True as far as it goes, and silent on what decides — part 3 of 5

“A current-carrying wire is neutral in the laboratory and charged in the frame of a moving charge, because the two lattices contract differently.”

That account is right and it is a story about a particular arrangement. The general statement is that (cρ, J) transforms exactly as (ct, x) does, so a wire with ρ = 0 and J ≠ 0 in one frame has ρ ≠ 0 in every other — at exactly one rapidity out of all of them, which the figure counts. No lattice appears in the derivation and the answer is the same.

Tested in Charge and current are one thing · the field transformation reading path

“E·B and E² − c²B² are invariant under a Lorentz transformation.”

They are, and the stronger statement is that they are the only two. An antisymmetric rank-two tensor in four dimensions has exactly two independent scalars — its contraction with itself and its contraction with its dual — so any other invariant anybody constructs is a function of those. That is why the three-way classification of fields is a classification and not a first attempt at one.

Tested in Six numbers, one object · the field transformation reading path

“Residual entropy is the entropy of a degenerate ground state.”

That is the ice case, and it is a count that nobody can change. A glass's residual entropy depends on the cooling rate — 15.6, 13.0 and 11.0 joules per kelvin per mole for the three rates computed here, for the same substance — because what is frozen in is whatever the liquid had when it stopped being able to relax. One number is a property of a spectrum and the other is a property of an experiment.

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

“The third law says the entropy of any substance approaches zero as the temperature approaches zero.”

It approaches k ln g, with g the degeneracy of the ground state, and that is zero only when the ground state is unique. Ice's 3.41, carbon monoxide's 4.6 and nitrous oxide's 5.8 joules per kelvin per mole are the values of R ln g for their respective counts, and the figures here compute each from the count and set it beside the calorimetry.

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

“An absorption coefficient says how much energy a wave loses.”

It says how much momentum it loses as well, and it is the same coefficient. Absorbing a fraction of the momentum flux per unit length gives a body force 2αI/c in the medium, so the force and the heating rate are one measurement read twice — their ratio is ρc_p/c, which contains no intensity and is checked here at two intensities four decades apart.

Tested in Where the loudness goes · the attenuation reading path

“Iodine is used as a contrast agent because it is dense and has a high atomic number.”

Those help and they are not the reason. What matters is where its K edge is: two per cent by mass of iodine adds 47 per cent to water's attenuation just below 33.17 keV and 213 per cent just above it, from a change of a third of a kiloelectronvolt in the photon energy. An element of the same density whose edge sat at 5 keV or at 300 would be useless, because a diagnostic tube produces neither.

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

“Optical molasses traps the atoms it cools.”

The force depends on velocity and not on position, so nothing pulls a displaced atom back. A cooled atom random-walks through the light and eventually leaves it; it lingers only because its steps are short. Holding atoms in place needs a force that depends on where they are, which is what adding a magnetic field gradient to the same beams supplies.

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

“Turning the light down keeps lowering the temperature.”

Only while the light-shift wells can hold the atoms being cooled. Below about a hundred recoil energies of depth the straight line fails and the temperature climbs — 8.25 µK at 50 and 16.8 µK at 25 in the runs here — and underneath everything sits the recoil of one photon, ħ²k²/mkB, which is 198 nK for caesium and is not a property of any mechanism that absorbs and re-emits light.

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

“The best shot at a target is the one that reaches it with least effort.”

The least-speed shot is also the one whose landing point does not move to first order when the launch angle is wrong, because at that speed the two launches that reach the target have merged into one. A free throw at its least speed, 7.17 m/s, arrives 1.5 cm low for a 2° aiming error; at 8 per cent more speed the flatter of its two solutions misses by 11 cm for the same error. What it does not forgive is an error in speed.

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

“Drag lowers the best launch angle because it steals range.”

Every angle loses range to drag, so the loss by itself moves nothing. What moves the answer is that drag bends the curves of equal range on the velocity map, outward most for steep launches, so the unchanged circle of throws touches them lower down. With drag at 2.4 times the weight the tangency found by integrating the curves is at 38.1°, the angle a direct search of the range also gives.

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

“The water condensed in a pore is ordinary liquid water at atmospheric pressure.”

It is under tension, at a pressure below the air's by (RT/Vₘ) ln RH, which is exactly the Laplace pressure of the meniscus holding it: −1.4 MPa at 99 per cent humidity, −14.5 MPa at 90, −95.1 MPa at 50. The homogeneous cavitation limit of about −140 MPa is reached at 36 per cent — and at that humidity the pores concerned are about a nanometre across, a few molecules of water.

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

“Raise the voltage far enough and any surface wets completely.”

The law says so, at 110 V for a 1 µm coating and 191 V for a 3 µm one. Real angles stop falling well before that, typically somewhere between 60° and 80°, and stay put however high the voltage goes. Charge trapped in the insulator, ionisation at the drop's sharp edge and the stability of the contact line have each been proposed as the cause, and none accounts for every experiment.

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

“Texturing the surface of a solar cell helps by reflecting less of the sunlight.”

It also randomises the direction of the light inside, and light in a randomised slab can leave only through an escape cone holding 1/n² of its directions. For weak absorption the path is lengthened 4n² times — 51 for silicon — so a 100 µm silicon slab with a mirror behind it absorbs 65.7 per cent of 1100 nm light instead of 6.76 per cent.

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

“A link designed to tolerate its fibre's mean polarisation delay will work.”

It will work most of the time. The Maxwell distribution has a tail: the delay exceeds twice its mean with probability 1.7 per cent and three times its mean with probability 4.2 × 10⁻⁵, about 22 minutes a year on a link in continuous use. Systems are designed to an agreed outage probability rather than to the mean, and three times the mean is the usual margin.

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

“Falling through a hole in the Earth takes forty-two minutes, and the centre is at about 170 GPa.”

Those are the uniform-Earth figures: 42.2 minutes and 173 GPa. With the real density a body falls through a straight shaft in 38.2 minutes, passes the centre at 9.92 km/s rather than 7.91, and the pressure there is 364 GPa — because the dense core raises the gravity above it and is itself heavy.

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

“The Josephson effect is a delicate curiosity of superconducting physics.”

It is how the volt is realised. At 70 GHz one junction's first step is 144.7 µV, and 69,085 junction-steps in series give 10 V with an uncertainty set by the microwave frequency. Since 2019 the constant 2e/h has been exact by definition, and a Josephson voltage together with a quantum Hall resistance is how a Kibble balance weighs a kilogram against Planck's constant.

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

“The larger the quantum numbers, the more classical the motion.”

At fixed anharmonicity a larger packet loses its classical look sooner — its collapse time falls as one over the square root of the number of quanta, from 30 periods at one quantum to 2.7 at a hundred — and the revival time does not change at all. What grows with the quantum number in a box or an atom is the number of orbits before the revival, 2n and 2n/3.

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

“A superposition leaking energy into its surroundings loses its quantum character in proportion to the energy lost.”

The coherence between two packets decays at 2α² times the energy decay rate, so a superposition of amplitude 3 loses its negative regions eighteen times faster than it loses energy at first. And for every amplitude the last negative value disappears at exactly γt = ln 2, when half the energy is gone, whatever remains of the stripes.

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

“Doubling a vehicle's power roughly doubles how fast it can go.”

At the top of the range the power needed grows almost as the cube of the speed, so doubling it raises the top speed by about the cube root of two. The car in the figures reaches 171, 220, 281 and 358 km/h at 50, 100, 200 and 400 kW, and a rider going from 250 to 400 W, 60% more, goes from 37.5 to 44.4 km/h.

Tested in The speed at which grip hands over to power · the energy reading path

“Knowing which events can influence which is enough to fix the geometry of spacetime.”

With one space dimension it fixes almost nothing. A warp of the two light-cone coordinates by different increasing functions preserves the causal relation of all 4,000 sampled pairs of events, yet bends straight worldlines and changes the proper time between timelike pairs by factors from 1.00 to 1.87. The claim becomes true, up to a uniform stretch, only with two or more space dimensions.

Tested in What the light cones alone can decide · the spacetime diagram reading path

“The far future of flat spacetime is one moment that everything reaches.”

For anything slower than light it is one point, i⁺, and because the past of that point is the whole diagram, an inertial observer who waits long enough can receive a signal from every event. For light it is a whole edge: rays that set out from x = −2, −1, 1 and 2 end at four different points along ℐ⁺, which is why radiation is defined there rather than at i⁺.

Tested in The five places infinity turns out to be · the spacetime diagram reading path

“A regenerator improves a gas turbine by recovering heat that would otherwise be wasted.”

It does, and the size of the improvement follows from where it moves the heat. In the ideal cycle at pressure ratio 12 and 1,500 K, the regenerator means heat is bought only above 737 K, at a mean of 1,074 K rather than 989 K, and thrown away only below 610 K, at a mean of 437 K rather than 486 K. The loop is unchanged and the efficiency rises from 50.8% to 59.3%.

Tested in The temperature an engine really takes its heat at · the heat engines reading path

“Hot water stores useful energy in proportion to the heat it holds.”

It stores heat in proportion to its temperature above the surroundings, and work in a much smaller and growing share. With the surroundings at 20 °C, a kilogram at 40 °C holds 84 kJ of heat and 2.7 kJ of work, 3.3%; at 60 °C, 6.3%; at 80 °C, 251 kJ of heat and 22.7 kJ of work, 9.0%.

