Illustrated Physics
physics, drawn
Concept

Time reversal — where it appears

The operation of reversing the sign of the time coordinate, under which the wave equation and Newton's laws are unchanged. Its invariance is what makes transmission through any passive medium the same in both directions, and the two things that break it — a flow and a magnetic field — are the only ways to build a one-way device.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

What the return journey does to each of them. The rotation of the plane after a beam has gone through a rotator, been reflected, and come back, against the length of the rotator — divided by the one-way rotation, so the two answers are 0 and 2 and nothing else can happen. A naturally active medium is handed with respect to the beam: reverse the beam and the sense of the rotation reverses with it, and the second pass undoes the first exactly, at every length and every wavelength. A Faraday rotator is handed with respect to the field, which does not care which way the light is going, so the second pass adds to the first and the round trip is twice the single one: 1 mm of it gives 21.7° out and 43.4° back; 2 mm of it gives 43.4° out and 86.8° back; 4 mm of it gives 86.8° out and 173.6° back; 8 mm of it gives 173.6° out and 347.2° back. That is a violation of reciprocity, and it is only available because a magnetic field is odd under time reversal. Everything a passive optical component can do — a lens, a mirror, a waveplate, a piece of quartz — looks the same run backwards, and none of them can be made into a one-way street. This can.

The rotation a return trip doubles

Quartz turns the plane of polarisation and so does glass in a magnetic field. The two look identical on the way through and are opposites on the way back — the crystal undoes its own rotation exactly, and the magnet adds to it. That difference is the whole of why a one-way street for light can be built at all, and why nothing passive will ever be one.

optics · Polarisation Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them.

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

waves · Wave motion Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them.

The gradient that drives the other thing

A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

thermodynamics · Diffusion Two solutions, and nothing in the equations to choose between them. A spherical pulse leaving a point and a spherical pulse arriving at one, each drawn at three times 0.35, 0.6, 0.85 in units where the speed is one. Both are exact solutions of the same wave equation, which is checked here by differencing the drawn samples twice in space and twice in time and requiring the residual to vanish for each. The one on the left is what is always used; the one on the right is discarded, and the equations do not do the discarding. The incoming pulse grows as it converges for the same reason the outgoing one decays as it spreads — the same energy through a smaller sphere — and it is as consistent with conservation as its mirror image is.

The solution that is thrown away

Maxwell's equations admit a field that converges on a charge exactly as readily as one that leaves it, and nothing in them prefers either. Retardation is a boundary condition rather than a law. Which boundary condition is right has been argued about for a century, one of the answers makes the arrow of time a property of there being absorbers, and the laboratory version of the question — whether an atom emits at all — has a measured answer that depends on what is listening.

electromagnetism · Retardation Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

optics · Refraction

Named alongside it

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

ReciprocityBoundary conditionIrreversibilityRefractive indexAdvanced solutionAntennaArrow of timeCausalityChiralityCircular birefringenceDetailed balanceDetector

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