True as far as it goes, and silent on what decides — part 4 of 5

“The temperature profile can be computed from the measured spectrum by inverting the radiative transfer.”

Seven readings with 0.25 K of noise do not determine 25 levels. Fitted to the readings alone the retrieval swings by tens of kelvin, with a root-mean-square error of about 12 K; with a smoothness constraint added it is about 1.6 K. The extra information comes from outside the measurement.

Tested in The spectrum that is a thermometer at every height · the blackbody reading path

“Neutralising half of a beam's charge with ions halves the force that spreads it.”

Ions cancel the electric push in proportion to their number and leave the magnetic pull untouched, so the net force changes sign — the beam starts to pinch — once the neutralised fraction exceeds 1/γ²: 69 per cent at γ = 1.2, 11 per cent at γ = 3, 1 per cent at γ = 10.

Tested in The beam that stops pushing itself apart · the field transformation reading path

“A plasma wave breaks when its field reaches the cold limit mcω_p/e.”

That is the limit for a slow wave. A wave whose phase moves at γₚ holds √(2(γₚ − 1)) times that before its own electrons overtake it — 4.2 times at γₚ = 10, 14 times at 100 — because relativity caps the electrons' speed below the wave's.

Tested in The wave an electron rides to a gigaelectronvolt · the relativistic dynamics reading path

“The diffusion-limited rate is a constant from the moment the reactants are mixed.”

The partners already close to each reactant are used up first, so the rate starts high and falls as 1 + R/√(πDt) to its steady value. For small molecules in water the unit of time is about 125 ps and the rate is within ten per cent of steady after about 4 ns.

Tested in The speed limit that does not care how big the molecules are · the diffusion reading path

“The thermal jiggle of a mechanical sensor is a nuisance and nothing more.”

Because ⟨x²⟩ = kT/k holds regardless of material or damping, measuring a soft spring's thermal jiggle calibrates it: a 0.1 N/m cantilever jiggles by 0.2 nm rms at room temperature and optical tweezers of 10⁻⁵ N/m by 20 nm, and stiffnesses are routinely found this way.

Tested in The noise a high Q moves out of the way · the equipartition reading path

“Pressure-ionised hydrogen is a very hot, very dense plasma.”

At the density where the ground state disappears the freed electrons outnumber the cells of room their thermal wavelength allows by 70, 25 and 9 at 5,000, 10,000 and 20,000 K. Below about 85,000 K they form a degenerate Fermi sea: pressure-ionised hydrogen is a metal.

Tested in The pressure that ionises · the chemical potential reading path

“A thin feature fades from a seismic image as it thins.”

A thin layer whose faces reflect with opposite signs does, falling to zero with its thickness. A thin transition whose two faces reflect with the same sign grows towards twice one face's echo as it thins, and simply looks like one sharp boundary.

Tested in The layer too thin to see · the impedance reading path

“Once a wave train has broken into groups it is permanently disordered.”

The modulated train gives the energy back: after its crest at t ≈ 5.6 it returns almost exactly to an even train by t ≈ 11, with its highest point at 1.01, and grows a crest again near t ≈ 17 — a recurrence of the kind Fermi, Pasta and Ulam found in a chain of oscillators.

Tested in The even wave train that cannot stay even · the wave packets reading path

“Recovery from jet lag takes a time proportional to the number of time zones crossed.”

For a body clock the daylight can pull by up to two hours a day, Adler's equation gives 4.8 days to recover from a 3-hour shift, 6.5 from 6 and 11.7 from 11.5: the pull vanishes at a half-day shift, and a clock near it lingers.

Tested in The clock that is pulled into step · the resonance reading path

“Free neutrons are unstable, so a neutron star must be slowly decaying.”

A neutron can only decay if its electron has an empty state to go to. The decay spectrum ends at 0.782 MeV. An electron sea filled to 0.6 MeV already leaves 4.2 per cent of the decays open, and one filled past 0.782 MeV, at 7.4 × 10³⁶ electrons per cubic metre, leaves none. At nuclear density the sea is seventy times deeper than that.

Tested in The protons a star cannot afford · the chemical potential reading path

“A paramagnet's magnetisation follows Curie's law, rising as 1/T as it is cooled.”

Curie's law has ∂M/∂T = −CB/T², infinite at zero, and ∂M/∂T equals ∂S/∂B, which the third law requires to vanish. Independent spin-half moments saturate instead. At zero field any coupling must order the spins below a temperature set by its strength, or they would keep R ln 2 of entropy at absolute zero.

Tested in The two laws absolute zero will not allow · the third law reading path

“A concentrated polymer solution's pressure follows the second-virial correction, rising as the square of the concentration.”

That is the mean-field answer, and Flory and Huggins's model gives a local slope of 2.10 in an athermal solvent. Scaling with the measured size exponent 0.588 gives 2.31, and measurements agree with it. The mean field averages away the correlation hole each chain carries around itself.

Tested in The pressure that stops counting molecules · the osmosis reading path

“Capillary rise between two walls is set by the gap and the liquid, as Jurin's law says.”

Only if the walls are rigid. Water between plastic sheets 1 mm apart, with a modulus of 3 GPa, stands 1.09 times Jurin's 14.7 mm when the sheets are 450 μm thick and 1.18 times at about 420 μm. Thinner than that, there is no height at which the climb stops and the sheets zip shut.

Tested in The walls a rinse pulls together · the capillarity reading path

“Friction dissipates energy only when surfaces slide over each other.”

A contact shaken below its sliding load slips back and forth in its outer ring on every cycle. It traces a hysteresis loop whose area is the energy lost: 0.028 at half the sliding load and 0.32 at 0.9, in units of μP times the sliding displacement.

Tested in The contact that slips before it slides · the friction reading path

“A molecule's properties reflect the detailed shape of the potential between its atoms.”

A weakly bound state's outside is the same exponential whatever the well. A square well and a Gaussian well tuned to bind at κR = 0.12 need depths of 2.72 and 3.06 in units of ħ²/2μR², and differ inside their range. Beyond it they are identical, and 89 per cent of the state lies out there.

Tested in The molecule that lives outside its own well · the harmonic approximation reading path

“Dispersion is a property of materials: the way a medium's refractive index varies with wavelength.”

A spread of colours in angle produces dispersion with no material at all. A 300-line-per-millimetre grating pair in air, passed twice at a 19.9 mm spacing, delays 840 nm light relative to 760 nm by nearly the same amount that 100 mm of fused silica delays 760 nm relative to 840 nm.

Tested in The dispersion made of angles · the dispersion reading path

“Light from a source in the water can enter the glass only within the critical angle, as the ray picture says.”

A molecule close to the glass also sends light beyond the critical angle, through its own near field. In contact, 49 per cent of the light it sends into the glass is there; at 50 nm, 37 per cent; at 150 nm, 20 per cent. The ratio of that light to the ordinary light halves in 71 nm, which makes it a ruler.

Tested in The mirror that lights a tenth of a micrometre · the total internal reflection reading path

“Pinning needs an interface covered in defects.”

It needs only a small fraction of the surface's cut bonds. The slope halves at 1.1 × 10¹³ states per cm² per eV and is 0.10 at 10¹⁴, against about 7 × 10¹⁴ surface atoms per cm² on silicon. One state for every ten to a hundred surface atoms, spread through the gap, pins the barrier.

Tested in The barrier the metal cannot choose · the bands reading path

“Adding more bars reflects proportionally more wave energy.”

Only at first. Peak reflected amplitude is 16 per cent for one bar, 56 per cent for four and 92 per cent for ten, and it saturates towards total reflection by twenty. Meanwhile the reflected band narrows from 74 m of wavelength to 13 m, roughly as one over the number of bars.

Tested in The sandbars that reflect the sea · the periodic media reading path

“A wider pipe is simply a better waveguide, since its loss is lower.”

At 60 GHz TE₀₁ loss falls roughly as the cube of the radius, from 4.1 dB/km at 15 mm to 0.51 at 30 mm and 0.11 at 50 mm. The number of modes the pipe can carry grows as the square, 175, 709 and 1,969. Each is somewhere for the signal's power to leak at a bend or joint.

Tested in The loss that falls as the frequency rises · the guided waves reading path

“The attraction between two neutral atoms falls as the inverse sixth power of their distance.”

Only within about c/ω₀, the reduced wavelength of their strongest transition. From the Casimir–Polder integral, r⁶ times the energy falls to half of London's value at 3.8 c/ω₀, to 23 per cent at 10 and to 2.4 per cent at 100. There the energy goes as the inverse seventh power.

Tested in The attraction that weakens when light is too slow · the retardation reading path

“Between charges of one sign there is one point where the field cancels for each pair of neighbouring charges.”

Three equal charges on an equilateral triangle make four zeros, one at the centre and three 0.164 of a side from it towards the middles of the sides. Three in a line make two, a square of four makes five, and a regular pentagon of five makes six.

Tested in The zeros a handful of charges can make · the the field concept reading path

“Because the naive relation fails, angle and angular momentum are not constrained by each other at all.”

Commutators of L with cos φ and sin φ give ΔL² ≥ (ħ²/4)R²/(1 − R²), with R = |⟨e^(iφ)⟩|. No state violates it. Von Mises states lie close: at κ = 2 the spread is 0.591ħ against a bound of 0.487ħ. When the angle is sharp it reduces to Heisenberg's; when the angle is spread round the circle it allows a sharp angular momentum.

