The space that speeds live in

Speeds that refuse to add gives the composition law and the substitution that tames it: rapidity, the inverse hyperbolic tangent of the speed, which simply adds for boosts in the same direction. That is usually where the matter is left, as a convenient change of variable.

It is more than a convenience. Rapidity is a distance, in a space that is a genuine geometry with a genuine curvature, and once that is said several apparently separate facts about relativity turn out to be one fact.

A boundary that is not an edge #

The first thing the picture makes visible is that the disc has no edge.

In ordinary language, velocities occupy a ball of radius $c$, and $c$ is a limit that nothing reaches. That description makes the limit sound like a wall. In the geometry there is no wall: the distance from the centre to the boundary, measured in rapidity, is infinite, and the boundary is not part of the space at all.

That reframes the speed limit as something less arbitrary. A particle accelerating steadily is not approaching a barrier; it is travelling at a steady rate through a space that never ends. The apparent slowing of its speed towards $c$ is the same illusion as the rings crowding on the page.

The crowding is the composition law. A boost is a fixed displacement in rapidity, always, wherever it starts from. Applied at the centre it produces a large change in speed; applied near the rim it produces almost none. Nothing about the boost changed — the space did.

The rotation is a shape #

The turn that two pushes leave behind establishes that two non-collinear boosts compose to a boost and a rotation, which is the fact that makes the Lorentz group non-abelian and gives an electron in an atom its Thomas precession. It is usually derived by multiplying matrices, and the answer arrives as a formula with no obvious meaning.

In velocity space it has an obvious meaning. Draw the two boosts as sides of a triangle, and the composite as the third side. Add up the triangle’s three interior angles. In a flat plane they would total two right angles; here they total less, and the shortfall is the rotation.

That is Gauss’s theorem in a hyperbolic plane, arriving in relativity. On a surface of curvature $K$ the angle defect of a geodesic triangle is $-K$ times its area, and parallel transport of a direction around a closed loop rotates it by the same amount. The Wigner rotation is holonomy, and it is present for exactly the reason a vector carried around a closed path on a sphere comes back turned.

The drawing here is the Poincaré model, which is conformal — angles on the page are the angles in the geometry. So the three angles in the figure can be measured with a protractor and will come out where the caption says. What the model distorts is length, which is why the triangle looks lopsided when its sides are equal.

Why the two agree #

The identity is exact and it is worth being clear about what makes it so, because two quantities agreeing to a nanodegree across a whole range is the kind of thing that deserves a reason rather than a plot.

The Lorentz group’s boosts are not a subgroup — composing two of them leaves the set — and the amount by which they leave it is a rotation. The set of boosts, parameterised by velocity, is the quotient of the Lorentz group by the rotations, and that quotient carries a natural metric. Compute the metric and it is the hyperbolic plane; compute the failure of two boosts to compose to a boost and it is the holonomy of that metric.

So the two calculations in the figure are the same calculation in two languages, and their agreement is not evidence for anything — it is a check that both were done correctly. What the agreement buys is the interpretation: an effect that looked like an algebraic accident of matrix multiplication is the curvature of a space.

The vanishing at zero and at a straight angle is the same statement. Two parallel boosts give a degenerate triangle with no area, so no defect and no rotation, which is why collinear velocity addition is simple and why rapidity adds along a line.

The same triangle with unequal sides #

Redrawing with unequal boosts is worth the space for two reasons.

The first is that it checks the placement rather than the arithmetic. The vertices are put on the page by hyperbolic trigonometry, and the figure then measures the distance between two of them using the disc’s own metric and requires it to be the second boost’s rapidity. A drawing that merely illustrated a formula could not fail that test; this one could, and did while it was being written — placing the third vertex at the wrong one of the triangle’s two unknown angles gave a side eleven per cent short, and the check caught it.

The second is that it separates two things the symmetric case runs together. Making one boost longer does not necessarily increase the rotation. What increases the rotation is area, and a triangle with one long side and one short one at a shallow angle is thin. That is why a very fast particle given a small transverse kick picks up almost no rotation, and why Thomas precession depends on the product of the two rapidities and the sine of the angle rather than on the speed alone.

