Did space snap open?

What there is

Start with a seating chart. A dinner party can be drawn as who talks to whom: a dot for each guest, a line for each conversation. That drawing has no room in it. There is no left or right, no near or far, just a pattern of who is connected to whom. And yet if every guest talks to exactly their four nearest neighbours, and every four of them close up into a little square of conversation, you could reconstruct the whole table from the chart alone. The room is in the pattern.

That is what space is. Underneath everything is a network of relationships with no positions at all. Each point is linked to a handful of others, and the only question that means anything is which points are linked to which. Distance, direction, dimension: all of these are counts of small loops. When every link closes into short loops in two directions, the network is a flat sheet, and "here" and "there" mean something. When the links are wired any old way, every point is a few steps from every other, and there is no "where" at all.

This is not a metaphor for space. It is a working model of it, and the model earns the name by a hard fact: the rule that rewards small loops is the network's version of Einstein's own equation for gravity. Curvature on a network is measured by asking whether your neighbours' neighbours are close to your neighbours, which they are exactly when short loops run through every link. Add up the curvature and you have the energy. Reward the loops and space is what falls out, with no positions ever put in.

X, the curled arrangement

Now take a flat sheet of this network and roll it up, the way you roll a poster to post it. Every point still has its four links and its small loops. But one of the two directions is now tiny: walk four steps that way and you are back where you started. The network has one large direction and one curled one. Call this arrangement X.

Two things about X matter more than anything else in this account. First, it is not chaos. It is one specific arrangement, as particular as a crystal, and it sits still. Second, it holds more energy than the flat sheet does: a fixed amount for every point in it. Rolling the poster stores energy in the curl, and the curl cannot let it go on its own. X is a coiled spring with the catch on.

Two ways a thing can change

Cool a glass of water and, for a long while, nothing much happens. Then at zero degrees it freezes, and it does three things at once: it changes all at once rather than by degrees, it gives off a burst of heat as it goes, and it can be caught waiting. Very clean water stays liquid below zero until something jolts it, because between water and ice there is a small hill, and the water cannot get over the hill on its own. Physicists call this a first-order change. Its signature is the burst: the latent heat.

Warm a fridge magnet instead and its pull simply fades. There is no hill, no burst, and no moment at which half of it has changed. That is a continuous change. The universe began with the first kind.

The opening

X waits. It waits the way a radioactive atom waits: not counting down, but idle until a chance jolt happens to be large enough. The jolt it needs is remarkably small and remarkably fixed. Twelve units of energy, delivered anywhere in it, are enough to start it, and twelve is the number whether X has fifty points or five hundred. Below twelve, nothing ever happens. Sealed up with nothing to spare, X waits for ever.

Then, at one spot, a few links let go of the curl. Those points become flat. Their neighbours are now next to a flat patch and it costs them nothing to join it, so they do, and their neighbours after them. The change does not happen everywhere at once. It starts at a seed and sweeps outward as a front, with the old curled arrangement on one side and fresh open space on the other, and almost nothing in between. Caught halfway, ninety-nine points in every hundred are cleanly one or the other.

What the lump became

Here is the arithmetic that makes a universe. The push that starts the change is twelve units and stays twelve. The energy that comes out is one unit for every point converted, so it grows with the size of X without limit. A small X gives back five times what it took to start; a large one, hundreds of times, then millions. Once started, the change pays for itself and runs away. That is the Big Bang: not an explosion into space, but the sudden opening of space, with the energy stored in the curl pouring out at the front as it goes.

Where does the heat go? In the real universe it has nowhere to go but into the space just made. So seal the box. Let nothing in or out and watch what the heat does. It does one of two things and never a third. With enough room to spread, the front runs to the end and leaves clean, flat space behind, hot everywhere. With too little room, the heat is so concentrated that it melts the space just made: the points lose their two tidy directions and go back to being wired any old way. What it never does is stall, part tube and part sheet, sitting still. There is no temperature at which the two arrangements are content side by side, so there is no slush. A conversion is a bonfire, not a thaw.

And the melted patches are not damage. Look at what they are: small regions with no positions inside them, carrying energy, embedded in a space that does have positions. In the published model this is the definition of matter. Matter is the part of the network that the opening left without a place, and its energy is the lump. So the sentence "the energy released became matter and light" is not a figure of speech. In the model you can count it.

