Did Time Have a Beginning?

Your car's odometer reads 41,207.

Quick question: did it start at zero?

Of course it did. But open it up and look at what's actually in there. The rule inside is "add one for every mile." A gear clicks over each time the wheels roll a mile, and that's the whole machine. Nothing in that rule knows where the count began. It works exactly as well starting at 8, or at 41,206, or at minus a million.

So why are you so sure it started at zero?

Because an odometer isn't only a rule. It's a count of miles somebody actually drove. And nobody drives minus a million miles. (If you ever see minus a million on your dashboard, call a mechanic, not a philosopher.)

Hold on to that. It's the whole post.

Swap the car for the universe and you get one of the oldest questions there is. Did time have a beginning, or does the past go back forever? And if time began, what came before it?

People usually ask it as "what came before the Big Bang?" This post is about the very first page of reality's record. How that page connects to the hot, expanding universe astronomers measure isn't worked out yet, so the post stops at the first page, and I'll mark the spot when we get there.

The short answer

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

1. The start isn't nothing. A world where nothing differs from anything else can't hold a first event, and a comparison against nothing costs more than any number you can name. The start is a blank page: a record with no entries, owing nothing and costing nothing.
2. The laws can't tell. They accept a past with no bottom as readily as a first page. Add the same number to every count on the books and every law still holds. In a world of clocks and light, read every clock's ticks in reverse, turn the light around, and every law still holds.
3. The record decides. A record is a count of events that happened, and a count of events that happened can't go below zero. So the record has a first page, and time has a first moment. In the world of clocks and light, a record that keeps pace with the clocks can't reach back forever either.
4. The first page is blank, and it isn't at the bottom. In that world, exactly one state has no clock advanced and no light emitted. It has exactly zero energy, it sits above the floor, and the gap pays for the first ticks and the first light.
5. Forward is the way the record grows. In the world of clocks and light, the laws run both ways. The count runs one way.

Why this matters: it tells you where a beginning comes from. It isn't in the laws. It's in the record. And it turns "what came before?" from a riddle into a plain answer: "before" means "an earlier entry," and the first page has none.

One thing from Part 1 does all the work here. Reality keeps a permanent record of every event. Entries go in one after another, and nothing is ever erased. I draw the record as the Ledger, a thick book with a deadpan face and exactly one response to anything that happens: Noted.¹ If you haven't read Part 1, that's all you need. If you have: yes, the Ledger is back, and yes, it's still very thorough.

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

Step 1: The start isn't nothing

Yeah-But, the voice in the back of your head that never lets anything big go by, is already clearing his throat. "Yeah, but something can't come from nothing."

Right. It can't. So whatever the start is, it isn't nothing. And "nothing" can mean two different things here. Neither one is a place to start. Here's why, and here's what the start is instead.

The void. Picture a world where everything is exactly the same as everything else. Not a dark, empty room. A world with nothing to tell apart. Part 1's first plain fact was that things can be told apart: hot isn't cold, here isn't there. In the void that fact is false. There's no pair of things to compare, so no comparison can even be stated, and nothing can be written down. A first event has to change something, and in the void there's nothing to change.

Nothing on one side of a comparison. Every comparison has two sides, and a price: the more lopsided it is, the more it costs. Write the two sides as a ratio, x, how many times bigger one side is than the other. The price is

J(x) = (x + 1/x) / 2 - 1

The price isn't chosen. How the prices of comparisons have to combine, plus a unit to set the scale, leaves exactly one cost, and it's this one.² Here's what it does:

Equal sides cost nothing, and that's the only place the cost is zero. Every other ratio costs something. Shrink one side and the cost climbs, faster and faster, with no ceiling. Let one side shrink to nothing and the cost grows without limit.

So "nothing" fails twice. The void can't hold a distinction. And putting nothing on one side of a comparison costs more than any number you can name.

What the start is. The Ledger keeps two lists: the comparisons made, and the entries written. Before the first event, both are empty. Every comparison that does get made comes with its reverse: "A to B at ratio 2" always goes in with "B to A at ratio 1/2." The two cancel. That gives the start two zeros, and they're different kinds of zero:

• Nothing owed. Every comparison is cancelled by its reverse, so the net flow is zero before the first event and stays zero after every event.
• Nothing spent. The total cost of the comparisons is zero. It stays zero only until the first comparison that tells two things apart. From then on, every comparison that tells two things apart adds to the cost, and no comparison takes any away.