Tested in The angle the uncertainty principle cannot be written for · the uncertainty reading path

“A weak value is the value the measured quantity really had between preparation and post-selection.”

It depends on the coupling as well. At a coupling of 0.001 pointer-widths the kept shift is 10.00 times the coupling for a weak value of 10. At a coupling of 3 it is 0.20, back inside the range of the spin's real values. A particle in the three-box arrangement has weak values 1, 1 and −1, and −1 is not a value presence can take.

Tested in The value a gentle measurement returns · the measurement reading path

“A horizon, a surface from which nothing can return, needs gravity strong enough to bend spacetime.”

Any medium whose flow outruns its own waves has one for those waves. In a flow accelerating from 0.55 to 1.3 times the speed of sound, sound cones tip over at the point where the flow reaches the speed of sound. Beyond it, sound cannot reach anywhere upstream of where it was made.

Tested in The river that sound cannot swim up · the Velocity-addition reading path

“Any bound system, accelerated slowly enough, will simply adopt its contracted equilibrium shape.”

Only if nothing about the system is degenerate. A slow classical electron's Coulomb orbit is: a steady push deforms it secularly however slowly it acts, and in a direct calculation the electron is stripped off within a few orbits. At 0.32c the electron's relativistic precession removes the degeneracy, and the orbit follows 1/γ to within a per cent up to 0.7c.

Tested in The contraction the forces work out for themselves · the length contraction reading path

“Adding a relaxation time to Fourier's law is a small correction that changes nothing qualitative.”

It changes the character of short-wavelength heat. Diffusion damps a ripple as k² and never oscillates. With a relaxation time, ripples shorter than 4πvτ oscillate and travel at nearly v while their damping levels off at 1/2τ. Heat then moves as a damped wave, second sound, with a sharp front.

Tested in The heat that arrives before it could · the relativistic thermodynamics reading path

“Time-of-flight observations of gamma-ray bursts have ruled out energy-dependent light speeds at the Planck scale.”

Only at first order. At second order a 31 GeV photon from the same burst would be 6.5 × 10⁻¹⁸ s late, and even a hundred-TeV photon 6.7 × 10⁻¹¹ s. The 0.83 s window excludes a second-order scale only below 2.8 × 10⁻⁹ of the Planck energy.

Tested in The photons that raced for seven billion years · the planck scale reading path

“Gravitational-wave detectors can only hear violent events such as mergers.”

A spinning neutron star with a mountain radiates a steady tone at twice its spin frequency. The Crab could send a strain of up to 1.4 × 10⁻²⁴ at 59.2 Hz, and Vela 3.3 × 10⁻²⁴ at 22.4 Hz. Such signals are found by adding up months of data coherently, and searches for them have been run for every known pulsar in the detectors' band.

Tested in The mountain a spinning star is allowed · the gravitational waves reading path

“The gravitational redshift tests the geometry of spacetime near a mass.”

It tests the time part of the geometry and nothing else: every clock comparison reads only the rate of clocks. The space part — the one that lengthens rulers — was first measured by light, which is bent and delayed half by each part. The parameter γ that measures it is known from radar delays to the Cassini spacecraft to within 2 × 10⁻⁵ of general relativity's value of one.

Tested in The radius a mass adds that no circumference shows · the gravitational redshift reading path

“Torsion balances and satellites have tested the equivalence principle for gravitational mass to a part in 10¹⁵.”

They have tested the passive gravitational mass — the mass gravity pulls on — against the inertial mass. The active mass, the one that produces a field, is a separate quantity. Its equality with the passive mass was tested only to 5 × 10⁻⁵ in a laboratory, by Kreuzer in 1968, and to 4 × 10⁻¹⁴ by the Moon, a hundred times short of the passive test.

Tested in The pull that has to equal the pull back · the equivalence principle reading path

“Neutron stars are held up by the exclusion principle acting on neutrons, as white dwarfs are by its action on electrons.”

A star of free neutrons, held up by their exclusion pressure alone, has a maximum mass of 0.71 solar masses in general relativity. Neutron stars of about 2.1 solar masses have been measured. The difference is the repulsion between nucleons at short range, which makes nuclear matter far stiffer than a free Fermi gas.

Tested in The pressure that weighs down what it holds up · the Self-gravity reading path

“The symmetry between electric and magnetic fields in Maxwell's equations is broken only by the absence of magnetic charges.”

It is also broken by nearly all matter. A medium keeps it only if its relative permittivity equals its relative permeability, so that its impedance equals that of empty space. Ordinary glass, with ε = 2.25 and μ = 1, breaks it at every surface, and the breaking shows up as light whose handedness is changed.

Tested in The handedness a mirror cannot keep · the maxwell equations reading path

“A violation of Newton's third law between current elements would mean momentum is not conserved.”

Only if the field carried none. When two circuits change, the field between them holds momentum, and the forces on the matter need not balance while it is changing. For steady currents the imbalance between individual pieces always sums to zero over a circuit, so momentum is conserved whichever law assigns the pieces their forces.

Tested in The force between pieces of a current · the ampere law reading path

“Four contacts are used instead of two only to get a more precise number.”

They are used to leave out the contacts. With two contacts the voltage is measured across the same points the current flows through, and includes the resistance of the contacts and of the crowding of current into them. With four, the voltage is taken between points where no current enters, and neither appears in the reading.

Tested in The sheet resistance that forgets the shape · the conductors reading path

“A thixotropic material is one whose viscosity falls as it is sheared faster.”

That is shear thinning, and it is a property of the steady state. Thixotropy is a dependence on time and history: the paste's structure builds while it rests and is broken while it flows. In the model drawn here the viscosity on restarting is 2 Pa s after 10 s at rest, 34 after 100 s and about 1,000 after 1,000 s, while the steady viscosity at 1 per second is 2 Pa s whatever the history.

Tested in The paste that forgets it was stirred · the rheology reading path

“Poiseuille's law gives the flow of blood through a vessel if blood's viscosity is known.”

It gives the flow of a fluid with one viscosity. Blood in a vessel a few cells across has no single viscosity: modelled as a core four times as viscous as plasma inside a 2.5 μm plasma sleeve, a 30 μm tube passes as much flow as a uniform liquid 1.57 times as viscous as plasma, and a 100 μm tube as one 2.57 times as viscous.

Tested in The viscosity that belongs to the tube · the rheology reading path

“Energy conservation tells whether a process will happen.”

It tells what is available, not whether it is reached. A standing domino has 0.42 of its weight times its height available and stays standing indefinitely, because between it and that energy is a hill of 0.006. Whether anything happens is decided by the hill, which conservation of energy says nothing about.

Tested in The push that pays only for the hill · the energy reading path

“The 84.4-minute period of a gyrocompass is a design choice for convenience.”

It is forced. Any instrument that is to indicate the true vertical on a vehicle accelerating horizontally is undisturbed by the accelerations only if its equivalent pendulum length is the Earth's radius, which makes its period 2π√(R/g) = 84.4 minutes. A one-metre pendulum aboard a ship gaining 10 knots in a minute leans by 30 minutes of arc, a 100-km one rings by 17, and one of the Earth's radius not at all.

Tested in The compass that finds the axis the Earth turns about · the rotation reading path

“An X-ray lens cannot be made, because the refractive index is so close to one.”

One lens is weak, not impossible: a biconcave beryllium lens with an apex radius of 50 μm focuses 8 keV X-rays at 4.7 m. Thirty in a row focus at 15.7 cm. Such compound refractive lenses, first made in 1996, are standard at synchrotrons — concave, because the index is below one.

Tested in The index that falls below one · the refraction reading path

“Locking the phases of an array of lasers puts all their power into one beam.”

It puts all their power into one mode of the whole aperture, gaps included. For eight locked beams covering half their aperture, 46 per cent of the power lands in the central peak and the rest in side peaks set by the gaps; at 90 per cent fill, 82 per cent. And the phases must be held to about 0.32 radian RMS — 55 nm at 1,060 nm — to keep 90 per cent of what is left.

Tested in The beams that add only if they differ · the etendue reading path

“An approximation accurate to a few per cent for one system will be accurate to a few per cent for a similar one.”

Only if the quantity sought is large compared with the error. The one-orbital picture misses about 1.5 eV of correlation energy in both helium and H⁻. Against helium's 24.6 eV ionisation energy that is 6 per cent; against H⁻'s 0.754 eV it is twice the answer, and the sign comes out wrong.

Tested in The ion that exists because its electrons keep apart · the atomic structure reading path

“Colour is an invented quantum number with no observable consequence, introduced only to save the quark model.”

It is counted directly. The rate at which electron–positron collisions make hadrons is proportional to the number of colours (R = 2 below charm for three, 0.67 for one); the π⁰ → γγ width goes as its square (7.78 eV predicted for three, 0.86 for one, 7.80 measured); and the Z boson's hadronic width is proportional to it (1.737 GeV predicted for three, 1.744 measured).

Tested in The label three identical quarks forced · the exclusion reading path

“The thermal spectrum an accelerating observer sees is a purely quantum effect, with no classical counterpart.”

Its form is classical. A single plane wave of fixed frequency, seen along a uniformly accelerated worldline, has its frequency fall as e^(−aτ/c), and the power spectrum of that one fading wave, multiplied by the frequency, is exactly 1/(e^(2πνc/a) − 1): the Bose–Einstein occupation at kT = ħa/2πc. Quantum theory supplies the vacuum's fluctuations for the observer to see; the Planck shape comes from the Doppler shift.

