Simply false — part 2 of 5

“A chaotic system is one with a random element in it.”

There is none. Two double pendulums are integrated from the same equations with a fourth-order Runge–Kutta step, differing only in a release angle of 120° against 120.00000001°, and they end up on opposite sides of the apparatus. The energy of both is held to 2.6 × 10⁻¹⁰ of itself over the whole run, so nothing has been added and nothing has leaked.

Tested in The error that doubles on a schedule · the chaos reading path

“Water rises up a tree by capillary action.”

A capillary lifts 2γcosθ/ρgr, so a 20 µm vessel — which is what a tree's conducting cells are — lifts 70 centimetres, and reaching a hundred metres needs a radius of 0.1 µm. A tube that fine carries essentially no flow, since the throughput goes as the fourth power of the radius. The rise and the conduction cannot be done by the same pore.

Tested in The column that is pulled, not pushed · the capillarity reading path

“Warm air rises, so a column with warm air at the bottom overturns.”

Every column has warm air at the bottom — the standard atmosphere cools at 6.5 K per kilometre — and almost none of them overturns. What decides is whether the environment cools faster than g/c_p = 9.76 K/km, because a lifted parcel cools at that rate whatever the environment does. At 6.5 K/km a displaced parcel comes back, with a period of 9.9 minutes.

Tested in The layer a parcel cannot leave · the stratification reading path

“Clocks on a rotating platform can be synchronised with each other, since each pair is close enough that special relativity applies.”

Each pair can, and the set cannot. Synchronising neighbours all the way round a rim of radius 0.5 m at 12.566 rad/s accumulates 0.037 femtoseconds per step over twelve stations and arrives back at the first clock 0.439 femtoseconds out — which is 4AΩ/c², the same quantity the two beams measure. The procedure that works locally everywhere fails to close.

Tested in The ring where the two beams disagree · the simultaneity reading path

“A horizon is a feature of a black hole, so it requires a strong gravitational field.”

It requires an observer who never stops accelerating, and nothing else. The worldline x² − c²t² = (c²/a)² is asymptotic to the ray x = ct, so every signal released at or behind that ray chases the rocket for ever without arriving. The boundary is at c²/a — 0.97 light years for one gravity — in flat spacetime with no mass anywhere in the problem.

Tested in The wall of silence behind a rocket that never stops · the accelerated frames reading path

“A system is either regular or chaotic, and turning up the coupling flips it from one to the other.”

Both behaviours are present at every coupling above zero. The phase portraits here show regular islands and a chaotic sea side by side at every value drawn, and the transition the number 0.971635 marks is not a change in any orbit's character — it is the disappearance of the last barrier that kept the chaotic orbits local.

Tested in The last curve to go · the chaos reading path

“A scale reads the weight of whatever is resting on it.”

Not while anything is arriving. A chain dropped from rest with its lower end touching the pan reads exactly three times the weight of the length that has landed, throughout the fall — one part pile and two parts momentum flux, because v² = 2gx makes the second term exactly twice the first at every instant. The reading falls discontinuously to one the moment the last link lands.

Tested in The pile that lands heavier than it weighs · the momentum reading path

“The quality factor measured from a ringdown and the one measured from a resonance curve are two different quantities that happen to be close.”

They are one quantity. The figure takes a decay, transforms exactly those samples, and reads the half-power width off the result: 25.0 from the decay and 25.0 from the width, differing by two hundredths of a per cent, which is the resolution of the frequency grid. Any disagreement larger than that is evidence that the system is not one damped mode.

Tested in The width that is a lifetime · the resonance reading path

“A graded transition is a matching layer with a wider bandwidth.”

It is a different kind of device. A quarter-wave layer has a design wavelength and a band around it; a taper has no design wavelength at all and a cutoff below which it does nothing, above which it works for ever. The computed curves cross zero in the first case and are flat in the second, and no amount of widening turns one shape into the other.

Tested in The taper that matches every note · the impedance reading path

“A diffusing particle explores its whole neighbourhood given enough time.”

Only in one and two dimensions. In three the expected number of returns to any given site is finite, so a walk visits a vanishing fraction of the sites within its own range: the fraction of walks that have ever come home flattens at 0.3405 in the simulation and stays there however long the run.

Tested in The walk that comes home · the diffusion reading path

“The third law says the entropy of everything goes to zero at absolute zero.”

It says the entropy approaches a constant independent of the other variables, and that the constant is zero for a perfect crystal. Ice, carbon monoxide and nitrous oxide are measured to keep 3.41, 4.6 and 4.8 joules per kelvin per mole, and each number is the logarithm of a count of arrangements the substance froze into before it could reach its ground state.

Tested in The entropy that is still there at zero · the third law reading path

“A yield stress is just a very large viscosity at low shear rate.”

A very thick Newtonian liquid pushed gently flows slowly; a yield-stress fluid pushed gently does not flow at all. In a pipe below the threshold pressure gradient the computed velocity profile is identically zero rather than small, and above it there is a core moving as a rigid plug — a shape no viscosity, however large or however rate-dependent, produces.

Tested in The paste that holds up its own hill · the rheology reading path

“A dispersion relation tells you how the speed of a wave depends on its wavelength.”

This one contains no wavelength at all. ω = N cos φ relates the frequency to the direction of the wavevector and says nothing about its magnitude, so beams of any thickness travel at the same angle and a wave of any wavelength at a given frequency goes the same way. The frequency selects a direction, which is a kind of dispersion relation nothing else in this collection has.

Tested in The wave that picks an angle · the stratification reading path

“A mirror that focuses parallel light to a point is a good imaging mirror.”

It is a good imaging mirror for one point. The traced paraboloid here is exactly stigmatic on axis and its blur grows in proportion to the field angle immediately off it, so a Newtonian telescope's field of good images is a few minutes of arc wide however perfectly the mirror was figured. The quality of the axial image says nothing at all about the field.

Tested in The condition a lens must meet · the imaging reading path

“Coherence is a property of a light source — a laser is coherent and a lamp is not.”

Transverse coherence is a property of the source's angular size seen from where the light arrives, and of nothing else. The coherence radius is λ/πθ for every source in the figure, from the Sun at nineteen microns to a Sun-like star at ten parsecs at thirty-nine metres. The same lamp is coherent across a bench and incoherent across a room, and starlight is coherent over a whole telescope because the star is small.

Tested in The fringe that measures a star · the coherence reading path

“Dispersion means different colours travel at different speeds, so a pulse of one colour is unaffected.”

A pulse of one colour does not exist. A pulse of duration τ occupies a band of frequencies of width at least 1/τ, and the shorter it is the wider that band — so the pulses most damaged by dispersion are exactly the short ones a fast system needs. The broadening is the product of the dispersion, the length and the spectral width, and the third factor cannot be reduced below what the first two make necessary.

Tested in The wavelength a fibre does not smear · the dispersion reading path

“The vector potential is the field's underlying reality and the field is derived from it.”

Three different vector potentials are drawn here for one uniform magnetic field, and their arrow patterns share no feature at all — one circulates, two are unidirectional in perpendicular directions. The curl recovered from each is 1.000000 tesla. A quantity that can be changed arbitrarily without changing anything measurable is not more fundamental than what it produces.

Tested in The potentials that are not unique · the maxwell equations reading path

“There is no field inside a conductor.”

There is no *static* field inside a conductor, because the charges have had time to arrange themselves. An alternating field enters to a depth √(2/μσω), which for copper is 9.2 mm at fifty hertz and 2.1 microns at a gigahertz. Whether a conductor is thick or thin is a statement about that length rather than about millimetres.

Tested in How far a field gets into metal · the conductors reading path

“The circulation of B round a circle enclosing a current is μ₀I.”

For a finite segment it is μ₀I cos α, with α the half-angle the segment subtends — 0.2425 of μ₀I for a wire half as long as the distance. The remainder is supplied by the displacement current of the charge accumulating at the segment's ends, and the two terms, both integrated here, sum to unity at every length to better than a ten-thousandth of a per cent.

Tested in The law that is always true and rarely useful · the ampere law reading path

“Relativity limits how far anybody can travel in a human lifetime, because nothing goes faster than light.”

It limits the distance measured in *home* years and not in shipboard years. At one gravity the galactic centre is 19.8 years of the traveller's own time away and Andromeda is 28.6, both computed here. Nothing exceeds c at any point: the speed is the hyperbolic tangent of the rapidity, and it is the rapidity that grows without limit.

