Illustrated Physics
physics, drawn
Series

Relativistic dynamics — the series

18 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · relativity
  2. 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.

    part 2 · relativity
  3. 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.

    part 3 · relativity
  4. 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.

    part 4 · relativity
  5. 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.

    part 5 · relativity
  6. 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.

    part 6 · relativity
  7. 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.

    part 7 · relativity
  8. 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.

    part 8 · relativity
  9. 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.

    part 9 · relativity
  10. The orbits of an electron in a travelling wave. The possible motions of an electron in a plasma wave whose phase moves at γₚ = 10 and whose field is 0.2 of the cold wave-breaking field, drawn as its energy against its position in the wave over two wavelengths — the level curves of H = γ − βₚ u − ε cos ψ, which the electron's motion conserves. Inside the separatrix (thick) an electron is trapped: it slides back and forth through the wave, gaining energy while it outruns the wave and losing it while the wave outruns it. The separatrix reaches γ = 99 at its top, near 4γₚ²ε = 80, starting from electrons moving at the wave's own speed. Outside it, electrons that are too slow are overtaken by the wave and those that are fast enough outrun it; neither gains energy on average.

    The wave an electron rides to a gigaelectronvolt

    A laser pulse fired into a thin plasma pushes its electrons aside and leaves them oscillating behind it, a wave of charge travelling at nearly the speed of light. The field in that wave is a thousand times stronger than any metal accelerator can hold, and an electron caught in it at the right moment is carried along like a surfer, gaining a gigaelectronvolt in a few centimetres. Then it outruns the wave. The distance it can ride, and the energy it can gain, are both set by how nearly the wave keeps pace with it — which is to say, by relativity.

    part 10 · relativity
  11. The neutrino's energy against the pion's, at four angles. The energy of the neutrino from a pion's decay against the pion's energy, for neutrinos going exactly forward and at 1°, 2.5° and 4° to the pion's line of flight. Straight ahead the neutrino takes a fixed share, 0.427, of the pion's energy, so pions of every energy give neutrinos of every energy. Off axis the curve rises, flattens and falls: at 2.5° it peaks at 0.683 GeV for a pion of 3.2 GeV, and every pion between 2 and 10 GeV gives a neutrino between 0.61 and 0.68 GeV. The flat top is where γθ = 1. Its height, p*/θ, depends on the angle alone — the rest-frame momentum of the decay, 29.8 MeV/c, divided by the angle in radians.

    The neutrino energy an angle fixes

    A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.

    part 11 · relativity
  12. The speed that stops while the energy keeps coming. The speed of an electron, as a fraction of the speed of light, against the kinetic energy given to it, from special relativity (solid) and from Newton's ½mv² (dashed); dots at the five energies Bertozzi used in 1964. Newton's speed passes the speed of light at a quarter of a megaelectronvolt and is 7.7 times it at 15 MeV. Relativity's speed is 0.863 c at 0.5 MeV, 0.967 c at 1.5 MeV, 0.9948 c at 4.5 MeV and 0.99946 c at 15 MeV. Between 1.5 and 15 MeV the energy grows tenfold and the speed by three and a half per cent. The energy does not stop arriving; the speed stops answering it.

    The speed that stops while the energy keeps coming

    In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed.

    part 12 · relativity
  13. The share of its energy a particle can hand to one electron. The largest fraction of its kinetic energy a muon, a pion, a proton or an alpha particle can give a free electron in one collision, against its Lorentz factor γ, on logarithmic axes. A slow proton can give at most 0.22 per cent, 4m/M, which is why a proton slows down through tens of thousands of gentle collisions rather than a few hard ones. The share rises as γ grows and reaches a half at γ = 916 for a proton and 101 for a muon — near M/2m, where the electron's energy in the centre-of-momentum frame has grown comparable to the projectile's, so that the two meet as equals and a head-on reversal can hand the electron most of the energy.