Tested in The work left in two buckets of water · the heat engines reading path

“Disorder destroys the states that live at the edge of a lattice.”

It depends on which symmetry the disorder keeps. Scattering every coupling by up to 30 per cent leaves the edge level at zero in 160 random chains, with a median below a hundred-thousandth of the coupling. Adding random site energies of up to 0.2 of the coupling, which couples each sublattice to itself, moves it off zero to a median of 0.041.

Tested in The end that knows how the middle was cut · the periodic media reading path

“An omnidirectional mirror reflects every angle.”

It reflects every angle from a particular medium. The same stack's window is 43.6% wide from air, 15.0% from an index of 1.34 like water, and gone from an index of 1.44 upwards, because a denser medium lets light bring a larger component of momentum along the layers and reach the places in the band diagram where the gaps move apart.

Tested in The mirror that works from every direction · the periodic media reading path

“Whether temperatures can be negative is a settled question.”

The count of states decides which way energy flows between large systems, and there it sides with Boltzmann. But Gibbs's entropy satisfies some exact relations that Boltzmann's satisfies only approximately — equipartition, and the invariance of the entropy under slow changes — and for small systems, or for defining the efficiency of an engine with one reservoir above infinite temperature, which definition to use is still argued.

Tested in The count that decides which entropy is right · the third law reading path

“Radiation reaction is fully described by requiring the reaction force's work to equal the energy radiated.”

That requirement holds only between moments at which the product of acceleration and velocity takes the same value — over a whole cycle, or between two states of no acceleration. In between, the difference is the Schott term mτ a·v, which for an oscillating charge swings by the whole mean radiated power every quarter cycle.

Tested in The bill that arrives when the pushing stops · the radiating charge reading path

“Lunar ranging's bound is set by how precisely the distance to the Moon can be measured.”

With phases spread evenly, six thousand normal points at 2 mm each would bound η near 6 × 10⁻⁶ by statistics alone. Published bounds sit near 10⁻⁴ because the fit has to separate the signal from reflector heating, atmospheric delay, the lunar interior and a monthly term in the Sun's own perturbation of the orbit — and because new and full moon, where the signal peaks, are the hardest phases to range at.

Tested in The binding energy that has to fall too · the equivalence principle reading path

“The energy of the ions striking the wafer is the self-bias voltage.”

It is the sheath voltage averaged over each ion's transit, and the sheath voltage swings through a full cycle. Argon crossing a 1 mm sheath takes about 120 ns: at 1 MHz that is a tenth of a cycle and the ions land at two energies eighty electron temperatures apart; at 60 MHz it is seven cycles and they land within a few.

Tested in The bias no battery supplies · the plasma oscillation reading path

“Finding the potential at one point inside a region requires solving Laplace's equation everywhere in it.”

The potential at a point is the average boundary potential found by random walkers started there, so it can be estimated at that point alone. Four thousand lattice walkers from one point in a square give 0.405 ± 0.008 against 0.408 from relaxing the whole grid, and the error falls as one over the square root of the number of walkers whether the region is a square or a cube.

Tested in The potential is where the wanderers stop · the potential reading path

“The conformation of a chain in a flow is fixed by the flow rate.”

When a chain's drag grows as it unfolds, a coiled and a stretched chain are both stable over a window of rates — between 0.44 and 1 times the unfolding rate for a drag that grows fourfold. Raised slowly, the chain stays coiled up to 1 and jumps to 0.88 of its length; lowered, it stays stretched down to 0.44. Single DNA molecules showed the loop in 2003.

Tested in The stretch a chain cannot outrun · the rheology reading path

“Counterion condensation is a chemical binding of ions to the polymer.”

It is an electrostatic threshold with no chemistry in it. A single counterion beside a line charge has a Boltzmann weight that goes as distance to the power −2ξ, and whether the integral of that weight converges at the chain decides whether the counterion wanders off or is captured: at ξ = 0.9 it is within r₀ of a vanishingly thin rod with probability 0.063, at ξ = 1.1 with probability 0.996.

Tested in The counterions that never leave the chain · the osmosis reading path

“An internal wave loses energy to smaller waves at whatever rate its amplitude sets, wherever it is.”

The fastest route, parametric decay into waves at half its frequency, needs those waves to exist, and an internal wave cannot oscillate below the local inertial frequency. For the semidiurnal tide the half-frequency waves are forbidden poleward of 28.8°. A pump at strength 0.2 grows a subharmonic by 0.157 e-folds per tidal period at 20° and leaves it bounded at 33°.

Tested in The latitude past which a tide cannot split · the stratification reading path

“Just before a stable cycle appears, a system is nearly periodic with small deviations.”

The deviations are not small; the bursts cover every value the map can take. What is nearly periodic is the calms between them, and their length grows without limit as the cycle's birth approaches: 245 iterates at a distance of 10⁻⁵, 804 at 10⁻⁶ and 8,125 at 10⁻⁸, a fitted slope of −0.506 against the −½ a parabolic channel requires.

Tested in The calm that is the ghost of a cycle · the chaos reading path

“Strongly forward-scattering particles, like cloud droplets, let much more light through than particles that scatter evenly.”

Only when the slab is thin. Measured in transport mean free paths — the scattering length divided by 1 − g — slabs with mean scattering cosines of 0, 0.5, 0.85 and 0.95 transmit within 9.2 per cent of one another from four transport lengths onward, and on the same diffusion curve. A forward scatterer is an even scatterer with a longer step.

Tested in The cloud light has to walk through · the scattering reading path

“Reflection happens at a surface.”

Reflection happens wherever the medium changes, and the change can be in time. Switching the index of a whole ring of grid cells from 1 to 2 at one instant, with no surface anywhere on it, sends back a pulse of amplitude 0.125 — the value continuity of D and B requires, to three decimal places.

Tested in The reflection that needs no surface · the refraction reading path

“The laboratory energy spectrum of a two-body decay is flat.”

Only when the parent is unpolarised. Pions from taus moving at 0.8c fill the same interval, 313 to 2667 MeV, whatever the tau's spin, but a fully polarised sample is tilted: the slope read back from 60,000 decays is +0.993 for spin along the flight and −1.006 against it, where an unpolarised sample gives −0.010.

Tested in The slope a spin leaves in a spectrum · the relativistic dynamics reading path

“The rate of a decay through several intermediate states is the sum of the rates through each.”

Only where their bands do not overlap. Three resonances whose rates on their own are 67.0, 25.5 and 33.2 per cent of the total add to 126 per cent, and where two bands cross, the share of decays landing there moves from 4.98 to 13.10 per cent as the relative phase of the two amplitudes changes, with every mass, width and strength held fixed.

Tested in The plane in which three bodies are flat · the relativistic dynamics reading path

“With enough laser power a sail can be driven as fast as desired.”

Power sets the push; diffraction sets how long it lasts. A beam focused from an aperture D spreads to the sail's radius at a distance πDa/2λ, 3 million km for a 1 km aperture and a 4 m sail, and beyond it the light caught falls as the square of the distance. The 1 g sail on 100 GW coasts at 0.267c, and quadrupling the aperture from 1 to 4 km raises that only to 0.441c.

Tested in The rocket that leaves its fuel at home · the Mass-energy reading path

“A magnetic field splits a spectral line into three: one unshifted and two shifted by the same amount either side.”

That is the pattern only when the field is strong compared with the atom's own fine-structure field. In 1 T, sodium's D1 line becomes four components at ±2/3 and ±4/3 of μB·B and D2 becomes six at ±1/3, ±1 and ±5/3. Only far above sodium's crossover of 36.8 T does the strength gather into three bands at −1, 0 and +1 times μB·B.

Tested in The field an atom calls strong · the atomic spectra reading path

“A state with a large energy spread changes quickly.”

It may change quickly; the spread is permission, not obligation. Two levels driven off resonance by as much as the drive have the same spread as two driven on resonance, and their survival never falls below 0.500, while three equally spaced levels reach orthogonality only at 1.0887 times the limit and a coherent state never does, bottoming out at 0.0183.

Tested in The fastest a state can stop being itself · the uncertainty reading path

“A substance always diffuses from higher to lower concentration.”

Only in a mixture of two. In Duncan and Toor's two bulbs, computed from the Maxwell–Stefan equations, nitrogen flows from the bulb with less nitrogen into the bulb with more from 0.1 to 6.5 hours, opening a difference of 0.1468, because carbon dioxide leaving one bulb drags it along. In a welded steel, carbon crosses from the side with 0.478 per cent into the side with 0.441, leaving 0.336 and 0.562 per cent either side of the weld after 13 days.

Tested in The gas that flows towards more of itself · the diffusion reading path

“A mode at a frequency where waves can propagate away must leak and have a finite width.”

Only if nothing cancels its leak. Two resonances leaking into one channel at rates 0.1 and 0.05, coupled with strength 0.2, have a combined mode whose decay rate is zero to 10⁻¹² at a detuning of 0.1414, while the other mode takes the whole 0.15. The mode sits in the continuum, and its two routes out cancel.

Tested in The resonance that refuses to leak · the resonance reading path

“Doubling the bandwidth doubles the reflection that has to be accepted.”

It halves the logarithm of the match. The floor is exp(−π/RCω), so at RCω = 0.5 a perfect network could reach 0.0019 and three elements reach 0.0146, while at RCω = 8 the floor is 0.675 and three elements manage only 0.748. Broadband matching of a strongly reactive load becomes exponentially harder, and the difficulty is in the load.

