Illustrated Physics
physics, drawn
Concept

Proper time — where it appears

The time a clock records along its own worldline, equal to the length of that line in the geometry the interval defines. Two clocks taking different routes between the same events accumulate different amounts of it, which is not a paradox but the ordinary fact that two roads have different lengths.

Named by 26 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

relativity · Spacetime diagram 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.

relativity · Time dilation 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.

relativity · Mass-energy Where a clock gains, and where it loses. The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.

The clock that runs slow lower down

Two identical clocks, one on the floor and one on a shelf, do not keep the same time — and the difference is large enough that a satellite navigation system which ignored it would be useless within a morning. The derivation needs nothing but a photon and a conservation law.

astrophysics · Gravitational redshift How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical.

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

astrophysics · Horizons What the distant observer actually receives. The frequency of a signal from a clock falling into a horizon, as received far away, against the receiver's own time. It is a straight line on a logarithmic axis, which means the fading is exponential: the e-folding time fitted to the drawn curve is 2.01 rs/c, which for a 10-solar-mass hole is 198 microseconds. Nothing hovers. The image reddens, the photons arrive at an exponentially falling rate, and within a millisecond there is nothing left to see.

Two clocks that disagree about the fall

A clock falling into a horizon crosses it in a few milliseconds by its own reckoning and never crosses it at all by a distant one. Both accounts are right, and the thing everybody remembers about the second — that the image hangs there for ever — is wrong.

astrophysics · Horizons 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.

relativity · Time dilation 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.

relativity · Spacetime diagram 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.

relativity · Time dilation The bigger the hole, the gentler the horizon. The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present.

The horizon that nothing marks

A falling body is torn apart by the difference in gravity between its ends. At the horizon of a black hole that difference goes as the inverse square of the mass, so a large enough hole can be entered intact — and nothing local happens at the crossing to say it has occurred.

astrophysics · Horizons 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.

relativity · Length contraction The figure a static field is not allowed to close. A spacetime diagram of two clocks held at fixed heights 22.5 metres apart, drawn as though spacetime were flat: time upward, height to the right, light at forty-five degrees. The lower clock sends two pulses; the upper clock receives them. Because the field does not change with time, nothing about the second pulse's journey differs from the first's, so the two null lines are congruent and the four worldlines bound a parallelogram. Opposite sides of a parallelogram in flat spacetime have equal length, so the proper time between emissions must equal the proper time between receptions, and the two clocks must agree. They do not: the measured fractional difference across a tower this tall is 2.455e-15, which Pound and Rebka established in 1960 and Pound and Snider confirmed to one per cent in 1964. Every step above is either a definition, an assumption of staticity, or a theorem of flat geometry — so the measurement refutes the flatness. No field equation has been written down, and none is needed: a laboratory result twenty-two metres tall is already incompatible with a flat spacetime.

The parallelogram that will not close

Two clocks twenty-two metres apart in a lift shaft run at different rates, by two parts in a thousand million million. That measurement, on its own, is enough to prove that spacetime cannot be flat — and the proof needs no field equation, no curvature tensor and no astronomy. It needs one drawing and the fact that opposite sides of a parallelogram are the same length.

astrophysics · Gravitational redshift 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.

relativity · Simultaneity 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.

relativity · Accelerated frames 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.

relativity · Accelerated frames 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.

relativity · Time dilation 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.

relativity · Doppler 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.

relativity · Accelerated frames 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.

relativity · Time dilation 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.

relativity · Time dilation A relativistic mass on a spring, at four amplitudes, scaled to one size. The displacement of a mass on an ideal spring against time, over two periods, with the mass's inertia growing with its energy as relativity requires, at amplitudes of 0.2, 1, 3, 10 times c/ω₀ — the distance light travels in one radian of the slow oscillation. Each curve is divided by its own amplitude and its own period so that shape alone is compared. The small swing is the familiar sine. The large one is a triangle: the mass spends almost the whole cycle moving at nearly the speed of light, turning round abruptly at each end. Periods: 1.007, 1.175, 2.132, 6.431 times the slow period.

The spring that becomes a light-clock

A mass on an ideal spring keeps the same period however far it swings — that is what makes it a clock. Give the mass the inertia relativity demands, pull it back far enough, and the clock breaks. The speed can never pass c, so a large swing spends almost all its time moving at nearly the speed of light, turning round abruptly at each end: the sine becomes a triangle, the period grows until it is simply the time light takes to cross the swing four times, and a clock riding on the mass ages a small fraction of what one beside it does.

relativity · Relativistic dynamics Two straight worldlines that meet twice. A spacetime diagram of a universe whose space is a circle 10 light-years round, drawn unrolled: the shaded strip is the whole of space, and the strips beside it are the same space again, so the dashed lines are the same stay-at-home twin A seen in the copies. Twin B leaves A at 0.6c and never changes speed or direction. After 16.67 years by A's clock B has gone once round and meets A again — the line reaches A's copy at 10 light-years. Neither twin has accelerated at any moment; both worldlines are straight. A has aged 16.67 years and B 13.33, less by the factor 1/γ = 0.800. The usual resolution of the twin paradox, that the travelling twin is the one who turned round, is not available: here nobody turned.