Tested in The ship that never arrives at c · the accelerated frames reading path

“A clock flown at altitude runs fast, because it is higher up.”

It depends which way it went. Height gains 3.5 nanoseconds an hour at nine kilometres, and motion loses a quantity quadratic in a speed that changes sign with the direction, so an eastward flight comes out at −61 nanoseconds and a westward one at +304 in this model, against measurements of −59 and +273. Both terms act on the same clock and only their sum is observable.

Tested in The two clocks that flew in opposite directions · the time dilation reading path

“A heavier star is a bigger star.”

Not when it is held up by degeneracy. The integrated structure here gives a radius falling as the mass rises — about the Earth's radius at a solar mass and a few thousand kilometres near the limit — because adding mass compresses the electron gas and the pressure it gains grows more slowly than the weight it must carry.

Tested in The mass no cold matter can hold up · the Self-gravity reading path

“A wave below the plasma frequency cannot get into a plasma.”

That holds for an unmagnetised plasma. Add a magnetic field and a wave circularly polarised the same way the electrons gyrate travels happily below both the plasma frequency and the gyrofrequency — 4 kilohertz through a medium whose plasma frequency is 90 kilohertz, in the figure here. The magnetic field gives the electrons a second frequency of their own and the response is no longer a simple cutoff.

Tested in The whistle that arrives sorted · the plasma oscillation reading path

“An electron's energy depends on its principal quantum number.”

Only in hydrogen. The solved levels here for a nucleus of charge 19 come out 3s at −10.500, 3p at −10.346 and 3d at −10.032 hartree, ordered by angular momentum, and the ordering appears the moment the potential is anything other than a bare Coulomb one. Hydrogen's degeneracy is a special property of the 1/r potential and not a general feature of atoms.

Tested in The order the shells fill · the atomic structure reading path

“Two spin-halves can be up-up, up-down, down-up or down-down, so there are four equivalent states.”

There are four states and two of them are not those. One spin up and the other down does not specify a total spin, because it does not say which spin is which; the states that do are the sum and the difference, and applying S² as a matrix gives 2ħ² for one and 0 for the other. Three of the four are one kind of object and one is another.

Tested in Four states, and one of them is odd · the spin reading path

“How much a material absorbs and how much it bends light are two independent properties, to be measured separately.”

They are one property looked at twice. Handed nothing but the absorption at every frequency, the Kramers–Kronig integral returns the refractive index at every frequency, and the computation here reproduces the model's own dispersion curve to two parts in ten million. No parameter of the material is used and none is fitted.

Tested in The answer that cannot come first · the attenuation reading path

“A wave in two dimensions is a wave in three with one direction ignored, so it behaves the same way with a different spreading law.”

The spreading law is the smaller difference. A three-dimensional impulse arrives as the pulse that was sent; a two-dimensional one arrives and then keeps arriving, with a tail falling as one over the time that is still at a tenth of its peak twelve pulse-lengths after the front. Both are exact solutions of the same wave equation with the same source.

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

“The ultraviolet catastrophe came from a mistake about how much energy each mode holds.”

The energy per mode was the standard classical result and was not the error. What made the integral diverge was the number of modes: it grows as ω² in three dimensions, without limit, so any fixed energy per mode gives an infinite total. Planck's constant enters as a bound on the energy of the high-frequency modes, not as a bound on how many there are.

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

“A trajectory with drag is a parabola with a smaller range.”

It is not a parabola and not any closed-form curve. The horizontal and vertical equations are coupled through the total speed, and at a drag of 2.4 weights the launch spends 45 per cent of its flight going up, arrives at 64° after leaving at 45°, and lands at 48 per cent of the speed it started with.

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

“A force that averages to zero over a cycle has no average effect, so shaking the pivot cannot hold the pendulum anywhere.”

The average of a force is not the force on the average when the two are correlated. The fast force is proportional to sin θ and the fast displacement it produces is proportional to the same thing, so their product has a non-zero average — which comes out as a term ¼(a/L)²(Ω/ω₀)² sin²θ added to the potential, and turns the maximum at 180° into a minimum.

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

“A particle takes the path of least action.”

It takes the path of stationary action, and past the kinetic focus that path is a saddle. For a harmonic oscillator the second variation of the action under a deformation sin(πt/T) is (λ²T/4)[(π/T)² − ω²], which is negative for every trip lasting more than half an oscillation — so nearby paths with the same endpoints have less action, and they are drawn here.

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

“A bigger heap of the same sand is steeper, because there is more weight pressing down.”

There is more weight and it acts on both sides of the comparison. The driving stress on a surface layer is ρgh sin θ and the frictional resistance is μρgh cos θ, so the thickness, the density and gravity all cancel and the criterion is tan θ = μ. Checked over three decades of layer thickness the ratio of the two stresses differs by zero to machine precision.

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

“A reversible separation is a theoretical idealisation with no bearing on how a plant is built.”

It is the target every stage is added to approach. One stage at half recovery costs 1.584 kWh/m³, two cost 1.320, eight cost 1.149, against a reversible 1.098 — the excess falling as one over the number of stages. That arithmetic is why real plants run two or three passes and not one, and why they stop at three.

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

“Helium II is an extraordinarily good thermal conductor.”

It does not conduct at all. Heat is carried by the normal component physically flowing toward the cold end while the superfluid flows back underneath it, and the transport is convective rather than diffusive — which is why its effective conductivity depends on the heat flux and on the channel width, and why it fails abruptly above a critical velocity.

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

“The instability is a property of the particular cloud — its size, its shape, how much gas it holds.”

None of those appears in it. Measured in each gas's own units, the dispersion relations of four densities six decades apart lie exactly on top of one another to 10⁻¹⁶: the curve is the same for a diffuse cloud and a protostellar core, with different numbers written on the axes. What the gas supplies is one sound speed and one density.

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

“A plasma is any ionised gas.”

It is an ionised gas in which the collective response works, and the test is the number of particles inside a Debye sphere. That number is 10⁹ for interstellar gas, 10⁵ for the ionosphere and about 8 in the core of the Sun — which is why the solar core is on the edge of being strongly coupled and the ordinary theory is applied to it with care.

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

“The Eddington limit is a hard ceiling that nothing can exceed.”

It is a ceiling on a spherical flow of fully ionised hydrogen, and both conditions can be broken. A flow confined to a disc lets radiation escape along the poles while matter arrives in the plane; and where the gas is dense enough for radiation to be trapped and carried inward, the limit is evaded outright. Observed sources exceeding it by factors of hundreds are known.

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

“The rainbow is brightest exactly at the rainbow angle, where the deviation is stationary and the rays pile up.”

The ray calculation gives an infinite intensity there, which is the signature of an approximation failing rather than of anything bright. The wave answer is the squared Airy function: finite, and peaking at ζ = −1.0188 in the scaled variable — displaced inside the geometric angle by an amount that depends on the drop size and the wavelength.

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

“The geometric phase is a quantum effect, discovered by Berry in 1984.”

Pancharatnam published the optical case in 1956, and it is elementary interference: the phase of the product of three overlaps between polarisation states. Berry's contribution was to recognise the same structure in adiabatic quantum transport, and the optical result had been sitting unread for twenty-eight years.

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

“The gas coming out of a small hole is a fair sample of the gas inside.”

It is biased by one factor of speed, because a molecule's chance of reaching the hole is proportional to how fast it is moving. The beam's distribution goes as v³ rather than v², its mean speed is 3π/8 times larger, and its mean kinetic energy is 2kT against 3/2 kT — a third higher, with no heating anywhere.

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

“Nucleation switches on at a critical supersaturation.”

The rate is smooth, finite and computable everywhere. It spans eighty decades over the range drawn here, so a ten per cent change in supersaturation moves it by ten orders of magnitude — which is why every experiment sees a sharp onset. Nothing switches; a continuous function is simply very steep.

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

“The Unruh effect is an analogy with Hawking radiation.”

It is the same formula. T = ħa/2πck for an accelerated observer and T = ħκ/2πck for a horizon of surface gravity κ, and evaluating the first at a black hole's surface gravity reproduces the second to a part in 10¹² — which is not an analogy but the equivalence principle applied to a temperature.

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

“Two clocks can be synchronised by moving one slowly from the other, which measures the one-way speed without assuming it.”

Slow transport gives the same answer as the radar convention only if the moving clock's rate change is assumed isotropic — which is the assumption being tested. The procedure is self-consistent and not independent: it reproduces whatever anisotropy was built into the description of the transported clock's timekeeping.