Tested in The heat hidden in a Doppler shift · the accelerated frames reading path

“Anything with energy E attracts other bodies as a mass E/c² would.”

The source of gravity is energy density plus three times pressure. Radiation, with p = ρc²/3, pulls a slow body beside it twice as hard as a static mass of the same energy; a beam of light pulls a ray running against it four times as hard, and a ray running alongside it not at all. Only for a system whose internal stresses sum to zero — a box of light including its walls — does E/c² describe its gravity.

Tested in The light that pulls twice as hard as its mass · the Mass-energy reading path

“Following one particle for long enough gives the same diffusion law as averaging over many particles.”

That is ergodicity, and it fails here. For α = 0.5 each single trajectory's time-averaged displacement grows nearly in proportion to the lag (mean slope 0.85) while the ensemble's grows as the square root, and repeated trajectories disagree about the diffusion coefficient with a relative variance of 0.66 that does not shrink as the records lengthen.

Tested in The diffusion that slows the longer it is watched · the diffusion reading path

“Increasing the pressure on water lowers its freezing point.”

Only along ordinary ice, up to 209 MPa, where the freezing point bottoms out at −22 °C. Beyond that water freezes into ices III, V, VI and VII, each denser than the liquid, and the freezing point rises with pressure: back to 0 °C at 632 MPa, 82 °C at 2.2 GPa, and above 400 °C at 20 GPa.

Tested in The melting line that turns round · the phase change reading path

“An absorption spectrum is a fixed property of a molecule.”

The same molecules show a flatter spectrum when packed into grains, because the strongest bands lose the most: for a two-band pigment the red band's height relative to the blue rises from 0.56 dissolved to 0.73 in 50 μm cells. A spectrum measured in cells reports the molecule and the packaging together.

Tested in The pigment that absorbs less for being packed · the attenuation reading path

“The middle ear amplifies sound.”

It adds no energy. It transforms: the eardrum's working area is about 17 times the stapes footplate's and the ossicles' lever about 1.3, so pressure rises about 22-fold while velocity falls by the same factor. That makes the inner ear's fluid look 500 times less stiff from the air side, and in a plane-wave picture raises the power crossing from 0.11 per cent to 43.

Tested in The lever that lets air speak to water · the impedance reading path

“Anything that comes within the event horizon is swallowed, and anything that stays outside it survives.”

The surface that decides is the unstable circular orbit, at 4 GM/c² for slow matter and 3 GM/c² for light — one and a half to two horizon radii. A particle that crosses it inward is lost, though it is still well outside the horizon; and whether it crosses it at all is settled far away, by its aim and speed.

Tested in The target that grows as the traveller slows · the horizons reading path

“A star becomes unstable to gravitational radiation only when it spins near break-up.”

That is true of ordinary oscillations, whose crests run backward through the star faster than a slowly spinning star can carry them forward. The r-modes run backward at a speed proportional to the spin itself, so seen from outside they move forward at 2/3 of the spin at every spin. In a fluid model they are unstable at any rotation that viscosity does not overcome — down to about 40 Hz at five billion kelvin.

Tested in The wobble that radiating makes larger · the gravitational waves reading path

“Gravitational waves from r-modes are what stops accreting neutron stars spinning faster than about 700 Hz.”

It is one of three proposed explanations, with crustal mountains and magnetic torques. The simple fluid model puts many accreting stars inside the instability window, which would make them unstable now — so either the damping in real stars is far stronger than the model's, or the modes saturate at amplitudes so small that they could not limit anything. No r-mode signal has been detected.

Tested in The wobble that radiating makes larger · the gravitational waves reading path

“The mass of a Cooper pair is twice the electron mass, corrected by known effects.”

The one precise measurement, in niobium in 1989, found 84 ± 21 parts per million above twice the free-electron mass where theory predicted 8 below it — a difference over four times its uncertainty. No accepted explanation exists and the measurement has not been repeated at that precision.

Tested in The field a spinning superconductor makes of itself · the superconductivity reading path

“Testing Coulomb's law means testing that the exponent is exactly 2.”

There are two distinct ways for the law to fail. A departure in the exponent, 1/r^(2+ε), has no length in it, and a shell test sees it as a sag of ε(1 − ln 2) whatever the shell's size. A photon mass gives a Yukawa potential with a length in it, and the same test sees it growing as the square of the size. Bounds on one are not bounds on the other.

Tested in The mass that would make Gauss's law leak · the Gauss's law reading path

“A grating sends a beam into a spread of orders.”

Only if it acts briefly. A short strong pulse is a thin grating and fills orders as Bessel functions squared, at most 33.9 per cent in each first order. A long weak one notices that only one exchange of photon kicks conserves energy, and at Bragg's angle can move 99.9 per cent of the particles into a single order — a mirror, or at half the time a beam splitter.

Tested in The grating made of light · the matter waves reading path

“Heat radiated across a gap is carried by light, whose wavelength is set by the temperature.”

In the far field, yes. Across a narrow gap the heat is carried by ripples along the surfaces whose lateral wavelength is set by the gap itself — half the heat across a 10 nm gap is carried by waves with wavenumber times gap below about 2 — so the transfer grows as the inverse square of the gap, with no limit set by the temperature.

Tested in The heat that crosses a gap too narrow for light · the blackbody reading path

“On a tilted surface a ball rolls downhill.”

Not if the surface turns. On a turntable tilted by 0.2° and turning at 33 rpm a released ball rolls at most 4.9 cm downhill and drifts across the slope at 2.5 cm/s — 5g sinα/2Ω — in a chain of loops, exactly as a charge in crossed electric and magnetic fields drifts perpendicular to the electric field.

Tested in The ball a turntable will not throw off · the circular motion reading path

“A mass M on a spring of stiffness k oscillates at √(k/M).”

Only for a massless spring. A spring a tenth of the mass lowers the frequency by 1.7 per cent, and one of equal mass by 14 per cent — the textbook formula is then 16 per cent too fast. Counting a third of the spring's mass, √(k/(M + mₛ/3)), is good to 0.01 and 0.66 per cent in the same two cases.

Tested in The third of itself a spring carries · the harmonic approximation reading path

“The Earth's rotation deflects moving objects to the right in the northern hemisphere.”

That is the horizontal part. The vertical part, largest at the equator where the horizontal part vanishes, deflects rising bodies west and falling ones east, and makes eastward shells carry further and westward ones fall short: an 800 m/s shell fired east at 45° lands 257 m beyond where it would on an Earth that did not turn, and one fired west 260 m short.

Tested in The ball thrown up that lands to the west · the projectile reading path

“de Sitter's binary-star argument proves the speed of light is independent of its source.”

For visible light it proves nothing. Interstellar electrons absorb and re-emit light over an extinction length of about 0.6 parsec at optical wavelengths, and re-emitted light travels at the gas's speed, not the star's. The argument needs photons that keep their speed for the distance of the binary: X-rays, whose extinction length is kiloparsecs, from binaries whose pulses are timed round their orbits.

Tested in The binary star that would be seen twice · the Velocity-addition reading path

“No matter can have a sound speed above c/√3, since the densest matter behaves as free quarks.”

It was a common conjecture, and at asymptotically high density it is believed true. Neutron stars contradict it at the densities they reach: in a model that trusts nuclear theory to twice nuclear density, the 2.08-solar-mass pulsar J0740+6620 needs cₛ² above 0.394 somewhere in its core; capping it at one-third holds up only 1.92.

Tested in The sound speed a neutron star needs · the relativistic thermodynamics reading path

“Causality is the only limit on how stiff matter can be.”

Causality caps the sound speed at c, which in the same model allows 2.97 solar masses. Nothing observed requires matter that stiff. The measured masses sit between the conformal value and the causal one, which is what makes them informative about the forces in dense matter.

Tested in The sound speed a neutron star needs · the relativistic thermodynamics reading path

“The electric fields that focus particles in a storage ring are irrelevant to the spin.”

At most speeds they add their own turning, proportional to a − 1/(γ² − 1), and would spoil a measurement of g. That coefficient vanishes at one Lorentz factor, √(1 + 1/a) = 29.30 for the muon: the magic momentum of 3.094 GeV/c at which every muon g − 2 ring since 1977 has run.

Tested in The spin that runs ahead of its own motion · the field transformation reading path

“Cold water always sinks below warm water.”

Fresh water is densest at 3.98 °C. Below that, colder water is lighter, which is why a lake cools to 4 °C throughout, then keeps its coldest water at the surface, where it freezes. Seawater saltier than 24.6 g/kg has no density maximum above its freezing point, so in the ocean colder water does always sink — and ice cannot form until the whole mixed layer reaches freezing.

Tested in The mixture heavier than either water · the stratification reading path

“A heavier log floats more stably than a lighter one.”

Stability is not monotonic in density. A square log's metacentric height lying flat is 1/12s − (1 − s)/2 of its side: positive for very light and very heavy logs, negative in between, most negative at half density. A log of density s and one of density 1 − s float at exactly the same angle.

Tested in The log that floats on its corner · the buoyancy reading path

“The depth resolution of optical coherence tomography is set by the lens that focuses the light.”

The lens sets the lateral resolution. The depth resolution is the coherence length, (2 ln 2/π)λ₀²/Δλ, fixed by the spectrum alone: 6.2 µm for 50 nm at 840 nm, 2.1 µm for 150 nm. The two are independent, which is why OCT can have fine depth resolution with a weak lens and a long depth of field.