Work it through. Make a comparison at ratio 2, with its reverse at ratio 1/2. Each costs 1/4, so the cost goes from 0 to 1/2, and the net flow is still zero. Make a comparison at ratio 1, where the two sides are equal, and the cost stays at 0.

That's the start. A blank page: nothing compared, nothing written, perfect balance, at the very bottom of the cost curve, nowhere near the cliff. I'll call it the first page.

So Yeah-But can relax. The start isn't nothing. It's a page.

Zoom out. Step 1 is done. "Nothing" can't be the start: a void can't hold a distinction, and putting nothing on one side of a comparison costs more than any number you can name. The start is a blank page, owing nothing and costing nothing. Next, and why: is there anything before the blank page? The first place to look is the laws. Step 2 asks them. They don't answer.

Step 2: The laws can't tell

Hand the laws a first moment or a past with no bottom, and they accept both. They're like the odometer's gears: they say how the count changes and say nothing about where it began. Here it is exactly, in two separate worlds.

The books

The entries written, oldest first, make up the record. The books are what the record adds up to. They're a small table: three rows, one for each direction of Part 1's cube, with two counts in each row. That's six counts. Every entry adds one mark to one of them.

The laws of the books are the rules for which entries are allowed, how the balances move and what each entry costs. Look at what those rules actually read: which entries are allowed, each row's balance (the gap between its two counts), how the totals change when an entry goes in, and what an entry costs. Every one of those reads a change or a difference. None of them reads the number sitting on the dial.

So add 1 to all six counts. Every entry that was allowed is still allowed. Every row's balance is the same. Every cost is the same. That's exact, and it holds for adding any whole number.³

Try it on the blank books, six zeros. Subtract 1 from every count and you get six minus-ones. Same balances, same allowed entries, same costs. Make six entries, one on each count, and you land on the blank books. So six minus-ones are books that come before the blank books, and the laws are perfectly happy with them. So are the books before those, six minus-twos. The chain runs on forever.

You don't even need the shift. Take any books at all and remove one mark from any count, and you have books exactly one entry earlier. Do it again. There's no bottom.

So the laws accept a past with no bottom.

Yeah-But, squinting into the fog at the left end of the chain: "Yeah, but a count of minus one is nonsense."

Yes. Keep hold of that. It's the whole next step.

The grid

The books are pure bookkeeping: no clocks, no light. So here's a second test, in a separate world that has both. The two worlds don't share a result, and I won't borrow anything from one to prop up the other. They give the same shape of answer, and that's the point.

Picture a three-dimensional grid of cells. In every cell sit four little clocks. Each clock counts through eight ticks, Part 1's eight, one for each corner of a cube. Each clock pushes on one shared field of light along an edge of its own, and the light pushes back. Clocks and light, nothing else. No atoms, no stars, no coffee. It's the first world in our work where clocks and light push on each other, and its laws are worked out exactly, for every finite patch of the grid and for the grid as a whole.⁴ I'll call it the grid.

The grid has settings we put in by hand: how strong the clocks are, how fast light moves, how hard a clock pushes on the light (always more than zero), and which edge each clock is wired to. Everything below holds at every value of them.

Start the grid blank: every clock at rest, no light anywhere. Call that moment zero. What was the grid like at moment minus one?

Everything the grid does, moment by moment, is a history. I'll call a history a film.

Look at a clock. It counts its eight ticks one way round: 7, 6, 5, and so on. Play the film backward and it would count 5, 6, 7. The light runs backward too: ripples that spread out now shrink in. You'd think that settles it: the film has a direction built in.

It doesn't. Read each clock's ticks in reverse order (call tick x "tick 7 minus x"), turn every ripple of light around, and the backward film obeys exactly the same laws as the forward film. For every measurement you can make, the grid at moment minus t is the mirror image of the grid at moment plus t.

So the blank moment sits in the middle of a picture that is its own mirror. Tell the laws "the grid is blank at moment zero" and they fill in every other moment, earlier and later. What they don't fill in is where the film starts. Here are three films that obey every law:

Film 1 runs both ways forever, with no first moment. Film 2 starts at the blank and runs on. Film 3 runs up to the blank and stops. Run time backward and film 2 becomes film 3. A rule that uses only the laws and the blank page can't pick film 2 over film 3. It can't tell a film from its mirror image, so it can't make the blank the first moment rather than the last.