The twin who never turns round

The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.

relativity · Time dilation The travelling twin's surfaces of now. Surfaces of equal radar time for a twin who leaves at 0.6c and turns round after 4 years of their own time, every half-year of the twin's time from 0.5 to 7.5, drawn in the home frame; each surface is every event the twin would time, by echo, at that moment. On the outbound leg, far from the turnaround's influence, the surfaces are the tilted lines of the outbound rest frame; on the way back, those of the return frame. In between they bend, with a corner where they cross the light rays from the turnaround, and they sweep up the home worldline smoothly: the home clock read at radar times 3.5, 4 and 4.5 is 4.00, 5.00 and 6.00. No two surfaces cross, so no event is given two times, and none is skipped.

The now a travelling twin can measure

The standard account of the twin paradox has the traveller's sense of “now on Earth” lurch forward by years at the moment of turning round. That lurch belongs to a recipe — borrow the simultaneity of whatever inertial frame the traveller happens to be in — and the recipe is not something the traveller can measure. Timing the home clock by echo, as a radar does, gives a now that the traveller can measure, that never jumps and never crosses itself, and that during the turnaround coincides, surprisingly, with the stay-at-home's own.

relativity · Simultaneity Thrown across a uniform field, a fast charge does not draw a parabola. The paths of a charge launched across a uniform electric field (which points up the page) at 0.30, 0.60, 0.90, 0.99 of the speed of light, from the exact relativistic motion (solid), beside the parabolas Newton's law gives for the same launches (dashed); lengths are in units of mc²/qE, the distance over which the field does work equal to the charge's rest energy. Near its launch each exact path is a parabola, but one with the inertia γ₀m in place of m, so it is deflected less than Newton says — 1.05, 1.25, 2.29, 7.09 times less for the four. Further out each turns steeply along the field: one unit across, the 0.3c charge is 11.6 units along where Newton says 5.6, because its sideways speed has fallen while the field kept pushing. Each exact path is a hyperbolic cosine. The field adds to the momentum along it while the sideways momentum stays fixed, and the charge's growing energy is a growing inertia that slows its sideways progress.

The throw a uniform field bends into a chain

Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock.

relativity · Relativistic dynamics Loops that swell as the electric field approaches cB. A charge starting from rest in crossed uniform fields — the electric field up the page, the magnetic field out of it — for E/cB = 0.3, 0.6, 0.85, integrated exactly; lengths in units of mc/qB. Each traces a chain of loops drifting to the right at E/B. Newton's law gives loops of height 2mE/qB² — 0.6, 1.2, 1.7 — whatever the speed. The exact loops are 0.66, 1.88, 6.13 high, larger by γ² of the drift speed, 1.10, 1.56, 3.60: in the frame moving at the drift the electric field vanishes and the magnetic field is weakened to B/γ, and the charge circles at the drift speed on a larger orbit than a weaker field would explain alone. As E approaches cB the drift approaches c and the loop grows without limit.

The orbit that opens when E reaches cB

A charge in crossed electric and magnetic fields drifts sideways at E/B, looping as it goes, and the textbook leaves it there. The textbook result has a speed in it that can be made larger than light, and something has to give. What gives is the loop. As the electric field approaches c times the magnetic one, the loops swell without limit, and at E = cB the charge stops looping and is carried off for ever. Above that ratio no frame can make the electric field vanish; one can make the magnetic field vanish instead, and in that frame the charge simply runs away. The transition from a bound orbit to a run-away is decided by a single quantity every observer agrees on, E² − c²B².

relativity · Field transformation A cloud thrown from one event into empty space. A spacetime diagram, time up, one space dimension across, light at 45° (the edges of the shaded cone). Particles leave one event at the bottom with 13 different speeds, equally spaced in rapidity from −2.10 to 2.10 — from 0.970c one way to the same the other — and coast in straight lines (thin). The curves are the hyperbolas on which every particle's own clock reads 1, 2, 3: the cloud's own instants. Measured along any one of them, the particles are evenly spaced — the cloud is uniform — and the spacing grows in proportion to the clock: it is expanding. Nothing here is curved and nothing is created; it is special relativity in flat spacetime, and every particle, the central one included, sees the same thing.

The universe one explosion makes in empty space

Throw particles out of a single event at every speed below light's and let them coast. Drawn on a spacetime diagram, nothing curves and nothing expands — it is special relativity in an empty box. Yet the particles, slicing their history by their own clocks, find themselves in a universe that is the same everywhere, expanding, obeying an exact Hubble law with no speed limit, and filling a space of constant negative curvature. Edward Milne built a cosmology on this in 1933, and it shows that 'space expanding' and 'galaxies moving' are two slicings of one set of worldlines.

relativity · Spacetime diagram

Named alongside it

The objects these essays reach for when they reach for this one.

Time dilationSimultaneityThe twin paradoxEquivalence principleInvariant intervalThe Lorentz factorGravitational redshiftLight coneReference framesWorldlineCoordinate timeEvent horizon

All concepts