Speeds that refuse to add, and the quantity that does

A train moves at $u$. A ball is thrown forward inside it at $v$ relative to the train. Everyone knows what the ground observer measures, and everyone is wrong: it is not $u + v$.

$$w = \frac{u+v}{1+uv/c^2}.$$

At everyday speeds the correction is unmeasurable — for a ball thrown at 30 m/s on a train at 50 m/s, the denominator differs from one by two parts in $10^{16}$. At relativistic speeds it dominates, and it does so in a way that has one striking property: however large $u$ and $v$ are, as long as both are below $c$, the result is below $c$.

The postulate as a special case #

The most economical thing about the formula is what it does when one of the speeds is $c$.

Set $v = c$. The numerator is $u + c$ and the denominator is $1 + u/c$, which is $(c+u)/c$. Dividing gives exactly $c$, for every $u$ whatever.

So “light travels at $c$ for every observer” — the postulate that the whole theory was built on — is not an extra rule sitting alongside the composition law. It is one line of the composition law’s own arithmetic. The top curve in the figure is flat at $1$ for that reason: a light ray composed with any boost is a light ray.

That is worth noticing as a matter of theory construction. The postulate was assumed at the start, the transformation was derived from it, and the composition law was derived from the transformation — and the composition law then contains the postulate as a special case. The theory closes on itself, which is a good sign and is not automatic.

Why the correction is a denominator #

The formula can be derived by algebra from the Lorentz transformation, and the algebra explains nothing. The picture does.

A velocity is the slope of a worldline, and changing frames tilts both axes toward the light line. A slope measured against tilted axes is not the slope measured against the original ones, and the denominator in the composition law is exactly the correction for that tilt. Nothing is being added imperfectly; a ratio is being recomputed against different axes.

A speed is a slope on this diagram: so much distance per so much time. Composing two speeds means measuring a worldline’s slope in a frame whose own axes are tilted. If only the space axis tilted — which is the Galilean picture — slopes would add. Both axes tilt, the time axis toward the light line as well, and the extra tilt is exactly the denominator.

The denominator is therefore the relativity of simultaneity wearing different clothes. The ground observer and the train observer disagree about which events are simultaneous, so they disagree about what “the same moment” means when measuring how far the ball got — and that disagreement, not any effect on the ball, is what makes the speeds compose rather than add.

The two observers also slice time differently, and a speed is a distance divided by a time. So even after the tilt of the space axis has been accounted for, the time the distance is divided by is a different time — which is where the second half of the denominator comes from and why the correction involves the product $uv/c^2$ rather than either speed alone.

Two speeds that nearly reach it #

Reading numbers off the curves is worth doing once, because the shape of the approach to $c$ is not obvious from the formula.

Compose $0.5c$ with $0.5c$ and the Galilean answer is $c$ exactly; the correct answer is $0.8c$. The correction is a fifth, at speeds that are not extreme. Compose $0.9c$ with $0.9c$ and the Galilean answer is $1.8c$ while the correct one is $0.9945c$ — the composition has moved the result less than five per cent of the way from $0.9c$ to $c$, having been given a boost that in Galilean terms would have doubled it.

That is the characteristic behaviour: near $c$, adding speed stops buying speed. A body at $0.99c$ given another $0.99c$ arrives at $0.99995c$. Each composition closes half the remaining gap several times over and never crosses it, in the way a geometric sequence approaches its limit.

There is a symmetry worth noticing in those numbers. The gap to the speed of light multiplies rather than adds: composing two speeds whose gaps to $c$ are small multiplies the gaps together, up to a factor of two. That is the shadow of the exponential structure that rapidity makes explicit in the next section, and it is why the natural way to describe a beam is by how many factors of ten it sits below $c$ rather than by its speed.

The variable in which it is addition #

The composition law is associative and has an identity and inverses, which is a strong hint that it is ordinary addition in a disguised variable. It is.

Define the rapidity $\varphi$ by $v = c\tanh\varphi$. Then composing two velocities corresponds to adding their rapidities:

$$\varphi_{\text{total}} = \varphi_1 + \varphi_2,$$

exactly, with no correction term at all. The awkward formula is what the simple one looks like after being pushed through a hyperbolic tangent.

Two routes to one number is this site’s standing test, and the figure makes it rather than asserts it: the marked point is placed at $\tanh(\varphi_1+\varphi_2)$, and the caption’s cross-check is $(u+v)/(1+uv)$ evaluated independently. They agree to four decimals because they are the same statement.

Rapidity is more than a trick. The tanh flattens toward one, so equal increments of rapidity buy less and less speed — the first unit of rapidity takes a body to $0.76c$, the second to $0.96c$, the third to $0.995c$. A rocket under constant proper acceleration gains rapidity at a constant rate, which makes rapidity the natural measure of “how much accelerating has been done”, and it is why an accelerator’s engineers think in terms of energy rather than of speed: past a certain point the speed stops being a useful description of the beam.