Tested in The depth a broad spectrum can see · the coherence reading path

“A wet shirt becomes see-through because water makes the fibres transparent.”

The fibres were transparent all along; a white shirt is white because each fibre surface turns a little light aside, and many surfaces turn most of it back. Water between the fibres cuts the index contrast — at a quartz-like surface the reflection falls eightfold — so the layer scatters less, transmits more, and skin behind it returns as much light as the cloth.

Tested in The sand that darkens when it is wet · the scattering reading path

“Burnout in a boiler happens when the water runs out.”

It can happen with the tube full. A surface heated at a set power that is raised past the critical heat flux cannot stay on the nucleate branch of the boiling curve, and jumps at the same flux to the film branch — in the correlations here, from 20 K to over 1,700 K above boiling, past the melting point of the tube.

Tested in The boiling curve that turns back · the phase change reading path

“Hot water freezes faster than warm water.”

Sixty years of reports have not produced a reproducible demonstration under controlled conditions: careful experiments, reviewed in 2016, found differences explained by where the thermometer sat, by supercooling that varies from run to run, and by evaporation. The effect has been demonstrated cleanly in other systems — a colloidal particle in a tilted double well in 2020 — where the mechanism can be computed.

Tested in The hotter start that cools first · the entropy reading path

“Time reversal only works if the medium is time-reversible, so it contradicts the arrow of time.”

The wave equation for sound and light in a lossless, stationary medium is time-reversible, and the mirror exploits that. It does not reverse entropy: it records a finite, bounded portion of the wave, re-emits it with ordinary energy from its own amplifiers, and fails exactly where the medium absorbs, moves or has changed since the recording.

Tested in The mirror that sends a wave back to where it began · the huygens reading path

“Every beam of light spreads by diffraction as it travels.”

In a uniform medium, yes. In a periodic row of guides the spreading rate is the curvature of the band, −2C cos k, which vanishes at a phase step of π/2 between neighbours: a beam launched there crosses the row without spreading, to second order — from 1.5 to only 2.1 guides wide over ten coupling lengths, against 6.5 for a beam launched straight.

Tested in The light that spreads in two beams · the periodic media reading path

“The Casimir force is a quantum effect: it vanishes if Planck's constant is set to zero.”

Only at separations well below ħc/kT, 7.6 μm at room temperature. Between ideal mirrors at 300 K the pressure at 10 μm is 3.05 times the zero-temperature value and within a part in ten thousand of ζ(3)kT/4πa³, a law with no Planck constant in it. Past about 3.3 μm the classical, thermal part is the larger.

Tested in The force that forgets Planck's constant · the retardation reading path

“More current always means better confinement.”

For hydrogen, bremsstrahlung against ohmic heating in Bennett equilibrium goes as 7.92·10⁻¹² I²/lnΛ with density, temperature and radius cancelled. Above about 1.1 to 1.6 MA radiation wins, the column cools and the current squeezes it further: a radiative collapse.

Tested in The current that squeezes what carries it · the ampere law reading path

“Pauli blocking suppresses scattering equally in all directions.”

The recoil is 2k sin(θ/2), so small-angle scattering, with small recoils, is blocked most and back-scattering, with a recoil of 2k, least. For a gas whose Fermi momentum equals the photon's at a tenth of the Fermi temperature, 29 per cent of atoms can scatter through 20° and all of them through 180°: the forward lobe of the pattern is removed.

Tested in The light a full sea will not scatter · the exclusion reading path

“Quantum effects of gravity on matter would be largest for the lightest particles.”

The natural height grows as m^(−2/3) — 0.88 mm for an electron against 5.87 μm for a neutron — but the electron's charge makes a stray field of a twentieth of a nanovolt per metre as strong as its weight. Atoms are neutral but their states are a fraction of a micrometre high, where the attraction of the surface is stronger than gravity. The neutron is the particle where the staircase is both large enough and clean.

Tested in The bounce that comes in fixed heights · the matter waves reading path

“Rolling resistance is a fixed coefficient of the wheel and the surface, like a friction coefficient.”

It depends on speed, load and size. On the computed viscoelastic bed it rises from zero at a crawl to a peak where the contact time matches the relaxation time — 0.0148 at 0.63 m/s for 1 kN per metre, 0.0306 at 1.26 m/s for 9 kN — and falls again; it grows with load roughly as W^(1/3) and falls with radius roughly as R^(−2/3).

Tested in The push a millimetre ahead of the axle · the friction reading path

“When gravity does work on a falling body with no air resistance, all of it becomes kinetic energy.”

Only if the body's mass is fixed. A drop gathering mist keeps 4/7 of gravity's work and loses 3/7 in the perfectly inelastic pick-up of each droplet; a chain paid out from a heap loses 1/3. A body whose moving mass grows as distanceⁿ loses n/(2n + 1).

Tested in The raindrop that falls at a seventh of g · the momentum reading path

“An undulator's wavelength is set by its magnet spacing and the electron energy.”

Also by the field strength and the angle of observation: λ = (λᵤ/2γ²)(1 + K²/2 + γ²θ²). At 6 GeV with a 3 cm period the forward photon energy falls from 11.4 keV at K near zero to 7.6 keV at K = 1 and 3.8 keV at K = 2, and at θ = 1/γ the wavelength is 1.67 times the on-axis value for K = 1.

Tested in The magnet period that comes out as an X-ray · the length contraction reading path

“Cherenkov light is blue because the particles are very energetic.”

The colour is set by the spectrum dN/dλ ∝ (1 − 1/β²n²)/λ², the same shape for every particle above threshold. Between 400 and 500 nm water yields 101 photons per centimetre, between 600 and 700 only 48; the count keeps rising into the ultraviolet until the water's index and transparency fall.

Tested in Why the glow of a fast charge is blue · the radiating charge reading path

“The ionosphere reflects radio waves below its plasma frequency and passes those above it.”

Only at vertical incidence. A ray at φ₀ from the vertical turns back where the plasma frequency equals f cos φ₀, so a layer with an 8 MHz critical frequency returns 11.3 MHz at 45° and 23.4 MHz at 70°, and over a single hop of 3,836 km up to 27 MHz, 3.37 times the critical frequency.

Tested in The frequency the sky returns only at a slant · the plasma oscillation reading path

“A photoelastic picture shows the stress in a part.”

It shows the difference of the two principal stresses, σ₁ − σ₂, as isochromatics, and their directions, as isoclinics, which move as the polarisers are turned. At the disc's centre the fringes say 4.24 MPa; the stresses are a tension of 1.06 MPa and a compression of 3.18 MPa, which needs the free-edge conditions or another measurement to separate.

Tested in The stress that polarised light can see · the polarisation reading path

“A telescope makes everything brighter in proportion to its collecting area.”

Only point sources. A star's collected light lands on one point of the retina and gains (D/7 mm)² — 816 times for 200 mm, 7.3 magnitudes — while an extended image is spread over an area that grows with the magnification, and its brightness per unit area never rises.

Tested in The nebula no telescope can brighten · the etendue reading path

“Corner darkening grows with how small the camera is.”

It depends only on the angle off the axis. A 50 mm lens on 36 × 24 mm, 23.4° to the corner, loses 0.49 stop; a phone lens with the same field of view on a far smaller sensor loses the same 0.49. What changes the loss is the field of view and the mapping, not the scale.

Tested in The four cosines at the edge of a photograph · the imaging reading path

“Airy beams do not diffract.”

Only the ideal one, which carries infinite energy. A real Airy beam, Ai(x)e^(ax), keeps its lobe's peak at 0.95 of its start by t = 2 for a = 0.03 against 0.43 for a Gaussian of the same width, but fades as e^(−at²/2): to 0.60 at t = 6 and below the Gaussian later for larger a.

Tested in The packet that bends with nothing pushing it · the wave packets reading path

“The rotational Doppler shift is a new effect, separate from the ordinary one.”

It is the ordinary k·v read locally. The helix tilts the wavefronts sideways by ℓ/r, the spinning surface moves sideways at Ωr, and their product is ℓΩ at every radius from the axis — so the shift is the same wherever on the surface the light lands.

Tested in The Doppler shift with nothing coming closer · the doppler reading path

“Adding water raises a bottle's pitch because the water shortens the column of air.”

It raises the pitch because it shrinks the air spring's volume: f goes as 1/√V. Half full the bottle sounds 155 Hz, up by √2; three-quarters full, 219 Hz, an octave above empty. Reaching each octave takes a quarter of the remaining air.

Tested in The note a bottle sings whatever its shape · the resonance reading path

“Capillary attraction between floaters is a short-range contact force.”

It reaches a few capillary lengths, √(γ/ρg) = 2.71 mm for water. In the linear theory the force goes as K₁(ℓ/ℓc): as ℓc/ℓ well inside ℓc, like the force between two charged lines, then down to 0.602 at ℓc, 0.040 at 3ℓc and 1.9 × 10⁻⁵ at 10ℓc.

Tested in The cereal that gathers at the edge of the bowl · the capillarity reading path

“A rising balloon's lift falls as the air thins.”

Not while its envelope is slack. A fixed amount of helium at the air's pressure expands exactly as the air thins, so it displaces the same mass of air at every height: the computed lift stays at 3.69 or 5.28 tonnes until the envelope fills at 39.9 or 37.5 km. Only then does it fall with the density.

