Illustrated Physics
physics, drawn
Field

Relativity

Space and time, drawn on the same axes.
A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Two axes, one speed, and a diagram that does the arguing

Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive.

 A light clock at β = 0.6. The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.

The clock that has to slow, and why no clock can refuse

One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising.

 Simultaneity at β = 0.5. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

Now is a choice of slicing

Two events happening at the same time is not a fact about the events. It is a fact about who is asking, and different observers slice spacetime at different angles.

 A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

The length that depends on when, and is not really about length

A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.

 Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it.

Mass is a form of energy, which is not the same as a source of it

The famous equation is usually read as a promise that mass can be turned into energy. It says something stricter and stranger — that a mass is an energy already, sitting there, whether or not anything ever releases it.

 Composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.

Speeds that refuse to add, and the quantity that does

Run at half the speed of light, throw something forward at half the speed of light, and the result is not the speed of light. It is four-fifths of it, and there is a variable in which the arithmetic is still simple addition.

 A spacetime diagram at β = 0.6. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light.

The twin who comes back younger

If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.

 Three answers where sound has two, and one where it has none. The factor by which an approaching source's frequency is raised, against its speed as a fraction of the wave speed. For sound it matters which of the two is moving: a moving source gives 1/(1 − β) and a moving observer gives 1 + β, and at 0.5 of the wave speed those are 2.000 and 1.500. For light there is one answer, 1.732 — the geometric mean of the other two, exactly — because there is no medium to be moving with respect to. The fourth curve is the transverse shift, which happens at closest approach when the distance is not changing at all: 0.866, and nothing classical predicts it.

The shift that survives at right angles

For sound it matters which of the two is moving, and the two answers differ. For light there is one answer — their geometric mean — and a term with no classical counterpart at all: a source going past at closest approach, with its distance not changing, is still shifted.

 What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The quantity nobody argues about

Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.

 The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Magnetism is electricity seen sideways

The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.

 What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The invariant that survives a boost

Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.

 Simultaneity at β = 0.866. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

The pole that fits and does not fit

A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.

 Where the light that was sideways ends up. The direction a photon is seen to travel in the laboratory, against the direction it was emitted in the frame of the source, for a source moving at 0.5c, 0.9c, 0.99c. The straight diagonal is what would happen if a boost only changed frequencies; every curve lies well below it, which is aberration. The number that matters is where the emitted right angle lands, because half of everything emitted is on that side of it: 60.0° at 0.5c, 25.8° at 0.9c, 8.1° at 0.99c. The usual shorthand for that angle is 1/γ, which gives 49.6°, 25.0°, 8.1° — good to a few per cent only once the source is genuinely relativistic, and wrong by 17% at 0.5c. Nothing is emitted differently in any of these cases: the source is radiating exactly as it always did, and it is the map from its angles to ours that has changed.

The sky that crowds into a cone

A boost does not only shift frequencies. It remaps directions, so half of everything a fast traveller can see is squeezed into a forward cone of half-angle about 1/γ — and because brightness carries four powers of the Doppler factor, what lies ahead is overwhelming and what lies behind has effectively gone.

 Two corrections, opposite in sign and different in size. How fast a clock in a circular orbit runs compared with one on the ground, in microseconds a day, against the height of the orbit — with the two effects drawn apart rather than added. Being high speeds a clock up, by an amount that saturates: the potential term is bounded because there is only so much potential to climb out of. Moving slows it down, and a higher orbit is a slower one, so that term shrinks toward zero. They cancel at 3186 km — a radius of exactly 1.5 Earth radii, which follows from setting the sum to zero and contains neither G, nor the Earth's mass, nor the speed of light. At 20200 km the gravitational term is 45.7 µs a day and the speed term −7.2, leaving 38.5. Left uncorrected, that is 11.5 km of position error a day, growing without limit, from a clock that is working perfectly.

The clock that is wrong in two directions

A satellite clock loses 7.2 microseconds a day to its speed and gains 45.9 to its height. The two effects have opposite signs, different sizes and different dependence on the orbit, so there is exactly one altitude where they cancel — and 38.6 microseconds a day, left alone, is eleven and a half kilometres of position error.

 The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones.