Tested in The mismatch no network can remove · the impedance reading path

“Near-field and far-field diffraction are two different theories, Fresnel's and Fraunhofer's.”

They are one multiplication at different distances. The same plane-wave spectrum of a five-wavelength slit, each component multiplied by the phase it gains, gives the slit's own shape at 0.5 wavelengths, a bright centre at 25 and the diffraction pattern at 100 — conserving the travelling power to 10⁻¹⁰ and agreeing with a direct Fresnel integral to 2.75 per cent where Fresnel's approximation holds.

Tested in The fan of plane waves inside every beam · the huygens reading path

“The mean field gets the Curie temperature, so it is the right description with the numbers slightly off.”

It is wrong at the other end of the temperature range by more decades than any fit could hide. Mean field permits one excitation — a whole spin turned over against its neighbours — so it predicts a magnetisation deficit that is exponentially small at low temperature. The real excitations are long spin waves with no energy gap at all, and at 20 K the two accounts differ by 31 decades, computed here over a Brillouin zone.

Tested in What holds a magnet together is not magnetism · the magnetisation reading path

“A particle below the single-domain diameter stays in one domain, and the diameter follows from comparing the wall's cost with the saving from dividing.”

That comparison is what the last figure computes and it answers itself for the soft materials. Permalloy's crossing comes out at 4.0 nanometres against an exchange length of 5.7, so the wall the comparison priced could not have existed at that size. The estimate is valid where the wall is much narrower than the particle, which is where the anisotropy is high, and it says nothing about soft particles — whose single-domain size is set by the exchange length instead and is tens of nanometres.

Tested in The first length that belongs to the substance · the magnetisation reading path

“The magnetic viscosity of a sample is a relaxation with a characteristic time.”

It has no characteristic time, which is the content of the observation rather than a gap in it. A single barrier gives an exponential with a definite time in it; a spread of barriers gives a sum whose logarithm is straight, matched here to 0.0125 of the magnetisation by a least-squares line across eight decades, against 0.3332 for a narrow spread. The straight line is a signature that no time exists, so a sample that has been watched for a year will drift as much again in the next ten.

Tested in Nothing keeps a magnetisation for ever · the magnetisation reading path

“Thermal stresses are relieved by allowing a little flexibility at the supports.”

A support ten times stiffer than the member — which is a loose connection by structural standards — still passes ninety per cent of the stress, and halving it needs a support as flexible as the whole member. The relief curve is drawn across eight decades of stiffness ratio and is flat exactly where a designer would want it steep. Either the connection is a mechanism that lets the member move freely, or the member takes essentially all of EαΔT.

Tested in The load nobody applied · the Free-body reading path

“Tightening a bolt harder always makes a joint safer.”

It reduces the fluctuation the bolt sees — from 20 kN to 4 kN for the joint computed here — and buys nothing beyond the separation load, which for a 30 kN preload and a load factor of 0.20 is 37.5 kN. Past that the clamping is used up, the bolt carries the whole external load, and the curves meet. A preload also consumes part of the bolt's own strength, so the useful range is bounded at both ends and is specified rather than left to whoever holds the spanner.

Tested in A state no load could reach · the Free-body reading path

“The collapse load is found by assuming a mechanism and equating work.”

That gives an upper bound, and a wrong mechanism gives an answer that is too high and therefore unsafe. Scanning the hinge position for a uniformly loaded propped cantilever shows the work balance returning anything from 11.657 to over 20 times Mₚ/L² depending on where the hinge is assumed; the true value is the least. Putting the hinge at midspan, which is the natural guess, overestimates by 2.9 per cent.

Tested in The one number the tolerances cannot touch · the Free-body reading path

“Étendue is conserved because rays cannot be squeezed together — it is a fact about geometry.”

Geometry gives no smallest value, because nothing in a ray has a size, so the geometrical statement permits an étendue of zero and a beam concentrated without limit. The floor comes from the wavelength: a patch of phase space smaller than λ in one dimension is forbidden by the same Fourier relation that gives a slit its diffraction pattern. Étendue divided by λ² is a count, and a count stops at one.

Tested in The invariant that is a count · the etendue reading path

“A solar cell's 33 per cent limit is a thermodynamic one.”

It is a consequence of having a single threshold. Detailed balance on a 5,762 K spectrum diluted to the solar constant gives a peak of 30.5 per cent at a gap of 1.25 eV, against a thermodynamic ceiling of 5 per cent for an unconcentrated heat engine and 85 for a concentrated one. The cell beats the engine and falls far short of thermodynamics, and the whole of the shortfall is that one gap cannot suit photons of every energy at once.

Tested in The work a diluted beam will not do · the etendue reading path

“The best profile is a parabola.”

The best exponent measured by tracing six rays through each of seventy profiles is 1.96, not 2, because the cancellation is exact only to first order in the index contrast. The minimum is sharp: an exponent wrong by a tenth costs a factor of ten in delay spread, which is why the profile of a graded fibre is measured on every preform. In a real fibre the material's own dispersion moves the optimum again, by a wavelength-dependent amount.

Tested in The same cone, and a different arrival · the etendue reading path

“The chain's thermalisation rate is a power of the nonlinearity, and the power can be measured.”

A power can be measured over the range a computation reaches, and it is 1.90 here — found by scanning for the exponent that collapses four runs onto one curve, which reduces their spread fourfold. Those runs are all above the knee. The weakly nonlinear regime the 1955 calculation sat in is decades of amplitude below them, its predicted exponent is far higher, and no run of a length anybody will wait for reaches it.

Tested in The condition three modes never meet · the equipartition reading path

“Averaging for longer always improves a measurement.”

It improves it as the square root of the time, so a hundredfold gain costs ten thousand times as long, and it only helps against noise that is white. Against a baseline that itself wanders, the total has a least value and averaging past it makes the measurement worse — computed here at 6.6 nanovolts after 16 seconds for a ten-kilohm source with a drift of three nanovolts per root second.

Tested in Half a kT in a piece of wire · the equipartition reading path

“The same method gives the mass of any gravitationally bound structure.”

It gives the mass of a structure that has had time to become virialised, which means several crossing times. A globular cluster has had fourteen thousand and Coma nine; a supercluster has had 0.14, so its parts have not completed one traverse, are still separating with the expansion, and their velocity spread is not a virial dispersion. The condition needs only a size and a speed and should be checked before the mass is quoted.

Tested in Weighing what cannot be put on a scale · the equipartition reading path

“The problem is that the vacuum energy is too small.”

There are two problems and only the first is about size. The second is that a density which does not dilute is comparable, now, with one that falls as the cube of the expansion: the crossing is located here by bisection at a redshift of 0.31, a few billion years ago out of fourteen. A hundredfold change in the constant would put that crossing far in the past or the future, and nothing about the estimate explains why it is here.

Tested in The estimate that misses by a hundred and twenty · the planck scale reading path

“Large extra dimensions solve the hierarchy problem.”

They remove the tuning and move the question. A Planck cutoff needs the Higgs mass's bare value and its top-quark correction to cancel to one part in 3 × 10³²; a cutoff at a TeV needs no cancellation at all. But the size of the extra dimensions then needs its own explanation — the two-dimension case requires a compactification radius thirty orders of magnitude above the fundamental length — and none has been given.

Tested in The scale that may not be where it looks · the planck scale reading path

“Losses in a fibre can be handled the way they are for classical signals, by amplifying.”

A quantum state cannot be copied, so there is no amplifier. A photon survives 22 km of fibre with probability 1/e, so a direct link over a thousand kilometres delivers 1.8 × 10⁻¹⁰ pairs a second from a source firing ten thousand million times a second — one every hundred and seventy years. Eight segments joined by swaps divide the exponent rather than the rate and deliver four million a second, a gain of 2 × 10¹⁶.

Tested in A link between two that never met · the entanglement reading path

“The area law is a statement about entanglement.”

It is also a statement about what is computable, about which correlations can exist, and about how sharply a phase transition announces itself. The block entropy of a gapped chain is a constant and of a critical one is a logarithm whose coefficient is the central charge — so measuring an entanglement identifies which universality class the transition belongs to, from a quantity that is not a correlation function of anything.

Tested in The corner of Hilbert space that is ever visited · the entanglement reading path

“Light waves in vacuum add exactly, because there is no medium to respond nonlinearly.”

Maxwell's equations are linear and quantum electrodynamics is not. A photon can make a virtual pair and the pair can absorb a second photon, so two beams scatter — with a cross-section of 10⁻⁶⁸ square metres at optical frequencies, which is the smallest in physics and is not zero. A magnetic field polarises the pairs and the vacuum acquires two refractive indices differing by 2 × 10⁻²¹ at sixteen tesla.

Tested in The one medium that was supposed to add exactly · the superposition reading path

“Stratification makes a submerged body stable at its neutral depth.”

Only if the fluid's density rises with depth faster than the body's own does. For a body as compressible as water the critical gradient is 4.62 parts per million per metre, computed here, and that is close to real ocean stratification — a strong thermocline exceeds it and the deep ocean does not. Both cases are drawn and their slopes have opposite signs.

Tested in The body that displaces two things · the buoyancy reading path

“A rotating observer has no notion of simultaneity.”