So the grid's laws accept a past with no bottom too.

Zoom out. Step 2 is done. In the books and in the grid, the laws accept a first page and a past with no bottom, and they can't choose between them. Next, and why: something other than the laws has to choose. The odometer knows where its count began because it counts something that happened. So does the record.

Step 3: The record decides

Time has a first moment because the record has a first page. A count of events that happened can't go below zero, and that puts a floor under time. Here it is in both worlds.

The books

Think about what the books are. They're what a record adds up to. Each count says how many entries of its kind have been written so far. You can't have written minus one entries, and the Ledger has no eraser to take one back. So every count is zero or more.

That single fact stops the chain. The books with every count at zero or more are exactly the books some record wrote. And among those, the blank books are the only ones with no earlier books: take a mark off any count of the blank books and you get a minus one, which counts nothing. So no books come before the blank books. They're the first page.

That's the odometer exactly. The gears don't know where the count began. The count does, because it counts something that happened.

Here's the part I want to be plain about. The laws didn't give time its first moment. The record did, and only because it's a count of events that happened. That's an ingredient, and none of the laws supplies it. Step 2 shows why: add one to every count and every law still holds, so no law that reads only changes and balances can tell books that began at zero from books that began at minus one. The beginning comes with the record. The laws don't force it.

The grid

The grid reaches the same answer by a different road, and this is the part I like best.

It needs two facts about the clocks. A running clock holds energy, and a clock at rest holds none. Add up the energy a clock holds over a stretch of time and call that its running total for the stretch.

Fact 1: the clocks never all rest together. Every stretch of time, however short, has a moment when some clock is running. Here's a drawing of that: the clocks are a band. The fact doesn't need any one player to keep going. The band never goes quiet. (Gary, the coin-flipping drummer from Part 1, can stop whenever his coin says so. He isn't in this band.)

Fact 2: the running has no ceiling. For at least one clock, the running total passes every number you can name as the stretch grows longer. It does that looking forward from the blank and looking backward from it.

Now take a record that keeps pace with the clocks: over any stretch, it grows by at least a fixed share of each clock's running total. And like any record, it's a count: never negative, and a finite number at every moment. Two things follow for every record like that.

The past can't go back forever. Say the share is one mark for every ten units of running total. Look back far enough for a clock's running total to reach 1,000 units and the record already holds at least 100 marks. Look back until it reaches a million units and the record holds at least 100,000. Fact 2 says the running total never stops climbing as you look further back, so there's no number the record stays under. If the past went back forever, the record at any one moment would be bigger than every number. But a record is a count, and a count at a moment is a finite number. So the past doesn't go back forever.

An empty record marks the first moment. If anything came before a moment, then somewhere in the stretch between, a clock was running (Fact 1), and the record grew over that stretch. So the record at that moment isn't empty. A moment where the record is empty has nothing before it. It's the first moment.

The film with no beginning is out. The laws allow it. The record can't carry it.

One step in this is still being worked out: showing that the Ledger's record is a record like that, one that keeps pace with the grid's clocks and reads blank exactly where no clock has advanced and no light has been emitted.

Yeah-But has one more. "Yeah, but what came before the first page?"

No earlier entry exists. "Before" means "an earlier entry in the record," and the first page has none.

Zoom out. Step 3 is done. The laws allow a first page and a past with no bottom. The record picks the first page: a count of events that happened has a floor, and in the grid a record that keeps pace with the clocks can't reach back forever. Next, and why: we know the first page is blank. We can say exactly what "blank" means and what's on it, and it turns out the blank page sits on the rim of a bowl, not at the bottom of it.

Step 4: The first page

A blank page of the record means nothing has happened yet: no clock has advanced and no light has been emitted. In the grid, exactly one state fits. It has exactly zero energy, it isn't the lowest state there is, and the gap pays for the first ticks and the first light. All of that is exact.

Exactly one state. "Blank" has two clauses. No clock has advanced: every measurement that reads a clock's running reads "at rest." No light has been emitted: every measurement of the field of light reads "empty." Exactly one state of the grid answers both ways, at every setting. You need both clauses. Turn one clock and leave the light empty, and the light reads "empty" but a clock is running, so that state isn't blank.

