Why Time Has a Direction

Milk poured into tea spreads until the whole cup is pale. The tea never separates back out.

Smoke rises from a chimney and thins into the air. It never collects itself and pours back down.

Film any of these and run the film backwards. Anyone watching spots it in a second.

You face one way in time as well. You remember last Tuesday. You do not remember next Tuesday. You can change what happens tomorrow. You cannot touch what happened yesterday.

So the world runs one way. That much is obvious. The puzzle is that the laws underneath it do not.

There is a second question about time, and this essay will not be answering it.

Philosophers have argued for a long time about time itself. Is the present moment objectively special? Are the past and the future equally real? Does time flow, or does it only seem to?

Those are real questions. They are not the question here, and nothing below takes a position on any of them. If you are waiting for an answer to them, it is not coming.

The question here is narrower, and it survives however that other argument turns out. It is not about time. It is about the things in time, and why they run one way.

The obvious explanation would be a law with a direction built into it. Some process at the bottom that only runs one way. Everything made out of it would inherit the one-wayness.

No such law has been found that does any of that work.

Gravity is the standard demonstration. Film a planet orbiting its star, then run the film backwards. What you see is another perfectly good orbit. Same ellipse. Same areas swept in the same times. Nothing in the law objects.

The general version is stronger. Take a system of any complexity whose forces depend on where the particles are. Reverse every particle's velocity at a single instant. The system now retraces its own history and undoes whatever it did. Reversing the motion reverses the whole sequence of changes. Hold on to that condition about the forces. It does real work later, and not every force meets it.

Now the part that needs care. Two different claims get run together here.

The first claim is that the dynamics can be run backwards. Give it the state at the end and it hands you the state at the beginning. Call this invertibility.

The second claim is stronger. It says the reversed history is itself lawful. If the theory permits a process forwards, it permits the process backwards. Call this reversal invariance. This is the one the essay needs, and invertibility does not give it to you.

Getting the second claim right means being careful about what a state is. A state has to describe the world at one instant. Positions plus velocities will not do. A velocity is a rate of change, and a rate is not a feature of a single instant.

So reversal is an operation on states. Reversal invariance is the claim that inverting a lawful sequence of states leaves it lawful.

Held to that standard, fewer theories pass than you would expect. Classical electrodynamics fails. So do quantum mechanics, quantum field theory and general relativity.

What they have instead is a weaker property. For each theory there is a transformation that leaves the particles' positions untouched. It turns lawful sequences into lawful reversed ones. Positions are all it preserves.

The magnetic field shows why the difference matters. A field is not the rate of change of anything. So no reversal transformation can simply flip it. The textbook recipe reverses it along with the velocities, and that step is never justified.

The weaker property is enough for this essay. A cup breaking, soup cooling, a body ageing — these are all matters of where the constituent particles are. Positions are exactly what the weaker invariance leaves alone. The conflict with everyday experience survives.

There is one measured exception. It is worth stating once and putting in its place.

Neutral kaons are unstable particles that oscillate into their antiparticles and back. The two directions do not run at the same rate. An experiment at CERN in 1998 found the beam decaying as kaon slightly more often than as antikaon, by about two-thirds of one per cent.

The weak interaction had form here. It was caught violating parity — the symmetry between left and right — in the 1950s. In 1964 the combination of charge and parity fell too, on these same particles. Apply all three inversions together, charge and parity and time, and as of the account this essay is drawing on the symmetry had survived every test put to it.

None of it does any work for the cup. Speaking, chemistry, the forces holding a table together: none of them lean on the decays where the asymmetry lives. The exception is real. It is measured. It explains nothing about breakfast. It does not come up again.

Relativity does not supply the missing direction either. A relativistic spacetime can carry an orientation. That means a consistent labelling, at every event, of which half of the light cone counts as future. Physics uses spacetimes where this labelling is possible. Fixing it is a matter of picking one class of cones and calling them the future ones.

