The universe one explosion makes in empty space

Two axes, one speed, and a diagram that does the arguing introduced the spacetime diagram as a tool for settling arguments about clocks and rods, and the diagrams since have been populated by a few observers at most: a rocket and the ground, a pair of twins, a train and a platform. What follows populates one with infinitely many observers at once and asks what they would say about the space they live in. The answer is a cosmology, and it uses nothing beyond special relativity: no gravity, no curved spacetime, no general relativity at all.

Start with a single event — an explosion, a firework, anything that throws debris in every direction — in otherwise empty space. Let the debris leave with every speed from nothing up to as close to light’s as one likes, and then let each piece coast forever in a straight line. On the spacetime diagram each piece is a straight line out of the event, steeper for slow pieces and closer to 45° for fast ones, and the whole cloud fills the inside of the event’s future light cone. That is all. The diagram is flat and the worldlines are straight; nothing is happening except that things are moving apart.

The instant on which every clock agrees #

Each piece of debris carries a clock, and the question that turns the cloud into a universe is which events count as “the same time” for the cloud as a whole. Now is a choice of slicing found that special relativity gives no unique answer: every inertial observer has a family of flat slices, tilted relative to everyone else’s, and none is privileged. But the cloud supplies a natural choice of its own. Every particle’s clock started at zero at the same event. So one can ask for the set of events at which every particle’s clock shows the same reading, $\tau$.

For a particle moving at speed $v$, the clock reading at an event a time $t$ after the explosion is $t\sqrt{1-v^2/c^2}$, and its position is $x = vt$. Putting those together, every event at which some particle’s clock reads $\tau$ satisfies $c^2t^2 - x^2 = c^2\tau^2$ — the interval from the explosion, which the quantity nobody argues about found to be the same for every observer. The set is a hyperbola, drawn three times in the first figure. It is not a flat slice of anyone’s simultaneity. It curves upward, and the curvature is exactly what is needed to catch the fast particles later in laboratory time, when their slower clocks have caught up to the same reading as the slow particles’.

Label each particle by its rapidity, the number $\eta$ with $v = c\tanh\eta$ that speeds that refuse to add found to be the quantity that does add. A particle of rapidity $\eta$ is at the event $(ct, x) = c\tau(\cosh\eta, \sinh\eta)$ when its clock reads $\tau$. The proper distance between two particles measured along the hyperbola — what a chain of rulers laid along it, each at rest with the particles beside it, would read — is $c\tau$ times the difference of their rapidities. The figures check that by integrating along the curve.

That one line carries the whole construction. If the particles are spread evenly in rapidity, they are spread evenly along the hyperbola: the cloud is uniform on its own instant. And the spacing is proportional to $\tau$: as the clocks advance, every separation grows in the same proportion. The cloud, described on its own instants, is a homogeneous universe expanding with a scale factor that is simply the time on its clocks.

One cloud, two censuses #

The second figure counts the same particles two ways. On the cloud’s own instant the count per unit length is flat: one particle in each step of rapidity, each step the same length. On the central particle’s instant — the horizontal line of the diagram, which is what “now” means for the particle that sits still at the middle — the count crowds towards the edge without limit. Near the light cone, a small range of distance holds an infinite range of rapidities, because the fast particles are all bunched just inside the cone in laboratory terms.

Both counts are correct. They are two answers to “how many particles are there here at one time?” with two meanings of “one time”, and how big now is found that the meaning of “now” far away is exactly what relativity leaves to choice. The point of the second figure is which choice makes the cloud look like a universe. On the central particle’s flat instant the cloud has a centre, an edge and a density that varies; it looks like what it is, debris from an explosion. On the cloud’s own curved instant it has no centre and no edge, and every particle — not just the one in the middle — finds itself at the centre of the same uniform distribution, because a Lorentz transformation to any particle’s rest frame maps the set of worldlines onto itself and leaves every hyperbola where it was.

Milne, who worked through this in 1932 and 1933, made that the starting point of his cosmology and gave it a name: the cosmological principle, the requirement that the universe look the same to every observer moving with it. He showed that the explosion satisfies it exactly. The principle has outlived the rest of his programme and is the founding assumption of every cosmological model in use.

A Hubble law with no speed limit #

Since the distance from the centre to a particle of rapidity $\eta$ is $D = c\tau\eta$, and $\eta$ does not change, the distance grows at the rate $dD/d\tau = c\eta = D/\tau$. That is Hubble’s law, recession rate proportional to distance, and here it is exact, with a Hubble constant equal to one over the age of the cloud. The third figure draws it as the straight line it is.

The line does not bend over at the speed of light. At a distance $c\tau$ the distance is growing at $c$; at three times that, at $3c$. Beside it the figure draws each particle’s velocity relative to the centre in the ordinary sense of special relativity, the slope of its worldline on the central particle’s diagram, $c\tanh\eta$, which flattens towards $c$ and never reaches it. Both curves describe the same particles. The first is the rate of change of a distance measured along curving instants, with respect to a clock that is not the central particle’s; the second is a relative velocity of two worldlines compared at one event. The spot that outruns light and carries nothing found the same lesson with a searchlight: a quantity built from positions at different times can grow faster than $c$ without anything travelling faster than light. In the cloud nothing does.