No global one. The failure of a synchronisation to close around a region is 2ΩL²/c², so a region is consistent to whatever precision is demanded provided it is small enough — 785 kilometres on the rotating Earth at the nanosecond level, 207 metres inside a laboratory centrifuge. The useful question is not whether such an observer has a now but over what distance, and the distance is computable.

Tested in How big now is · the simultaneity reading path

“The technique measures the structure of the material.”

It measures the first two or three shells around one chosen element and nothing else. The photoelectron travels about four and a half ångströms between inelastic collisions at the worst point of its range, so a neighbour six ångströms away contributes seven per cent of what one at two ångströms does. Beyond that the sample could be anything.

Tested in The ripple that counts the neighbours · the attenuation reading path

“A band gap is a number a material has.”

Four ordinary recipes applied to one spectrum of gallium arsenide return values spanning several tens of millielectronvolts — more than eight times its own Urbach energy — because the straight portion that is extrapolated is straight only well above the tail, and how far down the fit reaches decides the answer. The recipe is part of the number.

Tested in Below the gap, where there is nothing to absorb · the attenuation reading path

“Acoustic mixing works because the sound agitates the fluid.”

What mixes is the steady circulation, not the oscillation. A fluid parcel oscillating with amplitude U/ω travels about a hundred and sixty micrometres at a kilohertz and returns; the drift carries it away and does not bring it back. In water the drift beats diffusion for channels wider than sixteen micrometres at a metre-per-second amplitude, and that crossing is what the technique is bought for.

Tested in The drift a sound leaves behind · the viscosity reading path

“Viscosity is a function of temperature.”

It is a function of temperature and of pressure, and over the range a machine actually reaches the pressure term is much the larger. Heating a mineral oil from twenty degrees to a hundred divides its viscosity by about thirty; squeezing the same oil to the 1.32 gigapascals computed here for a loaded line contact multiplies it by more than ten to the eleven.

Tested in The oil that is a glass for a quarter of a millisecond · the viscosity reading path

“The bound is a theorem.”

It is a conjecture. It is proved within a family of strongly coupled theories that have a gravity dual, it is known to be violated in theories with enough species of field, and it has no proof for ordinary matter. What is not in doubt is the measurement: nothing anybody has measured lies below it, and the two systems that come closest reach it from far apart.

Tested in Whether a fluid can be made arbitrarily thin · the viscosity reading path

“Measuring the ringdown frequency measures the remnant's mass.”

It measures one combination of mass and spin, and the two cannot be separated by a pitch alone: a hole at 90 per cent of maximum spin rings 1.4 times higher than a non-spinning one of the same mass, so any pitch is consistent with a curve of masses. What breaks the degeneracy is a second number — how fast the mode dies, or the frequency of a second mode.

Tested in A few cycles that are only mass and spin · the gravitational waves reading path

“The memory is the trace left by the masses that flew apart.”

That is the linear part and it is the smaller one for a binary that merges. The larger contribution is sourced by the gravitational waves themselves: the radiation carries energy away, energy gravitates, and an unbound flux of energy leaves the same kind of permanent imprint an unbound mass would. The wave is the source of its own memory, which is why the effect is nonlinear in a theory whose waves are usually treated as linear.

Tested in The ring that does not come back · the gravitational waves reading path

“A magnetic circuit is closed, so a small air gap makes little difference.”

A one-millimetre gap in a three-hundred-millimetre core of relative permeability five thousand leaves under six per cent of the field the closed core had. The circulation of H is fixed by the winding; H in the gap exceeds H in the core by the relative permeability; so a gap one part in three hundred of the path takes nearly all of the available circulation.

Tested in The field that points against the magnet it is in · the ampere law reading path

“The circulation of H depends on the path taken.”

It depends on one whole number: how many times the path winds round each current, with a sign. A path that goes round twice counts twice, one that goes round and returns counts nothing, and one enclosing two opposed currents counts zero however it is shaped. Neither the path's length nor its distance from the wire appears anywhere in the answer.

Tested in A potential that does not come back to itself · the ampere law reading path

“Static friction accelerates a car, so it supplies the energy.”

It supplies the external force and none of the energy. The contact patch of a rolling tyre is instantaneously at rest, so the friction force acts through no displacement and its work is zero. The energy comes from the fuel. Both requirements are real and separate: an external force is needed to change the momentum, and an internal energy source is needed to change the kinetic energy.

Tested in The floor that does no work · the energy reading path

“A slow compression and a fast one differ only in how long they take.”

They differ in the answer. Compressed to a quarter of its length, the ball drawn here ends at 15.7 times its energy when the wall creeps and 41.0 times when the wall moves at nine tenths of the ball's speed — the fast piston does extra work, which is why a fast compression is irreversible and why it and a slow expansion are not a cycle.

Tested in The wall that moves while the ball is in flight · the energy reading path

“Apodisation improves an image.”

It trades. Every decibel of sidelobe suppression is bought with a wider core: a Hann taper is 18 decibels quieter than a hard edge and its core is 1.63 times wider, so two equally bright stars that were just resolved are no longer. Which is wanted depends on whether the second object is comparably bright or ten million times fainter, and the two cases want opposite ends of the curve.

Tested in The rings that belong to the edge · the diffraction reading path

“A mirror polished to twenty-five nanometres is a good mirror.”

It depends entirely on the wavelength it is used at. The same surface gives a Strehl of 0.32 at 400 nanometres and 0.998 at five micrometres, because a surface error costs twice itself in wavefront and the cost in waves is that divided by the wavelength. A surface specification quoted without a wavelength says nothing at all.

Tested in How accurate a mirror has to be · the diffraction reading path

“Whether a particle has two states or three is decided by whether its mass is exactly zero.”

Nothing observable turns on an exact zero. The third state's effect on any process scales with the rest energy over the total energy, which for a 91-GeV particle at 5 TeV is 0.018. So a very light massive photon would be indistinguishable from a massless one in any experiment above its mass, and every bound on the photon's mass is quoted with the length scale over which it was tested.

Tested in Two states where the counting says three · the spin reading path

“An electron's magnetic moment therefore cannot be measured.”

It is measured to twelve significant figures, by a completely different arrangement: a single electron in a trap, where the spin's precession is compared with the cyclotron motion of the same electron in the same field. What is impossible is separating a free electron's spin states by deflection, which is a statement about one method.

Tested in The experiment that defines spin and cannot be done on it · the spin reading path

“Raising the driving temperature raises the performance without limit.”

It approaches a ceiling. The engine factor is one minus ambient over the driving temperature, which tends to one, so the whole expression tends to the fridge coefficient alone — 10.1 for the case drawn. Doubling a 450-kelvin driving temperature buys 40 per cent, and there is nothing above the fridge coefficient at any temperature whatever.

Tested in A fridge with no work going into it · the heat engines reading path

“The figure of merit can be raised without limit by improving the material.”

The electronic thermal conductivity is tied to the electrical one by a ratio nearly the same for every conductor, so with the lattice contribution removed entirely the group can be no larger than the Seebeck coefficient squared divided by that ratio. Only the lattice part is genuinely free, which is why the last forty years of the subject have been about scattering phonons rather than about electrons.

Tested in An engine with one number in it · the heat engines reading path

“A moving body's temperature is lower by the Lorentz factor, as Planck and Einstein showed in 1907.”

Their derivation is internally consistent and so are the two later ones that give the opposite and no change at all. What the three differ over is what heat means under a boost, and since the first law ties heat to temperature, a choice about one is a choice about the other. All three agree at rest and none is refutable, because no thermometer reads any of them.

Tested in The body that has no temperature when it moves · the relativistic thermodynamics reading path

“The second law is frame-independent because entropy is.”

Entropy being invariant makes the statement well posed and does not prove it. What makes the second law frame-independent is that entropy is invariant AND that the events it compares are timelike-separated, so every observer agrees which came first. Two spacelike-separated entropy increases have no agreed order, and the law says nothing about their sequence — which is fine, because it makes no claim about it.

Tested in The count that no observer can disagree about · the relativistic thermodynamics reading path

“The drag is radiation pressure.”

It is the anisotropy of radiation pressure created by the motion itself. A body at rest in isotropic radiation feels pressure from every side and no net force. The force appears only because moving makes the radiation anisotropic in the body's own frame, so it is proportional to the speed rather than to the intensity alone, and it vanishes at rest however bright the bath.

Tested in The bath that pushes back · the relativistic thermodynamics reading path

“The action is a number that belongs to a path.”

It is also a function of the endpoint. Fixing where the motion starts and recording the action of the true path to every place and time gives one function whose gradient is the arriving momentum and whose time derivative is minus the arriving energy. The figures check both against trajectories integrated separately, to a part in ten thousand.

Tested in The action that knows where every path ends · the least action reading path

“Solving the Hamilton–Jacobi equation gives the motion everywhere.”

A smooth solution exists only until two trajectories of the family reach the same point. Past the envelope of a fan of throws the function has two branches, and at the focus of an oscillator it has no value at all; the figures draw both, computed from the trajectories rather than from the equation.

Tested in The action that knows where every path ends · the least action reading path

“The best free throw is the one that needs the least speed, because there an error in angle costs nothing.”

At the least-speed launch, 51.4° for a release 0.95 m below and 4.19 m short of the hoop, an error of angle does cost almost nothing — and the ball comes down at 38.5°, leaving its centre ±3.7 cm of room. For a thrower scattering ±0.05 m/s and ±1° the chance of a clean pass is 37 per cent aimed there and 58 per cent aimed at 58.5°, computed both from the exact window of speeds at every error of angle and by counting over a grid of both scatters.