The mass that is missing

A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

 Which happened first, asked of several observers. Two events on a spacetime diagram: one at the origin and one 3 light-seconds away and 1 second later, so that light leaving the first cannot reach the second. Through the second event runs a family of lines, each one the set of events some observer calls simultaneous with it; an observer moving at a fraction β of the speed of light has such a line of slope β on these axes. Where a line meets the vertical axis is the time that observer assigns to the second event. For a slow observer that meeting point is above the origin and the second event happens later; for a fast one it is below and the second event happens EARLIER. The changeover is at β = 0.3333, which is the time separation divided by the space separation, and it is a legal speed only because the separation is spacelike. So the order of these two events is not a property of the events. What every observer does agree on is that neither could have caused the other, because the two lie outside each other's light cones — drawn here as the diagonals — and that agreement is what causality rests on rather than on any shared notion of before.

Which came first, and who decides

Two events far apart can happen in either order, depending on who is asked, and both answers are correct. That is not a loophole in causality but the reason causality survives at all — because the pairs whose order is negotiable are exactly the pairs neither of which could have caused the other.

 A cube photographed at four speeds. The outline a camera records of a cube passing at β = 0, β = 0.4, β = 0.7, β = 0.9, moving to the right, seen at the moment it passes. The dashed square is what the usual picture shows: the cube flattened along its motion by the Lorentz factor. It is not what the camera gets. Light from the far face left earlier than light from the near face, by the time it needed to cross the extra distance, and the cube moved in that interval — so the far face appears displaced backwards and the side of the cube comes into view. The two effects together are exactly a rotation: at β = 0.4 the cube photographs as one turned 23.6°, at β = 0.7 the cube photographs as one turned 44.4°, at β = 0.9 the cube photographs as one turned 64.2°. The contraction has not gone anywhere — a ruler laid alongside still reads 43.6 per cent of the proper length at the highest speed drawn — but it is a statement about two positions at one time, and a photograph is not that.

The contraction no photograph shows

Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed. A camera would have shown it turned. The contraction is entirely real in the measurement that defines it, and a photograph is not that measurement.

 The rotation two boosts leave behind. The angle through which a frame's axes are turned after two boosts of equal size, against the angle between the two boosts, for 4 speeds. Two boosts in the same direction compose to a boost and nothing else, which is the zero at the left; two in different directions do not. What is left over is a rotation, and it is not small at large speeds: at β = 0.3 it peaks at 2.7° when the boosts are 91° apart, at β = 0.6 it peaks at 12.8° when the boosts are 96° apart, at β = 0.85 it peaks at 36.1° when the boosts are 108° apart, at β = 0.95 it peaks at 63.2° when the boosts are 122° apart. Each curve here is computed by multiplying the two boost matrices and pulling the rotation out of the product, not by evaluating a formula; the closed form for perpendicular boosts is used to check the extraction and appears nowhere in the drawing. The consequence is that the Lorentz boosts do not form a group by themselves — compose two and you leave the set — and that an object carried round a closed path in velocity space comes back turned.

The turn that two pushes leave behind

Two boosts in different directions do not compose to a boost. The product carries a rotation, so a frame carried once round a closed path comes back turned — and the size of that turn was the factor of two standing between the calculated and the measured splitting of a spectral line.

 Pushing one way and going another. The angle between an applied force and the acceleration it produces, against the angle between the force and the body's velocity, at four speeds. At 0° and 90° the two are parallel, because those are the two directions the γ³ and γ divisors do not mix. Everywhere between, they are not: at 0.99c the worst case is 73.9°, reached with the force at 8.0°. A body under a steady sideways-ish push does not travel along it.

The push that does not point where the body goes

Newton's second law survives relativity in the form F = dp/dt and in no other. Written as F = ma it fails outright, and not merely by a factor — at high speed a body pushed at forty-five degrees accelerates at eighty, because the same force is divided by γ³ along the motion and by γ across it.

 What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance.

The field nobody can transform away

A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter.

 Two rockets that keep their distance, and the string that does not. Two rockets 0.5 unit apart in the laboratory, given identical acceleration programmes there, drawn in units where the light speed is one and c²/a is one. Their laboratory separation is constant for ever — the two worldlines are the same curve shifted sideways, and every horizontal line meets them 0.5 apart. The slanted lines are the rockets' own lines of simultaneity, and the distance between the worldlines measured along those is what a string tied between them has to span: at 0.3c it is 0.512, a stretch of 2 per cent; at 0.6c it is 0.557, a stretch of 11 per cent; at 0.8c it is 0.631, a stretch of 26 per cent; at 0.9c it is 0.710, a stretch of 42 per cent. The γL that is always quoted — 0.524, 0.625, 0.833, 1.147 here — is the limit of that measurement for a vanishing gap, and at a gap of 0.5 in these units it overstates the stretch by up to 38.1 per cent; shrinking the gap a hundredfold brings the two within 0.46 per cent. Either way the string is stretched and breaks, while the gap in the laboratory never changes by a millimetre. Length contraction is not something that happens to a rod. It is a statement about which events count as simultaneous, and a rod that is not allowed to contract is a rod that is being pulled apart.

