Seven Brief Lessons on Physics (4 page)

Probability, Time, and the Heat of Black Holes

Along with the great theories that I have already discussed and that describe the elementary constituents of the world, there is another great bastion of physics that is somewhat different from the others. A single question unexpectedly gave rise to it: “What is heat?”

Until the mid-nineteenth century, physicists attempted to understand heat by thinking that it was a kind of fluid, called “caloric,” or two fluids, one hot and one cold. The idea turned out to be wrong. Eventually James Maxwell and the Austrian physicist Ludwig Boltzmann understood. And what they understood is
very beautiful, strange, and profound—and takes us into regions that are still largely unexplored.

What they came to understand is that a hot substance is not one that contains caloric fluid. A hot substance is a substance in which atoms move more quickly. Atoms and molecules, small clusters of atoms bound together, are always moving. They run, vibrate, bounce, and so on. Cold air is air in which atoms, or rather molecules, move more slowly. Hot air is air in which molecules move more rapidly. Beautifully simple. But it doesn’t end there.

Heat, as we know, always moves from hot things to cold. A cold teaspoon placed in a cup of hot tea also becomes hot. If we don’t dress accordingly on a freezing cold day, we quickly lose body heat and become cold. Why does heat go from hot things to cold things and not vice versa?

It is a crucial question because it relates to the nature of time. In every case in which heat exchange does not occur, or when the heat exchanged is negligible, we see that the future behaves exactly like the past. For example, for the motion of the planets of the solar system heat is almost irrelevant, and in fact this same motion could equally take place in reverse without any law of
physics being infringed. As soon as there is heat, however, the future is different from the past. While there is no friction, for instance, a pendulum can swing forever. If we filmed it and ran the film in reverse, we would see movement that is completely possible. But if there is friction, then the pendulum heats its supports slightly, loses energy, and slows down. Friction produces heat. And immediately we are able to distinguish the future (toward which the pendulum slows) from the past. We have never seen a pendulum start swinging from a stationary position, with its movement initiated by the energy obtained by absorbing heat from its supports. The difference between past and future exists only when there is heat. The fundamental phenomenon that distinguishes the future from the past is the fact that heat passes from things that are hotter to things that are colder.

So, again, why, as time goes by, does heat pass from hot things to cold and not the other way around?

The reason was discovered by Boltzmann and is surprisingly simple:
it is sheer chance
.

Boltzmann’s idea is subtle and brings into play the idea of probability. Heat does not move from hot things to cold things due to an absolute law: it does so only with
a large degree of probability. The reason for this is that it is statistically more probable that a quickly moving atom of the hot substance collides with a cold one and leaves it a little of its energy, rather than vice versa. Energy is conserved in the collisions but tends to get distributed in more or less equal parts when there are many collisions. In this way the temperature of objects in contact with each other tends to equalize. It is not impossible for a hot body to become hotter through contact with a colder one: it is just extremely improbable.

This bringing of
probability
to the heart of physics, and using it to explain the bases of the dynamics of heat, was initially considered to be absurd. As frequently happens, no one took Boltzmann seriously. On September 5, 1906, in Duino, near Trieste, he committed suicide by hanging himself, never having witnessed the subsequent universal recognition of the validity of his ideas.

In the second lesson I related how quantum mechanics predicts that the movement of every minute thing occurs by chance. This puts probability into play as well. But the probability that Boltzmann considered, the probability at the roots of heat, has a different nature and is independent from quantum mechanics. The probability in play in the science of heat is in a certain sense tied to our
ignorance
.

I may not know something with certainty, but I can still assign a lesser or greater degree of probability to something. For instance, I don’t know whether it will rain tomorrow here in Marseilles, or whether it will be sunny or will snow, but the probability that it will snow here tomorrow—in Marseilles, in August—is low. Similarly with regard to most physical objects: we know something but not everything about their state, and we can make predictions based only on probability. Think of a balloon filled with air. I can measure it: measure its shape, its volume, its pressure, its temperature . . . But the molecules of air inside the balloon are moving rapidly within it, and I do not know the exact position of each of them. This prevents me from predicting with precision how the balloon will behave. For instance, if I untie the knot that seals it and let it go, it will deflate noisily, rushing and colliding here and there in a way that is impossible for me to predict. Impossible, because I know only its shape, volume, pressure, and temperature. The bumping about here and there of the balloon depends on the detail of the position of the molecules inside it, which I don’t know. Yet even if I can’t predict
everything exactly, I can predict the probability that one thing or another will happen. It will be very improbable, for instance, that the balloon will fly out of the window, circle the lighthouse down there in the distance, and then return to land on my hand, at the point where it was released. Some behavior is more probable, other behavior more improbable.

In this same sense, the probability that when molecules collide heat passes from the hotter bodies to those that are colder can be calculated, and turns out to be much greater than the probability of heat moving toward the hotter body.

The branch of science that clarifies these things is called “statistical physics,” and one of its triumphs, beginning with Boltzmann, has been to understand the probabilistic nature of heat and temperature, that is to say, thermodynamics.

At first glance, the idea that our ignorance implies something about the behavior of the world seems irrational: the cold teaspoon heats up in hot tea and the balloon flies about when it is released regardless of what I know or don’t know. What does what we know or don’t know have to do with the laws that govern the world? The question is legitimate; the answer to it is subtle.

Teaspoon and balloon behave as they must, following the laws of physics in complete independence from what we know or don’t know about them. The predictability or unpredictability of their behavior does not pertain to their precise condition; it pertains to the limited set of their properties with which we interact.
This
set of properties depends on
our
specific way of interacting with the teaspoon or the balloon. Probability does not refer to the evolution of matter in itself. It relates to the evolution of those specific quantities we interact with. Once again, the profoundly relational nature of the concepts we use to organize the world emerges.

