Is There Another Universe Out There? Not Another Kind

In the movie, a door opens, and behind it is the universe next door.

Almost everything is the same. Same people, same coffee, same dog. One thing is off. In one universe the dinosaurs lived. In another, gravity is weak and everybody bounces when they walk. In the strangest, space has a fourth direction, and every knot slides loose.

We love these. Some physicists take versions of the idea seriously: other universes out there, near or far, each with its own settings.¹

So here's the question. Can another universe differ from this one in how it's built? Not a different history. Different rules, a different number of directions, a different price on everything.

Try to build one. I'll hand you the parts the movies use. You assemble a universe that isn't this one, and an inspector checks your work.

The inspector is the Ledger, my drawing of reality's permanent record. The record is real: every event that has happened is written down in it, nothing in it is ever erased, and the past still exists. The face is a drawing. The Ledger has one response to every entry: Noted. (It comes from Part 1 of this series.)

The short answer

There's no other kind of universe.

I'll say "world" for any candidate universe. Every world that can tell things apart and keep a record of it has the same shape as this one. The shape is four things, the four laws from Who Set the Dials on the Universe?:

1. One price for a difference. I'll call it J: J(x) = (x + 1/x) / 2 - 1, where x is how many times bigger one thing is than the other.
2. One ratio between levels. Draw reality's levels as the rungs of a ladder. After the two starting rungs, each new rung is built by joining rungs that already exist, and each rung is 1.618... times the one below. That's the golden ratio, written φ.
3. Three directions of space.
4. One full cycle of eight ticks. A tick is one step of the process that writes the record. Eight is 2 × 2 × 2: one doubling for each of the three directions.

What can differ from one world to another is the units: how big a unit of price is, and how big the bottom rung of the ladder is. Units aren't shape. A map in miles and a map in kilometres disagree on every number and agree on every street.

Here's the whole answer in five steps. The rest of the post takes them one at a time and shows how we know each one.

1. Nothing to tell apart, no shape. A world where no two things differ has no shape to be different from.
2. A world that forgets has no room. It can have a price and a ladder, and nothing in it stays done.
3. Two directions or four can't keep a record. Three can.
4. Everything else is units. Change the price, the ladder or the counting and you either break a rule or change only units and notation.
5. A universe in reach is part of this one. To detect it, compare with it or reach it, you need information about both, and that puts both in one record.

Yeah-But, the voice in the back of your head that has never once said wow, has a question. "So every universe is the same universe?"

No. The shape is the same. What a world records isn't part of the shape, and neither is how strong its forces are or what its particles weigh. The step from the shape to those numbers is still being worked out.

Why this matters: one picture of other universes is a catalog. Ours is one entry, with its settings, and the other entries have other settings. If reality worked that way, "why this one?" would have no answer but luck. There's no catalog of shapes. Every world that keeps a record has this shape, and the only thing left to differ in it is units. So "another universe" can't mean "another shape." It also gives you a test for any universe-next-door story: ask whether it keeps a record, and ask whether you can ever compare it with ours.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got, and why the next step comes next. The Ledger keeps the map.

Step 1: A world with nothing to tell apart has no shape

A world where no two things differ has no shape. The shape starts from a difference, and a world with none has nothing to start from.

Start with the simplest universe you can build: one thing, the same everywhere, forever. Nothing in it differs from anything else.

Is that a different universe? It's a universe with nothing to tell apart, and it has no shape. Here's why. The shape is what it takes to keep a record of differences, and a record of differences starts with a difference. To pick one thing out you have to pass another over, and that's two things that aren't the same. A world with one thing in it has nothing to pass over. Every comparison in it is the thing against itself, and equal things cost nothing: J(1) = 0. Every join in it is the thing joined to a copy of itself, which costs nothing and writes nothing. There's no difference to price, join or keep, so the shape has nowhere to start.

Give the world two things that differ and the shape has somewhere to start.² The rest comes from the next requirements, one at a time. Why Is There Something Rather Than Nothing? takes the empty world apart in full.

Zoom out. Step 1 is done. A world where no two things differ has no shape, because the shape starts from a difference. Next, and why: give the world differences, and a new question comes up. When the world completes a comparison, does the comparison stay completed?

Step 2: A world that forgets has no room

A world can price its differences and climb a ladder of levels and still keep nothing. A world that keeps nothing has no room in it.

