How to think about the future (Part 1)

Image: from Wikipedia, fair use
It turns out that the future is completely unpredictable.

That may seem like a trivial statement. But most people have no idea just how true it is. A simple physics calculation may shed some light on this subject.

In the film franchise "Back to the Future", the main characters Marty McFly and Doctor "Doc" Emmett Brown travel through time and engage in various hijinks. Their travels alter the timeline in various ways, causing sometimes subtle, and sometimes drastic changes. Their goal throughout much of the films is to preserve the timeline they originally knew, with perhaps some small positive improvements.

But how likely are they to actually succeed? Can Marty, for instance, expect to return to 1985 and find things more or less as he left it, with perhaps a nicer car in his garage and an improved family? Let's say that Marty travels back in 1955, disturbs absolutely nothing by his direct contact, stays for only one second, then comes back to 1985. What can he expect to find?

Well, even if nothing is altered by his direct physical contact, Marty still exerts an additional gravitational force on his surroundings simply due to his mass. Since this is going to be an order of magnitude calculation, let's say that Marty has a mass of 100 kgs.

Now, consider the air molecules on the other side of the Earth from Marty. The Earth's diameter is 12.7e6 m. That is 12.7 million meters - I'll be using scientific notation (using e for the power of 10) and meters-kilograms-seconds throughout this calculation. Knowing that the universal gravitational constant is 6.67e-11 in mks units, we can use Newton's law of Universal Gravitation (a = Gm/r^2) to find that Marty causes an acceleration of 4.13e-23 m/s^2 to these air molecules.

Wow, that's like, nothing, right? In the one second that Marty is in 1955, that additional acceleration causes these air molecules to move an additional 2.07e-23 m that they would have otherwise not traveled. That's 0.000 000 000 000 000 000 000 0207 meters. What difference could be made by some air molecules moving such a small additional distance?

Consider a surface of one square meter on that other side of the Earth. The air molecules within 2.07e-23 m of that surface will have then collided with that surface and bounced back, due to Marty's additional gravitational pull. One square meter with 2.07e-23 m of thickness gives a volume of 2.07e-23 m^3, and this is the volume of air we're concerned about. There is 1 mole (6.022e23) of air in 22.4e-3 m^3 of volume in typical conditions, so that amounts to perhaps 556 air molecules that have bounced back against the surface.

Now, the important point here is that these air molecules are now OUT OF PLACE. They're somewhere that they're not suppose to be. As far as anyone from 1955 is concerned, the states of these 556 air molecules are unpredictable. If Marty had not traveled back in time, they would be somewhere completely different on the atomic scale. That means that they would collide with other air molecules that they were not supposed to collide with, while missing collisions that they were suppose to make.

The focus on collisions here is important. As long as the change that Marty causes is all smooth - say, by moving everything over by 2.07e-23 m - then any predictions made from 1955 would only be off by that much. Only negligible changes would occur from moving things by a negligible distance. But collisions change all that. If an air molecule makes a collision that it wasn't suppose to make, or misses a collision that was suppose to happen, then your predictions about the trajectory of the molecules are not just slightly off, they're completely wrong. That's why we're focused on the air molecules that bounced off the surface (whose trajectories are now completely unpredictable), rather than all the other air molecules (whose positions are now just all off by 2.07e-23 m).

Now, each air molecule experiences something like 1e10 collisions with another molecule each second. Each collision changes the trajectory of BOTH molecules. That is to say, 1e10 time a second, the number of air molecules whose trajectory is now completely unpredictable DOUBLES.

If we ignore the fact that a molecules may run into the same molecule multiple times, the number of molecules whose trajectory is now completely unpredictable would be 2^1e10 in one second. This is an absurdly large number, even when you're counting molecules. For all practical purposes, you can consider the microstate of ALL the air molecules around these original 556 molecules to have become completely unpredictable, in far less than one second. This cloud of unpredictability would rapidly spread, limited only by the diffusion and mixing of the air.

Furthermore, remember that this happens to every square meter of surface on the Earth. The upshot is that, from Marty's one second stay in 1955, even if he affects nothing by his direct touch, the microstate of the Earth's ENTIRE atmosphere rapidly becomes COMPLETELY UNPREDICTABLE, completely different than how it otherwise would have been. Something similar must also happen to all the liquids on the Earth, although the calculation will of course differ in detail.