Tested in The balloon that floats at a density · the buoyancy reading path

“The colder the winter, the thicker the ice.”

Doubling the cold buys only √2 more ice, and what matters is the product of cold and time, the freezing degree-days: 500 of them, however delivered, grow 79 cm by Stefan's law. Snow matters more. With 0.2 W/m K against the ice's 2.2, each centimetre of snow insulates like eleven of ice: 60 days at 10 K grow 86 cm bare and 30 cm under 10 cm of snow.

Tested in The ice that grows more slowly the thicker it gets · the diffusion reading path

“Removing the 1 per cent of carbon-13 from diamond should barely change its conductivity, since the isotopes are chemically identical.”

Isotopes scatter phonons as ω⁴, by mass alone. Measured, enriched diamond conducts about 3,300 W/m K at room temperature against 2,200 for natural. A kinetic model that only adds scattering rates gets 5 per cent; the rest needs the phonon collisions that conserve momentum and keep feeding the high-frequency phonons the isotopes scatter.

Tested in The gas of sound that carries a diamond's heat · the kinetic theory reading path

“A solute shifts the freezing and boiling points by the same amount, in opposite directions.”

It lowers the liquid's chemical potential by RTx, and the crossing moves by RTx divided by the entropy change of the transition: 22 J/mol K for melting, 109 for boiling. Water's constants are 1.86 and 0.513 K kg/mol, so the freezing point always moves 3.63 times as far.

Tested in The degree a spoonful of solute buys · the chemical potential reading path

“Above a few GeV every singly charged particle looks the same in a detector, because all of them move at essentially the speed of light.”

Their speeds agree to a part in a thousand, but their Lorentz factors differ by the ratio of their masses, and transition radiation measures the Lorentz factor. At 2 GeV/c an electron (γ ≈ 3,900) yields about 1.1 X-rays above 4 keV per hundred surfaces crossed; a pion (γ ≈ 14) yields less than one in ten million.

Tested in The light a charge makes by changing medium · the radiating charge reading path

“CPT symmetry guarantees that antimatter falls like matter.”

CPT relates an antiapple falling on an anti-Earth to an apple falling on the Earth. It says nothing about an antiapple and the Earth, which is the case every experiment on this planet measures. That case was argued from energy conservation and from the virtual pairs inside nuclei, and measured by ALPHA-g in 2023 at 0.75 g with an uncertainty of about 0.2 g.

Tested in The antiatom that had to be weighed anyway · the equivalence principle reading path

“An electron microscope's resolution is set by the electron's wavelength, which at 200 kV is 2.5 picometres.”

Round electron lenses always have positive spherical aberration (Scherzer, 1936), and balancing it against diffraction gives a resolution of about 0.61(Cs λ³)^¼ — 0.22 nm for Cs = 1 mm at 200 kV, eighty-six wavelengths. Only non-round multipole correctors, outside the theorem, broke through after 1998.

Tested in The voltage that bends an electron like glass · the potential reading path

“Wireless power gets steadily better the closer the receiver is brought to the transmitter.”

Two resonators coupled more strongly than 1/Q split into two modes at ω₀/√(1 ± k), and a source held at the original frequency sits in the dip between them. With Q = 100 and k = 0.04 the peaks are at 0.981 and 1.021 of the resonance, and the power delivered at the centre falls as the coils approach.

Tested in The coupling a resonance rescues · the induction reading path

“Each joule of heat the ocean takes up raises the sea by a fixed amount.”

The yield depends on the expansion coefficient where the heat is stored. One watt-year per square metre lifts the sea 2.60 mm in tropical surface water at 27 °C, 1.50 mm in subtropical water at 12 °C and 400 m, 0.67 mm in Southern Ocean surface water at 2 °C and 0.40 mm in Arctic water at −1.5 °C.

Tested in The rise a joule of heat buys · the hydrostatics reading path

“The osmotic fragility test measures how fragile a patient's red cells are.”

Cells with much the same membrane strength burst at very different salt concentrations according to their ratio of area to volume. Modelled populations reach half-lysis at 0.42 per cent salt for normal cells, 0.54 for cells with an eighth less membrane, as in hereditary spherocytosis, and 0.34 for thin 70 fL cells.

Tested in The cell that bursts by geometry · the osmosis reading path

“Rising air in a cloud cools at a fixed moist adiabatic rate of about 6 K/km.”

The rate depends on how much vapour there is to condense, which grows by about 7 per cent per kelvin: 3.8 K/km for saturated air at 25 °C near the ground, 5.8 at 0 °C and 700 hPa, 7.7 at −20 °C and 500 hPa, approaching the dry 9.75 as the air dries out with height.

Tested in The cloud that cools more slowly than the air · the stratification reading path

“Insects jump well because their muscles are stronger for their size.”

Their muscle is ordinary. The energy a jump needs is the same per kilogram at every size, but a flea must deliver it in the 0.43 ms its legs take to straighten, which would need 140,534 W per kilogram of muscle — 562 times what muscle can do. Fleas, locusts and froghoppers load a spring slowly with their muscles and release it with a catch.

Tested in The jump that does not get higher with size · the energy reading path

“Heavy isotopes are harder to separate than light ones because their masses are proportionally closer.”

That is true of effusion, which separates by √(m₂/m₁): 1.0043 for ²³⁵UF₆ and ²³⁸UF₆. A centrifuge separates by the difference of the masses, so at a 500 m/s wall it separates ³⁶Ar from ⁴⁰Ar by 0.221 and ¹²⁹Xe from ¹³⁶Xe by 0.420 in the excess over one, far beyond effusion's 0.054 and 0.027.

Tested in The spin that sorts by a difference of mass · the circular motion reading path

“Reflection reverses the phase of a wave only at a denser medium, so light reflected from glass is reversed and light reflected inside glass is not.”

At normal incidence that is right for the s-polarised amplitude. But rₚ changes sign at Brewster's angle, and at grazing incidence both coefficients approach −1 for any pair of media: at 89° rₛ is −0.969 for glass and −0.996 for water at radio frequencies, which is why Lloyd's fringe at the mirror plane is dark.

Tested in The surface that cancels whatever skims it · the refraction reading path

“Focusing a camera moves the plane of sharp focus nearer or further while keeping it parallel to the sensor.”

Only when the lens is square to the sensor. With the lens tilted by α, every plane of focus passes through one line, J = f/sin α from the lens: for f = 50 and a 5° tilt, 574 units below it. Turning the focus swings the plane about that hinge, from crossing the axis at 3,164 units to 307.

Tested in The plane of focus a tilted lens tilts · the imaging reading path

“A longer scan gives a sharper image because more signal is collected.”

More signal lowers the noise; it is the duration of each readout under the gradient that sets the sharpness, because it sets the highest spatial frequency recorded. From the same 160 mm field, 16 samples resolve 10 mm, 64 resolve 2.5 mm and 256 resolve 0.63 mm.

Tested in The image made of frequencies · the spin reading path

“Measuring a long half-life precisely requires waiting for many decays.”

Only if decays are what is counted. In a day of counting one carbon-14 atom in three million decays; an accelerator mass spectrometer counts the atoms themselves, about 590,000 from a milligram of modern carbon in under an hour against twenty decays in a day, a gain of the mean life over the counting time.

Tested in The clock read by counting atoms · the decay reading path

“To get neutrinos of a lower energy, a beam must be made from lower-energy protons or pions.”

The energy can be chosen by angle. At an angle θ off the pion's path the neutrino energy cannot exceed p*/sin θ, where p* = 29.8 MeV/c is the momentum it carries in the pion's frame: 0.68 GeV at 2.5°, 2.04 GeV at 0.836°, from the same beam.

Tested in The neutrino energy an angle fixes · the relativistic dynamics reading path

“Equipartition gives each particle of a monatomic gas an average kinetic energy of 3kT/2.”

Only for slow particles. What equipartition distributes is ⟨p ∂E/∂p⟩ = kT per direction; for E = p²/2m that is twice the kinetic energy, giving 3kT/2, and for E = pc it is the energy itself, giving 3kT. A relativistic gas's kinetic energy per particle rises from 1.506 kT to 3.00 kT, and its heat capacity from 3k/2 to 3k.

Tested in The gas law relativity leaves exact · the relativistic thermodynamics reading path

“Resonant absorption works for any transition, since emitter and absorber are identical.”

It works when the recoil miss is small compared with the line's width. For sodium's 2.1 eV yellow line the miss is 2.1 × 10⁻¹⁰ eV, 197 times smaller than the natural width, and sodium vapour absorbs sodium light strongly. The miss grows as the square of the transition energy, and for gamma rays it is far larger than the width.

Tested in The gamma ray its own kind cannot absorb · the Mass-energy reading path

“Ice crystals in a cloud grow by collecting the water droplets they meet.”

That comes later. First they grow by vapour alone, without meeting anything: air saturated over the droplets is supersaturated over ice, by 10.5 per cent at −10 °C and 22 per cent at −20, so vapour diffuses to the crystals while the droplets evaporate to replace it. At −12 °C an ice particle grows from 5 to 100 μm in about 18 minutes while the droplets do not grow at all.

Tested in The ice that grows by stealing from the droplets · the chemical potential reading path

“Sound is carried by molecules, so it travels at the speed of the molecules.”

It travels at a fixed fraction of their mean speed, √(γπ/8): 343 m/s in air at 20 °C where the molecules average 463 m/s, 0.74 of it, and 1,007 m/s in helium against 1,245, 0.81. A compression moves only as fast as collisions can pass it on, and with no collisions there is no sound at all.

