensities*, not any interaction between photons.

Tested in The correlation that survives what the phase does not · the coherence reading path

“Negative charge is attracted to a positive conductor, so a negatively charged surface is pulled inward.”

The pull is outward whatever the sign, because the pressure goes as the square of the field. What is happening is like charges repelling within the layer, and two negative charges repel exactly as two positive ones do. A charged soap bubble expands; it does not matter which way it was charged, and reversing the charging makes no measurable difference to the radius.

Tested in The pressure a charge puts on its own metal · the conductors reading path

“The temperature wave in the ground travels downward at a speed, so the seasons arrive at depth after a delay set by that speed.”

There is no speed. A diffusive disturbance is nonzero everywhere the instant it starts, so nothing arrives at any time and there is no front to time. What the solution has is a phase lag proportional to depth — which looks like a speed and is not one, because it is proportional to the square root of the period. Halving the period does not halve the delay at a given depth; it multiplies it by the square root of two.

Tested in The summer that reaches the cellar in December · the diffusion reading path

“Water's melting curve leans backwards, which is unusual, so the phase rule must work differently for it.”

The rule counts and does not care. The backward lean is Clapeyron's relation with a negative volume change, because ice is less dense than water, and it gives −138 bar per kelvin — a slope so steep that it is drawn vertical on any diagram wide enough to show the other two curves. The counting that makes the triple point a point is untouched: it never asked which way anything leaned.

Tested in Why the triple point is a point · the phase change reading path

“The harmonic oscillator shows that quantum and classical predictions coincide, which is why the classical limit works.”

The harmonic oscillator is the one potential in which the average force and the force at the average cannot differ, because the force is linear. It is therefore the worst possible example from which to conclude anything about the classical limit: the agreement there is exact for every state, however wide, however far from classical — including states with a single quantum in them.

Tested in The average that obeys Newton · the correspondence reading path

“Making the barriers thicker makes the device work less well.”

It makes the peak sharper without lowering it. Every resonance reaches exactly one however opaque the barriers are, provided they are identical; what changes is the width, which falls from 5.2 × 10⁻¹ to 3.7 × 10⁻² as the barriers go from 0.6 to 1.2 wide. A better barrier buys selectivity and pays in time, because the linewidth is the escape rate of the state trapped between them.

Tested in Two walls that let more through than one · the tunnelling reading path

“The built-in potential can be measured by putting a voltmeter across an unbiased diode.”

It reads zero, and must. The junction's built-in potential is exactly cancelled by the contact potentials at the two metal–semiconductor contacts, because the whole loop is at one Fermi level and a voltmeter measures a difference in Fermi level. Getting energy out of a junction requires something that shifts the Fermi levels apart — light, or a temperature difference — and the impossibility of doing it with a loop of wire is the second law wearing a circuit diagram.

Tested in One level, and the field that bends the bands · the bands reading path

“The twin paradox needs general relativity, or at least the details of the acceleration.”

It needs neither. Counting signals settles it: over a round trip each twin receives the same number the other sent, and the asymmetry is that the traveller switches from receiving at 1/k to receiving at k halfway through the journey while the stay-at-home does so much later. The arithmetic gives the ratio of elapsed times as (k + 1/k)/2 with nothing about the turnaround in it beyond the fact that it happened.

Tested in Everything from an exchange of pulses · the doppler reading path

“Acceleration has no effect on a clock, so an accelerating observer sees the same physics as an inertial one.”

The clock hypothesis is about the rate of an ideal clock, not about everything an accelerating observer encounters. An accelerating observer has a horizon behind them, cannot receive signals from beyond it, and finds the vacuum to be a thermal bath at a temperature proportional to the acceleration. All of that is acceleration-dependent physics; none of it is the clock running at a different rate.

Tested in The clock that does not feel the turn · the accelerated frames reading path

“The two-fluid model says helium II is a mixture of two kinds of atom, one superfluid and one normal.”

Every atom is identical and none of them is either kind. The two densities are a bookkeeping device for a single quantum fluid whose excitations carry the entropy and the viscosity while the ground state carries neither. The test that the model is more than bookkeeping is that its one parameter, the ratio of the two densities, can be measured by a torsion pendulum and then predicts the fountain pressure, the counterflow velocity and this wave's speed with nothing further adjusted.

Tested in The heat that arrives as a wave · the superfluidity reading path

“Shaking a mixture must mix it, because mixing increases entropy.”

Entropy arguments require a temperature, and a granular material does not have a useful one. Lifting a one-millimetre sand grain by its own diameter costs about 10¹² times kT at room temperature, so thermal energy cannot rearrange anything and the equilibrium the second law describes is never explored. What decides the arrangement is the driving and the geometry, and both of those have a direction in them.

Tested in The big one comes to the top · the granular matter reading path

“A plasma is quasineutral, so the electron and ion densities are equal everywhere in it.”

Everywhere except where it meets something, which is the only place a plasma is ever used. Inside the sheath the ion density exceeds the electron density by tens of per cent of itself, and it has to: the layer's whole job is to hold a potential of several electron temperatures, which requires an uncompensated space charge. Quasineutrality is a statement about scales larger than the Debye length, and the sheath is a few of them thick.

Tested in The wall a plasma builds against itself · the plasma oscillation reading path

“A stronger trap needs a more powerful laser.”