The cold teaspoon heats up in hot tea because tea and spoon interact with us through a limited number of variables among the innumerable variables that characterize their microstate. The value of
these
variables is not sufficient to predict future behavior exactly (witness the balloon) but is sufficient to predict with optimum probability that the spoon will heat up.

I hope not to have lost the reader’s attention with these subtle distinctions . . .

Now, in the course of the twentieth century, thermodynamics (that is, the science of heat) and statistical mechanics (that is, the science of the probability of different motions) were extended to electromagnetic and
quantum phenomena. Extension to include the gravitational field, however, has proved problematic. How the gravitational field behaves when it heats up is still an unsolved problem.

We know what happens to a heated electromagnetic field: in an oven, for instance, there is hot electromagnetic radiation, which cooks a pie, and we know how to describe this. The electromagnetic waves vibrate, randomly sharing energy, and we can imagine the whole as being like a gas of photons that move fast and randomly like the molecules in a hot balloon. But what is a hot
gravitational
field?

The gravitational field, as we saw in the first lesson, is space itself, in effect space-time. Therefore, when heat is diffused to the gravitational field, time and space themselves must vibrate . . . But we still don’t know how to describe this well. We don’t have the equations to describe the thermal vibrations of a hot space-time. What is a vibrating time?

Such issues lead us to the heart of the problem of time: what exactly is the
flow
of time?

The problem was already present in classical physics and was highlighted in the nineteenth and twentieth centuries by philosophers—but it becomes a great deal
more acute in modern physics. Physics describes the world by means of formulas that tell how things vary as a function of “time.” But we can write formulas that tell us how things vary in relation to their “position,” or how the taste of a risotto varies as a function of the “variable quantity of butter.” Time seems to “flow,” whereas the quantity of butter or location in space does not “flow.” Where does the difference come from?

Another way of posing the problem is to ask oneself: what is the “present”? We say that only the things of the present exist: the past no longer exists and the future doesn’t exist yet. But in physics there is nothing that corresponds to the notion of the “now.” Compare “now” with “here.” “Here” designates the place where a speaker is: for two different people “here” points to two different places. Consequently “here” is a word the meaning of which depends on where it is spoken. The technical term for this kind of utterance is “indexical.” “Now” also points to the instant in which the word is uttered and is also classed as “indexical.” But no one would dream of saying that things “here” exist, whereas things that are not “here” do not exist. So then why do we say that things that are “now” exist and that everything else doesn’t? Is the present something that is objective in the
world, that “flows,” and that makes things “exist” one after the other, or is it only subjective, like “here”?

This may seem like an abstruse mental problem. But modern physics has made it into a burning issue, since special relativity has shown that the notion of the “present” is also subjective. Physicists and philosophers have come to the conclusion that the idea of a present that is common to the whole universe is an illusion and that the universal “flow” of time is a generalization that doesn’t work. When his great Italian friend Michele Besso died, Einstein wrote a moving letter to Michele’s sister: “Michele has left this strange world a little before me. This means nothing. People like us, who believe in physics, know that the distinction made between past, present and future is nothing more than a persistent, stubborn illusion.”

Illusion or not, what explains the fact that for us time “runs,” “flows,” “passes”? The passage of time is obvious to us all: our thoughts and our speech exist in time; the very structure of our language requires time—a thing “is” or “was” or “will be.” It is possible to imagine a world without colors, without matter, even without space, but it’s difficult to imagine one without time. The German philosopher Martin Heidegger emphasized our “dwelling in time.” Is it possible that the flow of time that
Heidegger treats as primal is absent from descriptions of the world?

Some philosophers, the most devoted followers of Heidegger among them, conclude that physics is incapable of describing the most fundamental aspects of reality, and they dismiss it as a misleading form of knowledge. But many times in the past we have realized that it is our immediate intuitions that are imprecise: if we had kept to these we would still believe that Earth is flat and that it is orbited by the sun. Our intuitions have developed on the basis of our limited experience. When we look a little further ahead, we discover that the world is not as it appears to us: Earth is round, and in Cape Town their feet are up and their heads are down. To trust immediate intuitions rather than collective examination that is rational, careful, and intelligent is not wisdom: it is the presumption of an old man who refuses to believe that the great world outside his village is any different from the one that he has always known.

As vivid as it may appear to us, our experience of the passage of time does not need to reflect a fundamental aspect of reality. But if it is not fundamental, where does it come from, our vivid experience of the passage of time?

I think that the answer lies in the intimate
connection between time and heat. There is a detectable difference between the past and the future only when there is the flow of heat. Heat is linked to probability; and probability in turn is linked to the fact that our interactions with the rest of the world do not register the fine details of reality. The flow of time emerges thus from physics, but not in the context of an exact description of things as they are. It emerges, rather, in the context of statistics and of thermodynamics. This may hold the key to the enigma of time. The “present” does not exist in an objective sense any more than “here” exists objectively, but the microscopic interactions within the world prompt the emergence of temporal phenomena within a system (for instance, ourselves) that interacts only through the medium of a myriad of variables.

Our memory and our consciousness are built on these statistical phenomena. For a hypothetically supersensible being, there would be no “flowing” of time: the universe would be a single block of past, present, and future. But due to the limitations of our consciousness we perceive only a blurred vision of the world and live in time. Borrowing words from my Italian editor, “what’s non-apparent is much vaster than what’s apparent.” From this limited, blurred focus we get our
perception of the passage of time. Is that clear? No, it isn’t. There is so much still to be understood.