Infinite Collisions, or Something from Nothing (Supertasks I)

Hello! It’s been over a year since my last post; I took some time off as I moved to a new city and a new job and became generally more busy. I missed writing this blog, though, and recently I found myself reading about supertasks, tasks or processes with infinitely many steps but which are completed in a finite amount of time. This made the finitely many things I was doing seem lazy by comparison, so I think I’m going to try updating this site regularly again.

The first supertasks were possibly those considered by Zeno in his paradoxes of motion (which we’ve thought about here before). In the simplest, Achilles is running a 1 kilometer race down a road. To complete this race, though, Achilles must first run the first 1/2 kilometer. He must then run the next 1/4 kilometer, and then the next 1/8 kilometer, and so on. He must seemingly complete an infinite number of tasks just in order to finish the race.

Achilles running a race. (Image by Martin Grandjean, CC BY-SA 4.0)

If this idea leaves you a bit confused, you’re not alone! It seems that the same idea can be applied to any continuous process; first half of the process must take place, then the next quarter, then the next eighth, and so on. How does anything ever get done? Is every task a supertask?

Supertasks reemerged as a topic of philosophical study in the mid-twentieth century, when J.F. Thomson introduced both the term supertask and a new example of a supertask, now known as Thomson’s Lamp: Imagine you have a lamp with an on/off switch. Now imagine that you switch the lamp on at time t = 0 seconds. After 1/2 second, you switch the lamp off again. After a further 1/4 second, you switch the lamp on again. After a further 1/8 second, you switch the lamp off again, and so on. After 1 full second, you have switched the lamp on and off infinitely many times. Now ask yourself: at time t = 1, is the lamp on or off? Is this even a coherent question to ask?

We’ll have more to say about this and other supertasks in future posts, but today I want to present a particularly elegant supertask introduced by Jon Perez Laraudogoitia in the aptly named paper “A Beautiful Supertask”, published in the journal Mind in 1996. Before describing the supertask, let me note that it will obviously not be compatible with our physical universe, or perhaps any coherent physical universe, for a number of reasons: it assumes perfectly elastic collisions with no energy lost (no sound or heat created, for instance), it assumes that particles with a fixed mass can be made arbitrarily small, it assumes that nothing will go horribly wrong if we enclose infinitely much matter in a bounded region, and it neglects the effects of gravity. Let’s put aside these objections for today, though, and simply enjoy the thought experiment.

Imagine that we have infinitely objects, all of the same mass. Call these objects m1, m2, m3, and so on. They are all arranged on a straight line, which we’ll think of as being horizontal. To start, each of these objects is standing still. Object m1 is located 1 meter to the right of a point we will call the origin, object m2 is 1/2 meter to the right of the origin, object m3 is 1/4 meter to the right of the origin, and so on. (Again, we’re ignoring gravity here. Also, so that we may fit these infinitely many objects in a finite amount of space, you can either assume that they are point masses without dimension or you can imagine them as spheres that get progressively smaller (but retain the same mass) as their index increases.) Meanwhile, another object of the same mass, m0, appears to the right of m1, moving to the left at 1 meter per second. At time t = 0 seconds, it collides with m1.

Our system immediately before t = 0.

What happens after time t = 0? Well, since we’re assuming our collisions are perfectly elastic, m0 transfers all of its momentum to m1, so that m0 is motionless 1 meter from the origin and m1 is moving to the left at 1 meter per second. After a further half second, at t = 1/2, m1 collides with m2, so that m1 comes to rest 1/2 meter from the origin while m2 begins moving to the left at 1 meter per second. This continues. At time t = 3/4, m2 collides with m3. At time t = 7/8, m3 collides with m4, and so on.

Where are we after 1 full second? By time t = 1, each object has been hit by the previous object and has in turn collided with the next object. m0 is motionless where m1 started, m1 is motionless where m2 started, m2 is motionless where m3 started, and so on. Our entire system is at rest. Moreover, compare the system {m0, m1, m2, …} at t=1 to the system {m1, m2, m3, …} at t=0. They’re indistinguishable! Our initial system {m1, m2, m3, …} has absorbed the new particle m0 and all of its kinetic energy and emerged entirely unchanged!

Things become seemingly stranger if we reverse time and consider this process backwards. We begin with the system {m0, m1, m2, …}, entirely at rest. But then, at t = -1, for no apparent reason, a chain reaction of collisions begins around the origin (“begins” might not be quite the right terms — there’s no first collision setting it all off). The collisions propagate rightward, and at t = 0 the system spits out particle m0, moving to the right at 1 meter per second, while the system {m1, m2, m3, …} is now identical to the system {m0, m1, m2, …} at time t = -1. The ejection of this particle from a previously static system is an inexplicable event. It has no root cause; each collision in the chain was caused by the one before it, ad infinitum. A moving object has seemingly emerged from nothing; kinetic energy has seemingly been created for free.

Cover Image: “4 Spheres” by Victor Vasarely