Zero energy. Energy has no built-in zero, so set one: an empty field of light with every clock at rest, before the clocks push on it, has energy zero. Switch the push on and the blank's energy doesn't move. Take any patch of the grid on its own, however big. The blank has exactly zero energy in it, and energy is conserved, so the patch keeps energy zero for the whole film. From then on, any energy the clocks and the light pick up is matched by negative energy in the clouds that form.

Not the lowest. Zero isn't the bottom. Each clock pushes on the field of light, and the field can settle into a cloud of light wrapped around the clock. A cloud lowers the energy. The blank's field is empty, so it holds no clouds. The blank is not the lowest state of any patch. The lowest energy a patch can have is no lower than minus 2a for every cell in it, where a is how hard a clock pushes on the light.

Here's a drawing of the real mechanism, exact in the simplest case, a single source of light. The blank is a ball sitting on the rim of a bowl. The floor of the bowl is where the cloud has formed.

The gap pays. The blank doesn't stay blank. A state at the very bottom of the bowl would sit still forever, but the blank sits on the rim, and from the rim it rolls: clocks start running, light starts filling the field, and clouds start to form. Like a ball rolling down a bowl, whatever it loses in height it gains in motion. In the grid, the energy the clouds release is exactly what the clocks and the light take.⁵

Here are the numbers. Take a patch of N cells on its own. The energy held by its clocks never goes above 2a times N. The energy in its light plus twice the energy in its clocks never goes above 8a times N. The light gets the bigger allowance because the field overshoots. In the single-source drawing it swings past the cloud's shape, out to as much as twice the cloud's size, and twice the size holds four times the energy. The total stays at zero the whole time. That's where the first ticks and the first light come from: the gap between the rim and the floor. Everything the patch does after the blank is paid for out of that one gap. Nothing is supplied from outside.

The Ledger, which has had nothing to note, is about to write Noted. for the first time.

That's where I said this post would stop: the first page of the record, what makes it blank, and what it pays for.

Zoom out. Step 4 is done. The first page is blank: one state, no clock advanced, no light emitted, energy exactly zero and above the floor, with the gap paying for the first ticks and the first light. Next, and why: one question is left. The laws can't say which way along the film is forward. The record can.

Step 5: Which way is forward

Forward is the direction the record grows. Here's how we know.

In the grid, the laws run both ways equally. Every film has a mirror twin, and the twin obeys every law too. Look back at the picture from Step 2. In film 2 the record grows to the right of the blank, and in film 3 it grows to the left. The laws can't say which of those two films is ours. They don't have to. The two are mirror images, and in both of them the record grows away from the blank.

So in the grid, "forward" isn't something the laws hand us. The count hands it to us. Entries are only ever added, so the record's count has a floor, zero, and no ceiling. That gives it a direction of its own: away from the floor. And it's a direction you can't flip. Take all the whole numbers, ..., -2, -1, 0, 1, 2, ... Negate every one and you get the same line of numbers running the other way, and nothing on the line says which way is "up." Now take the counting numbers, 0, 1, 2, 3, ... Negate them and you get 0, -1, -2, ..., which runs down forever. That isn't the counting numbers turned around. No relabeling turns a count that starts at zero around.

So the first page gives time its beginning and its direction in one move. Forward is away from the first page. It's the way the record grows. "Now," which Part 2 called the front edge of the record, is the place where the writing is happening, and forward is the way that edge moves. (Part 2 found the arrow you feel, which comes from the busy world around you. This is a different thing: the direction the books are written in.)

Zoom out. Step 5 is done. Time begins where the record begins, and it runs the way the record grows.

The whole trip, in one breath

Time has a first moment, and the record sets it, not the laws. The start isn't nothing. It's a blank page that owes nothing and costs nothing. The laws accept a past with no bottom: add the same number to every count on the books, or play the grid's film backward, and every law still holds. The floor comes from the record. A count of events that happened can't go below zero, and in the grid a record that keeps pace with the clocks can't reach back forever. In the grid, the first page is one state with zero energy, sitting above the floor, and the gap pays for the first ticks and the first light. Forward is the way the record grows.

So the universe has a first moment for the same reason your odometer starts at zero.

(The Ledger would like you to know that the first page is the only page in all of reality where it has never written Noted. There was nothing to note. It has made its peace with this.)