It also makes the composition law’s structure obvious. Rapidities run from $-\infty$ to $+\infty$ and add like ordinary numbers; velocities are their tanh and are therefore confined to $(-c, c)$ by construction. No amount of adding finite rapidities produces an infinite one, so no sequence of boosts reaches the speed of light. The speed limit is not a barrier that something pushes against; it is the range of a function.

What the limit is not #

Two readings of the speed limit are common and both are wrong in ways the figure can settle.

It is not a statement that light is fast. The composition law caps every relative velocity at $c$ whether or not any light is present. The constant is a property of spacetime’s geometry, and light travels at it because photons are massless — the speed would exist with the same value in a universe with no electromagnetic field at all.

It is not a limit on every speed that can be defined. Two particles approaching a detector from opposite directions, each at $0.9c$ in the laboratory, close the gap between them at $1.8c$ as measured in the laboratory frame, and there is nothing wrong with that number. It is a rate of change of a distance in one frame, not the speed of anything relative to anything. The composition law applies to the speed of one particle in the other’s frame, which comes out at $0.9945c$. Distinguishing the two is most of what makes collider kinematics confusing on first contact.

The same distinction covers the various things that exceed $c$ without carrying information: the spot of a laser swept across the moon, the intersection point of two closing blades, the crests inside a wave packet in a dispersive medium, and the phase velocity of a radio wave in the ionosphere. None of them is an object with a worldline.

What the law does not tell an observer #

There is a gap between what the composition law computes and what a camera records, and it is worth closing.

The law gives the speed measured in a frame, meaning the speed obtained from readings of clocks and rulers laid out at rest in that frame and read locally. It does not give what a single observer at one place sees, because light from the distant parts of a moving object takes different times to arrive.

Adding that travel time produces genuinely different phenomena: a rapidly approaching object appears rotated rather than contracted, an effect discovered half a century after the theory it belongs to; a receding object’s clock appears slowed by more than $\gamma$, and an approaching one’s appears fast, which is the relativistic Doppler shift rather than time dilation. The moving clock’s slowing is what remains after the travel time is taken out.

The construction underneath all of it is a light pulse crossing a moving gap. The composition law, the Lorentz factor and the length contraction are three consequences of that one geometric fact, and none of them is an optical effect: nothing here depends on how anything looks, only on what a measurement of a distance and a time returns.

Where the correction actually shows up #

The denominator is unmeasurably small at everyday speeds, and it is not unmeasurably small everywhere. Three places where it is routine:

Particle beams. In a collider, two beams meeting head-on at $\gamma$ of several thousand have a laboratory closing rate of nearly $2c$ and a relative speed, in either beam’s frame, indistinguishable from $c$. The useful quantity is neither: it is the total energy available in the centre-of-mass frame, and computing it from the beam energies is the composition law expressed in energy and momentum rather than in speeds. The whole reason colliders replaced fixed-target machines is that this quantity grows linearly with beam energy in a collider and only as its square root in a fixed target.

At 0.99c the Lorentz factor is 7.1, which turns a muon’s 660 m decay length into 4.7 km — and the composition law is what guarantees that no chain of boosts, however long, moves a muon past the right-hand edge of that curve. Every accelerator stage adds a rapidity and none of them adds a speed, so the factor grows without limit while the speed does not.

Muons in the atmosphere. A muon produced at 15 km with a lifetime of 2.2 microseconds should decay within about 660 metres. Enormous numbers reach the ground. In the ground frame the explanation is time dilation; in the muon’s frame it is that the atmosphere is contracted to a few hundred metres. Both frames agree on what arrives, and reconciling their two accounts requires composing velocities correctly at every step.

Satellite navigation. The corrections a positioning system applies are dominated by gravitational and dilation terms rather than by the composition law, but the underlying calculation is a chain of frame changes — satellite to Earth-centred inertial to rotating Earth to receiver — and each one composes velocities. Doing them by simple addition puts the answer out by a distance that grows through the day.

The pattern in all three is that the formula matters wherever frames are chained, rather than wherever something is fast. A single fast object measured in a single frame needs no composition law at all.

Fizeau’s experiment, forty years early #

The formula was confirmed before it existed, which is one of the better accidents in the subject’s history.

Fizeau measured in 1851 how fast light travels in moving water, by splitting a beam, sending one half with the flow of a pipe and one against it, and recombining them to read the phase difference. The expected answer under a straightforward ether theory was that the water would drag the light along at some fraction of its own speed, and the measured fraction came out at $1 - 1/n^2$ — an odd number that Fresnel had predicted by an argument nobody found convincing, involving the ether being partially dragged.

The composition law gives it in one line. Composing the speed of light in water, $c/n$, with the water’s speed $u$:

$$w = \frac{c/n + u}{1 + u/(nc)} \approx \frac{c}{n} + u\left(1 - \frac{1}{n^2}\right)$$

to first order in $u/c$. The mysterious coefficient is what the denominator produces when expanded, and no dragging of anything is involved.

That is a strong form of evidence, because the number was measured decades before the theory that explains it and was not adjustable. Einstein described the Fizeau result as one of the experiments that most influenced him, and it is a better piece of support for the composition law than any thought experiment about trains.