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

“A clever enough arrangement of fixed charges could hold another charge in stable equilibrium.”

No arrangement can, and the obstruction is not difficulty. In a charge-free region ∇²V = 0, so the average of V over any sphere equals its value at the centre — checked here to five parts in a hundred thousand — and a maximum or a minimum would require the average to differ from the centre. Every stationary point is a saddle.

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

“Decoherence explains why a measurement has a single definite outcome.”

It explains why the outcomes stop interfering and why one particular set of states is picked out, and it does not remove any branch. The state after decoherence is still a superposition — of system-plus-environment — and the step from that to one outcome is exactly the measurement problem, untouched. Every serious account says so; the claim persists anyway.

Tested in Where the interference goes · the measurement reading path

“Conservation of energy is an experimental fact, established by never having seen it fail.”

It is a consequence of the laws not mentioning the time, and it fails precisely when they do. A pendulum whose length is being changed by hand has no conserved energy, and the rate at which its energy changes is the explicit time-dependence of its Lagrangian — computable in advance rather than discovered by measurement. In an expanding universe the same statement applies to light, whose energy falls as it redshifts, with nowhere for it to go.

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

“A statically indeterminate problem is one that is hard to solve.”

It is one that has no solution as posed. Four reactions under a rigid top satisfy three equations, and the solutions form a line: every member of that line balances every force and every moment to machine precision, which is drawn here at five points along it. No amount of cleverness selects one, because nothing in the statement distinguishes them.

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

“A sound wave in air is described by the wave equation.”

It is described by the wave equation to first order in the amplitude and by nothing so simple beyond that. The speed of sound rises with the local temperature and the medium is itself moving with the wave, so a crest travels faster than a trough by (γ+1)/2 times the particle velocity. At an ordinary conversational level that correction takes kilometres to matter; at 194 decibels it takes metres, and every sonic boom is the correction having finished.

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

“A soliton is a wave that does not spread because the medium happens to be non-dispersive.”

The medium is strongly dispersive and that is the point. Switching the dispersive term off in the integration here leaves a pulse that steepens towards a shock; switching the nonlinear term off leaves one that spreads and sheds a tail. Only with both present does the profile survive, and it survives to a part in ten thousand billion of its own height.

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

“Viscosity is internal friction, so heating a substance always makes it flow more easily.”

Every gas gets more viscous as it warms. Air's viscosity rises by 55 per cent between 250 and 900 kelvin, and the exponent measured on the drawn curve is +0.77, where glycerol's over a comparable range is −23. Friction is the wrong picture for a gas: what is transported is momentum carried across a plane by molecules in flight, and a hotter molecule crosses more often.

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

“A granular material becomes rigid when it becomes dense enough.”

Density is a proxy and a poor one. The transition is in the contact count, and two packs of identical density can differ by a contact per grain if one was tapped and the other poured. The measurement here holds the grains, their positions and their density fixed and varies only which contacts are present: at a mean of three contacts each the pack has fifteen ways to rearrange for nothing, and at four it has none.

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

“A moon breaks up inside the Roche limit because the tidal force there exceeds the planet's pull on it.”

The tide is a *difference* of the planet's pull across the satellite and is far smaller than the pull itself everywhere. What it is compared against is the satellite's own surface gravity, which is what holds the satellite together; the drawn comparison crosses zero at 1.15 planetary radii for ice around a planet of Saturn's density, where the planet's own attraction is thousands of times larger than either.

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

“The sunlight arriving now was made in the core a hundred thousand years ago.”

No photon survives the journey. Energy is absorbed and re-emitted at every step — 10²⁰ of them on this estimate — and at each absorption the identity of the quantum is destroyed. What crosses the Sun is energy, and it arrives as a great many low-energy photons rather than as the gamma ray that started; the temperature falls from 15 million kelvin to 5,800 along the way, and the photon count rises to match.

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

“A halo is a rainbow made of ice instead of water.”

The mechanisms are different and the two look different in the way the mechanisms predict. A rainbow is refraction, one internal reflection, refraction, with the light coming back towards the sun and the deviation having a *maximum*; a halo is refraction through a 60° prism with no reflection at all, in the forward direction, with the deviation having a *minimum*. The consequence is visible: a rainbow's red is on the outside and a halo's is on the inside, and the two are drawn here from the same dispersion.

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

“Speckle is a texture of the illuminated surface, brought out by coherent light.”

The grain size is set by the aperture and not by the surface. Measured here over five apertures spanning a factor of four, the grain width goes as the aperture to the power −0.994 against an exact −1; a rougher surface, or one made of a different material, gives grains of exactly the same size through the same aperture. What the surface decides is where the grains fall, not how big they are.

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

“The photoelectric effect proves that light comes in quanta.”

A classical electromagnetic wave falling on an atom whose energy levels are quantised reproduces the threshold frequency, the instantaneous onset and the intensity-independent stopping voltage — the whole of the observed phenomenology. That calculation was done by Wentzel and Beck in 1926 and the effect has not been a proof of light quantisation since. What the effect demonstrates is that *matter* is quantised.

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

“Two levels avoid crossing because they repel one another, like two charges.”

Nothing repels. The eigenvalues of a two-by-two matrix with an off-diagonal element are plus and minus half the square root of the squared detuning plus four times the squared coupling, and that expression cannot vanish unless the coupling does. The separation at closest approach is exactly twice the coupling — measured off the drawn curves at 0.10, 0.30 and 0.70 eV for couplings of 0.05, 0.15 and 0.35 — and calling it a repulsion attributes to a force what belongs to an algebraic identity.

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

“The exchange interaction is a force that keeps identical fermions apart.”

There is no term in the Hamiltonian for it and no momentum carried between the particles. The pair correlation computed here has no interaction anywhere in it: the only difference between the three cases drawn is whether the two-particle state is symmetric, antisymmetric or neither, and that difference alone produces a hole reaching exactly zero at contact and a pile reaching exactly two.

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

“Field lines refract at a dielectric boundary in the same way light does, so the same law applies.”

The form is different and the difference is measurable. Light obeys a ratio of sines and a static field obeys a ratio of tangents, which agree only for small angles; at 68° incidence into a medium four times as permittive the tangent law gives 84° and a sine law would give an angle that does not exist at all. The two also bend in opposite senses relative to the normal for the same ratio.

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

“A large coil couples more strongly to a small one than the small one does to the large, because it intercepts more flux.”

It intercepts more flux and it also produces a weaker field per unit current over the region where the small one sits, and the two effects cancel exactly. The two couplings are computed here by separate surface integrals over a disc and a tilted rectangle differing by a factor of twenty-one in area, and they agree to 0.075 per cent across a range of separations.

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

“A superconductor is a metal whose resistance has fallen to zero, so its magnetic behaviour follows from that.”

Zero resistance predicts that whatever field was present when the resistance vanished stays trapped, because any change of flux would drive a current that never decays. A superconductor cooled *in* a field expels it instead. Those are different final states from the same history, so the magnetic behaviour is an independent property and not a consequence — which is what Meissner and Ochsenfeld found in 1933, twenty-two years after the resistance was measured.

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

“A negative temperature is colder than absolute zero.”

It is hotter than any positive temperature. Bring a negative-temperature system into contact with anything at all and energy flows out of it, because that is what raises the total entropy — the same criterion that decides every other direction of heat flow. The ordering is continuous in 1/T, which runs from large positive through zero to negative; the temperature itself has the discontinuity, and reading the ordering off T rather than 1/T is what produces the confusion.

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

“A gas cools when it expands, because it does work against its surroundings.”

In a throttling expansion no work is done on anything and no heat crosses any boundary; the enthalpy is what stays constant, and an *ideal* gas throttled through a plug does not change temperature at all. Whatever cooling occurs is entirely the departure from ideality, and it can be a warming: helium, hydrogen and neon all warm when throttled from room temperature.

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

“A rigid body is an idealisation that a sufficiently hard material approaches.”

It is a signal at infinite speed, and the light cone is a floor no material approaches. A 3-metre steel rod's far end waits 600 microseconds, against a light-crossing time of 10 nanoseconds — a factor of sixty thousand — and the stiffest material imaginable would still wait 10 nanoseconds. What separates steel from rigidity is not a large number but an infinite one.

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

“Relativity makes interstellar travel harder, because the mass increases as the ship speeds up.”

