How Will the Universe End? It Runs Out of Gas. It Never Stops Driving.

Your car's fuel gauge is on E.

The needle has been resting on the peg for a while, and the little orange light stopped being a hint a mile ago. Any minute now the engine coughs, and you coast to a stop at the side of the road.

Now look at the other dial. The odometer reads 41,207. It has never once run backward. It didn't lose a mile when the tank got low, and it won't lose one when the car stops. Every mile you ever drove is still in there.

Two dials, two jobs. One tells you what's left to burn. The other tells you what has happened.

The car is a drawing of something real. The universe has two dials like these. One is a gauge of what's left to burn. The other is an odometer of everything that has happened.

The usual story of how the universe ends reads the gauge. Every star burns through its fuel. Everything cools and spreads out toward one even temperature. Once it's all even, nothing flows from hot to cold, and with nothing flowing, nothing is left to run on. Physicists call that heat death. It's a gauge on E.

So here's the question. Is a gauge on E the end of the universe? Or is it only the end of one dial?

The usual story has other endings too, where space itself tears apart or collapses. This post follows the two dials. It says nothing about whether space keeps stretching or turns around.

The odometer is the record from the earlier posts in this series. 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 one response to everything that happens: Noted.¹ If you haven't met it before, that's all you need.

This post is the mirror of Did Time Have a Beginning? That one stopped at the first page of the record. This one goes looking for the last. (Same car. Still 41,207.)

The short answer

The universe has no last page. What runs out is imbalance, the thing you burn to do work. The record of what has happened doesn't run out: nothing in it is ever erased, and whatever its length, one more line can go on the end.

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 test worlds come up along the way, as in the beginning post. The books are pure bookkeeping, with no clocks and no light. This time they hold a record and one arrangement of comparisons. The grid is the same world as before, made of nothing but clocks and light. Steps 1 to 3 use the books. Steps 4 and 5 use the grid, and Step 5 also brings back the record's floor and missing ceiling from Step 3.

1. Nothing written is erased. The record only adds lines at the end, and every line already written stays exactly as it was. Even the last star going out would go in as one more line.
2. The gauge runs out. The world's arrangement has a cost, the price of its imbalance. Under the books' rule, that cost falls to a floor and stays there, and no run can burn it at a steady rate forever. That's the books' version of heat death.
3. The odometer doesn't. A record has a floor and no ceiling: whatever it holds, one more line can be written. Every tick writes one, so after enough ticks the record passes any number you name. And the gauge doesn't pay for the lines. A record paid for out of a finite tank would stop when the tank did.
4. The clocks never all rest. In the grid, every stretch of time, however late, has a moment when some clock is running, and some clock's running total never levels off. There's no final stillness. And on a grid whose clocks are all wired along one direction, the long run is never a heat bath.
5. No law names a last page. The grid's laws run both ways, so they can't tell a last moment from a first one. The floor under the record forces a first page. With no ceiling, nothing forces a last one.

Why this matters: it changes what an ending can be. Most people picture the end of the universe as a wall. The film stops, the lights go out, and it's all as if none of it had happened. The books show the reverse. A universe can run out of imbalance, which is what you burn to do work. It can't run out of what has happened. The best afternoon of your life is a line in the odometer, and the last star going out can't touch it.

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. The Ledger will keep the map.

Step 1: Nothing written is erased

Whatever the end is, it can't be an erasure. The record only adds lines at the end, and every line already written stays exactly as it was. Here's how we know.

The record is a list, and the only thing that can happen to a list in the books is a new line at the bottom, the way a new entry goes at the bottom of a diary. Take a record with three lines, A, B and C, and write a fourth, D. Then check three things.

• The old lines are unchanged. Read the first three back: A, B, C.
• Each one is still at its address. Line 2 is still B.
• The write head moved by exactly one. The write head counts the lines written, and the next line goes right after it. It was 3. Now it's 4.

That's all "nothing is erased" means in the books, and it's exact.

So say the last star does go out. That's an event, and an event goes in as a line, at the bottom. It doesn't take a single line above it along. Your seventh birthday is in the same ink as the first star. The past still exists. It's written down, and the end of the universe leaves every line of it where it is.