That structure is assumed by everything in this essay. It labels the two ends of time. It says nothing about why the world runs from one end to the other.

Here is the standard demonstration.

Fill half a tank with blue-dyed water and half with clear, with a divider between them. Slide the divider out. The colour spreads until the whole tank is the same pale shade. Then it stays that way. You can watch for as long as you like. It will not separate again.

Now look closer. Take any single collision between two molecules and run that fragment backwards. Measure everything about it. It obeys the laws of collision exactly as it did forwards.

So the film has a strange property. Every individual event in it is lawful in reverse. The reversed whole is absurd.

The way out is counting.

Start with the description you actually have. You cannot write down where every molecule is. What you can write down is that the top half is blue and the bottom half is clear. Or that the whole tank is uniform. That is a handful of numbers instead of Avogadro's number of them.

Call that coarse description a macrostate. Each macrostate covers an enormous set of exact microscopic arrangements. Any one of them would look identical to you. Call one of those exact arrangements a microstate.

Now count the microstates. Done properly this means measuring a volume, because the arrangements form a continuum rather than a list. The space you measure the volume in is called phase space. It has an axis for every position and every velocity of every particle, so a single point in it fixes the entire system.

The uniform macrostate occupies vastly more phase-space volume than the separated one. The ratio has no useful everyday comparison.

So start the tank separated and let the molecules do what molecules do. Almost every lawful path open to it runs into the uniform region and stays there for any stretch of time anyone will ever watch.

Entropy measures that volume. It belongs to the actual tank in front of you, by way of the macrostate its exact arrangement falls under.

Two claims often get attached to this argument. It supports neither of them.

The first is that entropy must increase. It does not say that. It says the paths on which entropy increases outnumber the rest so heavily that you will never see the alternative in a system of any size.

Separation is not forbidden. Every law involved permits it. It just does not happen. Shrink the tank until only a handful of molecules of each colour are left, and you will watch it un-mix from time to time. At those numbers the counting is not lopsided enough to settle anything.

The second is that entropy is disorder. It does not say that either. "Order" here is a technical word for a definite arrangement as against a mixed one. It carries none of its ordinary sense. An ordered arrangement need not be interesting, and an interesting one need not be ordered.

The argument is about how many arrangements answer to a description. It is about nothing else.

There is one more thing to notice about counting. Which arrangements count as the same depends on how coarsely you describe the system. Measure the tank on a finer grid and you are counting a different set. Track a property you were not tracking and the same thing happens. The number comes out different.

That dependence is a real feature of the definition. A position at the end of this essay is built on it.

Two objections landed on this argument while Boltzmann was alive. Neither has been withdrawn.

The first is Loschmidt's. Take the tank after it has mixed. Reverse every molecule's velocity at a single instant.

Nothing about being a tank of pale water fixes which way its molecules happen to be moving. So the reversed arrangement is just as much a tank of uniformly pale water as the original was. Run it forwards from there and it unmixes. The colours separate, lawfully, by the same dynamics that mixed them.

Reversing velocities does not shrink the phase-space volume you started from. The reversed set is exactly as large as the set it came from.

So whatever is doing the work in the counting argument, it is not the dynamics alone. For every path that mixes there is a partner that unmixes. The laws have no preference between them.

The second objection is Zermelo's. It stands on a theorem Poincaré proved in 1890.

Confine a classical system to a box. Give it a definite finite total energy, so its phase space is a finite region. Note that the evolution preserves phase-space volume. Under those conditions the system returns arbitrarily close to any state it has ever occupied. Then it does it again. Then it keeps doing it.

Those three conditions are not decoration. They are part of the statement, and they go wherever it goes. Anyone who quotes the theorem without them is claiming more than it says.

What does follow is narrow, and worth saying exactly. Entropy increasing for ever, without exception, is not something the dynamics can give you.

It does not follow that every isolated system recurs. It does not follow that anything interesting recurs on a timescale that matters. For a system of macroscopic size the fluctuations are smaller and rarer in proportion. They are real, and they are set at intervals with no bearing on anything.