Tested in The throw most likely to go in · the projectile reading path

“Aiming well is a matter of hitting the middle.”

Every aim in the figures is at the middle. What changes with the aim is the shape of the region of launches that succeed and how it lies against the thrower's scatter, and the best aim is where the most of the scatter falls inside — which depends on whether that thrower errs more in speed or in angle.

Tested in The throw most likely to go in · the projectile reading path

“The difference between an orbiting and a thrown clock is a weak-field curiosity.”

The exact Schwarzschild calculation keeps the same order down to 6.5 GM/c², where over one period the orbiting clock keeps 73.4 per cent of a distant clock's time, a held clock 83.2 per cent and a thrown one 88.4 per cent. Far out the three losses times the radius settle to 3/2, 1 and 0.68, matching the weak-field figures.

Tested in The orbit that ages less than a throw · the time dilation reading path

“Light that climbs out of a gravitational field loses energy, so thermal light arrives with the right colours but dimmer.”

It arrives as a Planck spectrum at a lower temperature, which is a different shape from a dimmed one. Carrying a 3000 K spectrum up a field that lowers every frequency by 0.8 gives a spectrum that matches Planck's law at 2400 K to one part in 10¹⁴, and the bottom spectrum scaled to the same peak height visibly does not.

Tested in The column that is hotter at the bottom · the relativistic thermodynamics reading path

“The equilibrium constant is a property of the molecules alone.”

K is, but how far a reaction goes is not. Dinitrogen tetroxide with ΔG° = +4.77 kJ/mol at 298 K is 88.6 per cent dissociated at 0.01 bar, 18.8 per cent at 1 bar and 6.0 per cent at 10 bar, because each gas's chemical potential rises with the logarithm of its own partial pressure and the dissociation makes more molecules.

Tested in The reaction that cannot go all the way · the chemical potential reading path

“Light from a light-emitting diode is not thermal radiation.”

Above its band gap an ideal diode at temperature T with a voltage V across it emits with exactly Planck's occupation, 1/(exp((ε − qV)/kT) − 1). The voltage multiplies every point of the thermal spectrum by exp(qV/kT) — 21.8 powers of ten at 1.3 V across a 1.42 eV gap at 300 K — and leaves its slope set by the lattice's kT.

Tested in The glow that carries a voltage · the chemical potential reading path

“The coincidence dip at a beam splitter shows that light is made of photons.”

Only a dip deeper than one half does. Two independent laser beams with a random relative phase, sampled over 40,000 trials, empty the dip to 0.499 of its far value, and two thermal sources to 0.669; no classical field can go below one half. Single identical photons reach zero.

Tested in The outcomes identical photons refuse · the photon reading path

“Identical photons in an optical network simply bunch together.”

In a symmetric three-way splitter with one photon in each input, identical photons never produce six of the ten possible outcomes — those whose output labels do not sum to a multiple of three — and put a third of their probability into one photon per output. Distinguishable photons produce all ten.

Tested in The outcomes identical photons refuse · the photon reading path

“Injecting squeezed light into a gravitational-wave detector lowers its noise.”

Squeezing in phase lowers the photon-counting noise at high frequency and raises the radiation-pressure noise at low frequency by the same factor. In a model interferometer whose two noises cross at 70 Hz, 6 dB of phase squeezing is worse than none below 99 Hz; only squeezing that rotates with frequency lowers the noise everywhere.

Tested in The noise pushed below the floor · the photon reading path

“The neutron interferometer measures the neutron's weight.”

It measures a phase that is 2πm²gλA sin α / h², which contains the mass twice, Planck's constant twice and the wavelength once. With the wavelength measured separately the experiment tests the combination m²g/h², so it is a test of quantum mechanics in a gravitational field as much as of gravity.

Tested in The fall that leaves the mass in the phase · the matter waves reading path

“The flattening of a planet's gravity field tells how flattened the planet is.”

It fixes J₂, and very different bodies share a J₂. A uniform body flattened by 0.34 per cent and a perfect sphere carrying 0.268 per cent of its mass as an equatorial ring both have J₂ = 0.001339; they differ at the next order, where J₄ is −3.85 × 10⁻⁶ for one and −0.001 for the other.

Tested in The field outside that cannot find the core · the Gauss's law reading path

“A disturbance that is not exactly a soliton spreads out and disappears.”

A hump of height 6 shaped as sech²x, integrated in the Korteweg–de Vries equation, leaves two crests of heights 8.00 and 2.00 running apart at speeds 16 and 4, with only a small ripple spreading behind. Read as a well, the same hump holds levels at κ = 2 and 1, and each level becomes a soliton of height 2κ².

Tested in The solitons a hump already contains · the wave packets reading path

“Light moves from one waveguide to its neighbour by leaking across the gap, and a longer coupler transfers more.”

Between identical guides the light beats: a coupler of two 0.5 µm silicon guides 250 nm apart transfers everything at 37.6 µm and gives it all back by 75 µm, oscillating with length for ever. Only between guides tapered so that their propagation constants cross does the transfer grow steadily with length — above 95 per cent from 195 µm.

Tested in The coupler that does not care about the colour · the guided waves reading path

“A photonic time crystal amplifies light without any source of energy.”

The energy comes from whatever modulates the index. A wave at the centre of the gap grows by a factor of 73 over thirty periods of a 0.2-deep modulation, and the equation contains nothing that limits the growth; every joule is work done by the drive changing the permittivity against the field already present.

Tested in The crystal made of moments · the refraction reading path

“A thick disordered medium transmits in proportion to one over its thickness, whatever the scattering.”

Only when the waves' phases are thrown away. The same random stacks of layers with every surface's reflection added as intensities transmit 1.0 per cent at 400 layers, falling as one over the thickness; computed with interference kept, by multiplying transfer matrices, the typical transmission falls by a factor of e every 17 layers and is below 10⁻¹⁰ at 400.

Tested in The walk that interference can stop · the scattering reading path

“More engine power always makes a ship go faster.”

Where the resistance has a hump, a steady thrust meets it more than once. A thrust that crosses the resistance curve at 0.94, 1.00 and 1.82 knots leaves a ship accelerating from rest at the first crossing, below the hump, although a steady state at 1.82 knots exists; only a thrust above the top of the hump, or a push past it, reaches the fast branch.

Tested in The wave that holds a ship back · the stratification reading path

“A perfectly conducting plasma conserves its field line topology, so it cannot relax at all.”

Perfectly, it cannot. A small resistivity lets field lines reconnect in thin sheets, which destroys the linkage of every individual flux tube while barely touching the total helicity: the helicity lost per unit time is bounded by the square root of the resistivity times the energy lost, so as the resistivity falls the energy can go while the helicity stays.

Tested in The twist that outlives the turbulence · the flux freezing reading path

“Confirming the size of the gravitational redshift confirms that it is a property of time.”

It confirms that one kind of clock shifts by the predicted amount. Universality is a separate claim: that every clock shifts by the same amount. The Sun's potential at the Earth swings by 3.3 × 10⁻¹⁰ each year, so two unlike clocks side by side would acquire an annual ratio term of Δβ × 1.65 × 10⁻¹⁰ if they responded differently — a quantity no measurement of a single clock's shift can see.

Tested in The clocks that must all slow together · the gravitational redshift reading path

“Doubling the charge of the counterions doubles how strongly they correlate.”

The coupling parameter Ξ = 2πq³ℓ_B²σ goes as the cube of the valence. At 0.5 elementary charges per square nanometre it is 1.6 for monovalent ions, 12.8 for divalent, 43.2 for trivalent and 102 for tetravalent — which is why sodium never produces the attraction and calcium, spermidine and spermine do.

Tested in The like charges that pull together · the osmosis reading path

“A plasma in a magnetic field leaves the field as it found it.”

It weakens it. The current its pressure gradient drives is diamagnetic, and with pressure balance B²/2μ₀ + p constant, a peak pressure of 0.6 of the outside field's magnetic pressure lowers the field at the centre to 0.63 of its outside value. The flux change through a loop round the plasma is −μ₀W⊥/B₀ at low pressure: 33 milliwebers for a column at 100 kPa in 3 T, which is how the stored energy of a fusion plasma is weighed.

Tested in The current no particle carries · the magnetism reading path

“Shielding a qubit from charge noise requires removing the stray charges.”

It can be done by making the circuit indifferent to them. The dependence of the levels on offset charge falls as exp(−√(8EJ/EC)), with EJ the Josephson energy and EC the charging energy: the ground level moves by 0.263 of a transition as the charge sweeps at EJ/EC = 1 and by 3.0 × 10⁻⁸ at 50, and the first excited level becomes 10⁵ times less sensitive.

Tested in The circuit that forgets its charge · the superconductivity reading path

“A relativistic particle in an inverse-square field moves on a slowly precessing ellipse.”

Only while the attraction is weak compared with angular momentum times c. The exact orbit is a rosette whose apsidal angle is π/√(1 − k²/L²c²), which grows without limit as k/Lc approaches one; the integrated orbit at k/Lc = 0.35 advances 24.3 degrees a turn, and at k/Lc = 1.05 there is no orbit at all — the body spirals into the centre within one revolution.

Tested in The orbit special relativity cannot close · the orbit stability reading path

“Dielectrophoresis can manipulate anything small enough, given enough field.”