It needs a tighter focus at least as much. The gradient force depends on the *gradient* of the intensity, which for a Gaussian beam scales as the power over the cube of the waist, while the scattering force scales as the power over the square. Halving the waist quadruples the stiffness and doubles the ratio of gradient to scattering force; doubling the power doubles both and improves the ratio not at all — which is why optical tweezers waited for high-aperture objectives rather than for bigger lasers.

Tested in The light that pulls rather than pushes · the radiation pressure reading path

“The contact between the balls is a spring, so the details do not matter.”

The same five balls with a linear spring between them send 0.942 of the speed to the end rather than 0.989, and the residual left behind is nearly five times larger. The figures sweep the exponent of the contact law from one to 2.4 and the far ball's share rises throughout. The details are the answer: everything else in the problem is identical between those runs.

Tested in Five balls, and the law that does not choose · the momentum reading path

“The capstan equation gives the tension in a wrapped rope.”

It gives the largest ratio the wrap can hold, which is a limit rather than a state. A belt transmitting a quarter of its capacity is not slipping over three quarters of its wrap and creeping over the rest at a quarter of the exponential rate; it has an idle arc where the tension is constant and an active arc where it climbs at the full rate. The figure computes how the wrap divides, and the active arc reaches the whole wrap only at capacity.

Tested in The part of the wrap that is actually gripping · the friction reading path

“Newton's corpuscular theory of light was refuted by the discovery of interference.”

Interference made a wave theory necessary; it did not by itself decide the corpuscular prediction about speed. Written as Jacobi did, mechanics is optics with an index proportional to the momentum, so a corpuscle entering glass must speed up while a wave entering it must slow down. Foucault measured the speed of light in water in 1850 and found it slower, which is the measurement that settled the sign of the exponent.

Tested in The principle that fixes the energy instead of the clock · the least action reading path

“Chaos in a mechanical system arrives by period doubling.”

That is the route for dissipative systems, which have attractors for a cycle to inherit stability from. A frictionless mechanical system has none: its route is the destruction of invariant curves as a coupling grows, which produces no cascade of doublings and no Feigenbaum constant. The two routes are drawn in different essays of this collection because they are different phenomena.

Tested in The map a dripping tap turns out to be · the chaos reading path

“Even harmonics come from a stronger nonlinearity than odd ones.”

They come from a different symmetry, not a different strength. A potential that is even in the displacement can only produce odd harmonics, at any amplitude, because the equation is unchanged under reversing the sign of both the displacement and the drive. The figures show a symmetric well producing even components below one part in ten thousand of the fundamental, and a well with a quadratic term producing a large second harmonic at the same drive.

Tested in The oscillator that answers at three times the question · the harmonic approximation reading path

“A solenoid's inductance is μ₀N²A/ℓ.”

That is the limit for a coil much longer than it is wide, and it is not a formula for a coil. Computed as a sum of Maxwell's mutual inductances between every pair of turns, a coil as long as its diameter has 68% of it and one a quarter as long has 36%. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computation here agrees with the table at three aspect ratios.

Tested in The circuit that fights its own change · the induction reading path

“The resistance of a joint is the resistivity times its length over its area.”

A constriction has neither. The current arrives from everywhere and leaves in every direction, so there is no path length to divide by and no cross-section that is being crossed. Solved exactly, the resistance of a circular spot of radius a between two half-spaces is ρ over 2a — a resistivity divided by a length. Doubling the size of the contact halves it; making the conductors ten times thicker does nothing at all.

Tested in The resistance that is a length · the conductors reading path

“Charge quantisation is explained by the standard model.”

It is an input to it rather than an output. Dirac's argument remains the only known reason electric charge must come in multiples: the string attached to a monopole is undetectable exactly when a charge carried round it gains a whole number of turns of phase, which requires the product of the two charges to be a multiple of Planck's constant. One monopole anywhere quantises every charge everywhere.

Tested in The symmetry one missing charge would complete · the maxwell equations reading path

“The critical field of a superconductor is where superconductivity stops.”

For a type-II material there are three fields. Flux begins to enter at the lower critical field, which falls as the ratio of the two lengths rises; the thermodynamic field is where the condensation energy equals the expulsion energy and marks nothing observable in the magnetisation; and superconductivity ends at the upper critical field, which for niobium–titanium is about eighty-five times the thermodynamic one.

Tested in Two lengths, and which one is longer · the superconductivity reading path

“The hidden momentum depends on the shape of the circuit.”

The figures compute it by summing the electric potential round each loop's own perimeter for four different shapes scaled to the same enclosed area — a square, a tall rectangle, a circle and an L — and the four agree with each other and with the closed form to the accuracy of the polygon. Only the area enters, because only the area enters the dipole moment.

Tested in The momentum of something that is not moving · the field energy reading path

“A lens in front of a fibre increases the light coupled into it.”

A lens conserves the product of area and solid angle. Demagnifying a source to fit the core opens its cone by the same factor, so what the core gains in area it loses at the wall, and the coupling is unchanged for any source already larger than the core. The figure computes the fraction coupled against source size and it is a ceiling, not a design target.

Tested in The cone a fibre will accept · the etendue reading path

“A microscope's depth of focus can be computed from the blur circle.”

The ray formula predicts a depth that goes to zero linearly with the aperture, and the real one goes as the inverse square, which at NA 1.4 is smaller by a factor of several. The correct criterion is a quarter of a wavelength of path error across the pupil, which the figure shows leaving eighty-one per cent of the peak intensity — a wave statement with no ray counterpart.