The string that breaks between two rockets

Two rockets a metre apart, given identical acceleration programmes, stay a metre apart in the laboratory for ever. A string tied between them breaks anyway. Nothing pulls on it, nothing in the laboratory moves relative to anything else, and the string is stretched — because the distance it has to span is measured on the rockets' slices of simultaneity and not on the laboratory's.

 What a collision has to spend, against what it is given. The energy available in a proton–proton collision, against the energy of one beam, on logarithmic axes. Against a stationary target the available energy is √(2mE) and the line has slope one half; head-on it is 2E and the slope is one. Bevatron at 6.2 GeV per beam reaches 3.7 GeV; SPS fixed target at 450 GeV per beam reaches 29.1 GeV; LEP at 104.5 GeV per beam reaches 209.0 GeV; Tevatron at 980 GeV per beam reaches 1960.0 GeV; LHC at 6500 GeV per beam reaches 13000.0 GeV. The gap is the whole architecture of the subject: the LHC's beams give 13000 GeV head-on and would give 110 GeV against a stationary proton, a factor of 118. Reaching the same 13000 GeV in fixed-target mode would need a beam of 90.1 million GeV. What the missing energy has gone into is not lost: it is the kinetic energy of the centre of mass, which every product has to carry away and which no experiment can use.

The collision that wastes most of the energy

The LHC's two beams carry 6,500 GeV each and 13,000 GeV are available. Fire one of those beams at a stationary block of copper instead and 110 GeV are available — the other 12,890 have gone into the motion of the wreckage and cannot be used for anything. The difference is a square root, and every accelerator built since 1970 is a consequence of it.

 The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales.

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

 One expression over 3 decades of area times rate. Sagnac time difference against the product of enclosed area and rotation rate, both logarithmic. The relation Δt = 4AΩ/c² is linear in that product — the fitted slope of the drawn points is 1.0000 — and it contains no refractive index, no shape of the loop and no position of the axis inside it. a 1 km fibre gyroscope on a 10 cm coil: 1.62e-19 s, 3.14e-5 fringes; Sagnac's own ring, 1913: 4.84e-17 s, 0.0666 fringes against a reported 0.07; a laboratory turntable at one revolution a second: 6.99e-17 s, 0.0331 fringes; Michelson and Gale, 1925: 4.50e-16 s, 0.2364 fringes against a reported 0.23. Michelson and Gale's rectangle in Illinois is the one that carries the check: 0.236 fringes predicted from its own dimensions and the vertical component of the Earth's rotation at its latitude, and 0.230 reported.

The ring where the two beams disagree

Send light both ways round a closed loop on a turntable and the two beams come back at different times, by 4AΩ/c² — an expression with no refractive index in it, no shape of the loop and no position of the axis. The same number is what a set of clocks round the rim fails to close by, which is the sharper statement — on a rotating platform there is no global simultaneity to be had.

 A horizon 0.97 light years behind, made by nothing but the motion. Position across and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.

The wall of silence behind a rocket that never stops

Hold a constant acceleration and the worldline is a hyperbola asymptotic to a light ray — so there is a light ray that never catches it. An observer who never stops accelerating has a horizon a distance c²/a behind, made by nothing but the motion, and at one gravity it sits 0.97 light years back. Nothing is there. No mass, no surface, no field.

 How far a 1-gravity ship gets, against its own clock. The distance covered by a ship accelerating steadily at 1 gravity for half the trip and braking for the other half, against the time on its own clock, on a logarithmic vertical axis. The curve is a cosine hyperbolic and therefore an exponential once the ship is relativistic, which it is after about a year: Proxima Centauri in 3.5 shipboard years, Sirius in 4.6 shipboard years, the Pleiades in 11.9 shipboard years, the galactic centre in 19.8 shipboard years, Andromeda in 28.6 shipboard years. Nothing about this violates anything. The speed never reaches c — it is the hyperbolic tangent of the rapidity and after 3.5 years it is 0.9495 at turnover — and every one of those journeys takes slightly more than the distance in years as measured from home. What is growing exponentially is not the speed but the length contraction, and the traveller's honest description of the trip is that the distance shrank. The rapidity is what accumulates steadily: it grows by one unit every 0.969 years of shipboard time, for ever, with no ceiling anywhere in the arithmetic. That is the whole reason the numbers come out survivable, and the reason the fuel does not.

