Into the Unknown, page 21
You might consider the particular order of your shuffled cards to be especially important for some reason and want to shuffle back to it; however, it is just as unlikely that you will ever get that particular card order again by shuffling as it is that you will shuffle back to the initial order in the pristine deck (although, after shuffling ~1068 times, I am not sure there would be much of a deck left to be shuffled). However unlikely, it is possible—which brings us to the Poincaré recurrence theorem: given enough time, any system that randomly moves between possible states will return to its initial state. For a full deck of 52 cards this would take a long time, but what if we only use the aces—meaning we have 4 cards. Calculating 4! doesn’t even require a calculator. With some efficient shuffling, you could revisit that state in just a couple minutes.
The metapoint of invoking the Poincaré recurrence theorem is that entropy does not always increase—it can and does decrease on small scales all the time. If entropy were causally connected to the arrow of time, that would require there to be micro–time reversals happening all around us every day and time would only move forward in a statical sense. I suppose we can’t rule out that these micro–time reversals are happening, but I find it much more likely that entropy is merely correlated and not causally connected with the arrow of time. As time goes forward, things in the universe continue a statistically random walk through all possible states, but some of those states are lower entropy than others.
There is one additional intriguing tidbit I want to add into the mix of our discussion on time, which may mean nothing, or it may be a clue about something more fundamental. As an undergraduate physics major, when I first learned about these things called “Feynman diagrams” (after the late physicist Richard Feynman), they utterly captivated me. If I hadn’t been hooked on physics before learning about them, they would have done the trick. These diagrams are a fantastically elegant way to conceptualize particle interactions. They made way more intuitive sense to me than endless equations describing the same phenomenon. At a fundamental level, these diagrams are just a two-dimensional plane of space and time that illustrates how particles interact. That is useful, but not necessarily mind-blowing.
I sat up in my seat and became riveted when my professor told the class that when using these diagrams, antiparticles behave exactly like their normal particle counterparts—but going backward in time. For example, in a Feynman diagram, a positron is equivalent to an electron going backward in time (and vice versa). Moreover, when we talk about particle-antiparticle annihilation, this is functionally equivalent to a particle going forward in time meeting up with one of its alter egos going backward in time.
At the time, I also happened to be taking a separate physics class for which our weekly homework involved coming up with three novel ideas involving physics. I don’t know how hard that might sound to you, but I promise it was both nontrivial and one of the best creative thinking exercises I’ve ever encountered in a physics class. The week we learned about Feynman diagrams, I wrote up a whole (naive undergraduate) theory about how the Big Bang could have been the result of a collision of a universe going forward and with a universe going backward in time, and I even worked out how this might also explain the matter-antimatter symmetry (which we will come to shortly). I’m now embarrassed by my (literally) sophomoric effort, but not too embarrassed to cop to it as a demonstration of how taken I was with Feynman diagrams. My homework came back with the comment “Interesting idea. Needs to be thought out more,” which was both disheartening and fair.
To be clear, I am not saying that positrons are in fact electrons going back in time. I can almost hear a legion of dyed-in-the-wool physicists groaning that I even brought this topic up—this feature of Feynman diagrams is almost universally dismissed as a “neat trick” with no physical meaning (akin to advanced waves and Maxwell’s equations, or imaginary numbers and Euler’s equation). However, there is also not clearly an empirical way to test the idea that positrons are electrons going back in time, as they are functionally the same. My philosophical position is that when we are asked to dismiss solutions as “unphysical” that are otherwise legitimate, we should take a very careful look at our justification for saying they are “unphysical.” We may well be cutting off promising avenues of thought before they even take root. Our human brains are limited enough—let’s not make it even more difficult to come up with new ideas by convincing ourselves that things are impossible just because they seem unlikely.
The apparent asymmetry of time is fundamentally perplexing. Symmetry is a fundamental part of modern physics, and many physicists have an almost religious belief in the foundational existence of symmetry in the laws of nature (I would count myself among them). In physics, what “symmetry” refers to is a thing being identical after you have done something to it. Except, in physics, instead of “thing” we say “system,” instead of “identical” we say “invariant,” and instead of “done something to it” we say “it has undergone a transformation.” So, restating this concept of symmetry in more physics-y jargon (you can break this out during small talk at your next gathering), symmetry refers to a system being invariant under a certain transformation. Transformations that often come up in the context of symmetry are reflection, rotation, and translation, but the list goes on and becomes increasingly abstract.
One of the great theories of the early 1900s came from Emmy Noether,3 and it directly connects underlying physical symmetries to conservation laws. As a concrete example, let’s take angular momentum. Lots of people have had conservation of angular momentum drilled into them at some point, so hopefully this concept is somewhat familiar to you, and you’ve seen (or you can imagine) an ice skater speeding up or slowing down their spin by moving their arms in or out. Conservation of angular momentum is tied to underlying rotational symmetry in physics—to put it bluntly, the laws of physics apply the same, regardless of how an object is oriented in space. We can do the same thing with conservation of normal (nonangular) momentum (which is connected to translation symmetry in space—moving an object linearly forward or backward along a dimension), or conservation of energy (which is connected to translation symmetry in time).