Tested in The focus that is a slab, not a plane · the imaging reading path

“A polarisation state is a two-component complex vector.”

That describes a fully polarised beam only. A partially polarised one cannot be written as any such vector, because the four measurable quantities it has do not satisfy the relation a pure state's do — its Poincaré vector is shorter than its intensity. The set of beams is a solid ball rather than its surface, and the interior is exactly what a two-component vector cannot reach.

Tested in The light with no direction of shaking · the polarisation reading path

“To find how much light a particle removes, add up what it scatters in every direction.”

That gives the scattering cross-section and misses absorption entirely. The optical theorem gives the sum of both from the forward amplitude alone, and the two routes disagree for any absorbing particle by exactly the absorbed part. The forward direction is the one place the scattered and incident waves are inseparable, which is why it is the only direction that knows about everything.

Tested in Everything a scatterer removes, from one direction · the scattering reading path

“An exact surface would remove aberration from an image.”

It removes it from one object point. The figure traces the same exact surface with the object moved sideways and the blur grows from nothing, close to the first power of the displacement — which is coma. A single surface has one shape, imaging one pair of points perfectly consumes it, and the rest of the field is left with whatever the shape happens to do.

Tested in The surface that images one point exactly · the fermat reading path

“Apodising an interferogram improves the spectrum.”

It removes the side lobes a sharp cut-off produces and widens every line by about half. Neither result is more correct than the other: both are choices about what to do with information the scan does not contain, and the choice depends on whether a weak line beside a strong one matters more than the last of the resolution.

Tested in The fringe and the spectrum are one measurement · the coherence reading path

“Two equal waves in antiphase cancel, so their combined intensity is zero.”

Their combined intensity is zero at that point. The intensity is a function of position and the statement is about one place; the same pair is delivering four times one wave's intensity half a fringe away. An energy argument has to be made over a region, and over any whole number of fringes the answer is the sum of the two intensities exactly.

Tested in What adding does to the energy · the superposition reading path

“At the transmission zero the system is absorbing everything.”

Nothing is absorbing. The figures decompose the amplitude into a constant background and a resonant part whose phase sweeps through half a turn, and at one detuning the two are equal and opposite. The energy is reflected or returned, not taken up, and a lossless system has a Fano zero in exactly the same place as a lossy one.

Tested in The resonance with a zero in it · the resonance reading path

“The backward wave is removed by taking the envelope on the forward side only.”

That is an instruction to ignore the answer rather than a reason it does not occur. The removal that works is a factor, one plus the cosine of the angle over two, which falls to exactly nothing at a hundred and eighty degrees — and it is not invented for the purpose. It comes out of solving the wave equation with Green's theorem, along with a quarter-period phase advance and a factor of one over the wavelength that the geometric construction contains no trace of.

Tested in The backward wave Huygens had to remove · the huygens reading path

“A chain of masses and springs is a discrete model of a string.”

It is a different medium, and the difference appears at short wavelength. A string's frequency rises without limit as the wavelength shortens; the chain's stops at twice the frequency of one mass held by two springs, reached when neighbours move in exact opposition. The two agree to second order in the wavenumber and nowhere else.

Tested in The frequency a lattice cannot carry · the periodic media reading path

“The pattern's depth measures how much of the wave is being absorbed.”

It measures the magnitude of the reflection, and the reflection tells nothing on its own about whether the missing power was absorbed or transmitted. The ratio of the extremes inverts exactly to the reflection magnitude, and the reflected power is its square; what happened to the rest is a separate question the pattern cannot answer.

Tested in The node that is not standing still · the standing waves reading path

“A bend loses light because rays hit the wall at a shallower angle.”

The ray account gives a threshold radius and no gradual behaviour, and it says nothing about why a fibre carrying a mode far from cutoff can be bent tightly with no measurable loss at all. The correct account is about the mode's evanescent tail: at a radius set by the bend, the field would have to travel faster than the cladding permits, and what leaks is the amount of field out there — which is exponentially small and exponentially sensitive.

Tested in The mode that will not turn a corner · the guided waves reading path

“A molecule at 300 kelvin has an energy of about kT.”

It has that energy on average over time, and at any instant it has whatever it has. The distribution of a six-degree-of-freedom system's energy is wide — the standard deviation is more than half the mean — so a statement about one molecule's energy at one moment is a statement about a draw from a broad distribution rather than about a property the molecule possesses.

Tested in The temperature a molecule does not have · the equipartition reading path

“Water is anomalous because its melting curve slopes backwards.”

The thermodynamics is not anomalous at all: the slope is the latent heat over the temperature and the volume change, exactly as for everything else, and the latent heat is positive as it always is. What is unusual is a structural fact — ice is less dense than water — and the backwards slope is that fact read through an entirely ordinary relation.

Tested in The melting curve that leans the wrong way · the phase change reading path

“The cross coefficients between different transport processes are independent quantities to be measured separately.”

They are equal. Onsager's relations, which follow from the microscopic equations running the same forwards and backwards in time, fix the coefficient coupling heat flow to a concentration gradient equal to the one coupling matter flow to a temperature gradient. Measuring either gives the other, and the equality has been checked in a dozen systems.