The relativistic mass ratio is *smaller* than the Newtonian one at every speed, because rapidity grows more slowly than velocity would need to. The difficulty is not relativistic at all — it is Tsiolkovsky's logarithm, which was there in 1903, and the exhaust speed of chemical propellant. A chemical drive needs 10^2968 even without relativity, and relativity reduces that number.

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

“The attractor of a chaotic system is a curve too complicated to draw.”

It is not a curve at all, and a number says so rather than an adjective. Counting boxes gives a dimension of 1.29, and the two Lyapunov exponents — measured separately, by a method that never counts anything — give 1.26 through Kaplan and Yorke's formula. A curve would give one and a filled region two. The two routes agree to 0.03, and neither can return a non-integer for anything a finite list of curves could describe.

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

“A pendulum is isochronous, so its period does not depend on the amplitude.”

It is isochronous only in the limit of no amplitude, and the same quadrature applied to a circular arc and to a cycloid separates the two cases completely: the cycloid's four release points give 1.003205 seconds each, agreeing to seven parts in ten million million, while the circle's spread over the same range of releases is eighteen per cent. The flat start of the circle's curve is what makes the error survivable and also what made it hard to notice — it is second order in the amplitude, so halving the swing quarters the error.

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

“Diffraction is what happens when the wave nature of light smears out the edge of a shadow.”

Smearing is a description and not a mechanism, and it predicts nothing. The split computed here is exact: the field behind a straight edge is the unblocked incident wave plus one further wave, and the two add back to the directly computed field to a part in 10¹². The second term is not a correction to the first. In the shadow it is the entire field, and in the light it is the thing the incident wave beats against — which is where the fringes come from, since a smooth decay has nothing to interfere with.

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

“A surface wave is a bulk wave trapped by repeated total internal reflection at the surface.”

Total internal reflection needs two media and an angle; a free surface has one medium and the surface wave has no angle at all. It is a separate solution of the same equations, with its own speed — 0.9194 shear speeds at Poisson's ratio 0.25 — obtained here by finding where the free-surface traction vanishes, and that speed is *slower* than every bulk wave in the material. A trapped bulk wave would have to travel at a bulk speed along the surface and cannot travel slower than one.

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

“A soap film always takes the shape of least area, so it breaks as soon as a smaller-area alternative exists.”

It takes a shape of *locally* least area, which is a weaker requirement. The film between two rings has a larger area than the two flat discs once the half-separation exceeds 0.5293 ring radii, and it survives to 0.6627 — 25 per cent further — because there is no continuous deformation that gets it to the discs without passing through larger areas. What ends it is not losing the competition but running out of solutions: past 0.6627 the equation fixing the catenoid has no root at all.

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

“A polymer solution climbs a stirring rod because it is very viscous and drags itself up.”

Viscosity produces one stress in simple shear and only one, at every rate, for any value of the viscosity: the shear stress. Glycerol at a thousand times water's viscosity produces exactly the same dip as water at the same rotation rate, because the surface shape depends on stresses the viscosity does not generate. What climbs the rod is the first normal stress difference, which grows as the square of the shear rate where the shear stress grows in proportion, and which is identically zero for every Newtonian liquid however thick.

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

“Light orbits the photon sphere, so a black hole is surrounded by a shell of trapped light.”

The effective potential for light has a maximum there and no minimum anywhere, so the circular orbit exists and is unstable: any ray on it leaves, inward or outward, on the smallest perturbation. Nothing accumulates there. What the photon sphere produces instead is the infinite sequence of ever fainter images at the shadow's rim, each from light that went round one more time before escaping, and the traced rays here wrap 1.06 times at an impact parameter two per cent above critical.

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

“A wave that damps must be losing energy to something dissipative.”

The equation integrated here has no collision term, no viscosity and no resistivity; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field falls by four decades over forty plasma periods, at 0.1534 per period against a published 0.1534, while the free energy of the perturbation is constant to 2.4 × 10⁻¹⁰. Nothing is dissipated. The energy is in the particles' ordered motion and can, in principle, be given back.

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

“Blocking half of an aperture can only make the image dimmer.”

Blocking the right half makes the axis brighter, and the arithmetic says by how much. The open rings all contribute with the same phase, so the amplitude grows in proportion to how many there are and the intensity as the square — measured here to within nought point one per cent of a parabola across twelve zone counts. The same apertures with nothing blocked do not grow at all: they oscillate between roughly four times the free intensity and almost nothing, according to whether the outermost zone they happen to contain is a contributing one.

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

“Making a quarter-wave plate requires a birefringent material.”

It requires a means of delaying one polarisation relative to the other by a quarter of a cycle, and a total internal reflection provides one with no material property involved beyond the index. Two reflections at 47.70 degrees inside a block of ordinary borosilicate give 90.0 degrees of retardation, and the block is not birefringent at all. Fresnel built one in 1817, before the wave theory that explains it was accepted.

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

“The image is a map of the surface's atoms.”

It is a map of the surface's density of electronic states near the Fermi level, at the energy the bias selects, convolved with the tip's own states. On graphite only half the carbon atoms appear, because the two sublattice sites have different densities of states there and one of them is nearly invisible; on some semiconductor surfaces the image inverts entirely when the bias is reversed, because filled and empty states sit in different places. What is measured is where the electrons are available, not where the nuclei are.

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

“A quantum state cannot be copied because any copier would disturb it, and a better-designed one might not.”

No design enters the argument. A copier is a linear machine, so a machine that copies two states perfectly has already had its behaviour on every superposition of them decided; the figure computes what it then does to an equal superposition, and the answer is an entangled pair whose overlap with the two copies wanted is exactly 0.500. There is nothing to improve, because there is no free parameter left by the time the contradiction appears.

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

“The carriers in a superconductor are electrons, so the flux quantum is h/e.”

h/e is 4.1357 × 10⁻¹⁵ webers and the measured quantum is 2.0678 — a clean factor of two, which no experimental accuracy is required to distinguish. Three candidate carrier charges give 4.1357, 2.0678 and 1.3786, and only the second matches. The experiment therefore weighs the carrier's charge without measuring a charge, and it did so in 1961, before any independent confirmation that pairing was what the microscopic theory required.

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

“The absence of magnetic monopoles is an experimental fact that the equations record.”

It is an experimental fact, and what the equations record is stronger and different: the divergence of B cannot change under Faraday's law whatever happens, so if it is zero once it is zero for ever. A monopole cannot be created by any process the equations describe. Adding one therefore means changing the equations, not adding a particle to them — which is why a monopole would force a magnetic current term into Faraday's law and would make the vector potential impossible to define globally.

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

“Radiation is a gas of photons, so it obeys the gas law with the photons as the particles.”

It obeys a different law, and the reason is that the number of particles is not an independent variable. The pressure of blackbody radiation is one third of its energy density; a molecular gas at the same energy density pushes twice as hard. The difference is that a photon of energy E carries momentum E/c while a slow molecule carries the square root of 2mE, and no amount of counting photons repairs it.

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

“The Seebeck and Peltier coefficients are two independent properties of a material, to be measured separately.”

They are one property measured two ways, related by a factor of the temperature. The figure computes both for the same conductor by two computations that share no step — one nulls a thermally driven current with a voltage, the other takes the ratio of two currents driven by a voltage at uniform temperature — and they agree to 3.8 × 10⁻⁷ of the sweep's own scale, through a sign change. A laboratory that measures both and finds them different has made an error, not a discovery.

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

“A phase transition is the breaking of a symmetry, so a system that cannot break one has no transition.”

A two-dimensional system with a continuous symmetry cannot develop long-range order at any temperature above zero, and it has a sharp transition all the same. What changes is topological: below the transition vortices exist only in bound pairs of opposite circulation, above it they are free. The free energy of a single vortex is (πJ − 2kT) ln(L/a), so what changes at kT = πJ/2 is the sign of a coefficient, and nothing anywhere becomes non-zero.

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

“A source whose components separate faster than light across the sky is moving faster than light.”

The apparent transverse speed is β sinθ/(1 − β cosθ), which exceeds one for any β above 1/√2 at the right angle, with the true speed below one throughout. At β = 0.99 the largest apparent speed is 7.02, reached at 8.1 degrees from the line of sight, and the maximum is γβ at arccos β — found here by searching the drawn curves rather than assumed. The excess comes from the source having closed on the observer between the two observations, so the light of the second had less far to go.

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

“Motion towards an observer makes a source brighter, so motion in any direction makes it brighter than at rest.”