There is one piece of history worth going through slowly, because it shows where the asymmetry actually came from.

In 1872 Boltzmann published a result that looked like a proof. Track a gas over time, he showed, and a certain quantity only ever moves in one direction. That quantity is essentially entropy. So entropy always increases, and the second law stops being a summary of experience and becomes a theorem of mechanics.

It is not a proof. Here is why.

To get the result, Boltzmann had to say something about two molecules that are about to collide. He assumed their velocities are uncorrelated. Knowing how one of them is moving tells you nothing about how the other is moving. This is called the assumption of molecular chaos.

For two molecules approaching each other, that is a reasonable thing to assume. They have not met. There is no reason for their motions to be related.

Now look at the same two molecules just after they collide. They are not uncorrelated any more. They have just bounced off each other, and how one leaves depends on how the other left. Correlating them is exactly what a collision does.

So the assumption is true before a collision and false after one. That is an asymmetry in time, and Boltzmann put it in by hand.

That is the whole problem. He fed a time-asymmetric assumption into the setup and drew a time-asymmetric conclusion out of it. The mechanics underneath the assumption is symmetric. The one-wayness was already sitting in the assumption before the proof started. It did not come from the laws.

The theorem does not prove the thing it appeared to prove.

The objection was pressed on him within a decade. A sharper version was revived in the 1890s. His own eventual position was symmetric, and he knew what that cost.

The reply that survives both objections is the counting argument restated with more care. The overwhelming majority of paths through any macrostate short of maximum entropy have higher entropy in their future. Loschmidt's reversed states are lawful and they exist. They are also an unimaginably thin set among the possibilities. Recurrence is real, and it sits beyond every interval anyone will ever care about.

That reply survives both objections. But it has a consequence nobody wanted, and the rest of this essay is about that consequence.

The majority argument has no direction in it. Run it the other way and it says the same thing with the same authority. The overwhelming majority of paths through any ordinary macrostate had higher entropy in their past as well.

Every ordinary macrostate is a local minimum for almost every path that passes through it. Counting arrangements is not the sort of operation that could know which way time runs.

Here is the same counting argument, pointed at yesterday.

Take a glass of water with a half-melted ice cube in it. Condition on that and nothing else. Then ask what the cube was doing five minutes ago.

The counting gives an answer. The answer is that five minutes ago the cube was more melted than it is now. Ice cubes in warm water are overwhelmingly likely to be closer to equilibrium a few minutes either side of any moment you pick. Equilibrium here means the state the system runs into and does not leave on any ordinary timescale — for the glass, all ice melted and the temperature even.

Run the reasoning forwards and it is right. Run it backwards and it says entropy has been falling all through the past.

Every record you have contradicts that.

Try patching it. Suppose the even-handed distribution held five minutes ago instead, over the macrostate of the fully unmelted cube. Now the present and the future come out right. The last five minutes match your memory.

Ten minutes ago then goes wrong in the same way, with ice spontaneously un-melting. Push the posit back another five minutes and the same thing happens further back still. Every local repair fails at the same point, one step earlier.

Now watch what the failure does to evidence.

The distribution being used here is a definite object. It is spread evenly, on the standard measure, over every microstate compatible with the present state of the world.

Everything you know is already inside that present state. Your memories are arrangements of your brain now. A photograph is an arrangement of paper and silver now. The book on the shelf, the fossil in the rock, the light arriving from a star — all of it is the world's present condition. All of it is already counted in.

So the distribution has taken your evidence into account. It still says the past ran the other way.

On its account the photograph was yellower and more worn before it was less so. It assembled itself out of drifting paper and dust rather than being taken of anybody. There almost certainly never was a child who looked like the one in the picture.

Fetching more photographs does not help. The likeliest story remains that they all arrived in this room together by coincidence.

