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

Everything we can see runs down. Stars burn through fuel they never get back. Heat flows from hot to cold and never the other way on its own, and an engine only works while heat has somewhere colder to go. Follow that to the end and you reach the usual story of how the universe ends: the last stars burn out, everything spreads toward one even temperature, and with no hotter side and no colder side, nothing is left to do work. Physicists call that heat death.¹

Heat death is the end of one thing: the supply of usable energy, the imbalance an engine runs on. The universe also holds something else, a record of everything that has happened.

Your car has a dial for each. The fuel gauge reads what's left to burn: when it sits on E, the engine coughs and you coast to a stop. The odometer reads what has happened. 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.

The car is a drawing of something real. The universe has these two dials. One is a gauge of what's left to burn: imbalance. The other is an odometer of everything that has happened: the record. The usual story of the end reads the gauge, and the gauge goes to 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.

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.

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. And running out of fuel doesn't stop the record being written, because the gauge doesn't pay for the lines. The answer rests on one premise: the writing of the record keeps going, step after step. Step 5 says how it enters the answer and what would break it.

The record. Reality keeps a permanent record of every event. Each entry in it is a comparison, one thing told apart from another. Telling two things apart takes information about both of them, and the least there can be is one yes or no. You're doing it right now: reading this tells one letter from another. Entries go in one after another. Nothing is ever erased: every event leaves a complete trace, and the past still exists. This is the record from What You Are, the first post in the series.

Physics already agrees on the part it covers: its basic equations never destroy information about the past. In quantum mechanics that is unitarity: the present fixes the past exactly, and running the equations backward recovers every earlier state. That's why the black hole information problem is a real problem. In 1976 Hawking calculated that a black hole destroys the information that falls into it, which clashed with unitarity.² Most physicists now expect the information to come back out in the radiation, and the open question is how. The record goes further than unitarity: it is reality's own record of every event, and it keeps every entry as written. Writing a new entry never alters an earlier one, as our institute has proved.³

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. Each of the five steps is an exact statement about a model world: a world with a short list of ingredients, so that what follows from them can be worked out completely. Two worlds come up. The books are pure bookkeeping: a record and one arrangement of comparisons, with no clocks and no light. The grid is 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. No run can burn that cost at a steady rate forever, whatever rule changes the arrangement, because a cost can't go below zero. Under the simple rule the books adopt, it also falls to a floor and stays there. 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. Each step in the writing adds one, so after enough steps 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, the even-temperature state the usual story ends in.
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 record, 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.

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 what that means exactly, in the books.

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. The laws that do the spreading never destroy information: the exact state at the end still fixes every earlier state. 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. No run can burn that cost at a steady rate forever, whatever rule changes the arrangement, because a cost can't go below zero. Under the simple rule the books adopt, it also falls to a floor and stays there. Here's how we know.

The price list. Every comparison has a price, and the price measures imbalance. Take two quantities of the same kind and write the comparison as a ratio x, how many times bigger one is than the other. The price has to be zero when the two match (x = 1), grow as they differ, and come out the same whichever one is put first (x or 1/x). Those three requirements, the rule for how the prices of combined comparisons add, and one calibration, how sharply the price rises near balance, leave exactly one formula:

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. The combining rule says: put two comparisons, x and y, together two ways, as xy and as x/y, and the prices of the two combinations have to satisfy J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y). Try it with x = y = 2. The left side is J(4) + J(1) = 9/8 + 0, and the right side is 1/2 + 1/2 + 1/8 = 9/8. Other formulas that look just as natural fail it. The squared logarithm, (ln x) × (ln x), is zero at balance, grows as the sides differ and treats x and 1/x alike, and it breaks the rule.⁴

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: if one side is 4 times the other, the other is 1/4 of it.) 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.

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

The rule. To watch a gauge fall, the books need a rule for how an arrangement changes. They adopt a simple one, so we can count.⁵ 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.

If there's nothing left to burn, doesn't that end everything that happens?

It ends the burning. Whether it ends what happens is a different question, and the books have a second dial for it.

Zoom out. Step 2 is done. No run burns the cost of the world's imbalance at a steady rate forever, and under the books' rule it falls to a floor and stays there. 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. Each step in the writing adds one line, so after enough steps 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, before anything has happened. 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.⁶

Each step writes a line. The world's basic process is the writing of the record, one step after another, and each step adds one entry. That's the process from What You Are, the first post in the series. So after n steps, the record is at least n lines longer than it was. Start with 3,000 lines: after a million steps there are at least 1,003,000. Name any number you like, and after that many steps 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 time the record does, and say every new line costs the gauge at least 0.01. A record that gets a line at every step then burns at least 0.01 at every step, 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 at every step, the gauge can't be what pays for them.

A line isn't bought out of the tank. A line is what an event is: two things are told apart, and that comparison is the entry. Its price goes on the running tally, which only grows, and nothing is drawn from the gauge. So the record runs on when the tank is empty.

Doesn't a car's odometer stop 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, as the argument above showed.

Zoom out. Step 3 is done. The record has a floor and no ceiling, each step in the writing adds a line, 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. Picture a three-dimensional grid of cells, with straight edges running along its three directions. In every cell sit four little clocks. Each clock counts through eight steps, one for each corner of a cube: three directions with two sides each make 2 × 2 × 2 = 8 corners, and a trip that touches every corner once takes eight steps. 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. This is the world of Did Time Have a Beginning?, and its laws are worked out exactly.⁷ 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 clocks share an edge, though two can sit on edges that point the same way. The next two facts hold at every value of every setting.

Start the grid blank: every clock at rest, no light anywhere. Call that state the blank. 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.

Why can't every clock rest? Count the clocks. 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 is rigid. It is analytic, which means a short stretch of it fixes all of it: 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.⁸

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, give or take a fixed handful of lines. Say the share is one line for every ten units, give or take five lines. A total of 1,000 puts at least 95 lines in the record. A total of a million puts in at least 99,995. Some clock's total passes every number, so the record does too. Whether the record of our universe keeps pace with the clocks is the open step that Step 5 names.

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, which physicists call polarizations. 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.⁹ 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. Play the grid's history backward, the way you'd run a film in reverse: run each clock in reverse, turn the light around, and the backward history obeys exactly the same laws. That reversal swaps beginnings and ends. A history that starts at the blank becomes one 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. (Time 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.)

One question is left, and it's the sharpest. Can a record with no ceiling still stop being written?

It can. Nothing in the laws forces the writing to go on, and nothing forces it to stop. That the writing keeps going, one step after another, goes in as a premise, the same way "a count of events that happened can't go below zero" goes in for the first page. A last step in the writing 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 writing keeps going goes in as a premise. 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. No run burns the cost of the world's arrangement at a steady rate forever, and under the books' rule it falls to a floor and stays there: 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, each step in the writing 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.