The collection

Every essay — page 22

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Mechanics

Motion, force, and the quantities that refuse to change.

A ball dropped from 1 m with e = 0.8, bouncing for ever and stopping anyway. The height of a ball dropped from 1 m onto a floor that returns 0.8 of its impact speed, against time. The first fall takes 0.452 s. Every flight after it is 0.8 times as long as the one before, so the flight times are a geometric series and their sum is finite: the ball has made infinitely many bounces by 4.064 s, t₁(1 + e)/(1 − e), and is at rest from then on. The stepped sum of the first 60 flights plus the geometric remainder agrees with the closed form to better than a part in a billion. The dashed line marks the moment it stops; the hops just before it are too small to draw.

The bounces that add up to a stop

A ball dropped on a hard floor bounces, and bounces again, each time a fixed fraction lower. The bounces never run out — there is always another, smaller one — and yet the ball is lying still four seconds later. The flight times form a geometric series with a finite sum, the floor's infinitely many kicks add up to exactly the ball's weight times that time, and the series is only broken where each flight becomes shorter than the impact that launched it.

5 figures · part 6 on Momentum

A pendulum swung 30° with a sideways push, seen from above. The path of the bob of a 1 m pendulum free to swing in any direction, seen from directly above, released 30° from the vertical with a sideways push that would, for a small swing, make an ellipse 0.3 as wide as it is long. The path is an ellipse that turns: its long axis advances 5.30° every half swing, in the same sense as the bob goes round, where Airy's small-amplitude rate (3/8) ωab/L² gives 5.06°. Nothing outside the pendulum turns it — the pivot is fixed and there is no Earth rotation in this calculation. A pendulum on a turning Earth that is meant to show the Earth's rotation has this turning to contend with too.

The ellipse a pendulum turns by itself

Let a pendulum swing in any direction and give it a small sideways push, and its bob traces an ellipse. For a small swing the ellipse closes. For a larger one it turns, steadily, in the sense the bob is going round — with nothing outside the pendulum turning it. Airy worked out the rate in 1851, the same year Foucault hung his pendulum in the Panthéon, and the two effects are the same size for a pendulum a metre long whose ellipse is a twentieth of a millimetre wide.

6 figures · part 7 on Pendulum

Waves

Oscillation, and everything that turns out to be an oscillation.

A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵.

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

5 figures · part 6 on Wave packets $The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.$

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

5 figures · part 6 on Guided waves

Optics

Light, and the small number of rules it obeys.

A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

The crystal made of moments

A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

5 figures · part 6 on Refraction

With interference kept, transmission falls exponentially; without it, only as one over the thickness. Light through stacks of randomly thick transparent layers, alternating indices 1 and 2.6, with thicknesses scattered by ±50 per cent about a quarter wave, on a logarithmic scale against the number of layers. The falling line is the transmission with the waves' interference kept, computed exactly by multiplying transfer matrices and averaged as the logarithm over 40 random stacks and five wavelengths. It falls in a straight line: the transmission drops by a factor of e every 17 layers, however thick the stack, which is exponential decay — localisation. The upper curve is the same stacks with every surface's reflection and transmission added as intensities, so that no interference survives. It falls only as one over the thickness, checked to grow in exact proportion, which is the diffusion of light through a cloud: at 400 layers it still transmits 1.0 per cent where the coherent stack typically transmits 4.5·10⁻¹¹.

The walk that interference can stop

Light scattered many times walks through a cloud, and a walk always gets through eventually — a slab twice as thick lets through half as much. Keep the waves' interference instead of adding intensities, and in one dimension the same disorder does something a walk cannot: it stops the light exponentially, traps it in modes with nothing special about where they sit, and turns transmission from a number into a spread over powers of ten. Whether the same can happen to light in three dimensions has been claimed, retracted and argued for thirty years.