Stated flatly: there is nothing about the world's present condition that can count as evidence its entropy was ever lower.

And almost nothing you know about the past is known by running the statistics backwards. It is known from records. Records are exactly what has just stopped being trustworthy.

The experiments that confirmed the mechanics this argument is built from are among the things the argument now says probably never happened.

A photograph of a birthday party makes the same point in one image. Given the present arrangement alone, the least improbable history of that photograph is not a party. It is a fluctuation that produced a photograph. That follows from the same reasoning that correctly tells you the photograph will fade in the future.

What has gone wrong here is a double standard. The argument helps itself to a low-entropy earlier state whenever it faces the future. It refuses itself the same assumption when it faces the past. The asymmetry is an undeclared assumption, and the physics derived none of it.

The counting has no direction in it. Whatever reason it gives you to expect higher entropy later, it gives you the identical reason to expect higher entropy earlier. Once you stand outside the problem rather than inside your own memory, the question becomes why entropy is lower in one direction than the other.

One escape route was tried early. It fails hard enough to be worth closing.

Suppose the whole universe sits at equilibrium for eternity. Everything we see is then a large chance fluctuation on its way back down. The waiting time is not an objection, since only a fluctuation big enough to produce observers gets observed.

The picture can be checked, because it predicts something specific. If this patch of order is a fluctuation, then everywhere you have not yet looked should be a mess. The cheapest fluctuation is always the smallest one that does the job.

Every place anyone has looked has been orderly in the same way as everywhere else. Bones where the geologist predicted bones. The same date for the revolution in every book. Stars like the other stars. The prediction fails on the data.

It fails in a worse way too. If your memories are states of your brain, then a brain fluctuating into existence complete with those states is cheaper than a whole universe containing that brain. On that hypothesis your memories are almost all false — including the ones you would use to check the hypothesis.

Tightening it does not rescue it. Assume only that you are typical among observers exactly like you, and it gets worse. Observers with your memories fluctuate alone far more often than they occur inside a real low-entropy history.

A hypothesis that destroys the evidence for itself is not one you can hold.

That leaves the problem where the last section put it, now with the bill attached. The counting argument is correct about the future and catastrophic about the past. Nothing inside it distinguishes the two.

Every local repair failed one step further back. The pattern does not stop.

Push it all the way and there is only one place left for a posit to sit. It has to be about the entire universe, at the earliest time there is. Anything smaller goes bad earlier than the thing it was brought in to fix.

So posit it. The universe began in a macrostate of extraordinarily low entropy. The statistics are then conditioned on that as well as on the present. The name it goes by is the Past Hypothesis.

Feynman arrived at the same move in a lecture in 1964, without the machinery and with no way to make it precise. He said the physical laws needed an extra statement added to them: that the universe in the past was more ordered than it is now, in the technical sense of order and nothing else.

He was also clear about what sort of thing he had added. It is lopsided in time in a way the rest of physics is not. And it is not a law in the ordinary sense. It describes the condition the world was in, rather than the rule by which the world develops. He filed it under astronomical history, and allowed that it might become physics one day.

Boltzmann had floated a version of it too, as one option among several. He called it unprovable when he did.

What the posit buys is the past back.

Condition the statistics on a low-entropy beginning as well as on what is in front of you. The conditional probabilities over histories change. The likeliest past compatible with a half-melted cube is no longer the more-melted one. The histories that would have delivered it are mostly cut away by where the universe started.

The photograph is likeliest to have been taken of somebody. Records recover their standing as evidence. The thermodynamic arrow becomes the direction that leads away from the special beginning.

The precise form matters, because the obvious version is wrong. It is not a matter of conditioning on the beginning and reading forwards.

Start with the even-handed distribution over the present macrostate. Condition it on everything we believe about the world's large-scale past. That comes to the same thing as conditioning on the posit. Now evolve forwards. Out comes what we expect of the future.

Now run the mirror operation. Condition instead on everything we believe about the large-scale future, and evolve backwards. Out comes far less than we credit ourselves with knowing.