Okay, so the future trajectory of all these molecules are completely unpredictable at this point. But really, can these molecular changes actually affect anything? What does it matter whether some molecules - or all molecules, for that matter - are here instead of there?

In the short term, it won't affect much. But there would be small changes. Microscopic differences can often affect the macroscopic world. The twinkling of stars, for example, is caused by the fluctuations in the momentary, localized density of the air. Photomultiplier tubes are devices specifically designed to cause macroscopically detectable measurements from microscopic events. Cosmic rays travelling through the atmosphere can be affected by the microstate of the air molecules, and they can cause neurons to fire when they hit your brain - perhaps causing a thought or a feeling you otherwise would not have had. The microscopic collisions of the molecules with other small particles causes Brownian motion - the random, jiggling movement of small particles suspended in water. So, a pollen grain would take a different random path in water, because Marty traveled back in time, for one second.

Now, do you think these changes still can't make any real differences? Try replacing "pollen grain" in the previous paragraph with "sperm".

Each sperm carries a different genetic code. Unless a specific sperm made it to the egg at the moment of your conception, you would not have been born. And the trajectory of that sperm would have been different, because the molecular motions in the fluids would have been different, all because Marty was in 1955 for one second and exerted a little bit of extra gravity. Forget about trying to get his parents to fall in love with one another - Marty was in all likelihood doomed to not exist the moment that he popped into 1955.

If you go review the argument and the calculations above, you'll see that the situation is actually far worse, in terms of predictability, than what I have presented. For example, altering the path of a sperm is not a particularly special way of affecting the future. In general, small changes - even microscopic changes - will eventually grow to cause macroscopic differences in time. This is called the butterfly effect: the name comes from an example where a butterfly flapping its wings can cause a hurricane on the other side of the Earth some time later.

The magnitude of the initial perturbation is also unnecessarily large in the example above. Remember, this doesn't affect just Marty; similar changes would happen all over the entire Earth. As if that wasn't enough, I once carried out another calculation which showed that even at INTERSTELLAR DISTANCES, a small change at another star will easily affect the microstate of things on Earth, which will then soon grows to make macroscopic differences.

So, that is the magnitude of the future's unpredictability: your very existence hinged on small changes happening in other parts of the galaxy.

But I don't want this post to be one of those "see how wrong the movies are" articles. I like the "Back to the Future" trilogy. I don't like tearing things down just for the sake of tearing them down. So I'll end this post with the following:

I think the truest thing in those movies is what Doc says at the very end: "your future hasn't been written yet. No one's has. Your future is whatever you make it. So make it a good one." That may or may not be philosophically true in the deepest sense. But what is absolutely certain is that Doc's advice is effectively true in a practical sense. What the above calculation shows is that you cannot hope to indefinitely predict the details of your future. No one can. In that sense, your future is not yet written. What you can do, however, is act in the present to control the short moments ahead, and focus on the bigger picture to extend the lifetime of our planning before the butterfly effect kicks in. We then adapt and adjust, one step at a time. That is how we make the future.

The next post will walk though the calculation I mentioned earlier, about how this all works even at interstellar distances.

You may next want to read:
How to think about the future (Part 2) (Next post of this series)
An analysis of "Let It Go" in Disney's "Frozen"
The dialogue between two aliens who found a book on Earth
Another post, from the table of contents

1 comment :

  1. The other easy way to find massive differences deriving from trivial ones is to think back through your life to a time when you changed your plans because of inclement weather. How would your life be different if you hadn't turned back for your umbrella? Maybe that led to you missing a bus and catching the next one, which means you met a completely different set of fellow-travellers. I know a lot of people who met their husbands/wives under such trivial circumstances - which are, in a scientifically provable way, derived from all those tiny tiny changes in the atmosphere.

    Yes, that's right. When you turn your head before exhaling, you're affecting whether people meet each other on the other side of the world...

    Of course, emacs has a command for that. Ctrl-X, Meta-C, XKCD-378.