5 figures · part 6 on Scattering

Every ray from one point, meeting again at another. Rays from a point half a radius from the centre of Maxwell's fish-eye, n = n₀/(1 + r²/R²), traced in 11 directions round the full circle. Every one is an arc of a circle, and every one passes through the point on the other side of the centre at distance R²/r from it — 2.00 radii — which is the image. The worst miss is 8.8e-4 radii, the size of the integration's own error. Rays leaving in opposite directions, rays leaving at right angles, rays that go the long way round: all arrive. The faint circles are contours of the index, which falls from n₀ at the centre to n₀/2 at one radius and n₀/5 at two.

The medium that images every point

A single refracting surface can be shaped to image one point perfectly and no other. A medium whose index falls smoothly away from a centre can do better: in Maxwell's fish-eye every ray from any point, leaving in any direction, arrives at one image point, and every such path takes exactly the same time. It is a perfect instrument for all of space at once — and the image it forms is sharp everywhere and a faithful copy nowhere.

5 figures · part 5 on Fermat

A parallel beam brought to a point on the far side. Parallel rays sent into Luneburg's sphere, whose index falls from the square root of two at the centre to one at the rim, from the left. Each beam is brought to a single point on the rim diametrically opposite the direction it came from; the worst of the 11 rays misses by 2.0e-3 radii. The sphere has no axis — every direction is the same to it — so a beam from any direction is focused as well as any other, and there is no such thing as an off-axis aberration.

The lens with no axis

Every ordinary lens has an axis, and every one of its hardest aberrations is a penalty for looking away from it. A sphere has no axis, so a sphere cannot have off-axis aberrations — but a glass ball has spherical aberration instead, because one index and one curvature are too few to focus every ray. Luneburg's sphere, whose index falls from √2 at the centre to 1 at the rim, keeps the symmetry and removes the aberration: it brings parallel light from every direction to a perfect point on its own surface.

5 figures · part 6 on Fermat

Electromagnetism

Charge, field, and the lines drawn between them.

Three bodies of one mass, one field outside, three fields inside. The gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.

The field outside that cannot find the core

Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.

5 figures · part 4 on Gauss's law

Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

5 figures · part 5 on Magnetism

A pendulum with unequal steps. The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.

The circuit that forgets its charge

A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.

5 figures · part 5 on Superconductivity

A living cell and a dead one, told apart by a frequency. The real part of the dielectrophoretic factor for a living cell and a dead one, ten micrometres across, in a liquid of conductivity 0.01 S/m, against frequency. The cell is a conducting interior inside a membrane five nanometres thick. The living cell's intact membrane blocks slow fields, so at low frequency it behaves as an insulator and is pushed out of strong field; above 42 kHz the membrane is short-circuited by its own capacitance and the conducting interior is felt, so the cell is pulled in; above 140 MHz the permittivities take over and it is pushed again. The dead cell's membrane leaks, its interior has lost ions, and it changes sign only at 8.8 MHz. In 2 bands — below 42 kHz and from 8.9 MHz to 140 MHz — the same field pushes the two in opposite directions.

The force whose sign a frequency chooses

A neutral particle in a non-uniform field is pulled towards strong field if it polarises more than the liquid around it and pushed away if it polarises less. With conduction in the picture, which of the two it does is decided by how fast the field alternates — and a living cell, a conductor wrapped in an insulator five nanometres thick, changes sign twice, where a dead one changes once. The same electrodes can then send the living cells one way and the dead ones the other.

6 figures · part 5 on Dielectrics

Thermodynamics

Heat, disorder, and the one law with a direction in it.

$How full a surface is, against the pressure of the gas above it. The fraction of adsorption sites occupied on a surface that binds a molecule with 0.35 eV, against the gas pressure on a logarithmic axis, at 250, 300, 350 K. Each curve is θ = p/(p + p½) — Langmuir's isotherm — and each is the same S-shape moved along the axis: on a logarithmic pressure scale the coverage is a function of ln p alone, shifted by the binding energy over kT. The half-full pressure is 5.58 Pa at 250 K, 132 Pa at 300 K, 1341 Pa at 350 K. Warming the surface does not change the shape; it slides the curve to higher pressure by a factor that is a Boltzmann factor, e^(ε/kT), which is how a binding energy is read off a family of isotherms.$

The site that fills like an electron level

A patch of catalyst holding a gas molecule, a haem in a muscle cell holding oxygen, an energy level in a metal holding an electron: each is a place that can hold one particle or none, in contact with a reservoir it trades particles with. Each fills according to the same function of one variable — how far the reservoir's chemical potential lies above the energy of the site. Langmuir's isotherm and the Fermi function are not analogous. They are the same law, and haemoglobin works because its sites break it.