The repair does not make the two directions symmetrical.

One more consequence, and it is the reason this section is not only about thermodynamics. The thing that makes the second law hold statistically across the history of the world is the same thing that makes records of the past possible. That is one fact, seen from two sides.

There is no matching hypothesis about the end of the universe. We condition on a beginning and not on a finish. That asymmetry of conditioning is where causes sitting before effects comes from. A future boundary condition would be perfectly lawful. From the inside it would feel like a long run of conspiracies.

Now the honest part. Naming the posit is not explaining it.

In the finished account it sits as a contingent empirical fact, next to the dynamical law and the statistical rule. Nothing in the dynamics picks it out or requires it.

The question has moved. It used to be a puzzle about every kitchen in the world. It is now a puzzle about the first instant of the universe. That is a real gain. It is not an answer.

It also raises a problem it cannot pay. A low-entropy beginning sounds wrong the moment you look at what the early universe was actually like. Hot, dense, smooth, close to uniform, close to thermal. That is a description of a system at equilibrium. Equilibrium is maximum entropy, not minimum.

The next section pays it.

The early universe was a hot gas in expanding thermal equilibrium. Thermal equilibrium is the maximum-entropy condition. So on the face of it the universe began at an entropy maximum, and the second law needs it to have begun near a minimum.

The reply most people reach for is about the ceiling. The maximum was small back then and has been growing ever since as the universe expanded. Entropy has spent its whole history chasing a rising ceiling.

That reply does not survive being run backwards.

Apply it to a universe that eventually recollapses. During the contraction the same reasoning runs in reverse. As the permitted maximum shrinks, entropy would have to come down with it. The second law would fail wholesale at the far end.

You can try to escape by hoping the universe never recollapses, or that the turnaround is too far off to matter. But a collapsing region is precisely what a black hole is. If entropy could reverse in a collapse, it would reverse near black holes. There, instead, the second law holds firmly.

The real answer is that once gravity is in the accounting, the accounting runs the other way.

For a gas in a box, high entropy means spread out. A clumped gas is a low-entropy gas.

For matter under its own gravity that reverses. Gravity attracts universally, and there is no opposite charge to screen it. So clumping raises entropy. Matter that starts smooth and ends up in stars, galaxies and black holes has been climbing in entropy the entire way. Collapse to a black hole sits at the top of the scale.

So a smooth early universe is not the equilibrium state it resembles. Measured against what gravitating matter could have been doing instead, smooth is extraordinarily special.

The trail is visible from where you are sitting. The Earth does not gain energy from the sun on balance. It radiates back as much as it takes in. What arrives is a small number of high-frequency photons. What leaves is a much larger number of low-frequency ones. That difference in the count is the low-entropy supply that everything alive runs on.

Trace it back. The sun is a low-entropy source because it condensed out of a cloud that was smoother still. That goes back to the smoothness of the early universe.

Now the number, with what it counts and what it assumes.

The instrument is a formula for black-hole entropy, due to Jacob Bekenstein in 1972 and Stephen Hawking in 1975. Entropy is proportional to the area of the hole's horizon — the surface past which nothing returns — and the constant is a quarter, in the appropriate units.

The calculation assumes the universe contains about 10^80 baryons. Baryons are the heavy particles ordinary matter is made of, protons and neutrons.

Penrose is explicit that this is an assumption rather than a measurement. And the uncertainty runs in a safe direction. If the true number is larger than 10^80, the final result comes out more extreme.

On that basis the entropy in the background radiation comes to about 10^88. That was once thought to be the largest single contribution in the universe.

A universe with all its mass in solar-mass black holes reaches 10^100. A realistic estimate built from galactic core masses gives 10^101. Applying the same formula to the entire mass of the universe gives a figure of 10^123 for the final crunch. That last number carries a condition worth keeping in view: it is computed for a closed universe that recollapses, which is the case the calculation assumes. It is the ceiling on that assumption, not a general result about any universe.

