Illustrated Physics
physics, drawn
Concept

Reference frame — where it appears

A system of coordinates from which positions and times are assigned, together with the state of motion it is attached to. Quantities that change between frames are components and those that do not are invariants, and confusing the two produces most of the apparent paradoxes in relativity.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

relativity · Simultaneity Six observers, six energies, one loss. The kinetic energy of the same collision before and after it, as measured by observers moving at 6 different speeds. No two of them agree about how much energy there was: the totals here range from 5.50 to 25.50 in the same units. Every one of them agrees about how much was lost — the gap between the two curves is 3.413 for all of them, varying by 6.2e-15. Energy is a quantity an observer owns; a change in it is not, and that is why heat, deformation and sound can be counted at all.

The energy that depends on the observer

A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.

mechanics · Energy 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.

relativity · Field transformation 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 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 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.

relativity · Simultaneity The height a clock can see. The fractional difference in rate between two clocks against how far apart in height they are, on logarithmic axes — a straight line of slope one, since the shift is gh/c² and comes to 1.09e-16 per metre near the ground. a caesium fountain, good to 1e-16, resolves 91.6 cm; an optical lattice clock, good to 1e-18, resolves 0.9 cm; the best clocks built, good to 8e-19, resolves 0.7 cm. The caesium fountains that define the second reach about a metre. The optical clocks that will replace them reach a centimetre, and the best of them a few millimetres. That is the whole of why this has stopped being a test of relativity and become a way of measuring the ground. A shift once so small that it took a Mössbauer experiment in a tower to see at all is now large enough to be a nuisance: two clocks in the same building disagree, and the disagreement has to be corrected for before either can be used to keep time.

The clock that measures a height

A clock a metre higher runs faster by a part in ten thousand million million million. That was once so small it took a tower and a Mössbauer source to see; the best clocks now resolve a centimetre of height, at any distance, without a line of sight. What began as a test of general relativity has become a surveying instrument that measures the quantity surveying actually wants.

astrophysics · Gravitational redshift The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.

The deflection that closes on itself

The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

mechanics · Circular motion 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

Ask what a boost leaves alone and the answer is 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.

relativity · Field transformation 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.

relativity · Field transformation Range as a map of launch velocities. The plane of launch velocities — horizontal component across, vertical up — with the curves of equal range drawn on it. Launched and landing at one height, the range is 2vₓvᵧ/g, so every curve of equal range is a hyperbola vₓvᵧ = constant, drawn here at ranges of 0.25, 0.50, 0.75, 1.00 v²/g. A thrower who can produce one speed in any direction can reach any point on the half-circle of radius v, and the best throw is where that circle touches the highest hyperbola it meets — at 45°, where the hyperbola vₓvᵧ = ½ is tangent to it, because a circle centred on the origin is symmetric about the diagonal and so is the hyperbola. The famous angle is a property of the shape of the set of throws.

The best throw is a tangency

Shot putters release at about 37°, long jumpers take off at about 20°, a ball thrown forward from a moving truck should be aimed steeply and flies flat, and a golf ball's drag alone moves its best angle to 38°. Each is usually explained as an exception to 45°. None of them is. Drawn as a map over launch velocities, range has curves of equal value, a thrower has a set of throws they can make, and the best throw is always where the set first touches a curve.

mechanics · Projectile 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.

relativity · Simultaneity 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.

relativity · Relativistic thermodynamics 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.

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

The balloon that leans the wrong way

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

fluids · Buoyancy Equal masses, one at rest, struck head on at four energies. A particle striking an identical particle at rest and scattering elastically, with the two leaving at equal angles either side of the incoming line, at four incident kinetic energies measured in units of the particle's rest energy. At 0.001 mc² the two tracks open at 90.0°. At 0.5 mc² the two tracks open at 83.6°. At 2 mc² the two tracks open at 70.5°. At 10 mc² the two tracks open at 44.4°. Slowly, the angle between them is a right angle, as every billiards player knows. Fast, it closes, because the incoming particle's energy is part of its inertia and the pair is carried forward. A proton at 2 mc² — about 1.9 GeV — and an electron at 1 MeV are both well inside the narrowing.

The right angle a fast collision closes

When one billiard ball strikes an identical one at rest, the two always leave at right angles — a fact every snooker player uses without knowing its proof. The proof is Newton's, and it fails for fast particles. Two protons leaving a collision at 1.9 GeV open at 70 degrees, not 90, and the narrowing follows one law, tan θ₁ tan θ₂ = 2/(γ + 1), in which the right angle is the slow limit. The circle that forced the right angle has been squashed into an ellipse by the energy the incoming particle carries as inertia.

relativity · Relativistic dynamics The drift is the area under the height. Height against time for the two balls: the throw (dashed curve) rising to 100 m in 4.52 s and falling back in as long again, and the drop (solid) falling from 100 m. For vertical motion the east–west velocity is always −2Ω cos λ times the height above the starting point, so the east–west drift is −2Ω cos λ times the area between the path and the starting height. The throw's area (lower shading) is (4/3)·h·t = 602 m s, above its start, so it drifts west; the drop's (upper shading) is h·t/3 = 151 m s, below its start, so it drifts east. The ratio of the two areas is four.

The ball thrown up that lands to the west

Drop a ball from a tower a hundred metres high and, because the Earth turns, it lands a centimetre and a half east of the point directly below. Throw a ball straight up so that it just reaches the same height, and it lands six centimetres west of where it was thrown — four times as far, the other way. The asymmetry has a one-line explanation that turns a problem in rotating frames into a problem about areas: for anything moving straight up or down, the sideways drift is proportional to the area under its graph of height against time. A dropped ball spends its flight below where it started, a thrown one above, and the thrown one's area is four times as large.

mechanics · Projectile 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

Named alongside it

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

InvarianceEquivalence principleRotating frameSimultaneityLight coneProper timeRapidityClock synchronisationField transformationThe Lorentz transformationMeasurementRelativity

All concepts