Tested in The gradient that drives the other thing · the diffusion reading path

“Jarzynski's equality makes free energies easy to measure from fast processes.”

It makes them possible and not easy. The exponential average is dominated by rare trajectories in the low-work tail, so the number of runs needed grows exponentially with the dissipation — the figure shows an estimate still a kilotesla-worth of kT high after ten thousand runs at three kT of dissipation. The equality is exact and the estimator is poor.

Tested in The second law, with a probability attached · the entropy reading path

“A gas obeying the ideal gas law is behaving ideally.”

At the Boyle temperature the first correction vanishes identically, so a real gas with strong attractions and a finite molecular size follows pV = nRT to first order in its density while doing nothing ideal at all. The second correction does not vanish there, so the agreement is a statement about one term rather than about the gas.

Tested in The first correction to the gas law · the kinetic theory reading path

“The uncertainty principle says that two quantities cannot be measured at once.”

Most pairs of quantities can. The relation has the commutator on its right-hand side, and for a pair that commutes the right-hand side is zero and the relation permits both spreads to vanish together. The grid of commutators computed here has more zero cells than non-zero ones, and each zero is a pair a single measurement can pin down exactly.

Tested in The questions that can be asked together · the uncertainty reading path

“The effect proves the vector potential is physically real in the way a field is.”

The potential is not unique — a gauge change alters it everywhere and shifts nothing measurable. What is measurable is its line integral round a closed contour, which is gauge-invariant and equals the enclosed flux. The figures compute that integral round four different contours and get the same number, which is the invariant statement the effect actually makes.

Tested in The phase a magnet leaves on a path it never touched · the matter waves reading path

“The classical limit is the limit of large quantum numbers.”

A stationary state of large quantum number does not swing at all: its probability density is time-independent and spread across the whole classical range, which is nothing like a pendulum. What swings is a superposition of many such states, and the classical limit is a statement about which superposition rather than about which energy.

Tested in The state that swings like a pendulum · the correspondence reading path

“The transmitted packet's peak arriving early means something crossed faster than light.”

The peak that emerges is not the peak that went in. A barrier attenuates by a factor that grows through the packet, so the late half is suppressed far more than the early half, and what comes out is a reshaped remnant of the leading edge. No point on the transmitted wave lies outside the light cone of the incident wave's own front, and the transmission at the thicknesses where the effect is dramatic is smaller than a part in a million.

Tested in How long the crossing takes · the tunnelling reading path

“Entanglement is a quantity a state has more or less of.”

Two three-qubit states with the same total have it in different places. GHZ has no pairwise entanglement whatever and all of it three-way; W has all of it pairwise and none three-way. The figures compute both splits from reduced density matrices, and no local operation converts one state into the other.

Tested in What two have they cannot give a third · the entanglement reading path

“An atom's size is set by the orbit its electron happens to be in.”

The size is the radius that minimises the sum of a confinement energy rising as one over the square of the radius and an attraction falling as its first power. The minimum is located here by searching the drawn curve and agrees with the Bohr radius to a part in a million, with no orbit, no quantisation condition, and no angular momentum anywhere in the argument.

Tested in Why an atom is the size it is · the atomic structure reading path

“If each observer measures the other's metre stick as shorter, one of them must be wrong.”

Both measurements come out at exactly 1/γ, to twelve decimals, and they are made across different sets of events. Each observer measures the two ends of the other's stick at what they call one time, and the two notions of one time pick out different pairs of events on the diagram.

Tested in The diagram a ruler cannot read · the spacetime diagram reading path

“Every observer agrees on where a system's centre is, and disagrees only about its coordinates.”

The weights in the average are energies, and energies are frame-dependent. Boosting a two-body system moves the share of the total carried by each body, so different observers put the centre at genuinely different events — not at different labels for one event.

Tested in The centre that is not a place · the relativistic dynamics reading path

“Charge density is the same in every frame, since charge is.”

Charge density rises by the Lorentz factor, because the same charges occupy a contracted length. The invariant is the product of the density and the volume, and the figures verify the cancellation across the whole speed range. Confusing the two is what produces the paradox that a neutral wire cannot become charged.

Tested in The one quantity a boost leaves alone · the field transformation reading path

“The rotation two boosts leave behind is a separate effect that has to be derived from the transformation.”

It is the angle by which the boost triangle's interior angles fall short of two right angles. The figures compute both — one from hyperbolic trigonometry, one from composing two Lorentz boosts — and they agree to better than a nanodegree at every angle between the boosts.

Tested in The space that speeds live in · the Velocity-addition reading path

“A straight line is the shortest path, so an inertial worldline minimises something.”

It maximises. Proper time is computed along a family of routes between the same two events and is largest for the straight one, checked by scanning the drawn curve; the ordinary length of the same drawn curves is smallest there. Both extremes sit at the same route and point in opposite directions.

Tested in The longest way round is the shortest clock · the time dilation reading path

“Viscosity communicates a wall's motion to the whole fluid.”

Only for steady motion. With the wall oscillating, the amplitude falls as exp(−y/δ) with δ = √(2ν/ω), and the envelope is checked against the drawn profiles to a part in a hundred million. Four skin depths out there is essentially nothing, however long one waits.

Tested in The shear that only reaches so far · the viscosity reading path

“Pressure is a scalar because it is defined as force divided by area.”

