The Order of Time

2 LOSS OF DIRECTION

If more gently than Orpheus

who moved even the trees

you were to pluck the zither

the life-blood would not return

to the vain shadow . . .

Harsh fate,

but its burden becomes lighter

to bear, since everything

that attempts to turn back

is impossible. (I, 24)

WHERE DOES THE ETERNAL CURRENT COME FROM?

Clocks may well run at different speeds in the mountains and in the plains, but is this really what concerns us, ultimately, about time? In a river, the water flows more slowly near its banks, faster in the middle—but it is still flowing. . . . Is time not also something that always flows—from the past to the future? Let’s leave aside the precise measurement of how much time passes that we wrestled with in the preceding chapter: the numbers by which time is measured. There’s another, more essential aspect to time: its passage, its flow, the eternal current of the first of Rilke’s Duino Elegies:

The eternal current

Draws all the ages along with it

Through both realms,

Overwhelming them in both.⁹

Past and future are different from each other. Cause precedes effect. Pain comes after a wound, not before it. The glass shatters into a thousand pieces, and the pieces do not re-form into a glass. We cannot change the past; we can have regrets, remorse, memories. The future instead is uncertainty, desire, anxiety, open space, destiny, perhaps. We can live toward it, shape it, because it does not yet exist. Everything is still possible. . . . Time is not a line with two equal directions: it is an arrow with different extremities.

And it is this, rather than the speed of its passing, that matters most to us about time. This is the fundamental thing about time. The secret of time lies in this slippage that we feel on our pulse, viscerally, in the enigma of memory, in anxiety about the future. This is what it means to think about time. What exactly is this flowing? Where is it nestled in the grammar of the world? What distinguishes the past, its having been, from the future, its not having been yet, in the folds of the mechanism of the world? Why, to us, is the past so different from the future?

Nineteenth- and twentieth-century physics engaged with these questions and ran into something unexpected and disconcerting—much more so than the relatively marginal fact that time passes at different speeds in different places. The difference between past and future, between cause and effect, between memory and hope, between regret and intention . . . in the elementary laws that describe the mechanisms of the world, there is no such difference.

HEAT

It all began with a regicide. On January 16, 1793, the National Convention in Paris sentenced Louis XVI to death. Rebellion is perhaps among the deepest roots of science: the refusal to accept the present order of things.¹⁰ Among those who took the fatal decision was a friend of Robespierre called Lazare Carnot. Carnot had a passion for the great Persian poet Saadi Shirazi. Captured and enslaved at Acre by the Crusaders, Shirazi is the author of those luminous verses that now stand at the entrance of the headquarters of the United Nations:

All of the sons of Adam are part of one single body,

They are of the same essence.

When time afflicts us with pain

In one part of that body

All the other parts feel it too.

If you fail to feel the pain of others

You do not deserve the name of man.

Perhaps poetry is another of science’s deepest roots: the capacity to see beyond the visible. Carnot names his first son after Saadi. Sadi Carnot is thus born out of poetry and rebellion.

As a young man, he develops a passion for those steam engines that at the start of the nineteenth century are beginning to transform the world by using fire to make things turn. In 1824, he writes a pamphlet with the alluring title “Reflections on the Motive Power of Fire,” in which he seeks to understand the theoretical basis of the functioning of these machines. The little treatise is packed with mistaken assumptions: he imagines that heat is a concrete entity—a kind of fluid that produces energy by “falling” from hot things to cold, just as the water in a waterfall produces energy by falling from above to below. But it contains a key idea: that steam engines function, in the final analysis, because the heat passes from hot to cold.

Sadi’s pamphlet finds its way into the hands of a fierce-eyed, austere Prussian professor called Rudolf Clausius. It is he who grasps the fundamental issue at stake, formulating a law that was destined to become famous: if nothing else around it changes, heat cannot pass from a cold body to a hot one.

The crucial point here is the difference from what happens with falling bodies: a ball may fall, but it can also come back up, by rebounding, for instance. Heat cannot.

This is the only basic law of physics that distinguishes the past from the future.

None of the others do so. Not Newton’s laws governing the mechanics of the world; not the equations for electricity and magnetism formulated by Maxwell. Not Einstein’s on relativistic gravity, nor those of quantum mechanics devised by Heisenberg, Schrödinger, and Dirac. Not those for elementary particles formulated by twentieth-century physicists. . . . Not one of these equations distinguishes the past from the future.¹¹ If a sequence of events is allowed by these equations, so is the same sequence run backward in time.¹² In the elementary equations of the world,¹³ the arrow of time appears only where there is heat.* The link between time and heat is therefore fundamental: every time a difference is manifested between the past and the future, heat is involved. In every sequence of events that becomes absurd if projected backward, there is something that is heating up.

