Between ourselves and the rest of the world there are physical interactions. Obviously, not all the variables of the world interact with us, or with the segment of the world to which we belong. Only a very minute fraction of these variables does so; most of them do not react with us at all. They do not register us, and we do not register them. This is why distinct configurations of the world seem equivalent to us. The physical interaction between myself and a glass of water—two pieces of the world—is independent of the motion of the single molecules of water. In the same way, the physical interaction between myself and a distant galaxy—two pieces of the world—ignores what happens in detail out there. Therefore, our vision of the world is blurred because the physical interactions between the part of the world to which we belong and the rest are blind to many variables.
This blurring is at the heart of Boltzmann’s theory.93 From this blurring, the concepts of heat and entropy are born—and these are linked to the phenomena that characterize the flow of time. The entropy of a system depends explicitly on blurring. It depends on what I do not register, because it depends on the number of indistinguishable configurations. The same microscopic configuration may be of high entropy with regard to one blurring and of low in relation to another.
This does not mean that blurring is a mental construct; it depends on actual, existing physical interactions.94 Entropy is not an arbitrary quantity, nor a subjective one. It is a relative one, like speed.
The speed of an object is not a property of the object alone: it is a property of the object in relation to another object. The speed of a child who is running on a moving train has a value relative to the train (a few steps per second) and a different value relative to the ground (a hundred kilometers per hour). If his mother tells the child to “Keep still!,” she does not mean that they have to throw themselves out of the window to stop in relation to the ground. She means that the child should stop with regard to the train. Speed is a property of an object with respect to another object. It is a relative quantity.
The same is true for entropy. The entropy of A with regard to B counts the number of configurations of A that the physical interactions between A and B do not distinguish.
Clarifying this point, which frequently causes confusion, opens up a seductive solution to the mystery of the arrow of time.
The entropy of the world does not depend only on the configuration of the world; it also depends on the way in which we are blurring the world, and this depends on what the variables of the world are that we interact with. That is to say, on the variables with which our part of the world interacts.
The entropy of the world in the far past appears very low to us. But this might not reflect the exact state of the world: it might regard the subset of the world’s variables with which we, as physical systems, have interacted. It is with respect to the dramatic blurring produced by our interactions with the world, caused by the small set of macroscopic variables in terms of which we describe the world, that the entropy of the universe was low.
This, which is a fact, opens up the possibility that it wasn’t the universe that was in a very particular configuration in the past. Perhaps instead it is us, and our interactions with the universe, that are particular. We are the ones who determine a particular macroscopic description. The initial low entropy of the universe, and hence the arrow of time, may be more down to us than to the universe itself. This is the basic idea.
Think of one of the grandest and most obvious phenomena: the diurnal rotation of the skies. It is the most immediate and magnificent characteristic of the universe around us: it turns. But is this turning really a characteristic of the universe? It is not. It took us thousands of years, but in the end we managed to understand the revolving of the heavens: we understood that it is we who turn, not the universe. The rotation of the heavens is a perspective effect due to our particular way of moving on Earth, rather than a mysterious property of the dynamics of the universe.
Something similar might be true for time’s arrow. The low initial entropy of the universe might be due to the particular way in which we—the physical system that we are part of—interact with it. We are attuned to a very particular subset of aspects of the universe, and it is this that is oriented in time.
How can a particular interaction between us and the rest of the world determine a low initial entropy?
It’s simple. Take a pack of twelve cards, six red and six black. Arrange it so that the red cards are all at the front. Shuffle the pack a little and then look for the black cards that have ended up among the red ones. Before shuffling, there are none; after, some. This is a basic example of the growth of entropy. At the start of the game, the number of black cards among the red in the first half of the pack is zero (the entropy is low) because it has started in a special configuration.
But now let’s play a different game. First, shuffle the pack in a random way, then look at the first six cards and commit them to memory. Shuffle a little and look to see which other cards have ended up among the first six. At the start, there were none, then their number grew, as it did in the previous example, together with the entropy. But there is a crucial difference between this example and the previous one: at the beginning of this one, the cards were in a random configuration. It was you who declared them to be particular, by taking note of which cards were in the front half of the pack at the beginning of the game.
The same may be true for the entropy of the universe: perhaps it was in no particular configuration. Perhaps we are the ones who belong to a particular physical system with respect to which its state can be particular.
But why should there be such a physical system, in relation to which the initial configuration of the universe turns out to be special? Because in the vastness of the universe, there are innumerable physical systems, and they interact with each other in ways that are even more numerous. Among these, through the endless game of probabilities and huge numbers, there will surely be some that interact with the rest of the universe precisely with those variables that found themselves having a particular value in the past.
It is hardly surprising that there are “special” subsets in a universe as vast as ours. It is not surprising that someone wins the lottery: someone wins it every week. It is unnatural to assume that the entire universe has been in an incredibly “special” configuration in the past, but there is nothing unnatural in imagining that the universe has parts that are “special.”
If a subset of the universe is special in this sense, then for this subset the entropy of the universe is low in the past, the second law of thermodynamics obtains; memories exist, traces are left—and there can be evolution, life, and thought.
In other words, if in the universe there is something like this—and it seems natural to me that there could be—then we belong to that something. Here, “we” refers to that collection of physical variables to which we commonly have access and by means of which we describe the universe. Perhaps, therefore, the flow of time is not a characteristic of the universe: like the rotation of the heavens, it is due to the particular perspective that we have from our corner of it.