The depth of the trap grows as the cube of the radius and has to beat kT. With |Re K| = ½, a field of 10⁶ V/m makes a trap one kT deep at a radius of 12 nm; a protein a few nanometres across needs 10⁷ V/m, where the current through any real buffer heats the liquid and drives flows that swamp the force.

Tested in The force whose sign a frequency chooses · the dielectrics reading path

“In free fall, bubbles in a liquid rise slowly.”

They do not rise at all, and neither does anything sink: with no effective gravity the pressure in the liquid is uniform, and Archimedes' principle gives zero. What decides the shape and position of a bubble or a drop is then surface tension, over any size smaller than the capillary length, the square root of surface tension over density times effective gravity — 2.7 mm for water on the Earth and 2.7 m at a millionth of g.

Tested in The balloon that leans the wrong way · the buoyancy reading path

“A finite number of grains can only collide a finite number of times in a finite time.”

Three grains on a line with a restitution below 7 − 4√3 = 0.0718 collide infinitely often: the interval between collisions shrinks by a nearly fixed factor, the series converges, and the grains end touching and moving as one after 3.23 time units at e = 0.03. At e = 0.1 they trade six collisions and fly apart.

Tested in The gas that cools itself into clumps · the granular matter reading path

“The infinitely many bounces are a real feature of a real ball.”

Only the first few dozen are. A real impact lasts a time set by the elastic contact, which Hertz's theory gives as growing as the impact speed to the minus one-fifth, while the flight after it shrinks in proportion to the speed. For a steel ball 20 mm across on steel with e = 0.8 the two cross at an impact speed of 1.6 mm/s, at the thirty-sixth bounce, and past that the ball is touching the floor more than it is flying.

Tested in The bounces that add up to a stop · the momentum reading path

“Airy's formula gives the ellipse's turning for any swing.”

It is the leading term in the size of the swing. The integrated turn agrees with it to a few per cent up to about fifteen degrees; at 50° with b/a = 0.3 the integration gives 13.5° per half swing against Airy's 11.9°, and the difference is the next term.

Tested in The ellipse a pendulum turns by itself · the pendulum reading path

“Light from a point to its image takes the shortest path.”

It takes every path at once, and each is stationary rather than shortest. From a source half a radius from the centre the traced rays range in length from 2.51 to 57.7 radii, and their optical lengths all equal π n₀R to a part in ten thousand. Perfect imaging requires exactly that equality.

Tested in The medium that images every point · the fermat reading path

“A graded-index sphere is an engineered curiosity with no natural counterpart.”

The lens in the eye of a fish is a graded sphere. Modelled with an index falling parabolically from 1.52 at the centre to 1.38 at the rim, it focuses in water at 2.42 radii with a spread of 0.37 radii, where a uniform ball of 1.52 focuses its central rays at 4.06 and spreads them over 1.86. The ratio of about 2.5 between focal distance and radius was measured by Matthiessen in 1877.

Tested in The lens with no axis · the fermat reading path

“A binding energy is measured by how much gas a surface holds.”

At one temperature the amount held depends on the binding energy and on the pressure together. The binding energy is the slope of the logarithm of the half-full pressure against 1/T: for sites of 0.2, 0.35 and 0.5 eV the slopes read back 0.200, 0.350 and 0.500 eV, which is the same construction that reads an activation energy off reaction rates.

Tested in The site that fills like an electron level · the chemical potential reading path

“Once doped, a semiconductor's minority carriers are negligible.”

They are few and they are what devices run on. A forward voltage splits the two chemical potentials by qV and multiplies the product n·p by e^(qV/kT) — 1.2 × 10¹⁰ times at 0.6 V in silicon — and the injected minority carriers, recombining at a rate proportional to the product, are the diode current.

Tested in The product doping cannot move · the chemical potential reading path

“Mercury is a liquid because its atoms are heavy and its bonds therefore weak.”

Its lighter partners zinc and cadmium melt at 419.5 °C and 321.1 °C. A 2013 simulation of mercury put its melting point at about −23 °C with relativity and about 82 °C without, against −38.8 °C measured: the relativistically contracted 6s pair barely takes part in bonding, and that, not the mass, is what melts it.

Tested in The metal that is yellow because it is heavy · the atomic structure reading path

“Landau's criterion predicts the speed at which helium flowing through a tube starts to dissipate.”

It gives an upper bound that real flows almost never reach. Feynman's estimate for making a quantised vortex across a channel of width d, (ħ/md) ln(d/a₀), is 15 cm/s for a 1 µm channel and 0.22 cm/s for 100 µm, and measured critical velocities follow it. Only ions dragged through pressurised helium, too small to shed vortices easily, reach the roton limit.

Tested in The speed below which nothing can be made · the superfluidity reading path

“The opening angle of a collision depends on what the particles are.”

For equal masses it depends only on γ: a 1 GeV proton and a 1 MeV electron, both at about twice their rest energy, open at 77.9° and 70.8°. Measured in units of the rest energy, every particle follows one universal curve, which is why a single photograph of two tracks from an elastic collision measures the incident energy.

Tested in The right angle a fast collision closes · the relativistic dynamics reading path

“The relativistic oscillator is just a slightly slower harmonic oscillator.”

Its motion changes shape, not just speed. At large amplitude it is a triangle wave, whose harmonics at 3, 5, 7 and 9 times the fundamental fall as 8/(π²n²); at 30 c/ω₀ the integrated motion sits on that series. A slow harmonic oscillator has no harmonics at all.

Tested in The spring that becomes a light-clock · the relativistic dynamics reading path

“A laser's output is short or long depending on how wide a range of frequencies it emits.”

The range sets the shortest pulse possible, not the pulse there is. Twenty modes with random phases carry the same spectrum as twenty in step, and their output is irregular noise peaking near 90 times one mode's intensity; in step it is a clean train peaking at exactly 400. The width of the spectrum is necessary for a short pulse and nowhere near sufficient.

Tested in The phases that turn a glow into pulses · the superposition reading path

“More beams make a more stable pattern, because each one contributes less of the total.”

More beams add phase combinations that no slide can absorb. Four beams in a plane leave one: changing one beam's phase by half a turn turns a lattice of bright spots into a finer pattern of half the period and half the peak, and no shift of the first reproduces it to better than 15 per cent of the peak. Five beams leave two.

Tested in The lattice no phase can bend · the superposition reading path

“Superoscillation beats the diffraction limit, so a lens's resolution is a matter of design rather than physics.”

It beats the width of the spot and pays for it in light. A least-energy band-limited field with zeros every 0.25 wavelength puts 2.2 × 10⁻⁴ of its energy in the central spot and has side lobes 495 times brighter than it; at 0.5 wavelength, the ordinary spacing, the share is 0.93. Each tenth of a wavelength narrower costs orders of magnitude.

Tested in The wiggle faster than any wave in it · the huygens reading path

“Synchrotron radiation reaches high frequencies because it is beamed into a cone of angle 1/γ.”

The cone accounts for one power of γ: the charge's cone sweeps across the observer in 1/γ of a turn, and the computed emission width falls with slope −1.00. The flash arrives with slope −3.00, 18,870 times shorter than it was emitted at γ = 100, because the charge follows its own light across the whole of that interval.

Tested in The flash a circling charge sends once a turn · the radiating charge reading path

“X-ray diffraction reveals the positions of the atoms in a crystal.”

It reveals their electrons. Above every binding energy each electron scatters as a free one and a nucleus scatters as its charge squared over its mass, so the nucleus contributes between 6.4 × 10⁻⁸ and 2.9 × 10⁻⁷ of what the electrons do. A hydrogen atom, with one electron, scatters 64 times less than an oxygen atom.

Tested in The cross-section that forgets the colour · the radiating charge reading path

“The Hulse–Taylor pulsar's orbital decay is the best test of gravitational radiation, so it is also the best test for any extra radiation.”

Its eccentricity of 0.617 multiplies the quadrupole loss by 11.84 and a dipole loss only by 3.94, burying any dipole three times deeper. The cleanest dipole test is a circular orbit: J1738+0333, with an eccentricity below 10⁻⁶, measures −25.9 ± 3.2 fs/s against general relativity's −27.5, leaving room for no more than 8.0 fs/s of extra loss.

Tested in The radiation a difference in falling would make · the equivalence principle reading path

“The two momenta of light in a medium are nE/c and E/nc, so their product is always (E/c)².”

Only without dispersion. The kinetic momentum follows the group index, the canonical one the phase index. In BK7 at 400 nm the index is 1.5308 and the group index 1.5841, and the product is 0.966 of (E/c)²; at 1000 nm it is 0.991.

Tested in The momentum light carries into glass · the field energy reading path

“Energy and momentum always form a four-vector, so the rest mass of any system is the same in every frame.”

Only for a closed system. For a charged shell's field alone, √(E² − p²c²) is U√(1 − β²/9): 0.9860 U at 0.5c and 0.9539 U at 0.9c. The field is not closed — something must hold the charge together — and with Poincaré's stress included the invariant is 4U/3 at every speed.

Tested in The mass a charge's field gets wrong by a third · the field energy reading path

“Molecules with no charge and no dipole moment interact only through weak dispersion forces.”

Their quadrupoles act on a passing ion. On the axis at 0.35 nm a singly charged cation is attracted by −3.8 kT to nitrogen, −11.6 kT to carbon dioxide and −23.5 kT to the face of benzene, and repelled by +25.7 kT from hexafluorobenzene, whose quadrupole has the opposite sign. The point-quadrupole numbers overstate at contact; the ratios and signs do not.