Seen at right angles a moving source is *fainter* than the same source at rest, by γ⁴ — a factor of 5.8 × 10⁵ at β = 0.99. The Doppler factor there is exactly 1/γ, which is less than one, because the transverse redshift is a pure time-dilation effect with no line-of-sight component to cancel it. Only sources within about one over γ of the line of sight are brightened at all, and everything outside that cone is dimmed.

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

“A rigid body can be set spinning, since rigidity is an idealisation that only fails at extreme accelerations.”

It fails at every acceleration, because the requirement is geometrical rather than dynamical. Each circle of material must end shorter in its own terms than it began, by the local Lorentz factor, while its distance from the axis is unchanged — 8.3 per cent at a rim reaching four-tenths of light speed and 56.4 per cent at nine-tenths, and zero on the axis in every case. A body that keeps every internal distance fixed as measured by itself admits no change of rotation rate at all, which is Born's definition and Herglotz and Noether's theorem about it.

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

“A ball rolling off a sphere leaves the surface when the component of gravity along the radius can no longer supply the centripetal force, which happens at 45°.”

It leaves at 48.19°, where cos θ = ⅔, and the condition is not about gravity's component but about the sign of the surface's own push. The reaction is mg(3cos θ − 2cos θ₀) and it reaches zero there; past that point the surface would have to pull inward, which a surface cannot do. Forty-five degrees is where gravity's radial and tangential components are equal, which is not a condition on anything.

Tested in The force a coordinate cannot see · the least action reading path

“The boundary of everywhere a projectile can reach is the trajectory launched at 45°.”

The 45° launch reaches the greatest range along the ground and is nowhere near the boundary anywhere else. The boundary is the envelope of the whole family — a parabola of height v²/2g and half-width v²/g — and the 45° trajectory touches it at exactly one point, its own landing point, as every other launch does at exactly one point of its own.

Tested in Everywhere a throw can reach · the projectile reading path

“The nth mode of a drumhead divides it into n regions, as the nth mode of a string divides it into n pieces.”

It divides it into at most n, and usually fewer. Counting the sign regions of the computed modes of a circular membrane by flood fill gives 1, 2, 4, 2, 6, 4, 8 against ranks of 1, 2, 4, 6, 7, 9, 11 — the second radial mode is two rings where its rank would allow six. Courant proved the inequality in 1923 and Pleijel proved in 1956 that equality fails for all but finitely many modes of any shape.

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

“An echo from a moving object is Doppler shifted twice, so the shift is the square of the one-way shift.”

The two shifts are different functions. A moving observer multiplies the frequency by (c + v)/c because it runs into the fronts; a moving source divides by (c − v)/c because it chases the ones it has made. Their product is (c + v)/(c − v), which equals the square of either only in the limit v/c → 0. At a tenth of the sound speed the round trip is 1.2222 and the square of the one-way shift is 1.2100.

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

“A cleverly shaped structure — a funnel, a horn, a wedge of absorber — can let a wave through in one direction and not the other.”

No passive, linear, stationary arrangement of materials can. A stack of nine slabs of random thickness and random wavenumber, with no symmetry in it anywhere, transmits 0.669424094 from the left and 0.669424094 from the right, in modulus and in phase. The result depends on the wave equation being unchanged under reversing the sign of time, and no amount of shaping alters that.

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

“In total internal reflection no light enters the second medium.”

A field is there and it is not small. What is absent is a *direction*: the conserved tangential component exceeds anything the far medium's circle can supply, so the normal component is imaginary and the field decays instead of travelling — at 50° from glass to air it falls by a factor e in 0.21 vacuum wavelengths. Bringing a second piece of glass within that distance recovers a transmitted beam, which no theory of a light that stops at the surface could explain.

Tested in The law that only asks about one component · the refraction reading path

“Interference requires the light to be coherent, so an instrument that measures a source's size must preserve the phase.”

Correlating the *intensities* at two apertures measures |γ|², which contains the same information about the source as the |γ| a fringe contrast measures — one curve determines the other, and both vanish at the same baseline. No phase is preserved anywhere in the apparatus, and the two detectors need not even be looking at the same wavefront to better than a few metres.

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

“At constant voltage the stored energy increases as the slab enters, so the system is moving to a state of higher energy and the force must be outward.”

The force is inward in both cases and the same size. At fixed voltage the battery supplies V²ΔC while the field keeps only half of it; the other half is the work done on the slab. The quantity a system at fixed voltage minimises is not the field energy but the field energy minus the battery's contribution, and reading a force off the wrong potential is exactly how the sign is lost.

Tested in The force that lives where the model is not · the dielectrics reading path

“The force per unit area on a charged surface is σE, the charge density times the field it sits in.”

It is σE/2. The field is σ/ε₀ just outside and exactly zero just inside, and the surface layer feels neither — a charge exerts no force on itself, so what acts on the layer is the field the rest of the charge makes, which is the mean of the two sides. Using the full external field doubles the answer, and the same factor appears wherever a surface distribution is asked what force it feels.

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

“The three latent heats at a triple point are three independent properties of the substance.”

Two of them fix the third, because enthalpy is a state function and the three paths between three phases must close. For water at 273.16 K the measured values are 333.5, 2500.9 and 2834.4 kJ/kg, and the first two sum to the third within four parts in ten thousand. The same closure makes the sublimation curve steeper than the vaporisation curve by exactly the amount the fusion latent heat predicts — 5.92 Pa/K on water's numbers.

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

“The classical limit is what happens as Planck's constant becomes negligible.”

Planck's constant is a constant and does not become anything. What has to be small is the spread of the state compared with the distance over which the force changes appreciably — a condition involving the potential's third derivative and the packet's width, both of which are properties of the situation. A macroscopic object in a sufficiently curved potential loses the correspondence, and a microscopic one in a harmonic well never does.

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

“Two barriers in series transmit the product of their individual transmissions.”

That would be true if probabilities composed, and they do not. Two barriers of width 0.9 that each pass 3.9 per cent transmit 1.000000 at the resonant energy — 640 times more than the product, and 26 times more than either barrier alone. What compose are amplitudes, and the amplitudes for the many bounces between the barriers add in phase at particular energies.

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

“A silicon diode has a forward voltage drop of 0.7 volts.”

It has no fixed drop. The current is exponential in the voltage, rising a decade every 59.5 millivolts at room temperature, so any current a circuit is likely to pass lands in a window a couple of tenths of a volt wide. The constancy is an artefact of reading a very steep curve on a linear axis, and the same diode drops 0.4 V at a microamp and 0.9 V at an amp.

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

“The travelling twin ages less because of the acceleration at the turnaround.”

Make the turn gentler and gentler and its contribution goes to nothing while the age difference does not. At a tenth of a gravity the turnaround accounts for about half the difference; at a thousand gravities it accounts for a hundredth of a per cent, and the total tends to 2.67 years for the four-light-year trip drawn — which is exactly what the instantaneous-turnaround idealisation gives. The acceleration decides which twin's path is bent; the legs decide by how much.

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

“Liquid helium below the lambda point is an extraordinarily good heat conductor.”

It does not conduct heat at all in the ordinary sense. Heat is carried by a counterflow — the normal component moving one way and the superfluid the other — and disturbances in it propagate as a wave with a definite speed rather than diffusing. Measured as a conductivity the answer is not a large number but a meaningless one: it depends on the geometry, the heat flux and the length of the sample, because there is no conductivity there to measure.

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

“Large particles rise because they are lighter, or because buoyancy carries them up.”

The effect works with an intruder denser than the bed as well as lighter, and the standard demonstration uses a large steel ball in fine sand, which is denser than what surrounds it. Density enters the problem and does not drive it: the ordinary version of the effect is driven by small grains percolating *down* through gaps too small for large ones, and the large one rises because the material beneath it has been replaced.

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

“A surface immersed in a plasma sits at the plasma's potential.”

It floats several electron temperatures below it. The electron flux to an uncharged surface exceeds the ion flux by the square root of the mass ratio, so the surface charges negative until the Boltzmann factor of its own potential has cut the electron flux down to match — which for hydrogen is 2.84 kTe/e and for xenon 4.68. The dependence is logarithmic in the mass ratio, which is why the whole periodic table lands between three and five.

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

“Light can only push, since it carries momentum in the direction it travels.”

The momentum flux gives a push and is not the only force light exerts. A non-uniform field polarises a particle and pulls the induced dipole toward the strong field, which for a focused beam is *back toward the waist* — against the direction of travel, for a particle downstream of the focus. The two forces are computed from different quantities: one from the intensity, the other from its gradient.