That definition gives a different number for every orientation of the area until something makes them equal. What makes them equal is a force balance on a shrinking wedge, in which the surface forces go as the square of the size and the weight as the cube, so the weight drops out in the limit. The wedge's excess is computed here and falls in exact proportion to its size.

Tested in The push that has no direction · the hydrostatics reading path

“The discharge goes as the area of the opening, so as the square of its diameter.”

It goes as the five-halves power. The extra half comes from the exit speed, which is set by the size of the opening rather than by the head above it — grains fall freely through a region of about the orifice size, so they leave at about √(gD), and the rate is that speed times the area.

Tested in The hourglass that keeps time · the granular matter reading path

“A cell placed in a solution shrinks or swells to a new steady volume.”

Only if the membrane holds the solute. When the solute leaks in, the cell shrinks and then returns, because the gradient it was responding to is abolished by the solute that caused it. The integration here shows the dip and the recovery, and a measurement made in the first moments cannot tell a leaky membrane from a tight one.

Tested in The membrane that almost holds · the osmosis reading path

“Capillary rise is higher in narrower channels, so the narrowest part of a container fills first.”

A corner has no narrowest part, so there is no height to compute. What decides whether the liquid enters it at all is the sum of the contact angle and the corner's half-angle: below ninety degrees the liquid runs in without limit, above it the liquid stays out however sharp the corner is made.

Tested in The corner a liquid never stops climbing · the surface tension reading path

“The Laplace–Runge–Lenz vector is a formal device with no observable content.”

Its direction is the perihelion's direction and its length is the eccentricity, both verified here against the orbit itself. When an extra inverse-cube term is added it turns, at a rate that matches the perihelion's own motion measured independently from the integrated path to better than two per cent.

Tested in The arrow that says which way the orbit points · the orbit stability reading path

“Black holes are slowly evaporating.”

Every black hole anybody has observed is colder than the microwave background it sits in, so it absorbs more than it emits and is growing. The two temperatures cross at about 4.5 × 10²² kg — a hundredth of the Moon — and a stellar-mass hole will not begin to lose mass until the background has cooled below sixty nanokelvin, in something like 10¹² years.

Tested in The hole that outlives everything and then does not · the horizons reading path

“A charge radiates if and only if it is accelerating.”

Whether radiation is present is not a locally defined question. A detector riding with a uniformly accelerated charge finds none, because the radiation goes into the region beyond its own horizon — nearly a light-year behind it at one gravity. A detector at rest far away finds the full Larmor power. Both are right about what they measure.

Tested in Whether a charge on a table glows · the radiating charge reading path

“A higher-energy probe always resolves a smaller distance.”

Only until the probe's own energy makes a horizon wider than its wavelength. The two lengths cross at about 10⁻³⁴ metres, located here by bisecting the drawn curves, and beyond it more energy hides more. The achievable resolution is the larger of the two, and that has a minimum.

Tested in The length no experiment can resolve · the planck scale reading path

“The belt trick shows why the sign is there.”

The belt demonstrates that a 2π rotation of an object tied to its surroundings cannot be undone by deformation while a 4π rotation can — a fact about the topology of the rotation group, which is the same group whether the thing being rotated is a spin, a belt or a coffee cup. It makes the double cover memorable and predicts nothing: it does not say which systems pick up the sign, and integer-spin systems, obeying the same topology, do not.

Tested in The turn that has to be made twice · the spin reading path

“A perturbation formula can be used for any load by making the load small enough in the answer.”

The exact frequency of a loaded string is root-found here against the first-order prediction as the load grows. The linear estimate runs away without bound while the true frequency flattens onto the mode of the clamped pieces, one rung lower than the unloaded mode. The two agree to a per cent for a load a few per cent of the string's mass and disagree by everything at ten times that.

Tested in The dent that raises the note · the standing waves reading path

“Field lines pull like elastic bands, so a picture of the lines shows the force.”

The tension along a line is only half of the tensor; the other half is a pressure of the same size across it. Two like charges have no line running between them at all and repel, and the figure computes that repulsion as the pressure their sideways-lying lines exert on the plane between them. Reading only the tension gets the sign wrong for every repulsive configuration.

Tested in The force read off a surface that touches nothing · the field energy reading path

“The magnetic moment is conserved, so a trapped particle stays trapped.”

It is an adiabatic invariant, not a conserved quantity: its change is smaller than any power of the slowness, which is not zero. Anything that varies on the timescale of a gyration breaks it — a collision, a wave at the cyclotron frequency, a field null. Mirror machines leak for exactly this reason as well as through the loss cone, and the loss cone itself narrows only as the inverse square root of the mirror ratio.

Tested in The drift that does not care what the charge is · the magnetism reading path

“Grip is set by the coefficient of friction.”

The slope of the force-slip curve at the origin is computed here from the tread stiffness alone and contains no coefficient at all, because at vanishing slip nothing is sliding. The coefficient decides only where the curve stops rising. Two tyres with the same coefficient and different tread stiffness behave completely differently at the slips a driver uses.

Tested in The grip that needs a little slipping · the friction reading path

“The intermediate-axis theorem says which spins are stable, so a body may spin stably about either of its extreme axes.”

The theorem is a statement about a perfectly rigid body, where both extreme axes give oscillatory linearised motion and the middle one does not. It says nothing about dissipation. Adding any internal energy sink leaves exactly one stable spin, and the figures here integrate the flip from a spin the rigid theorem calls stable.