Turn that into a volume of phase space and you get the precision of the initial state. It is one part in 10^(10^123), measured against that same recollapsing ceiling. It barely matters which target volume you compare against. At these magnitudes, subtracting one exponent from the other leaves the answer essentially unchanged.

A qualification travels with all of this. Reading that ladder as though a black hole were simply the highest-entropy thing there is goes further than the argument will bear. A black hole maximises entropy within a fixed region. The actual high-entropy end state of a universe that keeps expanding looks more like empty space thinning out indefinitely. The specialness of the smooth beginning does not depend on the unqualified version, and this essay does not assert it. The qualification comes from inside the essay's own sources.

Two familiar ideas get offered as ways of making this problem disappear. Neither of them does.

The expansion of the universe does not cause the arrow. The universe getting bigger is not the thing that makes entropy rise. The attempt to make expansion the driver has been pressed and does not hold up.

Inflation does not dissolve the specialness either. Inflation is offered partly as an account of why the universe is so uniform. But nothing in the mechanism treats the two temporal directions differently. So it cannot by itself account for the difference between the two ends. And the conditions needed to get an inflating patch started are at least as restrictive as the uniformity it was brought in to explain.

There is a proposal on the table. It is a proposal rather than settled physics.

The suggestion is a constraint on the Weyl curvature. That is the part of spacetime curvature which describes tidal distortion and gravitational waves, as against the part fixed directly by the matter present. The constraint says it vanishes, or does something very close to vanishing, at initial singularities and not at final ones. Penrose calls it the Weyl Curvature Hypothesis.

If it holds, the big bang and the big crunch are different kinds of object. The asymmetry the Past Hypothesis merely records would then follow from something structural about singularities.

Penrose is explicit that this leaves the main question standing. We would still need to understand why a time-asymmetric constraint should apply in the first place. And the structure of singularities cannot be settled without a quantum theory of gravity, which nobody has.

Everything so far has been about one arrow. There are several. The claim worth examining is that they have a common parent.

The thermodynamic arrow is the one already covered. One feature of it belongs here rather than earlier. The direction it picks out is consistent throughout the observable universe. It is not a local convention that could have come out differently in another galaxy.

The cosmological arrow is the expansion. It distinguishes the two directions of time, so it belongs on the list. It is not the parent of the others, and this is worth saying plainly, because the temptation is strong. The universe getting bigger is not the thing that makes entropy rise. So it goes on the list with no causal claim attached.

The radiative arrow is that waves leave their sources and spread outward. The equations governing them permit incoming solutions just as readily. The world only shows you the outgoing ones.

The diagnosis that survives scrutiny does not derive this from the second law. Down at the level of single emissions and absorptions there is no asymmetry at all. Any one absorber does the job of a coherent sink perfectly well.

What is missing from the world is coherent sinks on a large scale. Matter clumps into things that emit in an organised way. It does not clump into their absorbing counterparts.

So this arrow is not a child of the thermodynamic one. It points the same way, for the same underlying reason. That reason turns into a question about why matter ended up clumped into emitters in the first place.

Then the arrow this essay opened with: records, causes, memory.

A record is a relation between the states of a device at the two ends of an interaction. That is what makes it cheap. The inference it licenses runs from two times to a third lying between them, rather than from one time to another.

To work out by retrodiction whether a particular billiard ball was struck ten seconds ago, you would need the complete present state of every ball on the table. A record of it gives you the same conclusion from a single bit.

The catch is that the device has to have been in its ready condition at the far end of the interaction. Asking what put it there starts a regress. The regress terminates exactly where the last section terminated.

Causation runs on the same machinery. We treat what lies ahead as hanging on our present choices and what lies behind as settled. The reason is not that the past has some special metaphysical fixity.

It is that almost any small present difference we can bring about translates into large differences later. Almost none of them translates into a difference earlier — except by way of records, and records are not among the things our hands are on. That account starts from an un-argued conception of what falls under our direct control, and the source is candid that it does.