The rotation is a property of the pair, and it is an area. Any statement about it that mentions only one of the two boosts has thrown away half of what determines it.

Reading the curvature off #

The curvature can be measured from inside the space, without any embedding, by the classical method: draw a circle, measure its circumference, and compare with $2\pi r$.

In velocity space the circumference of a circle of rapidity radius $r$ is $2\pi\sinh r$, which exceeds $2\pi r$ and does so more and more quickly. At a rapidity of three — a speed of $0.995c$ — a circle has more than three times the circumference a flat plane would give it. There is more room out there than the page suggests.

The number that comes out of the small-radius expansion is exactly $-1$, and the figure computes it rather than quoting it. That is not a free choice. It is the statement that rapidity is the natural unit of length in this geometry, and it is what fixes the constant in the defect–area relation: the rotation equals the area, rather than being proportional to it with some factor of $c$ in the way.

A dimensionless curvature of exactly minus one is a strong statement about how tightly the pieces fit. Velocity space has one natural length, rapidity; the speed of light is the unit that makes rapidity dimensionless; and everything else follows.

The distance between two speeds #

There is a practical quantity the geometry supplies that the ordinary language has no word for: how far apart two velocities are.

Asking how different $0.99c$ and $0.999c$ are has no good answer in speeds — the difference is $0.009c$, which is also the difference between $0.001c$ and $0.010c$, and nobody would call those pairs equally different. In rapidity the first pair is $0.83$ apart and the second is $0.009$, a factor of ninety, and that ratio is the one that matters for everything a physicist does with the pair.

It is the ratio that decides how much energy it takes to get from one to the other, since a steady proper acceleration covers rapidity at a steady rate. It is the ratio that decides how different the two look in a collision, since the invariant mass of a pair depends on the rapidity between them. And it is the ratio that decides how much aberration separates their views of the sky.

Particle physics has adopted this without usually saying why: momenta are quoted in rapidity, or in the closely related pseudorapidity, because differences of rapidity are invariant under boosts along the beam while differences of speed and of angle are not. An experiment reaches for the natural distance in velocity space because it is the quantity its geometry does not distort, and the choice was made on practical grounds decades before anybody presented it as geometry.

What else this explains #

Reading the composition law as a geometry pays for itself several times over.

Why velocity addition is not associative in the way one expects. Composing three non-collinear boosts in different orders gives the same boost with different rotations, which is a triangle inequality question in a curved space rather than an algebraic curiosity.

Why aberration is a Möbius transformation. The boundary of the disc is the sphere of directions, and a boost acts on it by a conformal map — which is exactly the aberration formula that makes the sky that crowds into a cone look the way it does. The boundary at infinity of a hyperbolic plane is a circle, and the boundary of hyperbolic three-space is a sphere: the night sky as seen by a fast traveller is that boundary.

Why rapidity is the right variable for accelerator design. Adding rapidities is adding distances along a geodesic, so a sequence of boosts is a walk in a straight line, and the total is a sum. That is everything from an exchange of pulses in a different currency.

And why the interval and this curvature are the same fact. The set of four-velocities is the unit hyperboloid $u\cdot u = 1$, which the quantity nobody argues about draws as a calibration curve. Velocity space is that hyperboloid, and a hyperboloid in Minkowski space has constant negative curvature. The two figures in that essay and the disc here are the same surface in two models.

The scale at which it matters #

Everything above is exact at every speed, and it is worth knowing where it stops being negligible, because the answer explains why nobody noticed the geometry for three centuries.

At small rapidity the defect goes as the product of the two rapidities and the sine of the angle between them — so it is second order in the speeds. For a satellite at eight kilometres a second the rapidity is $2.7\times10^{-5}$, and two such boosts at right angles leave a rotation of about $7\times10^{-10}$ radians. That is well below anything a gyroscope on such a satellite would resolve against the far larger geodetic and frame-dragging effects.

