Illustrated Physics
physics, drawn
Concept

Velocity-addition — where it appears

The rule by which speeds combine between frames, which returns less than the sum and never exceeds the speed of light. Written in rapidity instead of speed it becomes simple addition, which is why a constant push accumulates rapidity at a steady rate while the speed saturates.

Named by 16 essays across 2 fields — each of them below, with the objects they name alongside it.

Composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.

Speeds that refuse to add, and the quantity that does

Run at half the speed of light, throw something forward at half the speed of light, and the result is not the speed of light. It is four-fifths of it, and there is a variable in which the arithmetic is still simple addition.

relativity · Velocity-addition Where the light that was sideways ends up. The direction a photon is seen to travel in the laboratory, against the direction it was emitted in the frame of the source, for a source moving at 0.5c, 0.9c, 0.99c. The straight diagonal is what would happen if a boost only changed frequencies; every curve lies well below it, which is aberration. The number that matters is where the emitted right angle lands, because half of everything emitted is on that side of it: 60.0° at 0.5c, 25.8° at 0.9c, 8.1° at 0.99c. The usual shorthand for that angle is 1/γ, which gives 49.6°, 25.0°, 8.1° — good to a few per cent only once the source is genuinely relativistic, and wrong by 17% at 0.5c. Nothing is emitted differently in any of these cases: the source is radiating exactly as it always did, and it is the map from its angles to ours that has changed.

The sky that crowds into a cone

A boost does not only shift frequencies. It remaps directions, so half of everything a fast traveller can see is squeezed into a forward cone of half-angle about 1/γ — and because brightness carries four powers of the Doppler factor, what lies ahead is overwhelming and what lies behind has effectively gone.

relativity · Doppler 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 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.

relativity · Relativistic dynamics 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 Every speed there is, on one disc. The whole of velocity space drawn as a disc: the boundary is the speed of light and every possible velocity is a point inside. The rings are equal steps of rapidity — 0.5, 1, 1.5, 2, 2.5 — and they sit at speeds 0.4621, 0.7616, 0.9051, 0.9640, 0.9866 of light. Equal steps of rapidity crowd towards the edge, checked ring by ring, which is the same fact as speeds refusing to add: a boost is a fixed step in rapidity and a shrinking step in speed. The drawing is the Poincaré model, in which angles are true and distances are not — so a shape near the rim is drawn small and is not small, and the boundary is infinitely far away in the geometry although it is a finite circle on the page.

The space that speeds live in

Speeds do not add, and the reason is that the set of all possible velocities is not a flat space. It is a hyperbolic plane of curvature minus one in rapidity — and the rotation two boosts leave behind is exactly the area of the triangle they make in it.

relativity · Velocity-addition The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

relativity · Velocity-addition One boost, three constants, three pictures. The axes of a frame moving at 0.5 in units where the constant is one, drawn for the three signs the constant can have. The faint cross is the original frame's axes; the two heavy lines are the moving frame's, obtained by boosting them rather than by tilting them by hand. With a positive constant the two axes close in on one another symmetrically, and the line they are closing on is the invariant speed. With a zero constant only the time axis tilts and the space axis stays where it was, which is absolute simultaneity — every frame agrees which events are at the same time. With a negative constant the pair rotates rigidly, like a pair of axes turned in a plane. Nothing about light has been used to draw any of them. The three are the whole of what homogeneity, isotropy, the group property and the relativity principle permit, and choosing between them is a measurement rather than a postulate.

The transformation that never mentions light

Assume space and time are homogeneous, that space is isotropic, that two changes of frame compose into a third, and that the relativity principle holds. Those four leave exactly one free constant — and three possible worlds, one of them Galileo's and one of them Einstein's. Light appears nowhere in the derivation; it enters only when the constant has to be measured.

relativity · Spacetime diagram 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 Sound cones that tip over. Time against position for sound in a fluid that speeds up from 0.55 to 1.3 times the speed of sound, drawn as the cone of places a sound made at each point can reach: its edges move at v − c and v + c. Upstream, where the flow is slow, the cones straddle the vertical and sound can go either way. At x = 0.20, where the flow reaches the speed of sound, the upstream edge stands vertical. Beyond it both edges point downstream: sound made there can no longer reach anywhere upstream of where it was made. That is a horizon, drawn in the same way as the light cones of a black hole, and it arises from nothing but the ordinary addition of a sound's speed to the speed of the water that carries it.

The river that sound cannot swim up

Sound travels at a fixed speed through the air or water carrying it, so a flowing medium adds its own speed to the sound's: faster downstream, slower upstream. That is the plainest velocity addition there is. Where the flow speeds up past the speed of sound, upstream-going sound is carried backwards, and nothing made beyond that point can be heard before it. The point is a horizon. Sound cones tip over there exactly as light cones do at a black hole, and the rate at which the flow speeds up plays the part of the surface gravity. A kitchen sink under a running tap makes the reverse kind of horizon, a white hole for ripples.

relativity · Velocity-addition The orbit an observer would reconstruct. The star's velocity along the line of sight, measured from the Doppler shift of its spectrum, as a fraction of its true orbital speed, plotted against the time the light arrives, in orbital periods, if light left the star at c plus the star's velocity. With light at c (dotted) the curve is the sine wave of a circular orbit. At 0.6 times the critical distance (solid) the same circular orbit arrives as a lopsided curve — a slow rise and a steep fall — that an astronomer would fit with a highly eccentric orbit that is not there. At twice the critical distance (dashed) the curve doubles back on itself: at some moments the spectrum would show two or three velocities at once, as if the star were in several places. Willem de Sitter pointed out in 1913 that spectroscopic binaries show nothing of the kind.