Tested in The axis a leak of energy chooses · the rotation reading path

“Stopping down fixes soft corners.”

It divides the blur by the f-number and does nothing to the defocus that causes it, which is checked here across three apertures. Going from f/2.8 to f/16 improves the corner by a factor of six and leaves it hundreds of micrometres out of focus. On the figures here the corner at f/16 is still blurred over more than a dozen sensor pixels.

Tested in The flat scene that comes back curved · the imaging reading path

“A polarising filter darkens the sky.”

It darkens the band ninety degrees from the sun and does almost nothing towards or away from it, where the light is unpolarised and any filter can only halve it. The transmission curves computed here converge at both ends and separate by a factor of several only near a right angle. On a wide-angle lens the result is a dark stripe across the frame with pale sky either side.

Tested in The pattern the sky is written in · the polarisation reading path

“Making the pawl spring stiffer, or the notch deeper, would let it work.”

Both slow the device down exponentially and neither changes the balance, because the same energy appears in both rates. The net rate is computed here for notches from two to twenty thermal energies deep and is zero at zero load in every case. A rectifier cannot be improved into working; it has to be given a second temperature.

Tested in The engine a fluctuation cannot run · the heat engines reading path

“Any nonlinearity couples the modes, so a nonlinear system reaches equipartition.”

Coupling is necessary and not sufficient. The chain integrated here has coupled modes throughout — energy visibly moves between them — and returns to its starting distribution again and again for the whole run. What decides whether the sharing sticks is the energy density, not the presence of a nonlinear term.

Tested in The energy that refuses to be shared · the equipartition reading path

“An atmosphere warms the surface by trapping heat, like a blanket.”

A blanket works by suppressing convection and conduction; the atmosphere is where the convection happens. The mechanism computed here is different: absorbing gas raises the level from which the planet's radiation finally escapes, that level is colder because temperature falls with height, and the whole profile must shift to restore the balance. The lapse rate doing the shifting is set by convection, not by radiation.

Tested in The height a planet is seen from · the blackbody reading path

“The area theorem is a consequence of general relativity, so it holds whenever general relativity does.”

Its proof requires an energy condition — no observer measures a negative energy density — which ordinary matter satisfies and Hawking radiation does not. An evaporating hole's area falls to zero, computed here, without any failure of the field equations. What survives is the generalised statement including the entropy outside the horizon.

Tested in The area that is not allowed to shrink · the horizons reading path

“A clock measures the strength of gravity where it is.”

It measures the potential, which is a different quantity: the figure here shows gravity varying by half a per cent along sea level while a clock carried along the same surface reads one rate throughout, because sea level is by definition a surface of constant potential. A gravimeter reads the gradient and a clock reads the thing being differentiated.

Tested in The clock that measures a height · the gravitational redshift reading path

“Dropping two objects and seeing them land together tests the equivalence principle.”

It tests it to about a part in a hundred, which was settled before the principle was stated. Every advance since has come from a null instrument that measures a difference directly rather than two large quantities that are then subtracted, and from modulating the signal at a frequency where the apparatus is quiet.

Tested in The fall that does not depend on what is falling · the equivalence principle reading path

“The experiment distinguishes special relativity from its predecessors.”

It distinguishes partial drag from no drag and from complete drag, which differ by factors of two, and on that it was decisive. The difference between Fresnel's first-order coefficient and the exact composition is second order in the water's speed — a part in 10^17 here, computed in the figures — and no nineteenth-century measurement could see it.

Tested in The drag that was only an addition · the Velocity-addition reading path

“The lab energy spectrum of a decay product carries information about the decay dynamics.”

For a two-body decay that is isotropic in the rest frame the spectrum is exactly rectangular, whatever the interaction responsible. All the dynamics can do is make the rest-frame distribution anisotropic, and even then the edges are pure kinematics. The two edge positions give the parent's speed and the rest-frame momentum, and nothing else in the spectrum says anything.

Tested in The cone a decay cannot leave · the relativistic dynamics reading path

“Special relativity rests on the constancy of the speed of light.”

It can be derived from homogeneity, isotropy, the group property and the relativity principle alone, which leave a one-parameter family with an undetermined constant. The light postulate is one way of fixing that constant and is not the only one — any measurement of the composition of two speeds fixes it too, which is what the last figure shows.

Tested in The transformation that never mentions light · the spacetime diagram reading path

“Osmotic pressure counts the particles a membrane holds back.”

It counts the excess of particles on one side over the other, and a charged macromolecule brings counterions with it that are free to cross and stay anyway. The figures solve for that partition: at physiological salt a gel with 200 mM of fixed charge holds about 60 mM more mobile ions than its bath, and almost all of the pressure is theirs rather than the polymer's.

Tested in The swelling a membrane cannot stop · the osmosis reading path

“Surface tension is what holds a soap film together.”

Surface tension acts along the film and holds it taut; it does nothing to stop the two surfaces from approaching each other. What resists thinning is a pressure between the surfaces that has no analogue at larger separations — electrostatic repulsion at tens of nanometres and contact at a few — and the figures compute where it balances the suction pulling the water out.

Tested in The film that goes black before it bursts · the surface tension reading path

“The densest random packing of spheres is 0.64, so a tapped column reaches 0.64.”