If I watch a film that shows a ball rolling, I cannot tell if the film is being projected correctly or in reverse. But if the ball stops, I know that it is being run properly; run backward, it would show an implausible event: a ball starting to move by itself. The ball’s slowing down and coming to rest are due to friction, and friction produces heat. Only where there is heat is there a distinction between past and future. Thoughts, for instance, unfold from the past to the future, not vice versa—and, in fact, thinking produces heat in our heads. . . .

Clausius introduces a quantity that measures this irreversible progress of heat in only one direction and, since he was a cultivated German, he gives it a name taken from ancient Greek—entropy:

I prefer to take the names of important scientific quantities from ancient languages, so that they may be the same in all the living languages. I therefore propose to call entropy the quantity (S) of a body, from the Greek word for transformation: ἡ τροπὴ.¹⁴

The page of the article by Clausius in which he introduces for the first time the concept and the word “entropy.” The equation provides the mathematical definition of the variation of entropy (S—S0) of a body: the sum (integral) of the quantity of heat dQ leaving the body at the temperature T.

Clausius’s entropy, indicated by the letter S, is a measurable and calculable quantity ¹⁵ that increases or remains the same but never decreases, in an isolated process. In order to indicate that it never decreases, we write:

ΔS ≥ 0

This reads: “Delta S is always greater than or equal to zero,” and we call this “the second principle of thermodynamics” (the first being the conservation of energy). Its nub is the fact that heat passes only from hot bodies to cold, never the other way around.

Forgive me for the equation—it’s the only one in the book. It is the equation for time’s arrow, and I could hardly refrain from including it in my book about time.

It is the only equation of fundamental physics that knows any difference between past and future. The only one that speaks of the flowing of time. Behind this unusual equation, an entire world lies hidden.

Revealing it will fall to an unfortunate and engaging Austrian, the grandson of a watchmaker, a tragic and romantic figure, Ludwig Boltzmann.

BLUR

It is Boltzmann who begins to see what lies behind the equation ΔS ≥ 0, throwing us into one of our most dizzying dives toward understanding the intimate grammar of our world.

Boltzmann works in Graz, Heidelberg, Berlin, Vienna, and then in Graz again. He liked to attribute his restlessness to the fact that he was born during Mardi Gras. He was only partly joking, since the instability of his character was real enough, oscillating as it did between elation and depression. He was short and stout, with dark, curly hair and the beard of a Taliban; his girlfriend called him “my dear sweet chubby one.” It was he, this Ludwig, who was the luckless hero of time’s directionality.

Sadi Carnot thought that heat was a substance, a fluid. He was wrong. Heat is the microscopic agitation of molecules. Hot tea is tea in which the molecules are very agitated. Cold tea is tea in which the molecules are only a little agitated. In an ice cube, warming up and melting molecules become increasingly agitated and lose their strict connections.

At the end of the nineteenth century, there were many who still did not believe in the existence of molecules and atoms: Ludwig was convinced of their reality and entered the fray on behalf of his belief. His diatribes against those who doubted the reality of atoms became legendary. “Our generation were at heart all on his side,” remarked one of the young lions of quantum mechanics years later.¹⁶ In one of these fiery polemics, at a conference in Vienna, a noted scientist¹⁷ maintained against him that scientific materialism was dead because the laws of matter are not subject to the directionality of time. Scientists are not immune from talking nonsense.

Looking at the sun going down, the eyes of Copernicus had seen the world turning. Looking at a glass of still water, the eyes of Boltzmann saw atoms and molecules frenziedly moving.

We see the water in a glass like the astronauts saw the Earth from the moon: calm, gleaming, blue. From the moon, they could see nothing of the exuberant agitation of life on Earth, its plants and animals, desires and despairs. Only a veined blue ball. Within the reflections in a glass of water, there is an analogous tumultuous life, made up of the activities of myriads of molecules—many more than there are living beings on Earth.

This tumult stirs up everything. If one section of the molecules is still, it becomes stirred up by the frenzy of neighboring ones that set them in motion, too: the agitation spreads, the molecules bump into and shove each other. In this way, cold things are heated in contact with hot ones: their molecules become jostled by hot ones and pushed into ferment. That is, they heat up.

Thermal agitation is like a continual shuffling of a pack of cards: if the cards are in order, the shuffling disorders them. In this way, heat passes from hot to cold, and not vice versa: by shuffling, by the natural disordering of everything. The growth of entropy is nothing other than the ubiquitous and familiar natural increase of disorder.

This is what Boltzmann understood. The difference between past and future does not lie in the elementary laws of motion; it does not reside in the deep grammar of nature. It is the natural disordering that leads to gradually less particular, less special situations.