This is the source of a long-running confusion about real cosmology, where galaxies beyond a certain distance recede faster than light. The confusion is whether that contradicts relativity. The cloud shows that it does not, in the simplest possible setting: superluminal recession is what the Hubble law looks like when distance is measured along the cosmological instant, in a spacetime where special relativity holds exactly.

What the central particle sees #

Nobody observes an instant. What the central particle actually sees is everything on its past light cone, the two lines running down from its present event in the fourth figure. Each line crosses every worldline once, so the central particle sees every particle in the cloud, however fast — there is no particle beyond its view. But it sees each as it was at a different age. Light from a particle of rapidity $\eta$, emitted when that particle’s clock read $\tau_e$, reaches the centre when the centre’s clock reads $\tau_e e^{\eta}$. The fast particles are seen almost at the moment of the explosion; the slow ones nearly as they are.

The light arrives redshifted, and the redshift has two readings that turn out to be the same number. One is the Doppler shift. The particle is receding with rapidity $\eta$, and everything from an exchange of pulses found that the relativistic Doppler factor for that recession is $e^{\eta}$: the received period is longer than the emitted one by that factor. The other is the ratio of clock readings at reception and emission, $\tau_o/\tau_e = e^{\eta}$, which in a cosmological reading is the factor by which the universe has expanded while the light was in transit. Wavelengths stretched in proportion to the scale factor, and light shifted by the motion of its source, give the same answer in the cloud. They have to, because there is only one set of worldlines and one light ray.

The whole of the infinite cloud is visible, in finite time, inside one light cone. The cost is that the edge of what is seen is infinitely redshifted: particles near light speed are seen at the moment of the explosion, with their light shifted to arbitrarily long wavelengths. A universe that began at a point and has always contained infinitely many particles nevertheless presents to each of its inhabitants a finite sky with a dim, red edge, which is a reminder of what the sky that crowds into a cone found for a single fast observer: a moving observer’s view of a uniform sky is itself distorted, and the cloud is uniform only because every observer in it is moving with it.

Space that has more room in it #

The last figure asks what geometry the cloud’s instants have. Add the other two dimensions of space, so that the debris flies out in every direction, and the instant at clock reading $\tau$ becomes the three-dimensional surface $c^2t^2 - r^2 = c^2\tau^2$. Draw on it a circle of particles all at the same proper distance $R$ from one particle. Its circumference, measured on the surface, is $2\pi c\tau\sinh(R/c\tau)$ — more than the $2\pi R$ of a circle in flat space, by eighteen per cent at $R = c\tau$ and eighty per cent at twice that. The area of a sphere grows faster still. There is more room at a distance than Euclid allows, which is the defining property of negatively curved space, and every particle finds the same: the cloud’s space has constant negative curvature, with radius of curvature $c\tau$ growing as the clocks run.

It is worth being exact about what has curved. Spacetime has not; it is the flat Minkowski spacetime of special relativity throughout. The instant has, in the sense that it is a curved surface inside flat spacetime, as the surface of a sphere is a curved surface inside flat three-dimensional space. A hyperboloid in Minkowski spacetime carries the geometry Lobachevsky and Bolyai invented in the 1820s, in which the angles of a triangle add to less than 180° and parallel lines diverge. The space that speeds live in met this geometry in a different guise: the set of all possible velocities, with distance measured by relative rapidity, is a hyperbolic space. The cloud’s instant at $\tau = 1$ is exactly that space, laid out in positions instead of velocities, because each particle’s position on the instant is fixed by its velocity alone. The geometry of velocity addition and the geometry of the expanding cloud are one geometry.

The same grid, turned on its side #

The hyperbolas of the first figure have a twin, and the twin has already appeared in this collection under a different name. Inside the future light cone of the explosion the curves $c^2t^2 - x^2 =$ constant are the cloud’s instants, and the straight lines through the origin are its particles. In the wedge to the right of the light cone, where $x^2 - c^2t^2$ is positive, the same two families swap roles: the hyperbolas $x^2 - c^2t^2 =$ constant are the worldlines of observers who accelerate uniformly for ever, and the straight lines through the origin are their instants. The wall of silence behind a rocket that never stops drew those observers and found a horizon behind them, a light cone from which no signal can ever reach them.

So one coordinate grid laid on flat spacetime describes two physical situations. Turned one way, it is a family of inertial observers flying apart from an event, sharing curved instants and seeing an expanding, open universe with no horizon. Turned the other way, it is a family of accelerating observers sharing flat instants and seeing a horizon and, as the temperature of an acceleration found, a warm glow from the vacuum. The geometry is the same hyperbola; what differs is whether the hyperbola is a world line or an instant. Both wedges are pieces of the same empty spacetime, and an observer can pass from one description to the other only by changing what is meant by a clock and an instant.