Matter is what space is made of, left without a place to be. Its energy is the energy the opening let go.

The scrap, and what dark matter is

The front does not quite finish. Almost every conversion leaves one small piece of the old arrangement stranded inside the new space: four points, still curled, that the front swept past without opening. The scrap is the same size whether the space around it is small or vast. It is not made of matter's melted phase and it is not space. It is leftover X.

This is dark matter. It was made in the same moment as ordinary matter, out of the same substance, but it is a different arrangement of that substance, so it does not take part in the ways matter usually interacts. It carries energy, so it gravitates. And there is no problem with there being so little of it at the start, because the universe's own history multiplies it. As space expands, light thins out faster than matter does, so matter's share of the whole keeps growing. Run the sums backward from today, where dark matter outweighs ordinary matter five to one, and at a starting temperature of ten billion degrees the leftover needed to be only about one part in a million of the energy. One scrap for every seed, and a sprinkle of seeds across the opening X, is enough.

Why the sky is the same everywhere

The picture makes a promise about the early universe, and the early universe keeps it. Every point that converts drops by the same amount, so the heat is the same everywhere the front has passed: the new space is born at one temperature, set by the height of the drop. The oldest light we can see is the glow of that heat, and it is the same in every direction to one part in a hundred thousand.

A sharp beginning was proposed once before, by Alan Guth in 1981, and it failed for a stated reason: the bubbles of new space never merged, because the old space between them was expanding faster than they could grow. That failure was a race, and here there is nothing to race against. X is not an expanding space. It is a curled arrangement with no expansion in it, so fronts spread and meet at their own pace, with nothing outrunning them. And each new space opens elsewhere, into its own directions, so no two universes ever need to join.

Space stays put

A marble on a shelf is stable for now. A marble at the bottom of a bowl is simply stable: there is nowhere lower for it to go. Flat space is the marble in the bowl. Take the whole family of arrangements the network can make, every wiring of every kind of small loop, and ask which of them sit lower than the flat sheet. The answer is none. The rule that forbids two small loops from sharing more than one link also forbids every arrangement that would undercut the sheet. Space is the floor by the rules, not by luck, and nothing in the rest of this account asks it to decay.

That settles a question people have argued about for a century: is our vacuum on a shelf, waiting to fall? No. It is in the bowl. Everything that happens next happens on top of it, and the only way off the floor is to be lifted.

What gravity is

Einstein's account of gravity is one sentence long. Matter tells space how to curve, and curved space tells matter how to move. Picture a stretched rubber sheet with a bowling ball on it: a marble rolled nearby curves in towards the ball, not because anything pulls it, but because the surface it moves on is no longer flat. That account is exact at every scale we have tested it, from falling apples to the orbits of black holes, and it comes back out of this network, because the loop-counting rule is Einstein's equation written on a network. What the network adds is the answer to the question Einstein's sentence leaves open: why does matter curve space towards itself?

The answer is counting. Shuffle a deck of cards and it comes out disordered, not because disorder is pulling on the cards but because there are vastly more disordered orders of a deck than ordered ones. A drop of ink spreads through water for the same reason: there are more ways for the ink to be spread out than gathered. Anything that keeps being jiggled drifts, without any force acting, towards the arrangements that have more ways of being. Physicists call the number of ways entropy, and the drift towards more of it is the second law of thermodynamics, the most reliable law we have.

Now the fact that turns counting into gravity. Take any region of space and ask which object in it has the most ways to be arranged. The answer, known since the 1970s, is a black hole: for its size, nothing has more entropy, and the amount is set by the area of its surface. So the count of ways is not uniform across space. It rises towards concentration, and it rises steepest where matter is densest. Everything that is jiggled drifts down that slope. From the outside the drift looks exactly like an attraction, and it has every property gravity has: it is universal, because counting applies to everything; it only ever attracts, because concentration only ever adds ways; it acts on energy of every kind, because every kind of energy is arrangements of the same substance; and it is feeble, because counting is a slope and not a shove.