What about things that really do look erased? The Universe Has No Delete Key follows a wiped memory and finds the difference between the starting patterns in the heat the wipe made. Why Your Coffee Never Un-Stirs follows a stirred coffee and finds the stir still in the record. The usual end of the universe is that same move, everywhere at once: differences spreading out into an even warmth. Spreading is moving. It isn't erasing.

Zoom out. Step 1 is done. The record only adds lines at the end, so no ending can erase anything. Next, and why: the usual story of the end still has something running out. Step 2 finds out exactly what.

Step 2: The gauge runs out

One thing in the books runs out: the cost of the world's imbalance. Under the rule we adopt for how an arrangement changes, that cost falls to a floor and stays there, and no run can burn it at a steady rate forever. Here's how we know.

The price list. Reality charges for differences. Write the two sides of a comparison as a ratio x, how many times bigger one side is than the other. The price is

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

Equal sides, x = 1, cost nothing. Doubling, x = 2, costs 1/4. Quadrupling, x = 4, costs 9/8. Nobody picked this formula. How comparison prices have to combine, plus a unit, leave only this one.²

An arrangement and its cost. Here's a world small enough to count. Right now it holds one comparison at ratio 4, with its reverse at ratio 1/4. (Every comparison comes with its reverse, as the beginning post explains.) Call the pair (4, 1/4) the world's arrangement. Its cost is the two prices added: 9/8 + 9/8 = 9/4. The gauge reads that cost: 9/4. The cost is also the tank, the amount there is to burn.

This isn't the running tally from the beginning post, the total price of every comparison ever made. That tally only goes up, because a comparison that has been made stays made. It sits with the odometer. The gauge prices how things stand right now.

The rule. The books change an arrangement by one rule, the one we adopt for them.³ Each step, the arrangement moves to the cheapest arrangement that has the same product. The product is the ratios multiplied together, and for this world it's 4 × 1/4 = 1. Cheapest means as balanced as the product allows. Among all pairs that multiply to 1, the cheapest is (1, 1), at cost 0 + 0 = 0. So the rule takes (4, 1/4) to (1, 1). The arrangement (2, 1/2), at 1/4 + 1/4 = 1/2, goes to (1, 1) too. And (1, 1) goes to (1, 1): from there the rule hands back the same arrangement, and nothing is left to spend.

I'll say a step burns the cost it removes, the way an engine burns gas. Taking (4, 1/4) to (1, 1) burns 9/4. Taking (2, 1/2) to (1, 1) burns 1/2. At (1, 1) a step burns nothing. The gauge is on E.

The books count steps, not years. How long a step takes in the real world isn't something this post claims.

The tank is finite. Now a fact that doesn't depend on the rule at all. Say the gauge falls by at least 0.01 on every step. After n steps it has fallen by at least n × 0.01. It can't fall below zero, since no cost is negative. So a gauge that started at 1/2 can keep that up for at most 50 steps: 50 × 0.01 = 1/2. After that, it's dry. The same goes for any price per step and any starting tank. The most steps you get is the tank divided by the price. No run burns at a steady rate forever.

That's heat death in the books: a gauge on E, with nothing left to burn.

Yeah-But, the voice in the back of your head that has never once said wow, clears his throat. "Yeah, but if there's nothing left to burn, nothing can happen. That's the end."

It's the end of burning. Whether it's the end of what happens is a different question, and the books have a second dial for it.

Zoom out. Step 2 is done. The cost of the world's imbalance falls to a floor and stays there, and no run burns it at a steady rate forever. That's the books' version of heat death: a gauge on E. Next, and why: a gauge can't tell you whether anything is still happening. The record can.

Step 3: The odometer doesn't

The record has a floor and no ceiling. Every tick writes one line, so after enough ticks the record passes any number you name. And the gauge doesn't pay for the lines, so the record can go on when the tank can't. Here's how we know.