"Room" means one plain thing here: a place where a completed comparison stays completed.

Build the world. Give it the price J, so equal things cost nothing and a doubling costs 1/4. Give it the ladder, 1, 1.618, 2.618, 4.236, each rung 1.618 times the one below. That's the price and the ladder done.

Now add one rule: motion erases. Anything that gets written can be undone by moving things around, and the undoing writes nothing. Slide, bend or turn the pieces and the entry is gone.

The Ledger writes Noted. and the line slides off the page. It writes Noted. again. Same thing. The Ledger is not amused.

That's a world that forgets. It's a consistent world, and that's proved: its price works, its ladder works, and nothing in it stays done, all at once. Every requirement holds except one, that a completed comparison is kept. Nothing that happens in that world stays happened, so there's nowhere in it to write anything for good.

Keeping is a requirement of its own. It doesn't come free with a price or a ladder. Drop it and you get a world with no room. Keep it and the next question is where, because something that keeps needs a place to keep it.

Zoom out. Step 2 is done. A world can have a price and a ladder and keep nothing. Keeping is its own requirement, and a world without it has no room. Next, and why: the place a world keeps things is space, so how many directions can that space have?

Step 3: Two directions or four can't keep a record

A completed comparison can be kept in exactly three directions of space. In two or four it can't.

Here are the requirements. They're the ones from Why Space Has Three Dimensions, which takes them slowly.

1. A record holds. Nothing you do by sliding, bending or turning things, short of cutting one or passing it through another, wipes out what's written.
2. A mark and its reader are loops. A completed comparison is a pair, the mark and what reads it. Each is a cycle that comes back to where it started, and in space a cycle is a loop.
3. Space is the plain kind. It has no edge and no handles, like the skin of a balloon.

Two loops hooked through each other, like two links of a chain, can't be pulled apart without cutting one. A hooked pair is a mark that holds. Now turn the number of directions like a dial.

Two directions. Picture the skin of a balloon: two directions, no edge. A loop drawn on it cuts the skin into two patches. A second loop can't cross the first, so it sits in one patch and shrinks to a dot. It never hooks.

Four directions. Lift one loop of a hooked pair into the spare direction, slide it clear of the other, and set it back down. They're apart, and nothing was cut. (A dot inside a circle on a page can't get out without crossing the line. Lift it off the page and it can. The page is a drawing of the real mechanism, one dimension down.)

Three directions. There's no spare direction. Every move that frees one loop runs it through the other. Hooked stays hooked.

That's proved for any way of drawing the loop, however twisted: of all the sizes of space, only three lets two loops hook.³ (In four directions a loop and a bubble, the skin of a ball, can hook. A mark and its reader are two loops, and two loops can't.)

So a flat world with two directions and a world with four can't keep a completed comparison. The forgetting world from Step 2 fails by its rule. These two fail by their geometry. Either way the record doesn't hold.

Zoom out. Step 3 is done. A completed comparison is kept in three directions and no other number: in two, the second loop shrinks away, and in four, it lifts off. Next, and why: a world that keeps its record has three directions. Is anything else in it free to change? Its price, its ladder, its way of counting?

Step 4: Everything else is units

Any two worlds that keep a record have the same price of difference, the same ladder, the same three directions and the same eight-tick cycle. They can differ in units.

Take the parts one at a time and try to change each.

Try a different price. The price has three requirements, and Who Set the Dials on the Universe? lays them out: prices combine one way, a slightly different ratio has a slightly different price, and there's one scale, and it's positive (a tiny mismatch costs a small positive amount, and we say exactly how small). Try a wavy price, one that rises and falls as the mismatch grows. It combines the right way and has no jumps. But at x = 2 it comes to about -0.23. The Ledger would pay you to be mismatched, and the positive scale rules that out. Keep all three and exactly one price is left, J. Its scale is a choice of unit, the way meters and feet are: measure prices in units half the size and every price doubles. (The 2 in front of the product in that post's combining recipe belongs to the same choice.)

Try a different ladder. A ladder has one rule: every step up multiplies by the same number. And its first new rung is a real join of two earlier rungs, rung 0 and rung 1. Joining a rung to a copy of itself costs nothing, so the Ledger has nothing to write: pen up. Allow those joins and a doubling ladder, 1, 2, 4, 8, gets in. Don't, and the ladder climbs by 1.618... and by no other number. The size of the bottom rung is the one thing left to choose. (Why Does the Golden Ratio Keep Showing Up? takes this one slowly.)