6 figures · part 3 on Chemical potential

Electrons and holes in silicon against the donors added. The concentrations of free electrons and holes in silicon at 300 K against the concentration of donor atoms added, on logarithmic axes, solved from charge neutrality and the product law n·p = nᵢ². Below the intrinsic concentration of 10¹⁰ per cubic centimetre the donors change almost nothing. Above it the electrons follow the donors one for one and the holes fall in exact proportion: a hundred times more donors, a hundred times fewer holes. At 10¹⁶ donors the electrons outnumber the holes by 10¹² to one. The dashed line is √(n·p), which does not move at all across ten decades of doping — checked to a part in a billion.

The product doping cannot move

Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.

6 figures · part 4 on Chemical potential

Quantum

Where the continuous picture runs out, and what replaces it.

$The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.$

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

5 figures · part 4 on Atomic structure

Helium's excitations, and the line that decides how fast it can flow. The energy of an elementary excitation in superfluid helium-4 against its momentum, as a smooth fit through published neutron-scattering values: a phonon branch rising at the speed of sound, 238 m/s, a maximum near 13.9 K, and the roton minimum of 8.62 K at 1.92 Å⁻¹. A straight line from the origin has a slope that is a velocity. The steepest line that just touches the spectrum is Landau's critical velocity, 57 m/s, and it touches at 1.98 Å⁻¹, just past the roton minimum. A body moving slower than that cannot make any excitation at all and so cannot lose energy to the liquid; the dashed line is the speed of sound, far above it.

The speed below which nothing can be made

An object moving through an ordinary liquid always feels drag, because it can always pass some of its energy to the liquid. In a superfluid there may be nothing it can pass it to. Landau's argument turns the question into geometry: an excitation of momentum p and energy ε can be made only by a body moving faster than ε/p, so a liquid flows without friction below the smallest value of that ratio. For helium the smallest value is not the speed of sound but 57 m/s, set by the dip in its spectrum called the roton — and an ideal gas, with no dip and no sound, has no such speed at all.

5 figures · part 5 on Superfluidity

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

$A thrust that cannot push past the hump. The resistance a hull meets against its speed in knots, in two layers whose fastest interfacial wave travels at 1.02 knots: ordinary friction rising as the square of the speed, plus the drag of the interfacial waves, whose hump sits just below the wave speed. The horizontal lines are 2 steady engine thrusts. A ship settles where its thrust meets the resistance curve. Thrust 0.6 meets it at 0.81 knots; Thrust 1.3 meets it at 0.94, 1.00, 1.82 knots. A ship accelerating from rest reaches the first crossing and stops gaining speed there, below the hump, even when a faster crossing exists beyond it — which is the dead water sailors reported, a ship held to a fraction of its usual speed by a wave it cannot see.$

The wave that holds a ship back

In 1893 the polar ship Fram, which could make four or five knots, was held to about one in a calm Arctic sea with nothing visible in the water. The sea was layered — a metre or two of fresh meltwater over salt — and the ship was making a wave on the boundary between the layers, a wave that travels at about a knot and carries away almost all of a slow ship's power. Below that speed the drag is a hump no steady thrust can climb; above it the wave cannot keep up and the drag falls away.

