Illustrated Physics
physics, drawn
Concept

Relativity — where it appears

The requirement that the laws be the same for every observer, together with what follows once the speed of light is one of them. Nothing in it is about light in particular: c is the invariant speed of the geometry, and light travels at it because a photon has no mass.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside 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.

relativity · Field transformation 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.

relativity · Velocity addition 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 How long an orbit has before the waves take it. The time a circular orbit has left before gravitational radiation brings it together, against its separation, for four pairs of masses. Both axes are logarithmic and every curve has the same measured slope, 4.00: the lifetime goes as the fourth power of the separation, so halving an orbit shortens its remaining life by a factor of sixteen. The Earth's orbit is 13 decades above the age of the universe and the neutron-star binary is below it, which is the whole difference between a system that is losing energy and a system that is going to merge.

The orbit that has to shrink

Two masses in orbit radiate gravitational waves and lose energy, so the orbit tightens, so they go faster and radiate harder. The runaway takes 10²³ years for the Earth and the Sun and eight minutes for the last thousand kilometres of a black-hole pair — and the same one-line formula gives both.

astrophysics · Gravitational waves A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

electromagnetism · Field energy 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 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.

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

Named alongside it

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

InvarianceReference frameBlackbodyCosmic microwave backgroundDissipationElectric fieldEquilibriumField transformationMagnetismMomentumReference framesThermodynamics

All concepts