Tested in The note too high for thin air · the kinetic theory reading path

“A wave meeting a faster medium beyond the critical angle is simply reflected back.”

It is totally reflected, and it also drives a disturbance along the boundary at the faster speed, which sends wavelets back up into the slow medium. Their envelope is a plane front leaning at the critical angle, the head wave, which appears beyond twice the depth times the tangent of that angle — 79 km for the crust in the figures — and carries the lower layer's speed to the surface.

Tested in The wave that arrives first by going the long way · the huygens reading path

“Ultrasound is absorbed in tissue as sound is in air, increasingly with the square of the frequency.”

Water does absorb as f², 0.055 dB/cm at 5 MHz. Soft tissue absorbs nearly in proportion to f, about 0.5 dB/cm per MHz — 2.5 dB/cm at 5 MHz — the signature of many relaxation processes with a broad spread of times. It is the near-linear law that makes the depth–resolution trade exact; an f² absorber would lose resolvable depths as 1/f.

Tested in The depth an ultrasound image pays for its sharpness · the attenuation reading path

“A gravitational redshift measures how strong gravity is at a star's surface, so it cannot give the star's mass without its radius.”

True of any star whose radius is free. A white dwarf's radius is fixed by its mass through degeneracy pressure, so the redshift is a single-valued function of mass: 30 km/s at 0.6 solar masses, 78 at 1.0, 223 at 1.35.

Tested in The redshift that weighs a dead star · the gravitational redshift reading path

“A charge circling in a magnetic field radiates a continuous spectrum at its gyration frequency and harmonics, at whatever energy it happens to have.”

Only while the quantum ħeB/m is small against the circling energy. At 10⁸ T the quantum is 11.6 keV, an electron in the first Landau level radiates back to the ground in 2.6 × 10⁻¹⁶ s, after 722 turns, and every electron in the plasma sits in the lowest level. Radiation at the cyclotron energy is then resonant scattering, which removes photons from a narrow band and leaves a line.

Tested in The circling that comes in quanta · the radiating charge reading path

“A gap can be made to spark at any voltage if it is made small enough.”

Not through the gas. Air between flat electrodes has a minimum breakdown voltage of about 330 V, reached for a gap of about ten micrometres at one atmosphere; smaller gaps need more voltage, not less, until below a few micrometres electrons are pulled straight out of the metal, a different process.

Tested in The spark that needs room to start · the dielectrics reading path

“A spinning top floats above the base wherever its weight balances the magnetic push.”

Only between half a ring radius and √(2/5) of one, where the field strength curves upward both along and across the axis: 6.6 mm of height for a 10 cm ring. That fixes the top's weight to within ±2.4 per cent, and a temperature change of a few kelvin, weakening both ferrite magnets, moves it out.

Tested in The window a spinning magnet floats in · the magnetism reading path

“The force on a railgun's armature is the current times the field between the rails times the width of the gap.”

That uses the field between endless rails. The armature sits where the rails end, and there each rail gives exactly half its endless field; the push is ½L′I², with L′ the inductance per metre — 3.13 MN for 3.74 MA and 0.447 μH/m.

Tested in The push at the end of a pair of rails · the ampere law reading path

“Surface tension can support any object whose weight per unit length of edge is less than twice the surface tension.”

That is the limit for a vanishingly thin object. A real cylinder can carry more, because it also displaces water below the level: a 1.36 mm needle with a 100° contact angle carries up to 160 mN per metre against 2γ = 146. And larger ones carry less than their weight would need: a steel cylinder floats only if it is thinner than about 1.65 mm.

Tested in The water the dent displaces · the buoyancy reading path

“A water table is just more sand with water in it, so a seismic wave passes it with little change.”

A compressional wave sees water filling the pores resist being squeezed, and its speed jumps at the table from 611 to 1,737 m/s in the model drawn. A shear wave sees only the water's added weight, which slows it from 430 to 391 m/s; water has no shear stiffness to add. The table reflects one kind of wave strongly and the other hardly at all.

Tested in The sound sand carries faster the deeper it goes · the granular matter reading path

“A vibrating gyroscope measures the rate of rotation, which must then be added up to give an angle.”

That describes one way of running one. Left to ring freely, a symmetric resonator's pattern angle in its case is −0.4 times the angle it has been turned through, at every instant and whatever the rate: it integrates rotation physically, and its reading is the angle itself.

Tested in The vibration that turns part of the way · the rotation reading path

“A resonator's frequency shift gives the mass added to it.”

It gives the mass weighted by the square of the mode's displacement where the mass sits. A speck at a cantilever's tip lowers its first mode by twice its fraction of the beam's mass; the same speck halfway along, by a tenth of that; near the clamp, by nothing. Two modes, which weight positions differently, give both the mass and where it landed.

Tested in The weight a frequency can find · the harmonic approximation reading path

“A transparent solid becomes invisible in a liquid of the same refractive index.”

Only at one wavelength. Glass and liquid disperse differently, so their indices agree at one colour — 589 nm for N-BK7 in the liquid drawn — and differ elsewhere by up to 7 × 10⁻³ in the violet; the immersed glass reflects 6 × 10⁻⁶ of blue light and none of yellow, and its outline shows coloured fringes.

Tested in The colour at which glass disappears · the dispersion reading path

“A surface rough on the scale of the wavelength cannot reflect specularly.”

Its specular share is exp(−(4πσ sin ψ/λ)²): a surface rough by half a wavelength keeps 7 × 10⁻¹⁸ of its mirror reflection at normal incidence and 74 per cent at a grazing angle of 5°.

Tested in The rough surface that is a mirror at a glance · the refraction reading path

“The width of a spectral line measures the lifetime of the excited state.”

Only for atoms at rest. In rubidium vapour at room temperature the lines are 6.07 MHz wide by their lifetime and the absorption is 511 MHz wide, because each atom sees the light Doppler-shifted by its own velocity; the natural width is recovered only by selecting the atoms that are not moving along the beam.

Tested in The lines a warm vapour hides · the atomic spectra reading path

“Liquid flowing out of a beaker through its film drains faster when the beaker's level is further above the bath, as in a siphon.”

A siphon's flow is driven by the difference of levels and slows as it shrinks. The film's flow is limited by a critical speed at the rim and is the same whatever the difference: a beaker 1 cm across, 1 cm above the bath at 1.4 K, falls at a steady 180 μm a minute and reaches the bath's level in 56 minutes.

Tested in The film whose thinnest point sets the flow · the superfluidity reading path

“Raising the voltage on an electrode raises the emitted current in proportion.”

Field emission goes as E² exp(−bφ^(3/2)/E): for tungsten it rises by more than twenty orders of magnitude as the field goes from 1 to 4 GV/m, and the shape of the electrode matters as much as the voltage, since a tip of 50 nm radius at 1 kV sees 4 GV/m while a flat plate a millimetre away sees a millionth of that.

Tested in The electrons a field pulls from cold metal · the tunnelling reading path

“Timing fast electrons and finding them never faster than light shows that their speed is limited.”

It shows the speed is limited; it does not show what happened to the energy. Bertozzi measured both: electrons of 1.5 and 15 MeV crossed 8.4 m in 29.0 and 28.03 ns, and the heat they deposited in a target matched the accelerator's energy, while ½mv² from their measured speed could never exceed 0.256 MeV.

Tested in The speed that stops while the energy keeps coming · the relativistic dynamics reading path

“Accelerating muons to higher energy lets them survive more laps of a storage ring, because time dilation lengthens their lives.”

It lengthens their lives by γ and their laps by the same γ, since the orbit's radius and the revolution period both grow with momentum. A muon makes eBτ/2πm turns per mean life — 431 at 1.45 T — at every energy; more turns need a stronger field.

Tested in The turns a muon makes at any energy · the time dilation reading path

“The wettest, most porous soils heave most because they hold the most water.”

Clean sand holds plenty of water and barely heaves: ice enters its pores at half a millikelvin of supercooling, so it freezes in place and never sustains a suction larger than the weight above it. Clay sustains a large suction but passes water too slowly. Silt sits between, and in a capillary model the fastest-heaving grain size is 29 μm under half a metre of soil.

Tested in The frost that lifts by drinking · the chemical potential reading path

“In a vacuum there is nothing to slow evaporation, so water boils away almost at once.”

It starts that way and stops. Leaving at the ceiling would take 6.2 MW of heat per square metre, the surface cannot be supplied that fast, and it cools until the evaporation it can afford matches the heat that reaches it. Water in vacuum freezes; ice lit by full sunlight settles near −68 °C.

Tested in The puddle that could dry in a second · the kinetic theory reading path

“A colour photograph records the colours of the scene.”

Dye-based film records three numbers at each point, through three broad windows as the eye does, so sodium light at 589 nm and a mixture of 545 and 630 nm light register as one yellow. A Lippmann record stores the spectrum: replayed, it returns one peak at 589 nm for the first and two, at 544 and 631 nm, for the second.

Tested in The photograph that keeps the spectrum · the superposition reading path

“A waveguide cannot carry a wave much longer than its own width.”

That is true of a pipe, which has one wall and a lowest frequency. Two walls — ground and ionosphere — guide a wave with no cutoff at all, and the 80 km gap carries waves 38,000 km long. Cutoffs exist only for the higher modes, the first at c/2h, 1.76 kHz for an 85 km night-time ceiling.