For an electron in the innermost shell of a heavy atom the rapidity is a few tenths, the orbit is a continuous sequence of boosts, and the accumulated rotation over one orbit is a substantial fraction of a turn. That is Thomas precession, and it enters the fine structure with a factor of one half that was a genuine puzzle until the geometry was understood — the naive calculation of spin–orbit coupling gives twice the observed splitting, and the missing half is this rotation working against it.

So the curvature of velocity space has exactly one place where it is a matter of everyday consequence, and it is the shape of atomic spectra. A geometric fact about the set of all velocities is measured by a spectrometer, which is a longer reach than most geometrical arguments in physics manage.

Where the model stops #

Velocity space is hyperbolic, and spacetime is flat. These are different spaces and the curvature belongs to the first. Nothing here is about gravity, and the analogy with general relativity’s curvature is a genuine analogy rather than an instance — although the mathematics is identical and the holonomy language transfers exactly.

The Poincaré disc is two-dimensional. Real velocity space is three-dimensional hyperbolic space, and the triangles drawn here live in a plane inside it that happens to be totally geodesic. The whole argument survives; only the drawing is a section.

The rotation is a rotation of axes, not of an object. Nothing physically turns as a result of composing two boosts. What turns is the relation between two observers’ coordinate frames, and it becomes a physical rotation only when something is carried around a closed circuit — a spin, in an atom, which is why Thomas precession is observable and the abstract rotation is not.

And this describes the proper orthochronous Lorentz group only. Reflections and time reversal are not boosts, they are not in this space, and nothing about the geometry sees them.

And a boost has to be a boost. Everything here concerns pure Lorentz boosts, which are the transformations with no rotation of their own. A general element of the group is a boost combined with a rotation, and the space of those is six-dimensional and is not a hyperbolic plane at all. The clean geometry belongs to the four-velocities, and reading it as a picture of the whole group would be a mistake the drawing does nothing to discourage.

The metric was asserted rather than derived here. That the natural distance between two velocities is the rapidity between them follows from the invariance of the interval, and the derivation is short, but nothing in the figures performs it. What the figures check is that the geometry, once granted, reproduces the Wigner rotation — which is a consequence and not the foundation.

What the pictures cannot show #

The disc is drawn with a boundary and the boundary is not there. Every figure that draws hyperbolic space has this problem, and the Poincaré model has it in its most seductive form: the edge looks like a place and is a completion added to make the drawing possible.

The triangle figure draws a defect of twenty-odd degrees, which is visible. At the speeds anything in a laboratory reaches, the defect is a millionth of a degree and no drawing shows it. The figures use rapidities of order one, which are speeds above three-quarters of light’s, and the reader should carry that scale — the geometry is very nearly flat wherever ordinary physics happens, and that is the whole reason Galilean velocity addition works.

One more thing the drawings hide is how much of the disc is empty. Every velocity anything in the solar system has ever had sits within a rapidity of $10^{-4}$ of the centre — a region so small that on the scale of these figures it is a point, and inside which the geometry is flat to a part in $10^{8}$. The pictures are drawn at rapidities of order one because nothing is visible otherwise, which means every one of them is a portrait of a regime that has been reached by accelerators, by cosmic rays and by the inner electrons of heavy atoms, and by nothing else.

Where the ladder goes next #

The velocity-addition ladder began with speeds that refuse to add, continued with the turn that two pushes leave behind, and passed through the motion that measures faster than light, where an apparent superluminal speed is a projection. This rung asks what kind of space the velocities form and finds a curved one. The rungs after it: the Lorentz group as a whole, of which this disc is one slice; the conformal action on the sky, which is aberration written as a Möbius map; and the spinor representation, where the same disc reappears as the object a two-component object transforms over.

The habit worth carrying away is that a stubborn nonlinearity often means a curved space. Speeds do not add because they are not displacements in a flat space, and once the space is identified the composition law, the rotation and the speed limit stop being three facts and become one.