A floor and no ceiling. The record is a list of lines. The shortest list is the empty one: zero lines, the first page from the beginning post. That's the floor. (It's a different floor from the gauge's. The gauge's floor is where it ends up. The record's floor is where it starts.) Now take any record at all, however long, and write one more line. It's longer by exactly one. So there's no longest possible record.⁴

Every tick writes a line. A tick is one step of the world's basic process, and every tick writes one entry. That's the process from What You Are, the first post in the series. Even a silent tick, where nothing flows, writes a line saying so. So after n ticks, the record is at least n lines longer than it was. Start with 3,000 lines. After a million ticks, there are at least 1,003,000. Name any number you like. After that many ticks plus one, the record is longer than your number.

The gauge doesn't pay. In a car, the engine burns the gas that turns the wheels, so an empty tank stops the odometer. Is the record paid for out of the gauge? Try it. Let the arrangement take one step each tick, and say every new line costs the gauge at least 0.01. A record that gets a line every tick then burns at least 0.01 on every tick, forever. We just saw that no run does that. The tank of 1/2 pays for 50 lines, and then it's dry. So if a line lands every tick, the gauge can't be what pays for them.

A line isn't something the world buys. A line is what an event is: when two things bump into each other, the bump is the entry. The record can run on because nothing has to pay for it.

Yeah-But is back. "Yeah, but my car's odometer stops when the tank is empty."

It does. That's where the car stops being a good drawing, and it's the whole point. In a car, the engine ties the two dials together. In the books, no pipe runs from the tank to the lines.

Zoom out. Step 3 is done. The record has a floor and no ceiling, it gets at least one line every tick, and the gauge doesn't pay for the lines. The gauge runs out and the odometer doesn't. Next, and why: the books have no clocks and no light. To see whether anything keeps happening once there's nothing left to burn, we need a world that has both. That's the grid.

Step 4: The clocks never all rest

The books can't say whether anything keeps happening once there's nothing left to burn. A world with clocks and light can say for itself. In the grid, every stretch of time, however short and however late, has a moment when some clock is running, and the running total of some clock never levels off. So the grid has no final stillness. Here's how we know.

The grid, again. Picture a three-dimensional grid of cells. In every cell sit four little clocks. Each clock counts through eight ticks, 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. That's everything: clocks and light, nothing else. No atoms, no stars, no coffee. The beginning post builds it, and its laws are worked out exactly.⁵ A running clock holds energy. A clock at rest holds none.

Four settings go 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 (no two share one). The next two facts hold at every value of every setting.

Start the grid blank: every clock at rest, no light anywhere, the blank page from the beginning post. Then let it run.

Fact 1: no stretch of time is quiet. Pick two moments, as close together as you like and as late as you like. Somewhere between them, 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. Some player always has the beat. (Gary, the drummer from Part 1 who flips a coin before every beat, can stop whenever his coin says so. This band doesn't need him.)

Why can't every clock rest? Every cell has four clocks and the grid has only three directions, so two clocks in a cell are wired to edges pointing the same way. They push on the same light, and as time passes the light shifts how those two clocks line up with each other. Call that shift their twist. If every clock rested for a stretch of time, the twist would have to hold perfectly still for the whole stretch. But the twist has a rigid shape: if it held still for even a short stretch, it would hold still at every moment. And it doesn't hold still at every moment, because the longest, slowest waves of light keep turning it. So not every clock can rest.

Fact 2: the running total never levels off. Add up the energy a clock holds, moment by moment, from the blank onward, and call the sum its running total. For at least one clock, that total passes every number you can name. Say it reaches 1,000. Later it reaches a million. It doesn't stop at any number.

No final stillness. A final stillness would be a time after which every clock rests for good. Fact 1 says no stretch of time, however late, is like that.

A note for readers of Time: the hub there is the state where one consciousness has no arrow and no tick, so it has no before and no after. That describes one rhythm and how tightly it's linked to the busy activity around it. It isn't the grid's clocks stopping.

Clocks and odometer. Fact 2 ties the clocks to the record. 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. Say the share is one line for every ten units. A total of 1,000 puts at least 100 lines in the record. A total of a million puts in at least 100,000. Some clock's total passes every number, so the record does too.