Why it has to be counting

One might have hoped gravity was simpler than that: a pull carried by the energy itself, so that two lumps of matter cost less together than apart. The network rules this out, and it rules it out exactly. The energy of the network is a sum of small separate bills, one for each link, each depending only on the loops through that link. Two scraps that share no loop therefore cost precisely twice what one costs, at every separation, with no falling-off and no approach. Put them side by side so their damage overlaps and the shared loop is paid for once, which makes touching cheaper by a fixed amount. Move them one step apart and that vanishes, not gradually but entirely. There is no long-range pull in the energy, and by the form of the energy there cannot be one.

So gravity is not in the bills. It is in the count of arrangements, which the bills say nothing about and which the black hole's entropy says is there. The apparent weakness of gravity, the puzzle of why it is so much feebler than the other forces, is then no puzzle. The other forces are bills. Gravity is a slope in a count.

The one pull that never fades

There is a force in the network that does not fall off with distance at all, and it is worth knowing, because it is the one that matters at the end. It acts not between things but between regions: wherever space borders on curled X, the boundary is pulled towards the X, one unit for every point it sweeps, at any range. It is the same fact as the drop in the landscape, seen as a force. It is what drives the front at the beginning of the universe. And, run backwards and paid for, it is what a black hole does at the end.

The slow part

Then the universe does the thing it does for most of its life. Matter drifts down the slope. Gas gathers into stars, stars into galaxies, galaxies into clusters. Where stars burn out, their cores collapse, and the densest of them cross the line into black holes, which then grow by swallowing whatever drifts in and by merging with each other. Energy that began spread evenly across the new space ends up gathered into fewer and fewer, denser and denser places. It takes billions of years, and it is the calm middle of the story, and it is where we live.

Nothing gets out

Throw a ball upward and it comes back. Throw it fast enough and it does not. Every body has a speed of escape, and a body dense enough has one higher than the speed of light. Nothing outruns light, so nothing leaves: not matter, not light, not news of any kind. The surface at which escape becomes impossible is the horizon, and what lies inside it is cut off from the rest of the universe for good. That is a black hole, and general relativity has predicted its every visible property, down to the ring of light around its shadow.

Notice what a black hole is in the terms of Part I. It is a sealed box. Energy goes in and none comes out. That is exactly the setting in which the opening was studied, run in reverse: instead of energy pouring out of a change, energy pours into a region and has nowhere to go.

Space folds back up

Give a patch of flat space a large budget of energy that it cannot get rid of, and it does the only thing it can. It comes apart. The points lose their two clean directions and go back to being wired any old way: the ordered sheet becomes the phase with no positions in it, which is the phase with the most ways to be arranged. This is not a failure of the picture. It is the picture. The state that concentrated energy drives space into is precisely the state that ordinary physics says a black hole is: the object with more ways to be than anything else of its size. The interior of a black hole is the substrate with its geometry taken away.

Now the step that closes the loop, and the reason it goes through the mess rather than around it. Count every single move the rules allow out of perfectly ordered space and ask how many of them build a new small loop. The answer is zero. Not few: none. Order can only be broken, one move at a time. But count the moves out of the melted phase and the picture reverses: thousands of them build loops, and the cheapest run downhill. So the way from space to a curled arrangement is not a fold. It is a break, then a build. Melt first, and from the melt the substrate reassembles into whichever arrangement the conditions favour. In the crush at the centre of a black hole, with nothing able to leave and the whole weight of the star pressing in, what reassembles is the curled arrangement: fewer large directions, more energy stored per point. X again.

A black hole is the emergent world reaching down and remaking the thing it is made of.

It opens again, elsewhere

A region of X, once made, is at the left-hand edge of the landscape again: a curled arrangement, holding energy, waiting for a push of twelve units. Inside a black hole there is no shortage of pushes. A fluctuation in the crush, or the violence of two black holes merging, supplies it many times over. What decides whether the new opening survives its own birth is the same thing that decided ours: room. The heat given off needs somewhere to go. Where there is room, a front runs and a fresh, hot, flat space is born.

It is born into its own directions, not ours. The curled directions that open are the small ones inside the black hole, and they open outward from there into a space that is not connected to the one the black hole sits in. Our space does not grow; we see nothing; the new space sees nothing of us. The loop has closed one turn.