One popular claim here needs qualifying. "Records must point to the past" is not strictly true. The absence of records of the future is a contingent feature of this world, rather than a necessity of what a record is.

There is a well-known proposal for how one cosmological posit gets distributed down to every small system in the world. Treat each quasi-isolated subsystem as getting the even-handed statistical treatment at the moment it is formed.

The proposal has been examined closely and it does not hold up. There is no principled moment of formation, and no principled boundary either. The glass, the room, the building, the city: which you pick changes the distribution radically.

From the other side, the objection is that the move does not explain the alignment at all. It writes the alignment into the way it describes the world, then presents that as a finding.

What is left is the plainer account. The small systems all point the same way because they all inherit their condition from the same early universe.

Finally the quantum arrow, which is where this essay meets the previous one.

The distinction drawn earlier does the work here. Unobserved, a quantum state evolves smoothly and reversibly, conserving information. Observed, on the textbook account, it collapses. Collapse is not reversible, because two different states can collapse onto the same one. You cannot run the operation backwards.

That is a genuine directedness in the microdynamics, if collapse is a physical process.

It does not do the job people hope. Even granting a physical collapse, it is very hard to see how collapse by itself could account for the Past Hypothesis. Explaining why entropy rises was never the difficult part. Explaining why it was low to begin with is.

And on the no-collapse reading there is nothing to explain from. Observation is just entanglement. The whole procedure stays reversible. There is no intrinsically quantum arrow of time at all.

Whichever way that argument goes, the arrow is left to statistics and the boundary condition.

One proposal goes further, and it is worth reporting as a proposal. In the GRW theory — named for Ghirardi, Rimini and Weber — wave functions occasionally localise on their own, at random, with a fixed probability per particle per unit time. The rate is low enough to be invisible for a single particle and fast enough to be decisive for anything macroscopic.

The suggestion is that these jumps are exactly the small perturbations statistical mechanics needs. Normal microstates are stable under tiny disturbances and abnormal ones are not. So the jumps would drive systems toward equilibrium as a consequence of the dynamics alone, with nothing added by hand.

The resulting account has two fundamental laws and one contingent fact, where the standard one had three laws and a fact. It needs no separate statistical postulate.

The person making the proposal attaches his own hedge to it, and it is a large one. All of this follows only on a very big if.

What all of this leaves is the shape Feynman described. You do not get irreversibility by reading it off the laws. It sits a long way from them. It takes a great deal of analysis to connect the two. It matters more to how the world works than almost anything the laws state directly. Knowing them hands you no understanding of it.

Five positions. Each one buys something and pays for it. No row is clean.

The Past Hypothesis. It buys working retrodiction, records restored to the status of evidence, and one arrow where there were five. The same posit that makes the second law hold statistically across the world's history is the thing that makes knowledge of the past possible.

It pays with an unexplained boundary condition. That condition sits in the account as a contingent fact rather than a law. Its specialness is measured at one part in 10^(10^123), against the recollapsing-universe ceiling the figure assumes.

Two objections against it do not agree with each other.

The first says the posit is hard to have on the cheap. It is difficult to see how a time-symmetric physics could require one end of the universe to be special without requiring both ends to be.

The second says that objection misfires. A constraint explains only if you can specify it without reference to what the laws will make of it. You can say what is odd about a low-entropy beginning by describing the state itself. It occupies a vanishingly small share of the available phase space, and that description needs no reference to what the dynamics will do with it. A special ending can only be picked out by where its evolution leads.

This essay carries both objections and settles neither.

Penrose. It buys a constraint: the Weyl curvature vanishing, or nearly so, at initial singularities and not at final ones. If it holds, the big bang is a different kind of object from the big crunch and from the insides of black holes. That supplies the asymmetry the Past Hypothesis only records.