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

“The nod is a small imperfection in a real top and disappears in an ideal one.”

The nod is a property of the ideal equations, not of a real top's defects: the figures here have no friction in them and every path but one nods. What removes it in a real top is exactly the imperfection — friction at the pivot damps the fastest motion first, and the nod is the fastest motion. An ideal top released from rest nods for ever.

Tested in The top that nods before it settles · the rotation reading path

“Momentum and kinetic energy conservation require that one ball leaves a cradle when one ball strikes it.”

Three equal balls with two conservation laws leave three unknown final velocities constrained to a curve, drawn here as an ellipse. Every point of it conserves both quantities exactly, including one where the middle ball rebounds at two thirds of the striking speed. Restricting to outcomes in which the balls do not pass through one another leaves an arc, and an arc is still a continuum. The two laws are two equations in three unknowns and cannot have a unique solution.

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

“A belt drive slips a little all the time, and the speed it loses is wear or stretch.”

The loss is creep over the active arc, and it is present in a perfect belt at any load below capacity. The figures compute how the wrap divides: below capacity part of it rides at constant tension and does nothing, and the rest creeps at the full exponential rate. The belt genuinely leaves the driver slower than it arrived, and no improvement in the belt removes it.

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

“The two versions differ only in notation.”

They differ in what is varied. A deformed path at fixed duration carries a different energy from the true one, and the figure shows that energy changing at first order in the deformation; a deformed path at fixed energy takes a different time, and that time changes at first order too. Each principle is stationary in its own quantity precisely because it lets the other one move.

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

“The irregular values a chaotic map visits are noise, and a longer run would average them out.”

The map here has no random number in it and the same starting value gives the same sequence every time, to every digit the arithmetic carries. What is drawn past the accumulation point is the set of values a single deterministic orbit visits, and it is a band with structure — windows of stable cycles inside it, each with a cascade of its own, at every magnification examined.

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

“The steady response of a driven oscillator is a function of the drive frequency.”

Over a range of frequencies this oscillator has two stable amplitudes and which one it is in depends on where the drive has been. Swept up, the amplitude climbs the leaning peak and falls off its edge; swept down over the same frequencies it follows the low branch and jumps up somewhere else. Both sweeps are the same equation with the same coefficients.

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

“An inductor opposes current.”

It opposes change. The figures show the voltage across a coil falling from the full supply to nothing while the current rises to its steady value: at the end there is a large current and no voltage at all. A coil carrying a steady current is a piece of wire, and the whole of its behaviour is in the derivative.

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

“Pressing two flat surfaces together makes contact over their apparent area.”

Surfaces are rough, so contact happens at the highest asperities and the real area is a small fraction of the apparent one. The measured resistance says how small: it is set by the number and size of the spots, and it flattens onto a floor decided by the size of the patch they are scattered over. That measurement is one of the two independent routes to the real contact area, the other being friction, and they agree.

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

“The jumps are noise, and a better sample would not have them.”

Nothing in the model here is random in time: the pinning landscape is fixed, and the same crystal gives the same sequence of jumps every time it is swept. The jumps are a deterministic consequence of a wall having to cross a landscape with local maxima in it, and a sample with less disorder has larger jumps rather than fewer.

Tested in The curve that is really a staircase · the magnetisation reading path

“A magnetic monopole would need a new law.”

It would need a source term in two equations that presently have none, and nothing else. The difficulty is not with the field, which is the simplest field there is, but with the potential: the figure shows that any vector potential producing a radial field must be infinite along a line, because a potential regular everywhere would give zero net flux. That line is the whole of the theoretical trouble.

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

“A better superconductor carries more current.”

A perfect type-II superconductor carries almost none. A current exerts a force on the flux tubes, moving tubes induce a voltage, and a voltage with a current is dissipation — so a defect-free type-II material has resistance at any field above the first critical one. What makes a wire useful is defects strong enough to pin the lattice, which is why practical conductors are deliberately made dirty and cold-worked.

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

“An object at rest has no momentum.”

A current loop at rest in an electric field carries mechanical momentum of magnitude mE/c², where m is its magnetic moment. Its carriers are moving, and those on the high-potential side are more energetic than those on the low, so the same number of them per second carries unequal momentum round the two halves. Nothing about the loop as a whole moves, and the loop's momentum is not zero.

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

“A fibre with a bigger core accepts light from a wider angle.”

The acceptance angle is the square root of the difference of the squared core and cladding indices, and the core's diameter appears nowhere in it. A fibre a hundred times thicker accepts exactly the same cone; what it gains is area, and area times solid angle is the quantity that decides how much light gets in.

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

“Depth of field is a property of a lens.”

It is a property of a lens, a distance, an aperture and a decision about how closely the result will be examined. The figures compute the blur circle for one lens at three apertures focused at one distance: the acceptable range runs from a fraction of a metre to infinity depending only on which horizontal line is drawn, and the line is a choice about the viewer rather than about the optics.

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

“Unpolarised light is light whose direction of vibration changes rapidly and at random.”

That description is a model of one way of arriving at the same measurements, and there are others which are experimentally identical — a beam that is half horizontal and half vertical with no correlation between them behaves the same way and has no changing direction anywhere. What is measurable is the set of Stokes parameters, and every arrangement giving the same four is indistinguishable by any experiment.

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

“A large opaque particle removes exactly the light its shadow blocks.”

It removes twice as much. Half is the shadow; the other half is light diffracted out of the beam at the shadow's edge, and it exists for any obstacle large compared with the wavelength. The figure computes the extinction efficiency against size and it approaches two, through an oscillation that reaches beyond three for particles a few wavelengths across.

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

“Spherical aberration is unavoidable in a refracting surface.”

A Cartesian oval has none: the figures trace rays over a half-angle of twenty-six degrees and every one arrives at the image point, with the optical path lengths across the drawn surface agreeing to one part in a billion. The surface is generated by the equal-path condition and is then checked to refract by Snell's law, which is Fermat's principle tested rather than assumed.

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

“The resolution of a Fourier-transform spectrometer is set by the detector or the light.”

It is set by how far the mirror travels. A scan of length L convolves the recovered spectrum with the transform of the window, giving lines about 0.6/L wide whatever the source or the detector. To resolve a tenth of a reciprocal centimetre the mirror must move six, and no improvement to anything else substitutes for the distance.

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

“At a dark fringe the two waves' energy is destroyed.”

The cross term that makes the fringes averages to exactly nothing over a whole cycle of phase and over a whole number of fringes, so the total power on a screen is the same whether the sources interfere or not. The figure integrates the pattern and finds the excess above the no-interference line equal to the deficit below it. Nothing is destroyed; it has been moved from the dark places to the bright ones.

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

“The disturbance at a point comes mostly from the part of the wavefront directly in line with it.”

Each half-period zone contributes the same amount, because a zone's area grows exactly as fast as its distance. What makes the sum converge is the obliquity factor and the finite extent of the wavefront, and the total comes to half the first zone's contribution. Blocking every second zone leaves the remainder in phase and gives a far brighter result than the unobstructed wave, which is what a zone plate is.

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

“A wave driven above the cutoff is absorbed.”

There is no damping anywhere in the model. Driven above the cutoff, the chain has no travelling solution at all, and the only bounded response is one that falls by a fixed factor from each mass to the next while alternating in sign. The decay is the chain's own dispersion relation continued past the point where it has a real solution, and the energy is reflected rather than taken up.

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

“A standing wave has nodes where nothing happens.”

Only a completely reflected one does. With a partial reflection the minima sit at one minus the reflection magnitude rather than at zero, and the power crossing every point of the line — including the minima — is the same, equal to the fraction transmitted. Energy passes steadily through the places the amplitude is smallest.

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

“Bend loss sets in gradually as the radius is reduced.”

It falls exponentially with the radius, so over a millimetre or two it goes from unmeasurable to fatal. The figures compute the loss per turn against radius and it crosses six decades in a few millimetres, which is why a fibre tolerates a coil of one size and not one slightly smaller.

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

“Fluctuations are experimental noise and vanish for an ideal system.”

They are a property of equilibrium and are largest for the most ideal systems, since a system with fewer degrees of freedom fluctuates more. The mean square fluctuation is fixed by the heat capacity, which is a bulk thermodynamic quantity, and the relation runs both ways: measuring how much a small system's energy wanders is a measurement of its heat capacity.

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

“Skating works because the pressure of the blade melts the ice.”