Tested in The tone the whole Earth rings at · the guided waves reading path

“Differential rotation in a star or planet stretches magnetic field and therefore amplifies it.”

It amplifies the toroidal field it winds out of the meridional one, and nothing else. In an axisymmetric system nothing turns toroidal field back into meridional, so the meridional field decays at its free rate and the toroidal field, after peaking, follows it. Regeneration needs motion that is not symmetric about the axis.

Tested in The field no symmetric flow can keep · the flux freezing reading path

“Sharp images of quasars billions of light-years away prove that spacetime has no foam at the Planck scale.”

They rule out foam whose fluctuations add like a random walk, which would scramble visible light from 3.2 thousand million light-years by 277 radians. The cube-root rule would put 5 × 10⁻⁸ radians on visible light and 0.19 radians on 10 MeV γ-rays, and only if the whole distance uncertainty lands on every photon's path — the step that is disputed.

Tested in The clock that cannot be both light and good · the planck scale reading path

“A gravitational wave has two polarisations, plus and cross, because that is what a wave of space does.”

It is what general relativity's wave does. The tide any metric wave carries is a symmetric 3 × 3 matrix with six independent entries, and the most general metric theory allows all six: plus, cross, breathing, longitudinal and two vector shears. A massless spin-2 graviton with nothing added keeps two; a scalar field adds breathing; a graviton mass, however small, gives five.

Tested in The four shapes a wave of gravity leaves out · the gravitational waves reading path

“Adding more wires to a cage improves its shielding exponentially.”

With the wire radius held at a fixed fraction of the spacing, the leak falls as about N^(−0.86): 41 per cent for 6 wires, 26 for 12, 15 for 24, 8 for 48. The exponential e^(−N) would have given 6 × 10⁻⁶ at 12 wires and 10⁻²¹ at 48.

Tested in The field a ring of wires lets through · the potential reading path

“Inside a superconductor magnetised by an applied field, the field falls off over the penetration depth, a fraction of a micrometre.”

That is true of the Meissner screening currents. In a pinned type-II superconductor the field enters to a depth set by the critical current, H/Jc, which for a field of a tesla and Jc of 10⁹ A/m² is close to a millimetre, three to four orders of magnitude deeper.

Tested in The field a superconductor keeps after all · the superconductivity reading path

“A three-dimensional foam coarsens by the same rule, with cells of more than some number of faces growing.”

Von Neumann's law is exact only in two dimensions, where a cell's boundary turning is fixed by counting corners. In three dimensions the rate depends on the cell's size and the lengths of its edges as well as on its faces; many-faced cells tend to grow and few-faced ones to shrink, as a statistical tendency rather than a law.

Tested in The bubble that grows by counting its sides · the surface tension reading path

“A liquid surface is a sharp step a molecule or two thick.”

Far below the critical point it is. In the computed van der Waals interface the 10–90 per cent width is 2.0 molecular lengths at 0.70 Tc, 5.6 at 0.95 and 12.6 at 0.99, and in a real fluid the width tracks the correlation length, growing as τ^−0.63 — tens of molecules a thousandth of Tc from the critical point, thousands at a millionth.

Tested in The tension that vanishes where the phases meet · the surface tension reading path

“Kepler's third law is a fact about gravity that has to be found by solving for the orbits.”

It follows from gravity's potential being proportional to 1/r alone. Scaling lengths by α and times by α^(3/2) multiplies the Lagrangian by a constant, so a scaled orbit is an orbit. Integrated ellipses of sizes 1, 2 and 3 have periods 1.000, 2.828 and 5.196 — the size to the power 3/2 — with no orbit solved for in the argument.

Tested in The third law that falls out of a change of scale · the least action reading path

“Because gravity weakens with height, a long-range throw goes less far than v²/g predicts.”

It goes further. The ground falling away under the throw outweighs the weakening of gravity: at 3 km/s the round-Earth range is 8 per cent longer than v²/g, at 6 km/s 5.3 thousand km against 3.7, and at the orbital speed it reaches half the Earth's circumference, 20.0 thousand km, where the formula gives 6.4.

Tested in The parabola that is the top of an ellipse · the projectile reading path

“A wavefront sensor measures the shape of a wavefront.”

It measures only local slopes — the direction each small patch of light is travelling — and the shape is computed by adding them up. That works because the slopes of light from a single point are the gradient of one surface, so their sum round any closed loop is zero: 112 lenslet slopes rebuild a wavefront of 0.53 waves rms to 0.087.

Tested in The surface every ray is normal to · the fermat reading path

“Light cancels only along dark fringes, where two waves arrive half a cycle apart.”

That is the case for two waves. Three or more waves cancel at isolated points in a plane, lines in space, round which the phase turns through a whole cycle: three equal waves 120° apart make a lattice of them, alternating in sense, and 24 random waves make about 2.7 per square wavelength in the drawn window, against π for an isotropic random wave.

Tested in The dark lines a wave cannot avoid · the diffraction reading path

“A qubit holds a continuum of states, so it can carry an unlimited amount of classical information.”

It can be prepared in a continuum of states, but a measurement can extract at most one bit from it (Holevo's bound). With a shared entangled pair, one transmitted qubit carries two bits: the sender's four operations on her half of |Φ+⟩ produce four orthogonal states of the pair, found with probability one each by the receiver's joint measurement.

Tested in The qubit that carries two bits · the entanglement reading path

“With a long enough interrogation, a clock's frequency can be pinned down as finely as wanted.”

Each atom answers only yes or no, so N atoms locate a fringe to about 1/√N of its width. A caesium fountain with a million atoms and half a second of free time is held by that alone to 4 × 10⁻¹⁴ after one second of averaging, improving only as one over the square root of the time.

Tested in The wait that sets the precision · the measurement reading path

“If the whole rod is made to start falling at once, it stays a rigid straight rod throughout.”

In the frame where the starts are simultaneous it stays straight. In the rod's own frame the start runs back along it at a finite rate, and for 0.80 L/c the rod is bent — level behind the point where the fall has reached, sloping in front — before it is straight again at its tilt. No body is rigid, because the instruction to move cannot reach all its points at once in every frame.

Tested in The rod that is level for only one observer · the simultaneity reading path

“A light beam crossing an accelerating rocket follows a parabola, sagging like a thrown ball.”

Across a cabin the sag is the thrown ball's, aw²/2c² — 5.5 × 10⁻¹⁵ m over 10 m at 1 g. Followed further, every ray in the rocket's own coordinates is an exact semicircle centred on the horizon, of radius (c²/a)/cos α for a ray launched at α above the horizontal; a parabola is its top in the limit of a small cabin.

Tested in The rocket in which light travels on circles · the accelerated frames reading path

“A spot swept across a screen appears at one edge and runs to the other.”

Only when the sweep along the screen stays slower than light everywhere. Where the beam meets the screen obliquely the sweep outruns light whatever the rotation rate, and the light arriving there earliest is not the light sent first: the spot is born at the earliest-lit point — 1.25 screen distances off-centre for a 0.5c sweep, 0.32 for a 3c sweep — and splits into two running apart.

Tested in The spot that outruns light and carries nothing · the Velocity-addition reading path

“Trouton's rule holds for all liquids with an entropy of boiling near 88 J/(mol K).”

Liquids that boil cold fall below it — neon 63, nitrogen 72 — because their vapour at one atmosphere is unusually dense; compared at equal vapour concentration, as Hildebrand proposed, the ordinary liquids' spread falls from 8.1 to 3.5 J/(mol K). Hydrogen-bonded liquids lie above (water 108.9, ethanol 109.7), acetic acid below (60.6) because its vapour is pairs, and helium at 19.6.

Tested in The entropy every liquid gains on boiling · the phase change reading path

“A bigger cluster grown the same way is the same shape, scaled up, and just as dense.”

It is the same shape statistically, but not as dense: the mass grows as the radius to a power below two, so the mean density falls as r^−0.35 and the cluster widens as N^0.60 rather than N^0.5. The gaps between branches never fill, because they are screened.

Tested in The cluster that grows where the walkers arrive · the diffusion reading path

“A harbour resonates at the period that fits a whole number of half-waves into its length, as a lake does.”

Its mouth is held at sea level, a node, so it fits odd numbers of quarter-waves: the slowest period is 4L/√(gh), twice that of a closed basin of the same length. Narrowing the mouth lowers the frequency further — the peak moves from kL = 1.571 to 0.726 — as the harbour turns into a Helmholtz resonator whose water sloshes through the entrance.

Tested in The tide a lake makes by itself · the standing waves reading path

“Splitting a wall into two leaves of half the weight each cannot help, because the total mass is the same.”

Below the leaves' mass–air–mass resonance it does not; above it the pair's loss rises at about 18 dB per octave rather than 6. Two 8.75 kg/m² plasterboards 100 mm apart resonate at 91 Hz and, square-on and idealised, stop 36 dB more at 1 kHz than the same 17.5 kg/m² in one piece.

Tested in The wall that stops sound by its weight · the impedance reading path

“The limb of a star is where its surface turns edge-on and its brightness falls to zero.”

That is where a straight ray grazes it. With bending the ray that leaves edge-on comes from further round, and a spot at the geometric limb, 90° from the line of sight, still appears with a third of its face-on brightness on a star of 3 rₛ and half of it on one of 2 rₛ.

Tested in The star whose far side can be seen · the light deflection reading path

“Gas falling onto a mass accelerates smoothly, and its pressure only slows it down.”