There are three particular types of symmetry that get entwined in physics at a really fundamental level: charge (C), parity (P), and time (T). Taking them each in turn: C-symmetry means that if you could replace every particle in the universe by the same particle with the opposite charge, the universe would behave the same. We can think of a classic atom as an example; if the proton in the nucleus had a negative charge and the electron had a positive charge, the atom would have the same properties. P-symmetry is basically just a reflection symmetry—if you take an object and reflect it along each of the three spatial axes, it would be identical (P-symmetry essentially filters for chirality, meaning whether something has a handedness to it). An everyday example of P-symmetry being broken is a clock; if you look at the face of a clock in a mirror, the hands will move counterclockwise because clocks have a handedness to them. T-symmetry, as you might expect, means a system behaves the same whether you run it forward or backward in time—like the balls on a billiards table (ignoring friction…). Taken together, C, P, and T appear to be required to be symmetric as a group, which was a groundbreaking result in the 1950s.
You may well be wondering what it means for C, P, and T to be required to have a combined symmetry (referred to as “CPT symmetry”), which is not casually intuitive. This required CPT symmetry means that if I have a normal cat, moving forward in time with the rest of us, it would be identical to a cat made of antiparticles moving backward in time and reflected in a mirror. This required CPT symmetry is so important that, if we are wrong about this, much of our fundamental understanding of physics goes out the window. Which is not to say that we aren’t capable of throwing physics out the window (I tried many times as a student) if we have no other choice.
We are getting to something essential and mysterious about the nature of time, and no one said this would be easy, so stay with me just a little longer. If we could talk about this over coffee with a napkin to write on, that would be great. But for the moment we are stuck in this book. The upshot: the arrow of time appears to be related to CP symmetry violations, which are in turn related to antimatter, which is itself related back to the arrow of time. So, it seems like we are onto something, we just don’t know what.
Now we just need to remember how to multiply by 1 and −1, and we are home free. Let’s say that if one of these three symmetries (C, P, or T) is symmetric, we assign it a value of +1. On the other hand, if it is asymmetric, we assign it a value of −1. For CPT symmetry to hold, the following must be true:
There are, of course, different ways you can multiply +1 and −1 using these three variables and still end up with +1. For decades everything in physics was moving along and continually finding that C × P = 1 (shockingly called “CP symmetry”), which worked well for interactions involving the strong force or electromagnetism. If C × P = 1, then T also had to = 1.
Then, in 1964 along came this subatomic particle called a “kaon” (or “k-meson”) and said, “Look what I can do!” And it had a partner in crime—the weak force, which I have found to be the most challenging of the four known forces to get a grasp of (hold tight for the chapter on the laws of nature—or don’t and go ahead and read that now if you want). The weak force seems like this shy wallflower of the forces, coming across as unassuming and timid. To be fair, the weak force is weak (though not remotely as weak as gravity), and therefore trickier to study than the strong force or electromagnetism. Experiments got more sophisticated, and one day the weak force came out of its shell to show the world its talent; with the help of the weak force, the decay of the kaon violated CP symmetry. If CP symmetry has been violated (C × P = −1), and because we need C × P × T = 1, then T must be −1.
If those last few paragraphs read along the lines of “Blah, blah, blah,” here is the point: kaon decay was the first time in physics that we had ever found anything that appeared to be fundamentally asymmetric in time. In the years since the kaon had its coming-out party, we have found a few other interactions that have CP violation (and are therefore asymmetric in time), and they all involve the weak force. As far as we can tell, the weak force is the only thing in physics that seems to know and care about the arrow of time. And we don’t know why. If you have gone through your life blissfully ignoring the weak force as some esoteric force that only physicists care about, maybe it will pique your interest now.
It turns out that all of this is related to the fact that you exist, which you are presumably somewhat aware of.4 Your existence would not be possible without an asymmetry between matter and antimatter in the universe. Given the extent to which physicists adore symmetry in the universe, this crucial asymmetry between matter and antimatter is particularly vexing. From both our theories and experiments, we unfailingly find that particles and antiparticles are created in equal numbers—as required by conservation of charge. So, this asymmetry really peeves us.
In the very early universe, matter and antimatter went through an intense period of mutual annihilation, turning the vast majority of their mass-energy into just energy. However, given that you are here reading this book, apparently the mutual annihilation wasn’t quite complete. For reasons we don’t (yet) understand, after all the destruction was done, something like 1/1,000,000,000 normal matter particles survived. If they hadn’t, we wouldn’t be here wondering about this asymmetry.