The published number is the asymptote of a fitted curve, and no experiment has reached it. The hundred-thousand-tap experiments end measurably short of it. Three different quantities get the same name — a fitted asymptote, a value reached after a stated number of taps, and a geometric limit — and they are not the same number.

Tested in The pile that is never finished settling · the granular matter reading path

“Sunspots are dark because the magnetic field is strong there.”

The field strength does not enter the brightness. What the field does is take over some of the pressure, so the gas beneath it is thinner and convection is suppressed; the surface then cools to about 4,100 K against 5,780 K, and the Stefan–Boltzmann fourth power turns that into a brightness ratio of 0.253 against a measured fifth. The field is two steps upstream of the darkness, and the pressure balance is the first of them.

Tested in The same force whichever way the surface faces · the flux freezing reading path

“Advanced solutions are unphysical because they would let effects precede causes.”

A converging wave is not an effect preceding a cause; it is a field specified on a surface at an earlier time. Time-reversal mirrors build them routinely — record a wave on a surrounding surface, replay it reversed, and a wave converges onto where the source was. The figure computes the focus that a spherical cap produces and finds it narrows as the cap approaches a full sphere, which is exactly what surrounding the target means. What is expensive is the surface, not the causality.

Tested in The solution that is thrown away · the retardation reading path

“A wave focused onto a line has a large amplitude there.”

It has a small wavelength there, which is not the same thing. The contraction squeezes the wavelength, so the shear rises even where the energy does not, and viscosity acts on the shear. The attractor is a place where energy is destroyed rather than concentrated — which is why it matters for mixing an ocean and not for storing energy in one.

Tested in The reflection that changes the wavelength · the stratification reading path

“Geostrophic balance describes large-scale atmospheric flow.”

It describes large-scale flow away from the equator. The geostrophic wind is the pressure gradient divided by ρf, and f vanishes at the equator, so the expression runs away below about fifteen degrees of latitude — the same gradient that implies 32 metres a second at 45° implies over a hundred at eight. Tropical meteorology uses different balances for that reason, and the failure is visible in the formula rather than discovered afterwards.

Tested in The ratio that decides whether the planet is turning · the circular motion reading path

“Light entering a metal refracts almost along the normal, because the index is large.”

That holds when the real part of the transmitted normal wavenumber is large, which is a far-infrared statement. In copper at ten micrometres the phase fronts do sit within five degrees of the normal. In silver at 550 nanometres the real part is 0.053, so at sixty degrees of incidence the fronts sit at 86 degrees — nearly along the surface. Both are computed from measured optical constants, and they are opposite.

Tested in The angle that is two angles · the refraction reading path

“A negative-index slab is a perfect lens, so it has unlimited resolution.”

It has unlimited resolution only at exactly ε = μ = −1 with no loss. The transfer function computed here from the slab's own Fresnel coefficients with complex ε and μ falls through a half at transverse wavenumbers of 18, 25 and 33 times k₀ for losses of 10⁻², 10⁻³ and 10⁻⁴. Resolution is a logarithm of the loss divided by the thickness, so it is finite for every real material and improves only very slowly as materials improve.

Tested in The ray on the wrong side of the normal · the refraction reading path

“Radioactive equilibrium means the amount of each isotope is constant.”

Every amount is falling, at the parent's rate. Secular equilibrium is not a steady state; it is a state in which every member's decline is slaved to one exponential, the parent's, and it looks constant only because that exponential is slow. Over four and a half billion years the whole chain halves together.

Tested in The chain that runs at its slowest member's rate · the decay reading path

“Chemical effects on decay rates are a per cent, so they are a per cent for any nuclide.”

They are a per cent for beryllium and parts per million for anything heavy. The density of a 1s electron at the nucleus goes as Z³, so the two innermost electrons dominate more and more completely as the element gets heavier and the share belonging to the valence electrons — the ones chemistry rearranges — falls as Z⁻³. The exponent is measured off the curve here.

Tested in The half-life that chemistry can change · the decay reading path

“Cooling slowly enough would reach the Kauzmann temperature and settle the question.”

The glass transition moves down by about three and a half kelvin per decade of cooling rate, and the crossing is thirty kelvin below the ordinary transition. Reaching it needs of order 10⁴¹ seconds per kelvin, computed here from the same relaxation law that locates the transition — some 10³³ years. The experiment that would decide is not slow, it is impossible.

Tested in The entropy that depends on how fast it was cooled · the third law reading path

“Residual entropy is a violation of the third law that has to be explained away.”

It is what the law predicts for a degenerate ground state, and the law's statistical form contains it. What would be a violation is a system in a unique ground state with non-zero entropy, and none has been found. The cases that look like violations divide into degeneracies — where the count works — and systems that never reached their ground state, where the count is an upper bound the measurement falls below.

Tested in A law about spectra, not about heat · the third law reading path

“Lead is the best shielding material at every photon energy.”

It is spectacularly good below its K edge and much less special above a megaelectronvolt. At 100 keV lead's half-value layer is 0.011 cm against water's 4.06, a ratio of 369; at 1 MeV the same ratio is 11.4, because absorption there is Compton scattering off electrons and every material has about the same number of them per gram. A megaelectronvolt shield is chosen by mass per unit area, not by atomic number.

Tested in The steps in an absorption curve · the attenuation reading path

“Brighter beams cool an atom further.”