A loop that crosses levels

This loop is a particular kind, and the kind has a name. Douglas Hofstadter called it a strange loop: a hierarchy of levels, each made of the one below, in which the top level reaches back down and acts on the bottom. His example is a mind. Neurons make symbols, symbols make a self, and the self then decides what the neurons do next. You climb up through the levels and arrive at the bottom.

Here the lower level is the network and the upper level is space, with matter and gravity as its own laws. Space is made of the network. Gravity is a law of space. And gravity, in a black hole, remakes the network. The emergent world does not merely rest on its substrate; it reaches down and rebuilds it. That is what separates this loop from the other loop in cosmology, Lee Smolin's, in which black holes give birth to new universes with slightly different laws each time. Smolin's is a family tree: each turn makes a new individual and the line never comes back. This one comes back to the same substrate, by the same rules, every turn.

Why the loop does not wear out

Every proposal for an eternal cycle has had to answer one objection, first raised by Richard Tolman in the 1930s: disorder accumulates. Each turn of a cycle leaves the universe messier than the last, so the cycles should grow and finally stop. This loop answers it in the way the objection itself suggests. Disorder does not accumulate across the loop, because black holes are where the disorder goes. The melted phase, the phase with the most ways, is what collects inside a horizon, and it is spent there: it is the very thing that pays for the substrate to be rebuilt. What comes out the far side of the crush is a curled arrangement with fewer ways to be and more energy stored in it. The loop hands its mess to its black holes and gets back a wound spring. The books close every turn.

Three shapes a reality could have

Set aside for a moment what reality is made of and ask only what shape its history has. There are three. It can start and stop. It can start and never stop, but wind down until nothing happens in it any more. Or it can go round. The honest rival to a loop is not the first shape but the second: a universe that expands for ever into cold emptiness, which is what our own cosmology projects if nothing bends it back. That shape is eternal too. It just has nothing going on in it after a while.

Finite things must repeat

Here is the theorem, and it is older than relativity. Take any system with a finite number of possible states, and let it keep running by any fixed rule. Because the states are finite and the running is not, the system must sooner or later land on a state it has been in before. From that moment on, it repeats: the same states, in the same order, for ever. This is not a hope about the world. It is the pigeonhole principle, and Henri Poincaré proved the physical version in 1890: a closed system that keeps moving returns, again and again, arbitrarily close to where it has been. If the rule has chance in it, as reshuffling does, the conclusion is stronger, not weaker: every state the system can reach, it reaches again, endlessly.

The substrate has finitely many states. A network of a given number of points has a definite, countable number of wirings. At eighteen points the count is about 1.8 trillion, and every one of them has been listed. A reality made of such a network and kept running by a fixed rule is a loop with certainty. It does not need a beginning, because the theorem does not ask for one; it needs only that the running never stops, and the running never stops because there is no rule that stops it.

Where a loop spends its time

The four legs of the loop are not equal in length, and the inequality is the whole point. Boltzmann's law, the same one behind why water evaporates slowly and gunpowder burns fast, says that a jiggled system lingers in a dip for a time that grows exponentially with the height of the wall around it. X sits in a shallow dip and lingers for a while. Space sits on the floor, with the highest walls of all, and lingers longest of anything. The change between them is over in an instant by comparison: a seed, a front, done. The model measured the lingering law with nothing adjusted and found it to three parts in a thousand.

So a loop that passes through space spends most of its time being space. If you were dropped at a random moment into the history of such a loop, the floor is where you would land.

Where a question can be asked

One more constraint, and it is the cheap one. To ask where you are, you need to exist, and existing takes time and calm. It takes a cold, flat, settled space with matter in it that lasts long enough for stars to form, burn, seed the next generation with heavier elements, and light a planet for a few billion years. No other leg of the loop offers that. The opening is over in an instant and is hotter than anything can survive. The inside of a black hole is a melt with no positions in it. X is a coiled spring with nothing happening. Only the floor is a place where something can be built slowly enough to wonder about the rest.