The ship that never arrives at c

Accelerate at one gravity and never stop. The speed creeps toward light and never reaches it, and meanwhile the galactic centre is twenty shipboard years away and Andromeda twenty-nine. What makes the journey survivable is that rapidity has no ceiling; what makes it impossible is that the fuel goes as the exponential of the same quantity.

 The clock that gains going one way and loses going the other. The rate at which a flown clock gains on a clock left at 30° latitude, in nanoseconds per hour, against the aeroplane's ground speed, with east taken as positive. Two terms are drawn and then their sum. Height alone gives 3.5 nanoseconds an hour at 9 km and does not care which way the aircraft is pointed. Motion costs time, and because the ground is already moving eastward at 402 metres a second, flying east adds to that speed and flying west subtracts from it — so the kinematic term is much larger going east and can change sign going west. The sum crosses zero at 180 metres a second eastward, which is the ground speed at which an aeroplane's clock keeps the time of the airfield it left. Over the two flights Hafele and Keating actually made, this simple model gives -61 nanoseconds eastward and +304 westward, against their own predictions of -40 and +275 and their measurements of -59 and +273. The model here uses one average altitude, one average speed and one latitude, where the real prediction integrated the flight logs; getting the signs and the rough sizes out of three lines of arithmetic is the point, and the last twenty per cent is what the logs are for. What no amount of arithmetic supplies is the thing the experiment settled: that the effect is real, that it acts on a caesium clock in a passenger seat, and that a difference of a few hundred nanoseconds after two days is measurable.

The two clocks that flew in opposite directions

Two caesium clocks were flown round the world in 1971, one each way, and came back disagreeing with the clock left behind — one having lost 59 nanoseconds and the other gained 273. Height alone would have made both gain. The sign flip comes from the ground already moving eastward at 400 metres a second before the aircraft took off.

 The temperature of an acceleration. The Unruh temperature against proper acceleration, both axes logarithmic. An observer accelerating through empty space finds it is not empty: the state that an inertial observer calls the vacuum, an accelerated one finds populated, with a thermal spectrum at T = ħa/2πck — which is 4.06e-21 kelvin for every metre per second squared. The line is straight because the relation is exactly proportional, and the numbers on it are what make the effect so hard to see: one gravity gives 3.98·10⁻²⁰ K, a centrifuge at 10⁵ g gives 3.97·10⁻¹⁵ K, an electron in a strong laser gives 40.6 K, the surface of a solar-mass hole gives 6.17·10⁻⁸ K. Reaching one kelvin requires 2.5·10²⁰ m/s², which is 2.5·10¹⁹ gravities and beyond anything that can be sustained. What makes the effect worth taking seriously despite that is not its size but its structure: it says the number of particles present is not a property of the field alone but of the observer as well, and the same expression with a black hole's surface gravity in place of the acceleration is the Hawking temperature exactly, checked here to a part in 10¹².

The temperature of an acceleration

Empty space is empty for an observer who is not accelerating. For one who is, the same state of the same field is a thermal bath at a temperature proportional to the acceleration — and the constant of proportionality is 4 × 10⁻²¹ kelvin for every metre per second squared, which is why nobody has felt it. What the effect changes is not what can be measured but what a particle is.

 Where to put the far clock's zero. Two clocks three light-seconds apart, synchronised by radar: a pulse leaves the near clock at 0, bounces off the far one, and returns at 6 seconds. The far clock must be set to some time between those, and every choice is drawn. Einstein's convention puts it at 3 — halfway — and gives the same speed of light in both directions. Any other value is equally consistent with every measurement that can be made, because everything measurable involves a round trip and the round trip takes 6 seconds under every one of them: computed here across the five conventions, the round-trip times differ by 0e+0 seconds. The lines are the resulting surfaces of simultaneity, which fan out from the halfway choice. What each choice fixes is the one-way speed of light — ε = 0.25 makes it 2.00c outward and 0.67c back, ε = 0.4 makes it 1.25c outward and 0.83c back, ε = 0.5 makes it 1.00c outward and 1.00c back, ε = 0.6 makes it 0.83c outward and 1.25c back, ε = 0.75 makes it 0.67c outward and 2.00c back — and no experiment distinguishes them, because measuring a one-way speed requires two synchronised clocks and synchronising them requires the answer.