It was a brilliant intuition, and a correct one. But does it clarify the difference between past and future? It does not. It just shifts the question. The question now becomes: why, in one of the two directions of time—the one we call past—were things more ordered? Why was the great pack of cards of the universe in order in the past? Why, in the past, was entropy lower?

If we observe a phenomenon that begins in a state of lower entropy, it is clear why entropy increases—because in the process of reshuffling, everything becomes disordered. But why do the phenomena that we observe around us in the cosmos begin in a state of lower entropy in the first place?

Here we get to the key point. If the first twenty-six cards in a pack are all red and the next twenty-six are all black, we say that the configuration of the cards is “particular,” that it is “ordered.” This order is lost when the pack is shuffled. The initial ordered configuration is a configuration “of low entropy.” But notice that it is particular if we look at the color of the cards—red or black. It is particular because I am looking at the color. Another configuration will be particular if the first twenty-six cards consist of only hearts and spades. Or if they are all odd numbers, or the twenty-six most creased cards in the pack, or exactly the same twenty-six of three days ago. . . . Or if they share any other characteristic. If we think about it carefully, every configuration is particular, every configuration is singular, if we look at all of its details, since every configuration always has something about it that characterizes it in a unique way. Just as, to its mother, every child is particular and unique.

It follows that the notion of certain configurations being more particular than others (twenty-six red cards followed by twenty-six black, for example) makes sense only if I limit myself to noticing only certain aspects of the cards (in this case, the colors). If I distinguish between all the cards, the configurations are all equivalent: none of them is more or less particular than others.¹⁸ The notion of “particularity” is born only at the moment we begin to see the universe in a blurred and approximate way.

Boltzmann has shown that entropy exists because we describe the world in a blurred fashion. He has demonstrated that entropy is precisely the quantity that counts how many are the different configurations that our blurred vision does not distinguish between. Heat, entropy, and the lower entropy of the past are notions that belong to an approximate, statistical description of nature.

The difference between past and future is deeply linked to this blurring. . . . So if I could take into account all the details of the exact, microscopic state of the world, would the characteristic aspects of the flowing of time disappear?

Yes. If I observe the microscopic state of things, then the difference between past and future vanishes. The future of the world, for instance, is determined by its present state—though neither more nor less than is the past.¹⁹ We often say that causes precede effects and yet, in the elementary grammar of things, there is no distinction between “cause” and “effect.”* There are regularities, represented by what we call physical laws, that link events of different times, but they are symmetric between future and past. In a microscopic description, there can be no sense in which the past is different from the future.*

This is the disconcerting conclusion that emerges from Boltzmann’s work: the difference between the past and the future refers only to our own blurred vision of the world. It’s a conclusion that leaves us flabbergasted: is it really possible that a perception so vivid, basic, existential—my perception of the passage of time—depends on the fact that I cannot apprehend the world in all of its minute detail? On a kind of distortion that’s produced by myopia? Is it true that, if I could see exactly and take into consideration the actual dance of millions of molecules, then the future would be “just like” the past? Is it possible that I have as much knowledge of the past—or ignorance of it—as I do of the future? Even allowing for the fact that our perceptions of the world are frequently wrong, can the world really be so profoundly different from our perception of it as this?

All this undermines the very basis of our usual way of understanding time. It provokes incredulity, just as much as the discovery of the movement of the Earth did. But just as with the movement of the Earth, the evidence is overwhelming: all the phenomena that characterize the flowing of time are reduced to a “particular” state in the world’s past, the “particularity” of which may be attributed to the blurring of our perspective.

Later on, I will delve into the mystery of this blurring, to see how it is tied to the strange initial improbability of the universe. For now, I will end with the mind-boggling fact that entropy, as Boltzmann fully understood, is nothing other than the number of microscopic states that our blurred vision of the world fails to distinguish.

The equation that states precisely this²⁰ is carved on Boltzmann’s tomb in Vienna, above a marble bust that portrays him as an austere and surly figure, such as I don’t believe he ever was in life. Many young students of physics go to visit his tomb, and linger there to ponder. And sometimes the odd elderly professor of physics as well.

Time has lost another of its crucial components: the intrinsic difference between past and future. Boltzmann understood that there is nothing intrinsic about the flowing of time. That it is only the blurred reflection of a mysterious improbability of the universe at a point in the past.

The source of Rilke’s “eternal current” is nothing other than this.

Appointed a university professor at just twenty-five years old; received at court by the emperor at the apex of his success; severely criticized by the majority of the academic world, which did not understand his ideas; always precariously balanced between enthusiasm and depression: the “dear sweet chubby one,” Ludwig Boltzmann, will end his life by hanging himself.

He does so at Duino, near Trieste, while his wife and daughter are swimming in the Adriatic.

The same Duino where, just a few years later, Rilke will write his Elegy.