A seventy-kilogram skater on a blade a millimetre wide and twenty centimetres long produces about thirty-five atmospheres, and the melting point falls by 7.4 millikelvin per atmosphere — a quarter of a kelvin in total. Nobody skates at a quarter of a degree below zero. The lubricating layer is there because a crystal's surface is disordered whether or not anything is pressing on it.

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

“A mixture at uniform temperature and uniform composition is at equilibrium, and one at uniform composition with a temperature gradient will simply conduct heat.”

It will also separate. A temperature gradient drives a flux of each component, so the composition drifts until the ordinary diffusion back down the concentration gradient it has created cancels it. The steady state is a balance of two flows rather than the absence of any, and the separation is measurable — a few per cent across a hundred degrees in a liquid.

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

“A free energy can only be measured by a reversible process.”

The two work distributions from a fast process cross at exactly the free-energy change, whatever the dissipation, and the exponential average of the work is exactly e to the minus the free energy. Both relations hold arbitrarily far from equilibrium, and both are used routinely on single molecules pulled far too quickly for any quasi-static argument.

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

“Real gases depart from ideality because their molecules take up room.”

That is one of two effects and it has the opposite sign from the other. The repulsive core makes the second virial coefficient positive and the attractive tail makes it negative, and at ordinary temperatures the attraction usually wins — most gases are more compressible than an ideal gas, not less. Which effect dominates is a matter of temperature, and the figures compute the crossover.

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

“The uncertainty relation is a fixed number attached to a pair of observables.”

Robertson's bound is half the size of the average commutator, and an average is taken in a state. The figures follow the same pair of spin components round two great circles of states: on one the bound is an equality everywhere, and on the other it is exactly zero while the product it bounds is not.

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

“A charged particle responds only to the fields at its own position.”

Both paths run where the field is smaller than a twentieth of a per cent of the winding's, and the fringe pattern moves by a whole fringe for every flux quantum enclosed. A response that depends on what is inside a region the particle never enters is not a response to the local field.

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

“Quantum wave packets always spread.”

In a harmonic well one width does not. A Gaussian of exactly the ground state's width, displaced, is propagated here for more than two full periods and its width moves by a part in ten thousand, while packets sixty per cent and seventy per cent wider breathe visibly at twice the swing frequency.

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

“There is a well-defined time a particle spends inside a barrier.”

Four defensible definitions are computed here from the same exact amplitude, and they disagree without limit as the barrier thickens: the phase time saturates, one Larmor time grows in proportion to the thickness, the other saturates at a different value, and the free-flight comparison grows too. A quantity with four different values is not one quantity being measured badly.

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

“A particle can be strongly entangled with several others at once.”

The CHSH values a party shares with two others obey S² + S² ≤ 8, so the most either can be while the other also exceeds the classical value of 2 is exactly 2 — no violation at all. The trade-off region is drawn and the best simultaneous value is found on it numerically.

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

“Heavier nuclei make smaller atoms, since the attraction is stronger.”

Across thirty-four main-group elements the one-electron prediction a₀/Z spans a factor of 83 and correlates with the measured radii at −0.50. The measured radii span 10 and repeat rather than fall. Every added proton arrives with an added electron, and what the outermost electron feels is a screened charge that barely grows across a row.

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

“The relativistic centre of mass is the ordinary centre of mass, with relativistic masses substituted.”

Weighting positions by rest mass gives a point that does not move uniformly through an elastic collision, because the rest masses are unchanged while the energies are redistributed. Weighting by energy gives a point whose worldline is straight to a part in a million million, at exactly the total momentum divided by the total energy.

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

“Charge transforms like energy, growing with the Lorentz factor.”

If it did, the inner electrons of an atom of atomic number Z — moving at about Zα — would leave the atom with a charge imbalance of order (Zα)²/2. For helium that is a hundred-millionth of an elementary charge, and the measured bound is 10⁻²¹, excluding it by thirteen orders of magnitude.

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

“Velocity space is the interior of a ball of radius c, which is an ordinary flat region with a boundary.”

Its boundary is infinitely far away in its own metric, so it is not a region with an edge — it is a complete plane. Equal steps of rapidity crowd towards the drawn edge without ever reaching it, and the crowding is checked ring by ring.

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

“The travelling twin ages less because of the acceleration at the turnaround.”

Every route drawn here has the same two endpoints and none of them has any special event: the proper time falls off smoothly with how far the route wanders, and the fall is second order in the wandering. The deficit is an integral over the whole route, and acceleration enters only as the reason a route can differ from the straight one at all.

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

“The drag on an oscillating surface is in phase with its velocity, like any drag.”

The stress leads the velocity by exactly forty-five degrees, located on the drawn trace. That splits it into equal halves: one that dissipates energy and one that is an added mass and does not. Neither half exists without the other, and no fluid or frequency moves the split.

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

“The pressure at a point in any fluid is the same in all directions.”

Only at rest. Shearing the same fluid opens the single point into a circle of stresses whose radius is the viscous stress: some planes then carry shear and the normal stress depends on the orientation. What survives is the average normal stress, and calling that the pressure is a convention rather than a theorem.

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

“A hopper empties faster when it is full, like any container.”

The fill height does not appear in the discharge law. The walls carry the weight of everything above a depth of a couple of silo diameters, so the grains at the opening are pushed only by their immediate neighbours. A liquid through the same hole falls to a third of its rate by the time a tenth of the head is left; the grain rate is flat until the surface reaches the outlet.

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

“Two solutions with the same osmolarity are equivalent to a cell.”

Osmolarity counts particles; tonicity asks what the membrane does about them. At the same concentration a red-cell membrane develops the full pressure against sucrose and essentially none against ethanol — a factor of a hundred between the extremes drawn, with the same particle count.

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

“An unbounded capillary rise would need an unbounded amount of liquid.”

The filament's width falls as one over the height, so its cross-section falls as the square, and the integral converges. The volume held above any height is that height times the cross-section there — a fraction of a microlitre for a millimetre corner — checked against a numerical sum over four thousand times the starting height.

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

“Mercury's anomalous precession is a special case needing a special explanation.”

One formula with no free parameters — six pi times the gravitational radius over the semi-latus rectum, per orbit — reproduces the quoted residual advance for Mercury, Venus, Earth, Mars and Icarus alike. Mercury is the largest because it is closest and quickest, and both enter.

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

“An interferometer is sensitive to waves from anywhere in the sky.”

The plus pattern vanishes along four lines and the cross pattern along others, and the sky average of each is well under one — their squares add to two fifths over the whole sky, summed over a hundred and sixty thousand directions. A detector's quoted range is set by that average and is far below its reach for a source directly overhead.

Tested in What the instrument actually hears · the gravitational waves reading path

“Evaporation is a gradual fading away.”

The mass falls as the cube root of the time remaining, so it is nearly constant for almost the whole life and then collapses. When one second of life is left the hole weighs about 230 tonnes whatever it started as, and it releases some 2 × 10²² joules in that second — a burst, not a fade.

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

“The paradox could be settled by measuring a charge held stationary in a gravitational field.”

The Larmor power for an electron at one gravity is 5 × 10⁻⁵² watts. Held for the whole age of the universe it would radiate 2 × 10⁻³⁴ joules — nine orders of magnitude below a single quantum at any frequency it could be emitted in. The question has never been experimental and could not be.

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

“All the Planck quantities are absurdly far from anything human.”

The Planck mass is twenty-two micrograms — a visible speck — and the Planck energy is the chemical energy of about fifty-seven litres of petrol, computed here rather than quoted. That one quantity sitting in the human range is the statement that gravity is weak, and it is the reason the length and the time are so extreme.

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

“A neutron looks different after one full turn, so an object can be distinguishable from itself rotated by 360 degrees.”

Every prediction for a single spin comes from the density matrix, which is built from the state and its conjugate and is therefore blind to an overall sign. The figure measures both distances across one turn: the state moves the full width of the space it lives in, and the largest change in any component of the spin's direction is at the level of the integrator's rounding. No measurement on one neutron distinguishes before from after.

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

“Making a resonator smaller always raises its frequency.”

The map of a cavity wall computed here is positive over part of its area and negative over the rest, with a contour between them where a dent does nothing. Pushing the wall in at the middle, where the electric field peaks, lowers the frequency; pushing it in near the side walls raises it. The whole-cavity result is the sum of these, and the sum being positive for a uniform squeeze says nothing about any one place.

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

“The stress tensor is a way of writing Coulomb's law and adds nothing to it.”