Pressure decides which of many steady flows the gas takes. Only one joins gas at rest far away to supersonic infall, and it must pass the speed of sound at exactly GM/2c² for an isothermal gas; the others either never reach the hole or start supersonic. That one flow fixes the rate, λ = exp(3/2)/4 = 1.12 in Ṁ = 4πλ(GM)²ρ∞/c³.

Tested in The gas a hole catches from far away · the horizons reading path

“The electric field near a wire carrying a changing current comes from the charges on the wire.”

A neutral wire has none. The electric field there comes entirely from the rate of change of the current, and for a current switched on it comes from only two points of the wire's history: the two places where the switch-on is being seen at that moment.

Tested in The field a changing field does not make · the maxwell equations reading path

“Two coils one radius apart make a uniform field along the line between them.”

They make it uniform in every direction. The design condition, a vanishing second derivative of the axial field, is imposed on the axis alone; because the field obeys Laplace's equation, the sideways variation is then also fourth order, three-eighths of the axial at the same distance. The part-in-a-thousand region reaches 0.22 radii from the axis and 0.17 along it.

Tested in The two coils one radius apart · the magnetism reading path

“Absorption above a direct gap rises from zero as the square root of the excess energy.”

Only if electron and hole do not attract. With the attraction, the absorption is multiplied by the Sommerfeld factor πx·exp(πx)/sinh(πx), x = √(Ry*/(E − Eg)), which is 2.3 at ten Rydbergs above the gap and grows towards it so that the absorption reaches the gap as a finite step.

Tested in The hydrogen atom inside a crystal · the bands reading path

“Positronium annihilates into two gamma rays of 511 keV, as any electron and positron at rest must.”

Only the spin-singlet form does. The triplet form carries the opposite charge-conjugation parity, and an even number of photons cannot carry it away: its fastest allowed decay is to three photons, sharing 1,022 keV continuously, and it lives 142 ns rather than 125 ps.

Tested in The atom with no nucleus · the atomic structure reading path

“A wheel rolling at relativistic speed simply looks like a wheel squashed along its direction of motion.”

Its rim is the squashed ellipse, but its spokes are bent: in the ground's frame points ahead of the hub are caught at an earlier moment of the rotation than points behind it. At 0.8c the spokes that point forward and back in the axle's frame both have their tips 0.52 radii above the hub.

Tested in The wheel whose top cannot go twice as fast · the Velocity-addition reading path

“Shearing a suspension for long enough brings it to a steady state, with a steady viscosity.”

Not a dilute suspension of rigid rods in a Newtonian liquid. Every Jeffery orbit has the same period, so a population started isotropic regroups once every half-period, and the orientation factor that sets its extra viscosity oscillates between about 0.005 and 0.14 for ever. Steady states come from what the model leaves out: the rods' Brownian rotation, their collisions, or the liquid's elasticity.

Tested in The rod that tumbles for ever in a stream · the viscosity reading path

“Any liquid that wets the wall will run towards the narrow end of a tapered tube.”

Only if it wets it well enough for its length. The two menisci are tilted in opposite directions by the taper, and for a gentle taper the push vanishes at the contact angle whose tangent is half the plug's length over its radius: 27° for a plug as long as its radius, 56° for one three times as long. A short plug of a liquid that wets only moderately does not move.

Tested in The drop that runs to the narrow end · the capillarity reading path

“A peak in a material's heat capacity means a phase transition.”

A phase transition makes one, but so does any set of discrete levels with no transition at all. A paramagnetic salt in a field has a smooth hump whose position moves in proportion to the field — 0.56, 1.12 and 2.80 K at 1, 2 and 5 T — and whose shape is fixed by the levels alone. The area under C/T distinguishes them: k ln N for N levels.

Tested in The heat capacity that goes up and comes down · the equipartition reading path

“Metal feels colder than wood because it conducts heat better.”

Conductivity is half of it. The surface between two bodies settles at a temperature weighted by each one's effusivity, √(kρc), which includes the heat capacity per volume. Water conducts heat a fifth as well as granite and feels nearly as cold — meeting skin at 25.6 °C against granite's 24.3 — because it holds twice as much heat per volume.

Tested in The metal that feels colder than the wood · the diffusion reading path

“A disordered stack of layers behaves like a periodic one with its band edges blurred.”

A regular pile of 60 quarter-wave plates has pass bands where it transmits on average 0.92. The same 60 plates with thicknesses varied randomly by ±25 per cent have no pass band: averaged across those wavelengths they transmit 0.135, with sharp spikes where a chance resonance lets one colour through. For a long enough pile disorder removes the bands, not blurs them.

Tested in The pile of glass that stops every colour · the periodic media reading path

“A moving object leaves its waves behind it.”

It leaves its gravity waves behind. Its capillary ripples have a group speed greater than their phase speed — up to 3⁄2 of it — so their energy outruns the object and they stand ahead of it: 1.6 mm ripples upstream of a line in a 50 cm/s stream, with 15 cm gravity waves downstream.

Tested in The speed below which nothing on water leaves a wake · the wave motion reading path

“Cancelling a noise source everywhere is impossible, because a second source can only cancel at points.”

It is possible if the second source is close enough to the first. Two point sources a distance d apart can radiate as little as 1 − sinc²(kd) of the first's power: 9 dB less at a tenth of a wavelength apart, but only 2.3 dB at a quarter. A 100 Hz transformer hum can be silenced throughout a room by speakers within about 30 cm of it.

Tested in The silence that fits in a tenth of a wavelength · the superposition reading path

“Two oscillations with no friction and no forcing must each keep their own energy.”

Only if they are exactly independent. The spring's bounce changes the pendulum's length twice per swing, which pumps the swing parametrically when the frequencies are in the ratio 2:1, and energy passes completely from one motion to the other and back.

Tested in The spring that swings when it was only bounced · the chaos reading path

“A damped oscillator comes to rest at its equilibrium position.”

With dry friction it comes to rest anywhere in a band of half-width F/k around equilibrium, because inside that band the spring cannot overcome the friction. The rest position depends on the release point: a zigzag that sweeps the whole band as the release distance grows by two band-widths.

Tested in The swing that dies in a straight line · the friction reading path

“A centrifugal governor holds an engine at a fixed speed whatever the load.”

It holds it near one. The arms can only open the throttle wider by dropping, and they drop only if the speed falls, so a heavier load means a lower steady speed: 0.11 units lower under a tenth of a unit of extra load with a stiff linkage, 0.32 with a soft one. The error is how the governor works, not a defect of it.

Tested in The governor that needed friction to work · the circular motion reading path

“A sphere of glass, like a block of glass, can trap light by total internal reflection.”

Only light that starts inside it, and only from near the surface. A ray from a point at radius ρ keeps meeting the surface at an angle whose sine is ρ sin α, so a source nearer the centre than 1/n of the radius traps nothing, while one at the surface traps 66 per cent of its light in water and 91 per cent in diamond. Light that came in from outside always gets out again.

Tested in The drop that cannot hold the light it lets in · the total internal reflection reading path

“A laser cavity resonates at the frequencies where a whole number of half-wavelengths fits between its mirrors.”

Only for its fundamental mode in a cavity of flat mirrors. Each transverse pattern picks up its own Gouy phase on each pass, (m + n + 1) times the fundamental's, so its resonances are shifted: in a confocal cavity by exactly half the spacing between longitudinal modes.

Tested in The half cycle a focus adds · the diffraction reading path

“An object in the open can never be cooled below the surrounding air without a refrigerator.”

Not if it can see a clear sky, which radiates like a body about 23 °C colder than air at 15 °C. Even in sunshine a surface that reflects more than 86 per cent of the sunlight and emits strongly in the infrared falls below the air temperature, 8.5 °C below at 97 per cent reflectance under a dry sky.

Tested in The cold reservoir overhead · the heat engines reading path

“A heated layer is stable as long as its salt makes it denser downward.”

That stops ordinary convection, not the slow oscillation that grows because heat diffuses about a hundred times faster than salt. Stopping that needs a salt gradient larger by the factor (Pr + 1)/(Pr + τ): 18 per cent more at 30 °C and 45 per cent more at 80 °C.

Tested in The pond that is hottest at the bottom · the stratification reading path

“When ice floating in a glass of water melts, the level always stays the same.”

Only for ice floating in fresh water with nothing frozen into it. In seawater the level rises, in an oil denser than ice and lighter than water it falls, and a stone frozen into a cube lowers the level when it is released, by its weight in water minus its volume — 0.65 mm for 4 g of stone in a 7 cm glass.

Tested in The floating ice that still raises the sea · the buoyancy reading path

“Mixing river water into the sea releases about 0.8 kWh for every cubic metre of river water.”

Only for a trickle into an ocean that stays at full strength. Mixing equal volumes, the reversible limit is 0.55 kWh per cubic metre of river water and a single held pressure collects 0.39, because the seawater is diluted as the river water enters it.

Tested in The waterfall at every river mouth · the osmosis reading path

“Above the critical power a beam focuses to a point after one focal length of its own lens.”

The lens it writes is strongest at the centre: at four times the critical power, rays near the axis cross after 1.73 m in glass and rays from the 1/e² radius after 3.63 m. The exact width law forces a collapse no later than a Rayleigh range divided by √(P/PG − 1), and the real collapse comes earlier.

Tested in The beam that focuses itself · the fermat reading path