Tested in The shape that takes five numbers · the the field concept reading path

“Sound travels faster in warm water, so it is slowest where the ocean is coldest, at the bottom.”

Temperature is half the story. Squeezing raises the sound speed by 15.5 m/s for every kilometre of depth, and in a model ocean cooling from 20 °C to 2 °C the two effects meet in a minimum of 1483.4 m/s at 1,145 m — the axis of the deep sound channel — with the coldest, deepest water carrying sound faster than the surface.

Tested in The sea that stands lower because water gives · the hydrostatics reading path

“The narrower the pore, the better it lifts water.”

The narrower the pore, the higher the water will eventually stand and the slower it gets there. After one second the best radius for water is 170 µm, reaching 57 mm; after a day it is 3.8 µm, reaching 2.52 m. The best radius falls as the deadline to the −1/3.

Tested in The pore that lifts highest fills slowest · the capillarity reading path

“With the correction for turning points, the semiclassical rule gives exact energies.”

Exact for the box, the harmonic well and the half well at every level; approximate otherwise. The bouncer's ground state comes out at 0.9924 of its computed value and its third level at 0.9994; a quartic well's ground state at 0.8178. The rule improves as the levels rise and is exact only where the potential is special.

Tested in The quarter cycle a turning point costs · the least action reading path

“The centre of percussion is a property of the body, like its centre of mass.”

It is a property of the body and the pivot together. For a uniform rod it is at 2L/3 from an end pivot, runs past the far end once the pivot is within L/6 of the middle, and goes to infinity as the pivot reaches the centre of mass; for a free rod struck at distance s from its centre, the still point is k²/s beyond the centre, 0.167L for a blow at the end.

Tested in The point that takes the blow without a jolt · the pendulum reading path

“A thicker metal film absorbs more, so the dip gets deeper as the gold gets thicker.”

The dip is deepest at an intermediate thickness. It reaches 0.367 at 30 nm, 0.075 at 40, 0.014 at 50, 0.209 at 60 and 0.472 at 70; scanning in half-nanometre steps, it reaches zero at 47.0 nm, where the plasmon's leak back into the prism equals its loss in the gold.

Tested in The angle at which total reflection stops being total · the total internal reflection reading path

“A solid glass corner cube reflects totally, so it works as well uncoated as silvered.”

It returns the same light but not the same field. Each sector's three total reflections shift the two polarisations differently, the six sectors return six polarisations with degrees of circular polarisation from 0.44 to 0.95, and their interference leaves the far-field peak at 0.266 of a silvered cube's.

Tested in The corner that sends light home · the total internal reflection reading path

“Hydrogen's 21 cm frequency follows from Fermi's formula for the magnetic energy of the electron at the proton.”

Fermi's formula gives 1421.160 MHz, 531 ppm high. The electron's anomalous moment, the reduced mass, relativity and the leading radiative correction bring it to 1420.462 MHz, still 40 ppm above the measured 1420.405752, and the rest is the proton's finite size.

Tested in The line that takes eleven million years · the atomic spectra reading path

“Quantum jumps and classical orbital motion are different kinds of thing.”

The frequency of the photon emitted dropping from n to n − 1, over the classical orbital frequency of level n, is 3.000 at n = 2, 1.173 at n = 10 and 1.015 at n = 100, tending to 1 as 1 + 3/2n. High in the sequence of levels, the quantum jump radiates at the frequency the classical orbit would.

Tested in The atom the size of a bacterium · the atomic spectra reading path

“Pound and Rebka's measurement of gravitational redshift needed only a tall tower and a sharp gamma-ray line.”

It needed the source and absorber at the same temperature. Their gravitational shift across 22.5 m was 2.46 × 10⁻¹⁵; a temperature difference of 1.10 K between two iron-57 samples produces the same shift, which is why they had to measure and correct for it.

Tested in The clock that runs slow because it is warm · the time dilation reading path

“Only a real photon beam can make photon–photon collisions.”

A fast nucleus carries a cloud of equivalent photons whose number goes as the square of its charge. A lead nucleus at 2.51 TeV per nucleon carries 6,724 times a proton's photons below its cut-off of about 38 GeV — some 100 per logarithmic interval at 1 GeV — and two lead nuclei passing without touching collide them.

Tested in The charge that passes as a flash of light · the field transformation reading path

“A battery that holds a steady voltage as it discharges is simply a well-designed battery.”

It is one whose electrode is separating into two phases. For a lattice of sites whose lithium ions attract each other with 4 kT, the voltage is exactly flat between 0.021 and 0.979 filled, while one phase grows at the other's expense; with no attraction it falls through the whole range as (kT/e) ln((1 − x)/x).

Tested in The voltage that is a chemical potential · the chemical potential reading path

“Driving a process harder always makes it more regular.”

Only until it stops stepping backwards. A single-step clock counting to a hundred is spread by ±20.2 at 0.5 kT per step and ±11.5 at 2 kT, and at 8 kT it has reached ±10.0, the spread of a process that never steps back, for four times the heat. Past that, regularity has to come from splitting each tick into many steps: at 20 kT per tick the product falls from 20.00 for one step to 2.07 for thirty.

Tested in The precision a clock pays for in heat · the entropy reading path

“Stacking more resistive sheets makes an absorber better.”

It makes it broader and thicker in step. One sheet is below −20 dB from 8.73 to 11.27 GHz at 7.5 mm, two from 5.73 to 14.27 GHz at 15 mm, three from 4.00 to 16.00 GHz at 22.5 mm, and every one of the three meets the thickness bound exactly: the extra sheets only move a fixed amount of absorption to where it is wanted.

Tested in The absorber whose bandwidth is its thickness · the impedance reading path

“Moving the end of a vibrating string shifts the frequencies of its modes, and that is all it does.”

Only if the end moves slowly. Moved by 1% of the length at twice the fundamental frequency, the end pumps the string from its fundamental into a spread of modes whose energy-weighted mean is mode 47 after thirty round trips, and the energy grows 38.3-fold in forty.

Tested in The string whose end will not keep still · the standing waves reading path

“Mixing in a stable ocean needs turbulence to supply the energy.”

Fingers draw their energy from the salt's half of the stratification. The fastest-growing finger carries only 0.56 to 0.66 as much density upward as heat as it carries downward as salt, for density ratios between 2 and 20, so it lowers the column's centre of mass and releases potential energy with no stirring at all.

Tested in The salt that sinks through a stable sea · the stratification reading path

“The fluid far from a slowly moving body is undisturbed, so the body's drag cannot depend on anything far away.”

True for a sphere, whose flow is back to 90 per cent of the stream within 7.5 radii. Beside a cylinder the flow reaches 90 per cent only at 232 radii for a Reynolds number of 10⁻², and 15,000 at 10⁻⁴. Inside a concentric container the drag per length over μU is 9.50 when the wall is ten radii away and still 0.98 when it is a million radii away.

Tested in The drag a distant wall decides · the viscosity reading path

“Given the shaking, the state the grains end in is determined.”

Not with three compartments. Equal thirds are stable up to B = 9 and a single full compartment is stable from B = 6.55, so between the two the grains keep whichever state the history of the shaking left them in.

Tested in The sand that chooses a side · the granular matter reading path

“The jet stream's large meanders are carried round the planet by the westerly wind.”

They move west through the air at β/k² — 14.8 m/s for a 6,000 km wave at 45° — while the wind carries them east. A 15 m/s westerly holds a wave 6,048 km long still over the ground, 4.7 of them round the latitude circle, which is why the largest meanders sit roughly where the mountains put them.

Tested in The wave that can only travel west · the circular motion reading path

“Computing more accurately cures the problem.”

It postpones it logarithmically. The logistic map at r = 4 computed in double precision as 4x(1 − x) and as 4x − 4x² parts by 0.1 at step 52; each extra digit of precision buys about three more steps, because errors double at every step.

Tested in The wrong orbit beside a right one · the chaos reading path

“Extrapolating the measured gaps in a straight line to zero gives the dissociation energy.”

For a Morse well it does. For a Lennard-Jones well of the same depth and curvature, which has a −1/r⁶ tail and holds 27 levels, the same line through the first five gaps encloses 0.7988 against the true 0.9474, 15.7 per cent short; with 20 gaps it is still 1.4 per cent short.

Tested in The spectrum that weighs a bond · the harmonic approximation reading path

“A stronger local oscillator always gives a better heterodyne measurement.”

Only until its own shot noise swamps the amplifier's. For a 1 pW signal at 1.55 µm in a 1 MHz band, the signal-to-noise ratio rises from −3.3 dB with a microwatt of oscillator to 7.4 dB with a hundred, and then stops at 8.0 dB, ηPₛ/(hνB), because the oscillator raises the signal and its own noise together.

Tested in The faint light a bright one makes measurable · the coherence reading path

“Correcting chromatic aberration requires two kinds of glass.”

Axial colour does. Lateral colour in an eyepiece can be removed with one: two N-BK7 lenses of 60 and 20 mm spaced 40 mm apart, half the sum of their focal lengths, have a focal length that changes by 0.01 per cent between blue and red, against −1.03 per cent at 20 mm and 3.19 per cent at 60 mm.