    The kick a fast particle can give an electron

    A slow proton crossing matter can give any one electron at most a fifth of a per cent of its energy, because an electron struck head-on by something two thousand times heavier rebounds at no more than twice the heavier one's speed. That rule of thumb fails at relativistic speeds, and fails upward: the electron's speed is capped below light, but its energy is not, and the largest kick grows as the square of the projectile's Lorentz factor until, above a teraelectronvolt, one collision can take more than half of everything a proton carries.

    part 13 · relativity
  14. Pushed forward, and left at rest. The path of an electron, initially at rest, overtaken by a 6-cycle pulse of linearly polarised light travelling to the right, for peak field strengths a₀ = 0.5, 1, 2, where a₀ = 1 is the strength at which the electron's quiver becomes relativistic; lengths in wavelengths of the light, the vertical axis along the electric field. While the pulse passes, the electron quivers across the beam and is pushed along it. When the pulse has gone it is at rest again — not moving, with none of the light's energy — but displaced forward by 0.14, 0.56, 2.25 wavelengths respectively, a distance growing as a₀². The quiver's width grows only as a₀: at a₀ = 2 the electron is pushed forward several times further than it swings sideways.

    The light that moves an electron and pays it nothing

    A pulse of light strong enough to throw an electron forward at a large fraction of the speed of light passes over it, and the electron is left exactly as it was found — at rest, with none of the light's energy — except that it is somewhere else. The reason is two quantities the electron cannot change while a plane wave is passing, and the same two quantities fix the figure eight it traces inside the wave and the angle at which it leaves a real, focused beam carrying the energy it managed to keep.

    part 14 · relativity
  15. An electron that runs into light and loses its energy to it. The energy of an electron of 511 MeV (γ = 1000) as it passes head-on through a 10-cycle pulse of 0.8 µm light, as a fraction of its starting energy, against the number of laser cycles it has crossed, for normalised field strengths a₀ = 30 and 100: with radiation reaction (solid) and without (dashed). Without it, the electron is shaken sideways while in the light and leaves with exactly the energy it brought, as a plane wave requires. With it, each half-cycle near the pulse's peak radiates away a slice: the electron leaves with 76 per cent of its energy at a₀ = 30 and 22 per cent at a₀ = 100, the rest gone as gamma rays.

    The light that takes back an electron's energy

    An electron at rest in the most intense laser light ever made feels its own radiation as a force a millionth of the light's push — the textbook reason radiation reaction never matters. Send the electron into the light at nearly the speed of light and the arithmetic turns over. In its own frame the light is stronger by twice its Lorentz factor, the power it radiates grows as the square of that, and a pulse a few tens of femtoseconds long can strip it of most of its energy as gamma rays. That is where the classical theory of a radiating charge, argued over for a century, finally has to answer to experiment — and where it starts to fail for quantum reasons.

    part 15 · relativity
  16. 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.

    part 16 · relativity
  17. A longer road against a faster car. For a proton with 0.1% more momentum than the design proton in a ring 628 m round, the change in its time per turn, in picoseconds, against the Lorentz factor: from its longer orbit, which adds about 56 ps a turn at every energy; from its higher speed, which subtracts 605 ps at γ = 2 but only 5.2 ps at γ = 20, because near the speed of light more momentum buys almost no more speed; and the net, computed exactly. The speed wins below γₜ = 6.1 and the path wins above it. In special relativity a body can always be given more momentum, but its speed is capped, and that is the whole cause.

    The energy above which faster arrives later

    In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.

    part 17 · relativity
  18. The peak no pion spectrum can move. The photon spectrum from a population of neutral pions, summed from their boxes, for pion spectra falling as three different powers of the energy. Whatever the pion spectrum, each curve peaks at the same 67.5 MeV and is symmetric on the logarithmic axis, since every box contains that point and a box's height falls as it widens; checked here at the point and its mirror, the ratio is 1.000. This is the shape, known as the pion bump, that Floyd Stecker worked out in 1971. Gamma-ray telescopes found it in 2013 in two supernova remnants, which was taken as direct evidence that such remnants accelerate protons, because the electrons that produce most other gamma rays have no reason to make a peak here.

    The flat spectrum a fast pion must paint

    A neutral pion falls apart into two photons, each with half its rest energy, in a hundred-million-billionth of a second. Seen in flight, those photons could have any energy between two edges, and every energy in that range is equally likely, whatever the pion's speed. Add up a whole population of pions and something unexpected appears: a peak at exactly the 67.5 MeV that no individual decay shows. Astronomers took it as evidence of where cosmic rays are made.

    part 18 · relativity

All series