The important event this Dec. 25 isn’t celebrating the birthday of Isaac Newton or other historical figures, it’s the release of The Curious Case of Benjamin Button, a David Fincher film starring Brad Pitt and based on the story by F. Scott Fitzgerald. As you all know, it’s a story based on the device of incompatible arrows of time: Benjamin is born old and ages backwards into youth (physically, not mentally), while the rest of the world behaves normally. Some have pretended that scientific interest in the movie centers on issues of aging and longevity, but of course it’s thermodynamics and entropy that take center stage. While entropy increases and the Second Law is respected in the rest of the world, Benjamin Button’s body seems to be magically decreasing in entropy. (Which does not, strictly speaking, violate the Second Law, since his body isn’t a closed system, but it sure is weird.)
It’s a great opportunity to address an old chestnut: why do arrows of time have to be compatible? Why can’t we imagine ever discovering another galaxy in which entropy increased toward (what we call) the past instead of the future, as in Greg Egan’s story, “The Hundred Light-Year Diary”? Or why can’t a body age backwards in time?
First we need to decide what the hell we mean. Let’s put aside for the moment sticky questions about collapsing wave functions, and presume that the fundamental laws of physics are perfectly reversible. In that case, given the precise state of the entire universe (or any closed system) at any one moment in time, we can use those laws to determine what the state will be at any future time, or what it was at any past time. That’s just how awesome the laws of physics are. (Of course we don’t know the laws, nor the state of the entire universe, nor could we actually carry out the relevant calculation even if we did, but we’re doing thought experiments here.) We usually take that time to be the “initial” time, but in principle we could choose any time — and in the present context, when we’re worried about arrows of time pointing in different directions, there is no time that is initial for everything. So what we mean is: Why is it difficult/impossible to choose a state of the universe with the property that, as we evolve it forward in time, some parts of it have increasing entropy and some parts have decreasing entropy?
Notice that we can choose conditions that reverse the arrow of time for some individual isolated system. Entropy counts the “typicalness” of the system’s microscopic state, from the point of view of macroscopic observers. And it tends to go up, because there are many more ways to be high-entropy than low entropy. Consider a box of gas, in which the gas molecules are (by some means) all bunched together in the middle of the box, in a low-entropy configuration. If we just let it evolve, the molecules will move around, colliding with each other and with the walls of the box, and ending up (with overwhelmingly probability) in a much higher-entropy configuration.
It’s easy to convince ourselves that there exists some configurations from which the entropy would spontaneously go down. For example, take the state of the above box of gas at any moment after it has become high-entropy, and consider the state in which all of the molecules have exactly the same positions but precisely reversed velocities. From there, the motion of the molecules will precisely re-trace the path that they took from the previous low-entropy state. To an external observer, it will look as if the entropy is spontaneously decreasing. (Of course we know that it took a lot of work to so precisely reverse all of those velocities, and the process of doing so increased the entropy of the wider world, so the Second Law is safe.)
But a merely reversed arrow of time is not the point; we want incompatible arrows of time. That means entropy increasing in some part of the universe while it is decreasing in others.
At first it would seem simple enough. Take two boxes, and prepare one of them in the low entropy state with gas in the middle, and the other in the delicately constructed state with reversed velocities. (That is, the two boxes on the left side of the two figures above.) The entropy will go up in one box, and down in the other, right? That’s true, but it’s kind of trivial. We need to have systems that interact — one system can somehow communicate with the other.
And that ruins everything, of course. Imagine we started with these two boxes, one of which had an entropy that was ready to go up and the other ready to go down. But now we introduced a tiny coupling — say, a few photons moving between the boxes, bouncing off a molecule in one before returning to the other. Certainly the interaction of Benjamin Button’s body with the rest of the world is much stronger than that. (Likewise Egan’s time-reversed galaxy, or Martin Amis’s narrator in Time’s Arrow.)
That extra little interaction will slightly alter the velocities of the molecules with which it interacts. (Momentum is conserved, so it has no choice.) That’s no problem for the box that starts with low entropy, as there is no delicate tuning required to make the entropy go up. But it completely ruins our attempt to set up conditions in the other box so that entropy goes down. Just a tiny change in velocity will quickly propagate through the gas, as one affected molecule hits another molecule, and then they hit two more, and so on. It was necessary for all of the velocities to be very precisely aligned to make the gas miraculously conspire to decrease its entropy, and any interaction we might want to introduce will destroy the required conspiracy. The entropy in the first box will very sensibly go up, while the entropy in the other will just stay high. You can’t have incompatible arrows of time among interacting subsystems of the universe.