It pays by being a hypothesis. Its author says, in the same passage, that we would still need to understand why a time-asymmetric constraint should apply at all. He adds that the structure of singularities cannot be settled without a quantum theory of gravity nobody has. That deferral is the price, in his own statement of it.

Perspectivalism. It buys the specialness dissolved.

Entropy is defined relative to a coarse description. Which description you use is not arbitrary. It is fixed by which variables you physically interact with, and you interact with a minute fraction of them.

So the low entropy of the past may be a fact about the subset of the world we are coupled to, rather than about the exact state of the world. On this reading the specialness belongs to us and our coupling. The universe itself need not have been in any remarkable condition.

Among enormously many subsystems, some will happen to couple to variables that had a particular value early on. For those subsystems entropy rises, records accumulate, and there is an arrow. We are one of them.

It pays with its own author's hedging, and this essay reports that as his rather than dressing it as an outside objection.

The text that advances the position concedes in a footnote that the notion of a low-entropy initial state is a long way from being well understood. It points the reader at a critique of the question itself. Its author says he is not sure the story is plausible, and knows of no better one, and names the Past Hypothesis as the alternative. His own grading puts this claim among the ideas he finds attractive and that are far from confirmed or widely accepted.

That price is not a rebuttal, and this essay does not stage one. No source it works from contains a developed reply.

Primitivism. It buys the arrow as intrinsic structure, not reduced to entropy at all.

The position attacks the reduction at the root. The laws are not in fact time-reversal invariant, since CP violation plus the CPT theorem gives T violation. So the standard argument fails at its first step. Time reversal for electromagnetism was never mere sequence reversal anyway, and requires an operation on instantaneous states. That is the point the second section spent its length on, conceded here and turned against the reducer. And laws alone explain very little; almost all explanation runs through initial conditions.

Then the positive step.

From any state after the beginning, entropy increase is typical in both temporal directions. Toward the past, typical behaviour never actually happens.

If typicality explains anything, something has to say why it works in one direction and not the other. What does that work is a fact about the order of production. States at the earlier end bring the later ones into being, and not the reverse. That is a direction of time doing the explaining rather than being explained.

Turn the whole account over and the entropy gradient is explained by the direction. What has to be added to the mathematics is an orientation. CP violation appears to demand one already.

It pays an ontological bill its author states himself: primitivism about laws of temporal evolution, and primitivism about the direction of time's passage. Both are bought in order to have an explanation the opposing account has to do without.

There is an epistemic cost as well. An intrinsic orientation that shows itself only in rare and recondite circumstances is postulated rather than determined. Meanwhile the passage of time is supposed to be ubiquitous and manifest.

One disagreement this essay carries rather than settles. Does one event count as earlier than another because of overwhelming statistical preponderance, with any backward influence simply too well concealed to detect? Or is the ordering primitive? Both sides are argued at strength.

No double standards, and where that leads. This last position comes in two parts. They are worth keeping apart, because one is a rule about how to argue and the other is a claim about the world.

The first is a discipline, and this essay has been operating under it since the fourth section. Do not apply an argument in one temporal direction that you would refuse in the other. When you catch yourself doing it, the asymmetry you are getting out is the one you put in.

Recognising an asymmetric assumption does not discharge it. It moves the explanatory burden. And the same discipline fixes the target. What requires an account sits at the early end of the entropy gradient. The late end needs none.

It buys every explanation in this essay having to survive it. It pays nothing, because it is not a position about the world.

The second is a position. It is a research programme with a declared premise, rather than a result.

Symmetry is the default in science, and departures need justifying. So an interaction in the future is as good a reason to think two systems are not independent as an interaction in the past. Pursued into quantum mechanics, this points at models with advanced action — influences running from later to earlier.

It buys a route to the low-entropy past that does not need an unexplained one-ended posit. It pays by presupposing the block universe rather than arguing for it, by the author's own declaration. The position inherits an unargued premise. And the advanced-action programme is offered as a direction for further work, not as a finding.