5 figures · part 6 on Stratification

Mean field always pushes; correlations pull. The pressure between two planes of equal charge with only their own counterions between them, against their separation, in units of the Gouy–Chapman length μ for the separation and 2πℓ_Bσ²kT for the pressure. The upper curve is the Poisson–Boltzmann result, solved from k·tan(kd/2) = 1: it is the density of counterions at the midplane and is positive at every separation, falling from the ideal-gas 2/d at contact to π²/d² far apart — 1.71 at 1μ, 0.290 at 4μ. The lower curve is the strong-coupling limit, 2/d − 1, which the same ions reach when their valence and the surface charge are high. It crosses zero at d = 2μ and is negative beyond, tending to −1: the two like-charged planes attract, and the separation 2μ is where they come to rest.

The like charges that pull together

Two surfaces carrying the same charge, with nothing between them but the ions that neutralise them, ought to repel, and the standard mean-field theory proves that they always do. With calcium or spermine as the counterions they attract, and come to rest a fraction of a nanometre apart. The mean field misses it because it averages the ions into a smooth cloud, and multivalent ions are too strongly repelled by each other to form one. Each keeps a patch of surface to itself, and the pressure between the plates becomes a single ion's business.

5 figures · part 6 on Osmosis

A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

6 figures · part 6 on Buoyancy $A gas of grains that lose 51 per cent of their energy per collision, cooling into clumps. 2000 discs in a square box with periodic walls, covering 25 per cent of its area, started with random velocities and left alone. Each collision keeps a fraction e = 0.7 of the relative velocity along the line of centres. The panels are the same gas at three temperatures, where temperature means the mean kinetic energy of a grain: 0.85 T₀, 0.019 T₀, 0.00049 T₀. It starts uniform — the grain counts in a 10 × 10 grid have a variance 0.3 times their mean, more even than scattered points because discs cannot overlap — and ends in dense bands and clumps with empty space between, the variance 17 times the mean. Nothing attracts anything. The clumps form because a dense patch collides more often, cools faster, loses pressure and is squeezed denser by the hotter gas around it.$

The gas that cools itself into clumps

Shake a box of grains hard enough and they fly about like the molecules of a gas. Stop shaking and the gas cools, because every collision destroys a little of the motion — and it does not cool evenly. A denser patch collides more often, cools faster, loses pressure, and is squeezed denser by the hotter gas around it. With nothing attracting anything to anything, the gas gathers itself into clumps and bands, and in the limit three grains can collide infinitely many times in a finite time.

5 figures · part 7 on Granular matter

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.

The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

6 figures · part 5 on Flux freezing $Both clocks move, and their ratio does not. Two years of a clock's fractional frequency against a distant clock (upper panel), and of the ratio of two unlike clocks kept side by side (lower panel). Above, both clocks slow as the Earth nears the Sun, with an amplitude of 1.65·10⁻¹⁰, and if the redshift is universal the two curves are one curve. Below, the ratio: the flat line is what universality predicts, and the sinusoid is what a clock responding to the potential 10⁻⁶ more strongly than the other would produce — an annual term of 1.65·10⁻¹⁶, a million times smaller than the shift both clocks share and within reach of clocks that compare to parts in 10¹⁷.$

The clocks that must all slow together

Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.

6 figures · part 4 on Gravitational redshift

The half of the bend a slow body never feels

Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.

6 figures · part 4 on Light deflection

A rosette: the orbit of a body whose inertia is its energy. An orbit of eccentricity 0.45 under inertia from energy, Newton's pull, integrated for 7 revolutions from the closest approach of a body with k/Lc = 0.35. The dashed curve is the Newtonian ellipse from the same starting position and momentum, which closes on itself. The integrated orbit does not: its closest approach moves forward by 24.3° each revolution, against the exact 24.3° of 2π(1/Γ − 1) with Γ = √(1 − k²/L²c²). The dots are the successive closest approaches.

The orbit special relativity cannot close

An inverse-square orbit closes on itself because of a conserved vector that nothing else has. Give the orbiting body an inertia that grows with its speed, as special relativity requires, and the vector turns — the orbit becomes a rosette, advancing by a sixth of Mercury's famous 43 arcseconds a century. A different theory that respects special relativity just as well turns the rosette backwards by the same amount. The 43 is not special relativity plus a correction; it is a measurement of what gravity pulls on.

6 figures · part 4 on Orbit stability

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