Try a different way of counting. Counting is a start and a step: a first count and "one more." Three rules decide whether a counter is a copy of ours: the start is never one more of anything, two different counts never step to the same next count, and every count is reached from the start. A counter that obeys them lines up with ours in exactly one way. A 12-hour clock comes back around to 12, so it breaks the first rule and isn't a copy. Write the count as 3, III or ///: the notation is yours, and the count is ours. (Is Math Invented or Discovered? takes this one slowly.)

So what can a world change? Its units. Here are two worlds that keep a record.

World A prices a doubling at 1/4 and a quadrupling at 9/8. World B counts cost in units half the size, so it writes 1/2 and 9/4. Every price in B is twice A's, no more and no less. Divide B's prices by 2 and you get A's, entry for entry.

World A's ladder starts at 1 and climbs 1, 1.618, 2.618. World B's starts at 5 and climbs 5, 8.09, 13.09, five times A's at every rung. The numbers differ. But in both worlds rung 0 is 0.618 of rung 1 and 0.382 of rung 2. And both have three directions and a cycle of eight ticks.

Those are units, and units aren't shape. Think of two maps of one town, one in miles and one in kilometres. They disagree on every number: the school is 1 mile from the library on one and 1.6 km on the other. They agree on every street. The maps are a drawing of the real thing. The streets are the shape, and the miles and kilometres are the units.

That's the result behind this whole post. Any two worlds that keep a record have the same shape, and the only things left to differ in it are the unit of price and the size of the bottom rung.⁴

Zoom out. Step 4 is done. Any two worlds that keep a record have the same price, ladder, directions and cycle. Only the units differ, the way miles and kilometres do. Next, and why: every universe so far was one we built on purpose. What about the one next door that we only imagine? Can we ever tell that it's different?

Step 5: A universe in reach is part of this one

A universe you can detect, compare with this one or reach isn't a second universe. It's part of this one.

Say the universe next door is real, and it's different from ours. Different how? To say it's different, you have to compare it with ours.

Here's the second plain fact from Part 1: comparing two things needs information about both. Colder than what? You need the earlier sip and the later one. So when you compare the neighbour with this universe, information about both has to be in hand, side by side, long enough for the comparison.

The Ledger writes down every event, and a comparison is an event. It goes on the page like any other entry: this, that, and "they differ here." Now the neighbour's information is written in the record, on the same page as ours. A record that holds information about both is one record. (The page is a drawing of the record.)

There's no second sheet. "Next door" turns out to be down the hall.

That step, from "comparing needs information about both" to "so both are in one record," is an argument from the plain facts, and it hasn't been written out as a proof yet.

Detecting and reaching work the same way. To detect the neighbour, something here has to register a difference in it, and registering a difference is recognition. To reach it, you arrive and notice that it isn't home. Either way you've compared it with ours, and a comparison puts both sides in one record.

Yeah-But has one more question. "And a universe that never touches this one? No light, no signal, no way to compare?"

Then nothing here can ever observe it, and I can't tell you what's in it. What Step 4 says still holds: if it keeps a record, it has this shape.

Zoom out. Step 5 is done. To detect, compare with or reach another universe, you need information about both, and that puts both in one record. Next, and why: that's every attempt. Time for the whole trip.

The whole trip, in one breath

A world where no two things differ has no shape. A world that forgets can have a price and a ladder and still has no room. A world with two directions or four can't keep a completed comparison, and three can. Any world that keeps a record has one price for a difference, one ratio of 1.618... between levels, three directions and a cycle of eight ticks, and the only thing left to differ in that shape is units, like miles and kilometres on two maps of one town. And a universe you can detect, compare with or reach is part of this one, because comparing puts both sides in one record.

So there's no other kind of universe. There's no catalog of shapes to pick from. You can describe a universe with a different shape, but it fails something a record needs: it breaks a requirement, or it can't keep what it writes. Reality has one shape, and it's the shape a record needs. You're standing in the only kind there is.

(The Ledger would like it noted that you spent this whole post trying to build a different universe, and it has yet to write down a single one. Noted.)