This is a subject for biology and not for physics.
> In fact, there is no relationship between the second law of thermodynamics and the fact that most animals senesce.
> In fact, there is no theoretical reason why an animal could not get more robust, more fertile, and stronger with each passing year.
> In fact, there are some animals and many more plants that fit the definition of “negative senescence”. Mortality rates decline over time, rather than rising exponentially as in the usual process of senescence.
There is a review paper on this subject, “The Case for Negative Senescence”, from the laboratory of James Vaupel, who is probably the world’s most prominent demographer:
http://tinyurl.com/9vegsu
Theor Popul Biol. 2004 Jun;65(4):339-51.
Ben Button Says:
December 25th, 2008 at 8:27 pm
Slightly off-topic: I suddenly realised, from reading this, that I don’t understand the “Horizon Problem” which inflation supposedly solves.
Think of it this way. Go out with your handy horn antenna and measure the cosmic microwave background (CMB) background temperature in a random direction. Now measure the temperature in the exact opposite direction. You are seeing photons, greatly redshifted, from the last scattering surface of the so called recombination. If you make some simple assumptions about the topology of the universe (e.g., that spacetime isn’t a torus or a Klein bottle or some other tricky topology), then those photons come from regions of spacetime that had no causal connection with each other at the time of recombination – i.e., there is no way for a star or a nova or a intelligent being in one of those regions to communicate with or influence the other, at least using light or matter to send signals. These two regions only “came through the horizon” and became causally connected with you now, when you got their photons, and are still not causally connected with each other. (You are closer to them than they are to each other.)
This leads to the Horizon Problem : if these two regions have never been in contact, then how do they “know” to have the same temperature ? You could have weird topologies for the entire universe (where looking in opposite directions means that you are really looking at the same distant objects), but I am not aware of any interesting theories that require or assume weird topologies. A related problem is the smoothness of the universe – gravity will make small initial perturbation grow, and it is hard to see how an initial big bang would give us the smoothness of the microwave background we observe. Note that having weird topologies really don’t do anything to solve this problem.
In inflation, a small piece of very early space time greatly expanded, at much faster than the speed of light, making everything we see (even the entire cosmic microwave background) to have come from one tiny original patch of spacetime, which was causally connected (and quite smooth, because it was tiny). So, in a sense you have pushed the Horizon problem “up a step” to the theory of inflation (there is no need to postulate faster light travel now, but you have to postulate a rapidly expanding spacetime then), but the theory of inflation seems to hang together well, and it does solve the horizon and smoothness problems.
Isn’t there a purely geometric way of posing the problem? The direction of time on the worldline of a particle is defined by the worldcones – the volume of one is shrinking and the other is expanding. Spacetime events move from within the future lightcone to the past lightcone. If this is a good definition of past and future, then consider two observers, each of whose past lies in the other’s future. We’re not invoking entropy here nor the reversibility of the laws of physics, nor any physical property of the observer (e.g., memory). What rules this out?
“Notice that we can choose conditions that reverse the arrow of time for some individual isolated system. Entropy counts the “typicalness” of the system’s microscopic state, from the point of view of macroscopic observers.”
A system that you have contrived such that it will change over time to a “low entropy” state must itself be very highly ordered. It’s not “typical”. It’s just that the very special ordering is not as obvious as it is when all your particles are bunched up in one place.
To me this seems to be a way of deceiving yourself that you’ve created something different. All you’ve done is to create a system that shows the weakness of our method of calculating the macroscopic property we call entropy.
Sean: “It’s more likely that both entropies will remain high for most of the evolution, and one will suddenly go down at the end.”
It’s not a statistical thermodynamics measure of entropy that will “suddenly go down”. It’s a sloppy macroscopic measure of entropy that you’re deceiving yourself with.
Concepts like the Arrow of Time only work as a macroscopic approximation when you don’t have a deceptive intelligence contriving your starting conditions. As soon as you introduce a deliberate deceiver, you’re not talking physics, you’re talking parlor tricks.
And you’re right that interactions with other systems will break the illusion. A system designed to evolve so that it simulates a reversed Arrow of Time is so fragile that any interaction from outside the system will wreck the illusion.
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