One wiring that never settles. One more thing the grid shows. Think about where the usual story ends: a heat bath, everything settled to one even temperature, with light in it too. Light wiggles sideways to the way it travels, and for each direction it travels it can wiggle in two independent ways. At any temperature above absolute zero, a heat bath has some light in both.

Now wire every clock to an edge pointing the same way, say up-down, and take light traveling left-right. It can wiggle up-down or in-out. The clocks push the up-down wiggle, and no clock ever pushes the in-out one. Light traveling any other way has a wiggle that no clock pushes, too. That wiggle starts empty, because the blank has no light, and it stays empty at every clock strength. A heat bath needs light in both. So a grid wired this way never becomes a heat bath: not at any moment, and not on average over a long run.⁶ That's proved for one direction, at every clock strength. With several directions and the clocks on, it isn't worked out yet.

Here's a drawing of the real mechanism. Picture a house where one room is sealed off from the furnace and from every other room. The furnace runs for a billion years. That room stays cold, so the house never reaches one even temperature.

Zoom out. Step 4 is done. In the grid, every stretch of time has a running clock and some clock's total never levels off, so there's no final stillness. On a grid wired along one direction, the long run is never a heat bath. Next, and why: a last page would need something to put it there. The laws are the first place to look, and Step 5 asks them.

Step 5: No law names a last page

No law of the grid puts a last page in the record. The laws can't tell a last moment from a first one, and the record has a floor and no ceiling. Here's how we know.

The laws run both ways. The beginning post played the grid's film backward: read each clock's ticks in reverse, turn the light around, and the backward film obeys exactly the same laws. That reversal swaps beginnings and ends. A film that starts at the blank becomes a film that ends there, and the other way round. So the laws accept a last moment as readily as a first one. They can't be what gives time its first moment, and they can't be what gives it a last.

The record has a floor and no ceiling. What gave time its first page was the record. A count of events that happened can't go below zero. That's a floor, and the floor forces a first page: no entry comes before the empty record. Now the mirror question. Does the count have a ceiling, a number of lines the record can't pass? It doesn't. You saw it in Step 3. Write one more line and the record is longer. With no ceiling, nothing forces a last page.

That's also why "later" means something in the grid. Its laws run both ways, and the record runs one way, because it only grows. So "later" is the way the record grows, and the end of the universe would be the end of the record's growth. (Part 2 found the arrow of time you feel, which comes from the busy activity around you. This is a different thing: the direction the books are written in.)

Yeah-But has been waiting for this one. "Yeah, but a record with no ceiling can still stop being written."

It can. Nothing in the laws forces the writing to go on, and nothing forces it to stop. That the ticks keep coming goes in as an ingredient, the same way "the record counts events that happened" went in for the beginning, and a last line in the record would end this answer. One step is still being worked out: showing that our universe is the grid, with a record that keeps pace with its clocks. Steps 4 and 5 are exact statements about the grid.

Zoom out. Step 5 is done. The grid's laws can't tell a last moment from a first one. The record has a floor and no ceiling, so nothing forces a last page, and that the ticks keep coming goes in as an ingredient. Next, and why: all five steps are done, so here's the whole trip in one breath.

The whole trip, in one breath

Nothing written is ever erased, so even the last star going out goes in as one more line. What runs out is imbalance. The cost of the world's arrangement falls to a floor and stays there, and no run burns it at a steady rate forever: that's the gauge on E, the books' version of heat death. The record is the other dial. It has a floor and no ceiling, every tick adds a line, and the gauge doesn't pay for the lines, so the record can run on when the tank can't. In the grid of clocks and light, every stretch of time has a running clock and some clock's total never levels off, so there's no final stillness. The grid's laws can't tell a last moment from a first one, so nothing in them puts a last page in the record.

Your car runs out of gas. The universe runs out of imbalance. And its odometer, which has kept every mile since the first page, has no last page to turn to.

(The Ledger would like you to know that it keeps no stamp reading THE END. It looked. It noted that it looked.)