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.

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?"

One thing from Part 1¹ does all the work here. Reality keeps a permanent record of every event. It's made of entries that go in one after another, and nothing is ever erased.

The reason is in known physics. The basic laws of physics never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. That's why physicists treat the black hole information problem as a real problem: losing information would break the laws.²

The record has the same property, and it's proved: writing a new entry never alters an earlier one.³

That gives the universe both of the odometer's parts. The laws are the gears, the rules for how things change. The record is the miles actually driven: the events that happened.

This post is about the very first page of reality's record, the page with nothing written on it yet. 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. Two things come up along the way. The books are the tally a record adds up to: a small table of counts, each saying how many entries of one kind have been written. The grid is a separate test world made of nothing but clocks and light. Step 2 builds both.

1. The start isn't nothing. A world where nothing differs from anything else can't hold a first event, and as one side of a comparison shrinks toward nothing, its cost passes every number you can name. The start is a blank page: a record with no entries, in perfect balance and costing nothing.
2. The laws can't tell. They accept a past with no bottom as readily as a first page. In the books, add the same number to every count and every law still holds. In the grid, 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 grid, a record that keeps pace with the clocks can't reach back forever either.
4. The first page is blank, and it isn't the lowest-energy state. In the grid, exactly one state is blank: no clock has advanced and no light has been emitted. It has exactly zero energy, it sits above the lowest-energy state, and the gap between them pays for the first ticks and the first light.
5. Forward is the way the record grows. In the grid, 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. Time runs the way the record grows, so "before" means "an earlier entry in the record," and the first page has none.

Here's the route. I'll stop after each step to zoom out. Each zoom-out says where we are, what we have, and why the next step comes next.

Step 1: The start isn't nothing

The first objection is the obvious one: 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 is that things can be told apart: hot isn't cold, here isn't there. You're doing it right now: every measurement and every perception tells one thing from another. Telling two things apart needs information about both of them, and the least there can be is one yes or no. In the void, nothing can be told apart. 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, A and B, and a cost that measures imbalance: the more lopsided the comparison, the more it costs. Write the two sides as a ratio, x: how many times bigger A is than B. If A is twice B, x is 2. The cost is

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

The cost isn't chosen. It has to meet three requirements:

• It is zero when the two sides match (x = 1).
• It grows as they differ.
• It is the same whichever side is put first (x or 1/x).

Those three requirements, the rule for how the costs of combined comparisons add, and one calibration (how sharply the cost rises near balance) leave exactly one formula, and it's this one. The footnote states the rule and the calibration exactly and gives a check you can run.⁴ Try it at ratio 2: (2 + 1/2) / 2 - 1 = 1/4. Here's the cost at more ratios. Watch the bottom rows, where one side shrinks:

Equal sides cost nothing, and that's the only place the cost is zero. Every other ratio costs something. A ratio and its reverse cost the same: 2 and 1/2 both cost 1/4, and 4 and 1/4 both cost 9/8. Drawn as a curve, the cost is a valley with ratio 1 at the bottom. Shrink one side and you climb a cliff: the cost rises faster and faster, and as one side shrinks toward nothing it grows without limit.

So "nothing" fails twice. The void can't hold a distinction. And shrink one side of a comparison toward nothing and the cost passes every number you can name; a side of exactly nothing is never reached.

What the start is. Two lists matter at the start: the comparisons made, and the entries written in the record. Before the first event, both are empty, and the comparisons have two totals that are both zero. One stays zero. The other never goes down.

Make a comparison at ratio 1, where the two sides are equal. It costs 0, so the total cost stays at 0. Now make one at ratio 2: A is twice B. "A to B at ratio 2" always goes in with its reverse, "B to A at ratio 1/2," because a comparison of A with B is also a comparison of B with A: if A is twice B, B is half A. Each costs 1/4, so the total cost goes from 0 to 1/4 + 1/4 = 1/2. The two cancel: ratio 2 one way, 1/2 the other, so nothing is owed on balance. Call what is owed on balance the net flow. It is still zero.

• Net flow: zero, and it stays zero. Every comparison that does get made comes with its reverse, so the net flow is zero before the first event and stays zero after every event.
• Total cost: zero, and it never goes down. It stays zero only until the first comparison whose two sides aren't equal. From then on, every such comparison adds to the cost, and no comparison takes any away.

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. It's the first page of the record.

So 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 as one side of a comparison shrinks toward nothing, the cost passes every number you can name. The start is a blank page, in perfect balance 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 and the grid.

The books: add the same number to every count

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 of space's three dimensions, with two counts in each row: one for debits and one for credits, because an entry on a row goes in on one side or the other. That's six counts. Each count says how many entries of one kind have been written so far, and every entry adds one mark, a single tally stroke, to one of them.

Why three rows: space has three dimensions. Lasting structure needs a connection that no smooth motion can undo, and linking is that connection: two linked rings can't be pulled apart without cutting one. That holds only in three dimensions. In two, loops can't wind around each other at all, and in four or more any two loops slide apart.