What a faster signal would actually cost #

The composition law says no chain of boosts reaches $c$. It does not, by itself, forbid something that was never slower — and the reason such a thing is nonetheless ruled out is worth following, because it is a statement about the order of events rather than about speed.

Take two events far apart in space and close together in time, so close that no light could get from one to the other. They lie outside each other’s light cones. Boost to another frame and the tilt of the time axis changes which of them comes first: for spacelike-separated events, the order is not merely unknown but frame-dependent, and there is always a frame in which either one is earlier. Nothing is wrong with that, because neither event can have influenced the other.

Now suppose a signal could cross that gap. In the frame where it was sent it travels forward in time; in some other frame, reachable by a perfectly ordinary boost of less than $c$, the same signal arrives before it left. Two such signals, sent from two observers in relative motion, deliver a reply to a message that has not yet been sent — the arrangement usually called a tachyonic antitelephone, and it takes only the composition law and two subluminal frames to build.

So the light cone is not a fence around how fast things can go. It is the boundary between pairs of events whose order every observer agrees on and pairs whose order is a matter of frame, and the prohibition on superluminal signalling is what keeps causes on the agreed side of it.

The synchronisation the law quietly assumes #

One more thing is buried in the phrase “the speed measured in a frame”. A speed is a distance divided by a time, and if the distance is any real distance, the two times are read from two different clocks. Those clocks had to be synchronised, and synchronising distant clocks requires knowing how long a signal took to reach one from the other.

Which is the thing being measured. Einstein saw this and settled it by definition rather than by experiment: send a pulse from A to B and back, and declare that it arrived at the midpoint of the two readings at A. That is Einstein synchronisation, and it amounts to assuming the outward and return speeds are equal.

What experiment gives is the two-way speed — out and back, one clock, no synchronisation needed — and that has been measured to extraordinary precision and is invariant. The one-way speed is a different matter. Reichenbach’s observation is that a whole family of conventions, splitting the round trip unevenly, is consistent with every measurement ever made; they give an anisotropic one-way speed, awkward coordinates, and identical predictions for everything observable.

This is not a loose end in the theory so much as a statement of what a coordinate is. The composition law, the Lorentz factor and the simultaneity slices all live inside the standard convention, and they are exactly as physical as it is: the invariant content is the interval between events, and the frames are a way of labelling it.

What the figures cannot show #

Both figures draw speeds along one line, and the restriction is doing more work than it appears to.

When the two velocities are not parallel, the composition law is no longer a single fraction and, worse, it stops being commutative: composing a boost along $x$ with a boost along $y$ gives a different result from composing them in the other order. The difference is not merely a different speed — the two results differ by a rotation.

That rotation is real and is not an artefact of bookkeeping. A body carried round a closed loop of boosts comes back rotated, which is Thomas precession, and it appears in the spin of an electron orbiting a nucleus with a factor of exactly one half that was a genuine puzzle in atomic spectra until Thomas identified it in 1926. A one-dimensional figure cannot hint at it, because in one dimension boosts do commute and the effect is exactly zero.

The second omission is time. Both figures are relations between numbers, with no worldline in them; the composition of two boosts is presented as arithmetic rather than as a sequence of events. The spacetime diagram is where that gap is closed, and it is why the diagram remains the primary object of the subject and the formulas are consequences.

The third is that a curve of $w$ against $v$ says nothing about how a body got to $v$. Nothing on this page distinguishes a body that was always moving from one that accelerated, and the distinction is where the twin paradox lives — a question about proper time along a path, not about the speeds at its ends.

The Galilean law was never obvious either #

It is worth resisting the impression that the composition law replaced something self-evident with something contrived, because simple addition of velocities is itself a physical claim and was recognised as one.

Adding velocities assumes that the two observers agree about time and about distance — that a second on the train is a second on the ground and a metre is a metre. Under those assumptions the ball’s extra displacement per second simply adds to the train’s, and $w = u + v$ follows. The assumptions had never been tested at speeds where they could fail, and they are exactly what the light clock shows to be false.

So the two laws are not an intuitive one and a strange one. They are two consequences of two different assumptions about clocks, and each is a theorem given its premises. The composition formula reduces to the Galilean form whenever $uv \ll c^2$, which is the correct relationship between a general law and the special case that preceded it — the same relationship the relativistic energy has to $\tfrac12 mv^2$, and the same one every superseded law in physics has to its replacement when the replacement is right.

Where the ladder goes next #

The rungs from here: the Lorentz transformation derived rather than assumed; rapidity as the hyperbolic angle, and boosts as rotations through an imaginary angle; the composition of velocities that are not parallel, where the result depends on the order and leaves behind a rotation — Thomas precession — that has observable consequences in atomic spectra; the aberration of starlight, which is the composition law applied to a direction rather than a magnitude; four-velocity, which composes by ordinary matrix multiplication and removes the awkwardness entirely; and the relativistic Doppler effect, where the same tanh reappears as an exponential of rapidity.

The idea to carry forward is the change of variable. Velocities compose awkwardly and are bounded; rapidities add and are not. When a law looks ugly, it is often the right law written in the wrong variable — and finding the variable in which it is simple usually says something about what the law is really about.