Tested in The colour fringe no aperture can close · the imaging reading path

“The deeper a diamond is cut, the more light it returns, because the light has further to travel.”

For a ray entering straight down, the two pavilion reflections succeed for every angle between 24.4° and 51.9°, but the ray reaches the top again at |4θp − 180°| from its normal, which is inside diamond's 24.4° critical angle only between 38.9° and 51.1°. At 52° all four traced rays reach the top at 28° and leave through the bottom.

Tested in The window a diamond is cut inside · the total internal reflection reading path

“A hotter core has more pressure and so supports a star better.”

Only if the heat raises the pressure. Near two billion kelvin up to 29 per cent of the heat added to an oxygen core goes into the rest mass of new pairs, and the first adiabatic exponent falls to 1.222 at 10³ g/cm³ and 1.285 at 10⁵ — below the 4/3 a self-gravitating body needs to be stable.

Tested in The heat that turns into matter · the relativistic thermodynamics reading path

“A straight worldline is the one that carries the most proper time between its ends.”

Only among worldlines that wind round space the same number of times. Between two events 40 years apart on the stay-at-home's worldline there are seven straight worldlines, one for each number of laps from −3 to 3, carrying 40.00, 38.73, 34.64 and 26.46 years for zero, one, two and three laps.

Tested in The twin who never turns round · the time dilation reading path

“The early universe became transparent when it cooled enough for the typical photon to be unable to ionise hydrogen.”

The typical photon could not ionise hydrogen long before. With 1.6 billion photons per nucleon the rare energetic ones keep hydrogen half ionised, in Saha equilibrium, down to 3,734 K, where the binding energy is 42 times kT; in reality recombination lagged further and the light last scattered near 2,970 K.

Tested in The electron that leaves because there is room · the chemical potential reading path

“The third law is a statement about cooling — that absolute zero cannot be reached.”

It is also a constraint on every phase diagram. By Clausius and Clapeyron a coexistence line's slope is ΔS/ΔV, and ΔS → 0 forces the slope to zero: helium-3's melting pressure changes by 11.4 kPa between 0.9 and 4 mK, against 508 kPa between 1 mK and its minimum, and lead's critical field reaches absolute zero with zero slope.

Tested in The melting curve that has to arrive flat · the third law reading path

“Nuclear half-lives are properties of the nucleus that cannot be predicted from simple principles.”

The multipole order predicts them to within a factor of a few. Barium-137m's 661.7 keV transition joins states of spin 11/2 and 3/2 with opposite parity, so it must be M4; Weisskopf's M4 estimate is 5.6 minutes against a measured gamma half-life of about 2.8, while the E3 and E5 estimates are thousands of times away.

Tested in The spin a photon has to carry away · the decay reading path

“Cooling a system always slows its escape over a barrier.”

Only escape over the barrier. Below the crossover the escape temperature of a junction biased at 0.985 or 0.99 of its critical current levels off at 42–44 mK however cold the junction gets, because the phase then passes through the barrier at a rate that does not depend on temperature.

Tested in The circuit that goes through a wall · the tunnelling reading path

“Every pair of orbiting bodies has stable Trojan points sixty degrees ahead of and behind the smaller one.”

Only if the smaller holds less than 0.0385 of the total mass. The Sun and Earth, the Sun and Jupiter and the Earth and the Moon qualify; Pluto and Charon, at 0.1085, do not. Near L4, a body stays within reach for 95 orbits at mass ratios of 0.03 and 0.035, and is flung clear after 30 orbits at 0.042 and 7 at 0.05.

Tested in The hilltop that holds the Trojans · the orbit stability reading path

“Tides heat a moon by stretching it, so any moon close to a large planet is tidally heated.”

A steady stretch does no work. A moon on a circular orbit that keeps one face to its planet carries a fixed bulge and is not flexed at all; the heating goes as the square of the eccentricity, and at Io's distance falls fourfold if the eccentricity halves.

Tested in The moon kept molten by its neighbours · the Self-gravity reading path

“Any attractive central force allows stable circular orbits, since a body can always be given the right speed.”

Circular orbits exist for any attraction, and are stable only if κ²/Ω² = 4 − d is positive. A circular orbit nudged 5 per cent faster stays between 1.00 and 1.23 of its radius in three dimensions and between 1.00 and 1.10 in two, and leaves within 1.93 turns in four and 1.19 in five.

Tested in The number of dimensions a planet can orbit in · the Gauss's law reading path

“The field between the ground and the ionosphere is uniform, like the field between the plates of a capacitor, so the voltage is spread evenly over the height.”

The current is uniform and the conductivity is not. With the conductivity rising e-fold every 3 km, the field falls by the same factor; half the column's resistance, and half the 250 kV, lies in the lowest 2.1 km, and 96 per cent of the potential is reached by 10 km of a gap sixty kilometres deep.

Tested in The field that keeps Greenwich time · the potential reading path

“A single coil fed with alternating current makes no rotating field at all, so it can do nothing to a rotor.”

Its field is exactly two fields of half the strength turning opposite ways. A stationary rotor feels both equally and the torque is zero, so a single-phase motor cannot start; pushed either way, the rotor lags the field it follows less than the other, the balance tips, and it runs up to 99 per cent of synchronous speed in the model.

Tested in The field that turns with nothing turning · the superposition reading path

“The point-mass lens, which always makes two images, is the general case.”

It is the singular case. Its lens equation runs to infinity at the centre, so it is not continuous there, and the odd-number rule needs continuity. Any lens with a core, however small, restores a third image.

Tested in The fifth image a galaxy hides · the light deflection reading path

“A self-bias requires the powered electrode to be smaller than the grounded one.”

Area asymmetry is one source. Amplitude asymmetry in the waveform is another, and for a waveform of N harmonics with amplitudes falling as (N − k + 1)/N the bias between equal electrodes is −(N − 1)/(N + 1) of half the peak-to-peak voltage: −0.33 with two, −0.78 with eight.

Tested in The bias a waveform can choose · the plasma oscillation reading path

“The point-dipole model explains the strength of commercial electrorheological fluids.”

It gives 17.5 Pa at 2 kV/mm for a 30 per cent suspension with β = 0.8. Working fluids reach hundreds of pascals to kilopascals at such fields, because the field between two nearly touching spheres is concentrated far beyond the applied one, and in a conducting particle far more so.

Tested in The liquid a field turns solid · the dielectrics reading path

“An eddy-current brake can bring a vehicle to rest.”

Its force goes to zero with the speed, so the final approach is an exponential decay — in the drawn case the speed falls tenfold every 14 s — and at rest it holds nothing on a gradient. It is also weak at high speed, where its drag falls as one over the speed.

Tested in The drag that falls as the magnet speeds up · the induction reading path

“A suspension jams when its particles are packed as closely as they can go.”

That is 0.64 for frictionless spheres, and a suspension below it can still jam under stress. At φ = 0.60 the model's suspension stops flowing entirely above 2.4 times the onset stress, because frictional contacts, once switched on, jam at 0.58.

Tested in The suspension that seizes when pushed hard · the rheology reading path

“A water column is stable whenever the water below is denser than the water above at their own depths.”

Stability is about a parcel displaced into its neighbours' surroundings. In a lake whose deep water is at 3.4 °C, a parcel at 2.8 °C pushed down from 250 m is lighter than its surroundings at first and denser than them below about 450 m, because the temperature of maximum density falls with pressure, and there it sinks unaided.

Tested in The water that warms on the way down · the hydrostatics reading path

“An elastic collision between two bodies of the same material returns all the kinetic energy of the motion.”

Only if neither body is left vibrating. A rod striking a longer rod of the same section stops dead after 2L₁/c and the struck rod moves off at L₁/L₂ of the striker's speed, so its restitution is L₁/L₂ and the rest of the energy, two thirds at a length ratio of three, is a stress pulse travelling inside the long rod.

Tested in The bounce an elastic plate cannot give back · the momentum reading path

“Once started, a free wobble persists for ever, as the free precession of a spinning body does.”

The Earth's wobble loses energy to the mantle's internal friction and the oceans; with a quality factor of about 80 it would decay by a factor of e in about 30 years. It has been present at 0.1–0.25″ since the 1890s, so something — mainly shifting pressure on the ocean floor and the ground from the oceans and atmosphere — keeps exciting it.

Tested in The wobble that is slow because the Earth gives · the rotation reading path

“A pinhole camera's resolution is fixed by the size of its hole.”

The best hole grows as the square root of the camera's depth, so the angular blur falls as one over that square root: about 23′ for a camera 10 mm deep and 2.3′ for one a metre deep, with the f-number rising from about 95 to about 950.

Tested in The hole that makes the sharpest picture · the imaging reading path

“The opposition surge is a single effect with one width.”

Shadow hiding in a layer of grains produces a peak several degrees wide whose width grows as the grains pack more densely — 4.9° at 10 per cent and 14.2° at 50 per cent in the computed two-dimensional layers. Coherent backscattering adds a narrower, wavelength-dependent, polarised peak in the last degree or two.

Tested in The full moon that is too bright · the scattering reading path

“At a finite temperature, a particle at the Fermi level still cannot scatter.”

The Fermi level is smeared over about kT, and the summed collision rate at ε = 0 is (πkT)²/4 in the model drawn, agreeing with Landau's (ε² + (πkT)²)/2(1 + e^(−ε/kT)) to within half a per cent across the plotted range.

Tested in The collisions the exclusion principle forbids · the exclusion reading path