The age a Hubble law implies #

In the cloud the Hubble constant is exactly one over the age: a particle at distance $D$ recedes at $D/\tau$, so running every particle backwards at its present rate brings them all together a time $\tau$ ago, which is when the explosion happened. The age read off the expansion and the age on the clocks agree, because nothing has sped the expansion up or slowed it down.

In a universe with matter and nothing else, the expansion has been slowing, so the particles used to recede faster than they do now, and the true age is shorter than one over the Hubble constant — two-thirds of it in the simplest case. That mismatch was a real crisis in the 1990s, when globular clusters appeared older than a matter-dominated universe allowed. The measured age of the real universe, 13.8 billion years, turned out to be close to one over its present Hubble constant, about 14 billion years, the value the empty cloud gives exactly. That is not because the universe is empty. It is because its early deceleration and recent acceleration happen to roughly cancel over its history, which is the same near-cancellation that keeps the supernova distances close to the empty model’s, and whether it is a coincidence or a clue is one of the questions that the present epoch’s acceleration raises.

Why the real universe is not this one #

A universe that is homogeneous, isotropic, expanding, open, with an exact Hubble law, and needing no physics beyond 1905: it is tempting to stop. Milne did not stop; he tried to build all of physics on the kinematics of such observers, and that programme, called kinematic relativity, did not survive. The model fails as a description of the real universe for one reason, and the reason is gravity.

The cloud is empty. Its particles are test particles, too light to pull on one another. Give them mass, and every particle is attracted towards every other, the expansion slows, and the worldlines curve towards one another. In general relativity a homogeneous, isotropic universe with matter in it is described by the same kind of slicing — instants on which all comoving clocks agree — but the scale factor no longer grows in proportion to time, and the instants’ curvature is set by the density rather than being fixed at the empty value. The empty cloud is the limit of those models as the density goes to zero, the “coasting” universe with scale factor proportional to time, and it is the only one of them whose spacetime is flat.

The real universe is not empty, but it is not far from coasting in one specific sense. Its expansion slowed while matter dominated and has been speeding up since dark energy took over, a few billion years ago, and over the range of distances where distant supernovae are measured the two effects partly cancel. Plotted as brightness against redshift, the measurements lie not far from the empty model’s curve out to a redshift of about one, and the evidence for acceleration in 1998 was precisely that they lie on the far side of it from every model with matter and no dark energy. The empty cloud is, in that sense, the dividing line, and it is useful to have a model of it that can be drawn on paper.

Where the drawing stops #

The construction rests on two idealisations beyond emptiness. The debris must start from a single event, and it must contain every rapidity in equal measure out to infinity; a real explosion has a finite fastest particle and so an edge, and an observer near the edge would see a lopsided sky. And the particles must be test particles in every respect: no collisions, no pressure, nothing exchanged between them. Within those conditions the description is exact at every speed and every distance, because special relativity is.

What the cloud cannot be used for is any question in which the matter matters. How fast structure grows, how the expansion rate changes, what happens to light passing near a galaxy, how much helium the early universe made — all of these depend on gravity and on the contents of the universe, and the empty cloud has neither. It shows which features of cosmology are kinematics, and it is silent about the rest.

What the spacetime diagrams leave out #

The diagrams have one dimension of space, so they show the expansion along a line and not the negative curvature, which needs at least two dimensions of space to exist; the fifth figure gives its measure as numbers rather than as a picture, because a hyperboloid in Minkowski spacetime cannot be drawn faithfully on Euclidean paper. The hyperbolas look curved in the diagram for the same reason a circle drawn on a spacetime diagram would look like a hyperbola: the paper’s geometry is not spacetime’s. And none of the figures shows the particles’ own view, a sky on which every other particle appears redshifted by an amount that depends only on its rapidity relative to the observer — the same sky, particle for particle, wherever the observer is.

Still open: what the real universe’s instants are #

The empty cloud has a natural instant, and in a universe full of matter the same role is played by the frame in which the cosmic microwave background looks the same in every direction. That frame is defined by the contents of the universe, not by the geometry, and whether the matter and radiation agree on it is a live question. Counts of distant radio galaxies and quasars across the sky should show a slight excess in the direction of the Earth’s motion through the background, by the same aberration and Doppler boosting that make the background warmer ahead, and several surveys have found the dipole in the source counts to be two to three times larger than the background’s motion predicts. Whether that is an error in the surveys, a local structure, or a sign that the universe’s matter and its radiation do not share one cosmological instant is not yet settled.

The habit worth carrying away is to ask which slicing a statement about “the universe at one time” assumes. Particles thrown from one event into flat spacetime are, on the instants where their own clocks agree, a uniform universe expanding with scale factor $\tau$, obeying an exact Hubble law $dD/d\tau = D/\tau$ with no speed limit, redshifting light by $e^{\eta}$ — at once a Doppler shift and a stretching — and living in a space of constant negative curvature whose circles are 18 per cent too long at a radius $c\tau$. Expansion of space and motion through space are two slicings of the same worldlines; what tells them apart in the real universe is the gravity the empty cloud leaves out.