The speed that cannot be measured one way

Every measurement of the speed of light ever made has sent it out and brought it back. Measuring it one way needs two clocks that agree, and making two distant clocks agree needs a rule about when the far one should read what — which is a choice, not a discovery. The constancy of c is a fact about round trips; its isotropy is a convention, chosen because it makes the equations simple.

 The far end that has not been told yet. A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

Nothing is allowed to be rigid

A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.

 The fuel a starship needs, which is most of the universe. The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

The fuel a starship needs

Tsiolkovsky's logarithm survives relativity with one substitution: what adds is the rapidity rather than the velocity. The result is that a photon rocket reaches half light speed on a mass ratio of the square root of three, and a chemical one reaches a tenth of it on a mass ratio with three thousand digits.

 How fast something looks as it moves across the sky. The apparent transverse speed of a source, in units of the speed of light, against the angle between its motion and the line of sight, at β = 0.8, β = 0.95, β = 0.99. Every curve rises above one over a range of angles, reaching 1.33 at 36.8°, 3.04 at 18.4°, 7.02 at 8.1° — and those maxima are γβ at arccos β, found by searching the drawn curves rather than put into them. Nothing is moving faster than light. What has happened is that the source has come closer between the two observations, so the second flash had less far to travel and arrived sooner than it would have done; dividing the transverse distance by the interval between arrivals therefore gives too large a speed. Below β = 1/√2 no angle produces the illusion at all, so seeing it is a measurement: it puts a floor under the speed and a ceiling on the angle at once.

The motion that measures faster than light

Take two photographs of a jet a year apart, measure how far a blob moved across the sky, divide by a year, and the answer can be seven times the speed of light. Nothing has broken. The blob came closer between the two pictures, so the second flash had less far to travel and arrived early, and the interval between arrivals is not the interval between departures.

 The same source, seen coming and seen going. The observed flux of a moving source, relative to the same source at rest, against the angle between its motion and the line of sight, on a logarithmic scale, at β = 0.5, β = 0.9, β = 0.99. The curves span 8.1e+1, 1.3e+5, 1.6e+9 between the approaching and receding directions, which is ((1+β)/(1−β))⁴ exactly. The fourth power comes from the one quantity every observer agrees about: the specific intensity divided by the cube of the frequency. Three powers of the Doppler factor come from that invariance and the fourth from integrating over frequency. Two consequences are worth reading off. A source seen side-on is fainter than the same source at rest, by γ⁴ — a factor of 2.5e+3 at the fastest speed drawn. And half the light arrives inside a cone of 60.0°, 25.8°, 8.1°, which for a fast source is one over γ.

The brightness that is not the same for everyone

A moving source is not merely shifted in colour. Its light is concentrated forwards, so the same lamp is enormously brighter seen coming than seen going — by the fourth power of one number, which for a jet at ninety-nine per cent of light speed is a factor of a thousand million. The reason is that only one combination of intensity and frequency is the same for every observer, and everything else follows from it.

 A circumference that is more than 2π times the radius. The ratio of a rotating disc's measured circumference to 2π times its measured radius, against the speed of the rim, together with the rate of a clock carried on the rim. Rulers laid round the rim lie along their own direction of motion and are contracted; rulers laid along a radius lie across it and are not. So the circumference takes more of them than a stationary observer counts and the radius takes the same number, and the ratio is γ: 1.091 at β = 0.4, 1.400 at β = 0.7, 2.294 at β = 0.9. The geometry a rotating observer measures is therefore not Euclidean, and it is not Euclidean by an amount that depends on where on the disc the measurement is made. That is the observation Einstein said set him on the road to describing gravity with curved geometry: here is an accelerated frame, and here is a geometry in it that no choice of Cartesian coordinates can flatten. The rim's clock runs slow by the same factor, so a rotating frame has neither a common time nor a flat space.

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

 One factor, defined by an experiment rather than by a transformation. A observer A stays at x = 0 and flashes a light every 1 second by their own clock. B recedes at 0.6c. The flashes are the diagonal lines; where each meets B's worldline is where B receives it. B's clock reads a longer gap between arrivals than A's read between departures, by the factor k = 2.0000, and it is the same factor between every consecutive pair — measured here off the drawn meetings rather than assumed. That single number is the whole apparatus. Nobody has written down a coordinate transformation, chosen a convention for distant simultaneity, or drawn a tilted axis; the only thing used is that light travels on the diagonals and that neither observer is special, so B's flashes reach A stretched by the same k. From it: γ = (k + 1/k)/2 = 1.2500, and β = (k² − 1)/(k² + 1) = 0.6000.