You might be wondering whether there could just be pockets of antimatter somewhere in the universe that we don’t interact with. There could be, but they would have to be outside of our horizon (which doesn’t rule them out). Even in “empty” space, there is on average about one atom in each cubic meter, so there really isn’t likely to be a buffer of truly empty space between regions of matter and antimatter. As a result, the boundaries between these regions would be generating significant radiation from particles annihilating on the edges, and we haven’t detected anything like this. Moreover, for the pockets-of-antimatter hypothesis to work, we would also have to explain—statistically—how matter and antimatter were not mixed at a level requiring annihilation. Given that particle-antiparticle pairs are created together and are attracted to each other, it is very hard to explain how clumps larger than galaxies could become segregated.
Perhaps the most promising hypothesis to explain the fact that the universe is dominated by matter over antimatter is CP violation. In cases of CP violation, we see that—for reasons we don’t understand—mesons (like the kaon) have a very slight variation in how they decay that require C × P = −1. Current experiments being run at the Large Hadron Collider are pushing the envelope on this, and perhaps this section of the book will need to be revised in the coming years.
Topology of Time
While time apparently having a direction feels natural from our lived experience, why time appears to be structurally different from space is a fundamental question in modern physics. Not only does there seem to be only a single dimension of time, but it also appears to only go in one direction, not offering us any choice about whether or not that is the direction we want to go. Because we seem to just be along for the ride as far as time goes, it might seem bizarre to even consider whether time has a “shape.”
Insofar as most people even have cause to think about time at all,5 I think there is a common conventional view based on lived experience. If I were to define a “common sense” view of time, it would be something like: Time is infinite (at least into the future). Time is one-dimensional. Time only has a single branch. Time has a constant rate. Time goes in a straight line and doesn’t loop around. Time is continuous. Time is relational and doesn’t exist as a “thing.”
I will be the first to admit that this “common sense” view of time is well aligned with my own perception. I’ve just learned not to trust my perception. How would we know if any of these statements were not true? Since we are embedded in time and don’t seem to have much of a choice about that, it is very hard for us to get a solid grounding here. Nevertheless, let’s take these statements one at a time and give them a proper inspection.
Does time have to be infinite? In a standard view of time and space, reinforced by countless physics and math diagrams, we commonly envision time as being orthogonal to space and unidimensional (as in Panel A).6 Given the lack of constraints available to us empirically, there are literally an infinite number of topologies that time might have. For the sake of playing around with a couple toy ideas, let’s explore a few different geometries as a thought experiment.
If the universe is finite in space, but infinite in time, the universe could have a shape in space-time similar to that shown in Panel B—if you kept flying your spaceship far enough, you could, in principle, loop back around to the same location in space (although the path for this in space-time would look like a spiral on this cylindrical figure, because you would also be moving up in time while looping around). The Panel B version of space-time is something that astronomers commonly envision when thinking about the universe, and we are not uncomfortable with the idea that the universe could possibly be finite in the spatial dimensions.
Now let’s flip the script. If space could be finite, why not time? What if time were finite and space were infinite? This scenario would result in something like Panel C. Curiously, it seems easier for people (or at least for my students) to be comfortable with the idea of time having a beginning than an ending, which speaks to how differently we perceive time and space. Panel D is a version of the Hawking-Hartle no-boundary universe from the chapter on the Big Bang (you may want to flip back to that chapter for a refresher). In the “no-boundary” universe, the dimension of time becomes increasingly space-like the closer we get to a time = 0, and thus there is no “before” time = 0 as a result of the coordinate system.
Finally, the last permutation of these options: What if space and time were both finite? Then we might end up with something like Panel E. The situation in a universe like Panel E is interesting, because whether we view something as happening along the direction of time or space depends entirely on how the coordinate grid (latitude and longitude) is oriented; the symmetries embedded in this topology render the choice of coordinate somewhat arbitrary unless there is an anchor point—for example, because the Earth rotates, a practical coordinate grid is imprinted, and we get to have nice things like a north pole, south pole, and an equator.
To be clear, I am absolutely not advocating for or against any of these geometries. Rather, I think it is an interesting mental exercise to consider what it would mean if, for example, time were finite.
Does time have a single dimension? Next on the list of a “common sense” view of time is that it is one-dimensional. If time were two (or more) dimensions, you could—for example—turn “left” in time. One can imagine different incarnations of dimensionality in our universe, besides the 3 + 1 of space and time that we find agreeable. As a bit of a spoiler, the fact that we appear to have only a single dimension of time is probably critical for having a universe hospitable for life, as we will see in the chapter on fine-tuning. In a nutshell, if we had more than one dimension of time while also having more than one dimension of space, we end up with a mathematical situation that is probably not too good for life. In a scenario where you have multiples of one type of dimension (in our case, space), and also more than one of another type of dimension (in this case, time), you end up with what is called an “ultrahyperbolic equation.” Ultrahyperbolic equations are pernicious and do not have well-defined stable solutions. If our universe had such dimensionality, nothing would be predictable, making it pure chaos at a fundamental level. If there is another dimension of time, it is thankfully hiding away somewhere and not destroying the order of the universe.