The limit contains no intensity at all once the beams are weak, and above that it gets worse. The coldest temperature computed for sodium is 247 µK at a tenth of saturation per beam and 332 µK at full saturation, because the heating follows the total scattering rate while the friction saturates first.

Tested in The friction made of light · the radiation pressure reading path

“Colder atoms need beams tuned closer to resonance, where the cooling force is strongest.”

That is the Doppler mechanism's trend, and the mechanism that actually sets the temperature runs the other way. Its temperature is a straight line in the light-shift well depth, and at fixed intensity the depth falls as one over the detuning: for caesium at five linewidths the Doppler theory predicts 664 µK and the polarisation-gradient model 10.8 µK, and the second gets colder still as the beams are tuned further away.

Tested in The limit that belonged to a simpler atom · the radiation pressure reading path

“Forty-five degrees gives the longest throw.”

It gives the longest throw onto ground at the height of the hand, and nowhere else. Along a 20° hillside the best launch is 55° and 45° lands at 91 per cent of the best; along a 30° downward slope the best is 30° and the reach is twice the level range. Every one of these was found by searching the launches and agrees with 45° plus half the slope to a millionth of a degree.

Tested in One curve answers every slope · the projectile reading path

“Athletes who launch well below 45° are using a poor technique.”

Forty-five degrees is the best throw for a set of launches that is a circle centred on the origin, and no body produces that set. A thrower whose speed falls by 30 per cent per radian of elevation does best at 34.9°, and a long jumper whose horizontal speed comes from a 9.5 m/s run-up and whose vertical speed is a 3.2 m/s push does best at 18.6°. Both come from the same tangency as 45°, applied to a different shape.

Tested in The best throw is a tangency · the projectile reading path

“The water held by a porous material is fixed by the humidity of the air around it.”

It depends on which way the humidity last moved, even in perfectly smooth, perfectly wetting pores. A pore fills through a cylindrical film with one curvature and empties through a hemispherical meniscus with two, so it fills at a higher humidity than it empties at. For pores spread about 4 nm, half the volume is full at 87.7 per cent on the way up and still full at 76.9 per cent on the way down.

Tested in The pore that fills from dry air · the capillarity reading path

“Electrowetting works by lowering the surface tension of the liquid.”

The liquid's own surface tension is unchanged, and so is the angle at the very edge of the contact line. What the voltage adds is capacitor energy under the wetted area, and the force it produces is concentrated in the fringing field within about a coating thickness of the contact line — the same force that pulls a dielectric slab into a charged capacitor. At 110 V on a 1 µm coating that pull reaches γ(1 − cos θ₀) = 104 mN/m, which is what complete wetting would need.

Tested in The angle a voltage can set · the capillarity reading path

“A high-index covering on a chip lets more light out than the invariant allows.”

It lets more light across the chip's face — 10.5 per cent into epoxy against 4.35 into air for a gallium nitride chip — because the critical angle depends on the ratio of the two indices. The light is then inside the covering at that material's radiance and must still leave it. The covering is shaped as a dome much larger than the chip so that every ray meets its surface near the normal, and the étendue crossing each surface in turn is conserved throughout.

Tested in The cone light has to find to get out · the etendue reading path

“Twice as much fibre gives twice as much polarisation delay.”

The sections' delays add as the steps of a random walk in three dimensions, because each section rotates the polarisation it receives before adding its own. The ensemble delay grows as the length to the power 0.50: 5.18 ps at 100 km for a coefficient of 0.5 ps/√km, where the same sections laid with their axes aligned would have given 172 ps.

Tested in The delay that is a random variable · the dispersion reading path

“Inside a planet, gravity falls in proportion to the distance from the centre.”

That is true only of a planet of uniform density. The local form of Gauss's law gives dg/dr = 4πG(ρ − ⅔ρ̄), so gravity decreases going down only where the local density exceeds two thirds of the mean density enclosed. The Earth's mantle is lighter than that everywhere, and its core heavier, and the figures check the formula against the slope of the computed profile to half a per cent.

Tested in The pull that grows on the way down · the Gauss's law reading path

“The voltage produced by a Josephson junction depends on how well it is made.”

The mean voltage between steps does, through the critical current and the resistance. The voltage of a microwave-induced step does not: it is n times the drive frequency times h/2e, and the integrated steps sit at 0.60, 1.20 and 1.80 in units chosen so that the step voltage contains only the drive frequency. Junctions of different superconductors have been compared at the level of parts in 10¹⁶ and give the same steps.

Tested in The voltage that is a frequency · the superconductivity reading path

“The spreading of a wave packet in a potential well is a quantum effect.”

Spreading in an anharmonic well is what any cloud of oscillators with slightly different frequencies does. Through the collapse the quantum mean position and that of a classical ensemble with the same starting spread agree to a few per cent of the swing. The quantum signature is the return, which requires the energies to be discrete.

Tested in The return a classical cloud never makes · the correspondence reading path

“A negative Wigner function is the mark of a large or exotic superposition, such as a cat state.”

Every state of definite energy except the ground state goes negative, and each takes the value ±1/π at the origin — the largest magnitude a Wigner function can have. A single quantum already has a negative volume of 4/√e − 2 = 0.425, two-thirds of the 0.624 a superposition of two packets carrying six quanta reaches. Among pure states only the Gaussian ones never go negative.

Tested in The probability that goes below zero · the correspondence reading path