Everything from an exchange of pulses

Send a flash every second and ask how often the far observer receives them. That one measured ratio generates time dilation, the composition of velocities and the twin result, with no coordinate transformation written down anywhere and no convention chosen about what "at the same time" means far away.

 The turnaround, made gentler and gentler, and the difference that does not move. A round trip to a star 4 light-years away at 0.6c, with the turnaround done at nine different accelerations from a tenth of a gravity to a thousand. The upper curve is the age difference between the twins and the lower one is how much of that difference the turnaround itself contributes. At 0.1 g the turn accounts for 51 per cent of it; at 1000 g it accounts for 0.00 per cent, and it keeps falling. The total does not follow it down: it tends to 2.67 years, which is what the instantaneous-turnaround cartoon gives. So the acceleration is not what makes the twins differ. It is what makes one twin's path the bent one, and a bent path through spacetime is shorter for the same reason a bent path on a map is longer — but the amount is in the legs, not in the corner, and the corner's contribution can be made as small as anyone likes without the difference going away.

The clock that does not feel the turn

Proper time is the integral of dt over gamma, which presumes that a clock's rate depends on its speed and on nothing else — not on its acceleration, not on how long it has been accelerating. That is an assumption about clocks rather than a theorem about spacetime, and the twin result is empty without it.

 Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500.

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

 The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting.

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

 Two things that change and one that does not. How a boost treats a piece of charged matter: the charge density rises by the Lorentz factor because the same charges occupy a contracted length, and the length falls by the same factor. At β = 0.6 the density is 1.250 times what it was and the length is 0.800 times, and their product is one to fourteen decimals across the whole range drawn. So the total charge is the same number in every frame, and it is the only quantity in the transformation that is. Charge density is the time component of a four-vector and transforms like an energy; charge itself is a scalar, and nothing about the observer changes it.

The one quantity a boost leaves alone

Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.

 Every speed there is, on one disc. The whole of velocity space drawn as a disc: the boundary is the speed of light and every possible velocity is a point inside. The rings are equal steps of rapidity — 0.5, 1, 1.5, 2, 2.5 — and they sit at speeds 0.4621, 0.7616, 0.9051, 0.9640, 0.9866 of light. Equal steps of rapidity crowd towards the edge, checked ring by ring, which is the same fact as speeds refusing to add: a boost is a fixed step in rapidity and a shrinking step in speed. The drawing is the Poincaré model, in which angles are true and distances are not — so a shape near the rim is drawn small and is not small, and the boundary is infinitely far away in the geometry although it is a finite circle on the page.

The space that speeds live in

Speeds do not add, and the reason is that the set of all possible velocities is not a flat space. It is a hyperbolic plane of curvature minus one in rapidity — and the rotation two boosts leave behind is exactly the area of the triangle they make in it.

 Every detour costs time. 4 routes between the same two events, 10 seconds apart in the frame drawn, each swinging out and back 1 time on the way. The proper time each carries is the integral of the square root of one minus the speed squared, computed by Simpson's rule along each curve: the straight route, 10.0000 s; wandering 1 light-seconds, 9.7485 s; wandering 2 light-seconds, 8.9245 s; wandering 3 light-seconds, 7.0935 s. The straight one carries the most, and every other one carries less — checked, on each drawn route. That is the opposite of what a length behaves like on paper, where the straight line is the shortest, and the whole difference is the minus sign in front of the space term.

The longest way round is the shortest clock

Of all the routes between two events, the one with no acceleration in it carries the most time on its own clock. That is the opposite of the Euclidean statement about straight lines, it comes entirely from one minus sign, and in a gravitational field it is why a thrown ball follows the path it does.

 The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

 The one number the boost leaves alone. The mass of a lambda into a proton and a pion, reconstructed from the two products' laboratory energies and momenta alone, against how fast the parent was moving — for five different rest-frame emission angles. Every curve is the same horizontal line. The lab energies vary by more than a factor of ten across this range and the angles between the products vary from almost 180° to a few degrees; the combination E² − p² of the pair does not vary at all, to 3.2e-15, which the figure requires before drawing anything. The light curves are the energies of the two products, on the same axis and to a different scale, drawn to show how much is moving while the invariant does not. This is the whole method of particle physics. A parent that lives for 10⁻²³ seconds is never detected; what is detected is two tracks, and their invariant mass is computed and histogrammed over millions of events. A parent that exists shows up as a peak at its own mass, at the same place whatever the beam energy, which is what makes the peak believable.