Coulomb's law is a statement about two point charges in vacuum. The surface integral computes the force on the contents of a region whatever they are — a polarised dielectric, a conductor with unknown surface charge, a magnetised body — and it is how such forces are actually computed. The figures here recover Coulomb's law only as a check that the method is right.

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

“Heavier particles drift more slowly, so an electric field separates the species in a magnetised plasma.”

Three species with masses differing by four orders of magnitude and charges of both signs are integrated here in the same crossed fields. Their orbits differ in size and period by those same factors and their guiding centres move together to within a fraction of a per cent. An electric force is itself proportional to charge, so the charge cancels out of the drift and no current is produced.

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

“A rolling wheel has no relative motion at the contact, so a driven wheel rolls without slipping until it breaks away.”

Rolling without slipping is the zero-force case. The moment a torque is applied, the tread ahead of the contact is elastically stretched relative to the carcass, and the integration here shows the trailing part of the patch sliding at every non-zero force. Two per cent slip is normal cruising, and it is sliding over about a quarter of the footprint.

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

“The body flips because something is exerting a torque on it.”

The damping used here is constructed to be orthogonal to the angular momentum, so the momentum is conserved identically rather than approximately, and the integration reports its drift over the whole run. Nothing acts from outside; the momentum vector in space never moves. What changes is which body axis is lined up with it.

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

“Field curvature is an aberration, so a better-made lens has less of it.”

The Petzval sum depends only on the powers and refractive indices of the elements. Bending a surface, moving the stop, adding an aspheric profile and improving the polish change every other aberration and leave this one identically unchanged. It is removed only by adding elements of the opposite sign in a different glass, which is a change of design rather than of quality.

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

“The sky is completely polarised at ninety degrees from the sun.”

Ideal single scattering by isotropic dipoles gives a hundred per cent there. Air molecules are not isotropic, which costs a few per cent that no sky can recover, and light scattered more than once arrives having forgotten the geometry, which costs much more. A clear high sky reaches about eighty per cent and a hazy one about half of that.

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

“A ratchet in a gas turns slowly in one direction, because collisions in the allowed direction are kept and the others are refused.”

The forward rate needs a collision able to lift the pawl and is exp(−ε/kT); the backward rate needs the pawl to lift by its own fluctuation and is exp(−ε/kT) with the same ε and the same T. At one temperature they are identical for every notch depth, which the figures verify, so the net rate is exactly zero and not merely small.

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

“The recurrence shows equipartition failing.”

It shows equipartition being slow. The threshold figure runs the same chain across three decades of energy density and the spread rises steadily with it; below the knee the sharing time is longer than the run, above it the sharing happens while the run is going on. The original calculation sat well below the knee and ran for a few thousand periods.

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

“The airless bodies confirm the balance argument exactly.”

The Moon's mean surface is twenty kelvin below its balance temperature and Mercury's ninety below, because radiated flux goes as the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest. Only a body near isothermal sits on the line, and the deviation for an airless body is in the opposite direction from an atmosphere's.

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

“A black hole's mass can only increase, since nothing gets out.”

The merger computed here loses three solar masses to gravitational radiation, and a spinning hole can in principle give up twenty-nine per cent of its mass without anything crossing the horizon outwards. What cannot decrease is the area, and the two statements are different because area depends on the spin as well as the mass.

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

“Clock comparisons over long distances are limited by the length of the link.”

The uncertainty of an optical-fibre frequency comparison does not grow with distance the way a levelled line's does, because there are no accumulating set-ups in between. The two methods cross at around a hundred kilometres, beyond which the clock is the only one that does not degrade — which is the case for chronometric levelling.

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

“The satellite test worked because a spacecraft has a better accelerometer.”

Its differential resolution is within an order of magnitude of the best torsion balance's, as the figures here show. What it has is a driving acceleration five hundred times larger, because a body in free fall around the Earth is pulled by the whole field rather than by the thousandth of it that the Earth's spin leaves pointing sideways.

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

“Fizeau's experiment measured a partial dragging of the ether.”

It measured how fast light travels in moving water, which is a fact about light and water. The interpretation as ether drag was the framework of the time and required a coefficient depending on the refractive index that no ether model produced. Composing c/n with v relativistically gives the same coefficient in two lines and mentions no medium beyond the water.

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

“A fast enough parent sends all its decay products forward.”

Only those slower than the parent. A product whose own centre-of-mass speed exceeds the parent's can still be found going backwards, at any parent speed, and the figures compute the largest lab angle for both cases. The threshold is a comparison of two speeds, not a matter of the parent being fast in absolute terms.

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

“The four assumptions rule out Galilean relativity.”

They do not. The zero value of the constant satisfies every one of them, has an associative composition law, and gives absolute simultaneity — which the figures verify. It is the physically wrong answer and not the logically excluded one, and knowing which is which is the point of doing the derivation this way.

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

“A membrane potential requires a pump.”

The Donnan potential computed here is tens of millivolts and arises at equilibrium, with no current flowing and no work being done. A nerve cell's resting potential is a different thing of a similar size — maintained against a leak by pumps, and therefore not an equilibrium at all.

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

“A black patch in a soap film is a hole, or a film about to break.”

The reflectance computed here goes to exactly zero as the thickness does, because the two surfaces reflect equally and half a cycle out of step, so a film thinner than about a tenth of a wavelength is black while entirely present. A black film is also the most stable state available: it is where the thinning stops, and a film that has reached it can last for hours.

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

“A column of grains settles to its densest packing after enough shaking.”

The approach is logarithmic in the number of taps, so each further slice of the remaining gap costs a factor rather than an amount. The figures compute the number of taps to close successive fractions: eighty per cent takes tens of thousands, ninety-five per cent would take longer than the age of the Earth at one tap a second.

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

“A flux tube in pressure balance with its surroundings has the same density as them, so it stays where it is put.”

Total pressure balance is p_in + B²/2μ₀ = p_out, so the gas pressure inside is lower by exactly the magnetic pressure. At equal temperature the density follows, and the deficit is 1/(1+β) — one half at β = 1 and one part in ten thousand at β = 10⁴. The curve drawn has no root anywhere. A magnetised tube is never neutrally buoyant; it is only ever slowly buoyant.

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

“A wave in a plasma is a sound wave with magnetic corrections.”

One of the three is not. The shear wave's speed is B/√(μ₀ρ) and contains no temperature, no pressure and no sound speed, so it is unchanged if the gas is made ten times hotter or replaced with a different gas at the same mass density. The polar diagram drawn here shows it as a figure-of-eight sitting between the two magnetosonic branches, and it is the branch whose formula has no thermodynamics in it at all.

Tested in The wave that does not know what the gas is made of · the flux freezing reading path

“The near field is where the far-field formula has not become accurate yet, so it is a matter of a few per cent.”

The three terms fall as 1/r³, 1/r² and 1/r, so at a tenth of the crossing radius the near terms are a hundred and a thousand times the radiation term. That is not a correction; it is a different field with a different distance law, a different phase and a different energy budget. The figures show the total dipping below every individual term just inside the crossing, because the static and radiation terms are in antiphase and partly cancel — behaviour no small correction produces.

Tested in The distance where a field changes its mind · the retardation reading path

“The dipole approximation holds when the antenna is small compared with the distance to the observer.”

The distance to the observer does not appear in the condition. What matters is the extra path from one end of the source to the other, compared with a wavelength: a source a tenth of a wavelength long is down 1.6 per cent at the worst angle, one of 0.6 wavelengths is down 36 per cent, and one of 1.2 wavelengths has nulls in its pattern. All three statements hold at any observation distance in the far field.

Tested in When the source is not heard all at once · the retardation reading path

“Maxwell's equations say that fields propagate outward from their sources.”

They say a field satisfies a wave equation, and a wave equation is second order in time and therefore symmetric under reversing it. The converging solution f(r+ct)/r is drawn here beside the diverging one and both are checked by differencing the same samples the figure draws, twice in space and twice in time: the residual vanishes for both. Retardation is imposed on the solutions afterwards, and choosing it is choosing a boundary condition.

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

“The wavelength of a mountain wave is set by the width of the mountain.”

The vertical wavelength is 2πU/N, in which no dimension of the ridge appears. What the ridge's width decides is which horizontal wavenumbers are excited and therefore how much of the response is a propagating wave rather than a local disturbance — an amplitude question rather than a wavelength one. The figures compute the pattern for two different ridges at the same wind and stability and the vertical spacing is identical.

Tested in The wave that is required to stand still · the stratification reading path