The binary star that would be seen twice

If light thrown forward by a moving source travelled faster, the way a ball thrown from a moving train does, then light from a star swinging towards the Earth in a binary orbit would catch up with light it had sent out earlier. From far enough away the two would arrive together, and the star would be seen in two places at once. Willem de Sitter pointed out in 1913 that nothing of the kind is seen, and that this settles whether the speed of light depends on its source. The argument had a loophole that took half a century to notice — interstellar gas re-emits light at its own speed — and closing it needed X-ray stars.

relativity · Velocity-addition In the rod's frame: a long rod threads through, front first. The same events seen from the rod's own frame, computed by transforming each point's motion from the hole's frame. Here the rod is its full length and the hole is the contracted one, 0.60 of it, sweeping left at 0.8c — a long rod over a short hole. But the moment each point starts to fall is not one moment in this frame: the front end starts at t′ = −1.07, the rear end at −0.27, and each point then falls at γu = 0.50c. While the start of the fall runs back along it the rod is bent, level behind and sloping in front; once every point is falling it is straight again and tilted, front down, by 21.8°. It threads through the hole front first as the hole slides under it, each part dropping below the plate just before the plate's far edge reaches it. It passes through in both frames; only the order of events along it differs.

The rod that is level for only one observer

A metre rod slides at high speed over a metre-wide hole in a thin plate. In the plate's frame the rod is contracted and drops through. In the rod's frame the hole is contracted, the rod is far too long for it, and it drops through anyway — because the instruction 'every point starts to fall now' names a different now for the rod. Its front end starts first, the start runs back along it, and the rod tips into the hole like a spear. Whether a moving thing is level is a question about which events are simultaneous, and so it has an answer only for an observer.

relativity · Simultaneity Where the spot is, and when, for three sweep speeds. A spacetime diagram of the spot a turning beam makes on a flat screen at distance D: position along the screen across, the time at which light arrives there up, both in units of D and D/c, for beams sweeping the screen's nearest point at ωD = 0.5c, 1.0c, 3.0c. Each curve is the arrival time t(x) = atan(x/D)/ω + √(D² + x²)/c. Every curve has a lowest point on the approaching side — at x = −1.25, −0.79, −0.32 D respectively — because far out the beam strikes the screen obliquely and its sweep along the screen outruns light whatever ω is. Light reaches that point first: the spot is born there, as a pair, one running back up the screen and one forward. The faster the sweep, the nearer the centre the birth. The faint dashed lines are light signals, slope one. Wherever a spot's track is flatter than they are, it is a sequence of events no signal could connect.

The spot that outruns light and carries nothing

Flick a laser pointer across the face of the Moon and the spot it makes crosses the Moon faster than light. Nothing is wrong with relativity: the spot is not a thing, and no part of it travels from one side to the other. Each lit point is lit by its own light, sent from the hand. What makes the spot worth studying is how strange such a non-thing is allowed to be. It can be born in the middle of the screen as a pair running apart; its speed can be made infinite, or reversed, merely by watching it from a moving frame; and the composition of velocities that caps everything real has nothing to say about it.

relativity · Velocity-addition How fast each point of a fast wheel moves. The speed through the ground of each point on the rim of a rolling wheel, in units of the speed of light, against the point's angle round the rim measured from the bottom, for axles moving at 0.5c, 0.8c, 0.95c. The dashed curves are the everyday answer, 2v sin(α/2), which puts the top at twice the axle's speed and would carry it past light. Composed relativistically, the bottom is still at rest and the top moves at 2v/(1 + v²/c²): 0.800c, 0.976c, 0.999c, always below c. Every point of the rim stays slower than light, and the fast part of the wheel is flattened into a broad plateau over the top.

The wheel whose top cannot go twice as fast

The top of a rolling wheel moves at twice the speed of its axle, and the bottom is momentarily at rest — two facts every cyclist's spokes show, blurred at the top and sharp at the bottom. Roll the wheel at a large fraction of the speed of light and the second fact survives exactly while the first cannot: the top moves at 2v/(1 + v²/c²), never as fast as light. Caught at one instant of the ground's time the spokes are bent, and over one turn the wheel advances not the circumference measured from its axle but γ times it — the rim's own length, laid down on the ground by a contact point that does not move.

relativity · Velocity-addition 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.

relativity · Relativistic dynamics One spray of particles, seen from two frames. Massless particles sprayed evenly in every direction from a source, as distributions in two variables. Left: against the cosine of the angle from the beam axis, for the source at rest (flat) and the same source moving along the axis at 0.8c (peaked forward), which crowds the particles forward until 81 times as many go straight ahead as straight back. Right: the same two sprays against pseudorapidity, η = −ln tan(θ/2). In η the moving spray is the resting one moved over by 1.099, the rapidity of 0.8c, with its shape — ½ sech² η, a peak of 0.5 and a width of 1.76 at half height — unchanged. A boost along the beam that distorts every distribution in angle or velocity only translates a distribution in rapidity.

The angle a collider counts in logarithms

The two proton beams of the Large Hadron Collider meet head on with equal energies, so the collisions ought to happen at rest in the laboratory. They do not. Each collision is really between two fragments of protons carrying unknown, unequal shares of the momentum, and every one is moving along the beam at a different speed. Particle physicists therefore describe what comes out in a variable that a boost along the beam simply shifts: rapidity, the quantity in which speeds add. A detector cannot measure rapidity directly. It measures an angle, and converts it to a logarithm.

relativity · Velocity-addition

Named alongside it

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

The Lorentz factorThe Lorentz transformationRapidityLight coneRelativity principleCausalityReference framesSimultaneitySpacetime diagramDoppler effectLength contractionMeasurement

All concepts