The cone a decay cannot leave

A particle at rest breaks into two and they go opposite ways. Set the parent moving and the whole pattern folds forward — into a cone with a hard edge, beyond which nothing is emitted at any rest-frame angle at all. The energy spectrum that comes out is exactly rectangular, and the one number the boost leaves alone is how the parent is identified at all.

 One boost, three constants, three pictures. The axes of a frame moving at 0.5 in units where the constant is one, drawn for the three signs the constant can have. The faint cross is the original frame's axes; the two heavy lines are the moving frame's, obtained by boosting them rather than by tilting them by hand. With a positive constant the two axes close in on one another symmetrically, and the line they are closing on is the invariant speed. With a zero constant only the time axis tilts and the space axis stays where it was, which is absolute simultaneity — every frame agrees which events are at the same time. With a negative constant the pair rotates rigidly, like a pair of axes turned in a plane. Nothing about light has been used to draw any of them. The three are the whole of what homogeneity, isotropy, the group property and the relativity principle permit, and choosing between them is a measurement rather than a postulate.

The transformation that never mentions light

Assume space and time are homogeneous, that space is isotropic, that two changes of frame compose into a third, and that the relativity principle holds. Those four leave exactly one free constant — and three possible worlds, one of them Galileo's and one of them Einstein's. Light appears nowhere in the derivation; it enters only when the constant has to be measured.

 Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Charge and current are one thing

The rung below asks what a boost leaves alone and answers charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

 Six numbers, one object. The electromagnetic field tensor written out as the four-by-four array it is, for a field with E = (0.4, 1.2, 0) and cB = (0, 0, 0.7), and again after a boost of rapidity 0.9 along the first axis. The array is antisymmetric — checked entry by entry — so of its sixteen slots only six are independent, and those six are the three components of E and the three of cB. The boost does not act on E and B separately; it acts on the array, mixing the top row into the lower block, which is what 'the electric field in one frame is partly magnetic in another' means written down. Its two scalars, E·B = 0.000 and E² − c²B² = 1.110, are unchanged, and they are the only two an antisymmetric rank-two tensor has.

Six numbers, one object

Three components of E and three of B mix into each other under a boost and never into anything else. Six numbers that transform among themselves are the independent entries of a four-by-four antisymmetric array, and writing them that way is not notation — it turns Maxwell's four equations into two, makes the two invariants the only two there could be, and shows that "electric" is a choice of axes rather than a kind of field.

 A map that keeps every cone and bends every worldline. Left: a grid of inertial worldlines, drawn solid, lines of simultaneity, faint, and light lines at 45°, dashed, in one space dimension. Right: the same grid after a map that stretches the light-cone coordinates u = t − x and v = t + x by two different increasing functions, u + 0.45·tanh(1.5u) and sinh(0.45v)/0.45. Light lines go to light lines, still at 45° to within a part in a billion, and the causal order of every one of 4000 sampled pairs of events is unchanged: whatever could influence what still can, and nothing new can. But the straight worldlines are bent — the one through x = 1 by 0.19 across the window — and the lines of simultaneity are no longer straight. In one space dimension the light cones cannot tell this picture from the inertial one.

What the light cones alone can decide

Keep nothing of spacetime but its light cones — which events could influence which — and ask how much geometry survives. With one dimension of space, almost none: any pair of increasing stretches of the two families of light lines preserves every cone and bends every straight worldline. With two or more, almost all of it: the only maps that keep every cone are Lorentz transformations, shifts and a uniform stretch, and nothing about straightness has to be assumed.

 All of flat spacetime in a diamond. The whole of flat spacetime with one space dimension, squeezed into a finite diamond by applying arctan separately to the two light-cone coordinates u = t − x and v = t + x. Solid curves are the worldlines of observers at rest at x = −3, −2, −1, 0, 1, 2, 3; faint curves are the instants t equal to the same values; the dashed lines are the two light rays through the origin, still at 45°, as every light line is — checked to a part in a billion — and the causal order of 4000 sampled pairs is unchanged. Infinity is not one place. Every worldline at rest runs from the bottom corner i⁻ to the top corner i⁺; every instant runs between the side corners i⁰; and light rays begin on the lower edges ℐ⁻ and end on the upper edges ℐ⁺. Each of these limits is checked at ten million units out.