And an argument that needs no counting at all

Anything with a beginning owes you an account of the beginning. Whatever you name as its cause, you can ask what put that there, and what put that there, without end. A loop owes nothing outside itself. Every part of it is accounted for by the part before, the whole way round, and there is no outside to point at. It is like a dictionary: every word in it defined by other words in it, and not one of them needing a word from beyond the covers. Of all the shapes a reality could have, the loop is the only one that is complete.

One for each degree of freedom

Back to the seating chart. Suppose two guests have exactly the same conversations: each talks to the same four people, and nobody else at the table can tell which of the two said what. Swap their name cards. The chart is unchanged. Nothing at the party, and nothing that could ever happen at the party, would register the swap. The two guests are, for every purpose the party has, one guest with two names.

The points of the network are like that. A point is nothing but its relationships, so two points with the same relationships are interchangeable, and a version of the world in which they are renamed is not a second world. It is the same world written differently. Now recall where we are: inside a loop with no beginning, no end and no outside. There is no vantage point from which two versions that differ only by a renaming could be compared. So they are not two. They are one thing, in all of its namings at once, one naming for each degree of freedom the arrangement has.

Physics has known this for a century without saying it this way. When quantum mechanics describes two identical particles, it does not say "this one is here and that one is there". It writes the state so that both assignments hold at the same time, because no experiment can tell which particle is which. That rule, the symmetry of exchange, is what makes electrons stack into the shells of the periodic table and photons pile into laser light. It is the first thing this account asks for, already in the textbook.

The wave

Here, then, is what a superposition is. It is not a particle smeared out or undecided. It is the whole set of namings of one arrangement, all standing, none preferred, because nothing has happened that could prefer one. An isolated closed loop of four points can be named in eight ways: start anywhere, go either way round. Left to itself, the loop is in all eight. Ask which way round its vibration runs and the honest answer is both, because "which way" is a naming, and a naming is not a fact until something depends on it. That is why a lone particle passes through both slits and interferes with itself. It is not in two places. It is in every naming of one place.

The "most probable version" is what you would get if you were forced to pick: the naming that the most arrangements around it agree with. Until you are forced, there is no picking, and the world simply is what it is in every version at once.

Measurement

An interaction is a new relationship. Add a link between the loop and something outside it, and a swap that once changed nothing now changes who is joined to whom. The namings are no longer the same thing; they have become versions that differ, and you are in one of them. Nothing collapsed and nothing chose. A distinction came into existence that had not existed before, and the world was on one side of it.

The network puts a number on this, and the number has been counted. For a perfectly ordered sheet of 160 points, the renamings that preserve every relationship number 320. For a pile of separate closed knots, they number about a thousand trillion trillion. For a melted region, the phase of matter that has interacted with everything around it, they number exactly one. An interacting thing has one version. That is why the world of tables and planets looks classical: everything large is wired into everything else, and a thing wired into everything has no swap left that changes nothing. Time playing out is measurement done continuously, one interaction after another, each one turning a naming into a fact.

Nothing collapses. A distinction comes into existence, and the world is on one side of it.

Particles as vibrations

What, then, is a particle? A small closed loop of the substrate, sitting inside space, carrying energy. And a closed loop can vibrate only in certain ways, for the same reason a guitar string can: a wave that runs round a loop must meet itself when it gets back, so only whole numbers of waves fit. On a loop of four points there are exactly two allowed frequencies. The lower one comes twice, once for each way round, which is the two-slit doubling again. The higher one comes once. On a loop of six points there are three. Each allowed vibration is a kind of particle, and its frequency is the particle's energy, which is to say its mass. The families of particles are the families of ways a closed loop can hum.

Every version exists

Now put this beside Part IV. The loop has no start and no end, and a finite system that keeps running returns to every state it can reach. So every version of every degree of freedom that can come out of the lump does come out, not once but endlessly, and from inside there is no telling which turn of the loop you are on, or which naming of it you are. All of them are our world. Space is not different from one version to the next, because the versions differ only in names, and names are not part of space. What separates versions is not being in a different place or a different time. It is an interaction, or time playing out, which is the same thing done slowly.

This is the last piece of the strange loop. The upper level, space with matter in it, is where interactions happen and namings become facts. The lower level, the network, holds every naming at once. Space reaching down into its substrate is a black hole; the substrate reaching up into space is a measurement. Both are the same hierarchy, tangled, seen from opposite ends.