A row's balance is the gap between its two counts: a row reading 5 and 3 has balance 2. The laws of the books are the rules for which entries are allowed, how each row's balance moves, how the totals change when an entry goes in, and what an entry costs. Every one of those rules reads a change or a difference. None of them reads the number a count is showing, just as "add one for every mile" never reads the odometer.

So add 1 to all six counts. I'll call that a shift. Every entry that was allowed is still allowed. Every row's balance is the same: a row reading 5 and 3 becomes 6 and 4, and its balance is still 2. Every cost is the same. That's exact, and it holds for adding any whole number.⁵

Try it on the blank books, the books of a record with nothing written in it: 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.

Look at the left end of the chain, where the counts are minus one and lower. A count of minus one is nonsense. Keep hold of that. It's the whole next step.

The grid: play the film backward

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 are separate: each reaches the same shape of answer on its own, with nothing borrowed from the other.

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. The cube has eight corners because space has three directions, each with two sides, and 2 times 2 times 2 = 8. A cycle that visits every corner once takes eight steps. Each clock pushes on one shared field of light along an edge of the grid, an edge no other clock uses, and the light pushes back.

That's everything: clocks and light, nothing else. No atoms, no stars. Its laws are worked out exactly, for every finite patch of the grid and for the grid as a whole, in a paper the footnote names.⁶ I'll call it the grid.

The grid has settings that 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), which I'll call a;
• which edge each clock is wired to.

Everything below holds at every value of them, so the results don't depend on which edge each clock gets.

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

I'll call everything the grid does, moment by moment, a film.

Look at a clock and number its eight ticks 0 to 7. It counts them 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 which way is forward: 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." Then the backward count 5, 6, 7 reads 2, 1, 0, which counts down like the forward film. Turn every ripple of light around too, and the backward film obeys exactly the same laws as the forward film. For every measurement you can make, and for any time t, the grid at moment minus t is the mirror image of the grid at moment plus t. That is worked out from the grid's laws, at every setting.

So the grid is its own mirror around moment zero. 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 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: time runs the way the record grows, as Step 5 shows. 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: no count goes below zero

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 nothing is ever erased, so no written entry can be taken 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. 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 can't be a count. 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: entries actually written, only ever added, each kept with every earlier one. 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: a record that keeps pace can't reach back forever

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. Both are results worked out from the grid's laws, at every setting. A running clock holds energy, and a clock at rest holds none. Add up the energy a clock holds, moment by moment, over a stretch of time, and call the sum its running total for that 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.

Fact 2: the running total 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: it grows at least in step with how much the clocks run. In exact terms, keeping pace means this: over any stretch, the record grows by at least a fixed share of each clock's running total, give or take a fixed handful of entries. The handful is there because a count grows one whole entry at a time. And like any record, it's a count: never negative, and a finite number at every moment. One thing follows 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, give or take five marks. Look back far enough for a clock's running total to reach 1,000 units and the record already holds at least 95 marks. Look back until it reaches a million units and the record holds at least 99,995. Fact 2 says the running total never stops climbing as you look further back. So 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.

Film 1, 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 record's own count keeps pace with the grid's clocks in exactly this sense. What is worked out is the other half: for every record that keeps pace, the past can't go back forever.

One question is left: what came before the first page? Time runs the way the record grows, so "before" means "an earlier entry in the record." The first page has none. No earlier entry exists.

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. Step 4 says exactly what "blank" means and what it pays for. It turns out the blank page sits on the rim of a bowl, not at the bottom of it.

Step 4: The first page is blank

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, and it isn't the lowest state there is. The gap down to the lowest state 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 of whether a clock is 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. Advance 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 we set one. An empty field of light with every clock at rest, before the clocks push on it, has energy zero. Where zero sits doesn't change the gap: the gap is a difference between two energies, and a difference doesn't depend on where you put zero. Switch the push on and the blank's energy doesn't move. Now 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.

Not the lowest. Zero isn't the bottom. Each clock pushes on the field of light, with the strength a from Step 2, 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.

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. The blank sits on the rim, and from the rim it rolls: clocks start running. 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.⁷

Two numbers say how much the clocks and the light can take from the gap. Each is a ceiling, and each grows with the number of cells: a fixed allowance for every cell. 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, the field swings past the cloud's shape, out to as much as twice the cloud's size. Twice the size holds four times the energy, and four times 2a is 8a. The patch's total energy 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.

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. Its energy is exactly zero and above the lowest state, and the gap pays 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, ... They start at zero and climb forever. Negate them and you get 0, -1, -2, ..., which starts at zero and sinks 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. Forward is the way that edge moves. (Part 2 found the arrow of time 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. Next, and why: all five steps are done, so here's the whole trip in one breath.

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 in perfect balance, and it 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 record supplies the floor. 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 lowest state. The gap pays for the first ticks and the first light. Forward is the way the record grows.

The first moment comes from the record being a count of events that happened: a record can't hold minus one entries, so no count goes below zero. The books are what reality's record adds up to, and the blank books are the only books with nothing before them. The grid is a separate world of clocks and light, and it reaches the same shape of answer on its own.

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