The five places infinity turns out to be

Flat spacetime goes on for ever in every direction, and it can still be drawn whole on a page. Squeeze each family of light rays with a function that keeps their order and the infinite plane becomes a diamond with every light cone still at 45°. The price is distance, which the picture no longer shows. What it shows instead is that infinity is not one place: observers slower than light all end at a single point, instants end at another, and light ends along a whole edge of its own.

 The rectangle a spin tilts. The laboratory energy of the pion from a tau into a pion and a neutrino moving at 0.8 of the speed of light, for samples of 60,000 decays whose parents have αP = +1, 0, −1 along their line of flight. Every sample fills the same interval, 313 to 2667 MeV: the edges are set by the masses and the speed, and they do not move. Inside that interval an unpolarised sample is flat and a polarised one is tilted, its height at each energy 1 + αP times the cosine of the rest-frame angle that energy corresponds to. Reading the slope back from each sample gives +0.993, −0.010, −1.006, each with a standard error near 0.006. A parent spinning along its flight throws the pion forwards in its own frame, and the boost turns forwards into more energetic, so the tilt of an energy spectrum measures a polarisation without any angle being measured at all.

The slope a spin leaves in a spectrum

An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.

 Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

 What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

 How large now is, on this planet. By how much a synchronisation carried around a region of the rotating Earth fails to come back to itself, against the size of that region — from a metre to the whole planet, both axes logarithmic. The three horizontal lines are what three kinds of clock can resolve, and where each crosses the curve is where that clock can detect that 'now' is not a global notion: a good wristwatch at 785 thousand km, a quartz oscillator at 25 thousand km, a caesium clock at 785 km. Carried the whole way round the equator the defect is 207 nanoseconds, which is sixty metres of light travel and is the correction every satellite-navigation system applies. The effect is not small and not exotic; it is a routine engineering term, and the reason it was not an engineering term before 1955 is that nothing could measure it. A wristwatch's now is global out past the Moon; a caesium clock's reaches about the width of a large country.

How big now is

Three earlier arguments have established that a global now is a choice, that part of the choice is convention, and that for a rotating observer no consistent global choice exists at all. What survives is a size. Every observer has a local now, and how local is computable: on the rotating Earth it is 785 kilometres to the nanosecond, and a freely falling frame is inertial over the tolerance times the distance to the centre, divided by two.

 A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

 What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all.

The count that no observer can disagree about

A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

 The bath does push back, by an unmeasurable amount. The retarding force on a perfectly absorbing body moving through isotropic radiation at 2.725 kelvin, against its speed, for three areas. Moving through a bath of radiation is not free: the radiation arriving from ahead is blue-shifted and more intense and the radiation from behind is red-shifted and weaker, so the body absorbs more momentum from the front than from the back and decelerates. A square metre at half the speed of light feels 3.7e-14 newtons. That is the reason a preferred frame exists without relativity being violated: the laws are the same in every frame and the radiation is not — it is a physical system with a state, and its state picks out the frame in which it is isotropic, exactly as a body of water does.

The bath that pushes back

Moving through a bath of radiation is not free. The light arriving from ahead is blue-shifted and more intense and the light from behind is weaker, so a body absorbs more momentum from the front than from the back and slows down. That drag picks out the frame in which the radiation is isotropic — without violating relativity, because the laws are the same in every frame and the radiation is not.

 The paths in space: orbits of one period, and a throw straight up. The same free falls drawn in space around the Earth, which is the filled disc. All start at the marked point 2 Earth radii from the centre. The circle is the circular orbit. The ellipses, of eccentricity 0.2 and 0.4, have the same period, so they come back to the start at the same moment. The straight line is the thrown clock's path: straight up to 4.46 Earth radii and back down the same line, arriving as the orbits complete one revolution. The Earth's rotation is ignored and it is treated as a point mass for the paths that pass close to it.

The orbit that ages less than a throw

A clock in orbit and a clock thrown straight up leave the same point at the same moment and meet there again one period later. Both fall freely the whole way, so both follow paths of stationary proper time — and the thrown clock comes back 4.1 microseconds older. Even a clock held still by a rocket, which is not falling at all, beats the orbit. Free fall picks out a path that is stationary, not one that is longest.

 Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity.

The column that is hotter at the bottom

Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

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