Information and the nature of mass
I’ve had an idea bouncing around in my head since last fall when I had an interesting discussion on the nature of mass on LinkedIn. It actually follows up on some ideas that I outlined in Physics Today about seven years ago. It all centers on the concept of mass: what is it? There seem to be two schools of thought on this issue (aptly represented by myself and Frank Wilczek in the aforementioned article). The first school of thought assumes that mass is merely energy. The second school of thought says that, although it can be transformed into energy, it isn’t quite the same thing. One of the arguments against the former is that it (perhaps unintentionally) hides the significance of the spacetime metric, something Lev Okun has pointed out. To fully understand this, however, we need to understand how it is mathematically defined (something everyone agrees on). Specifically, mass is defined (mathematically) to be the magnitude of the four-momentum vector.
What’s tricky here is that the magnitude of four-vectors aren’t quite like that of other vectors – the signature plays a role. Specifically, the square of the temporal component of the four-momentum (called the relativistic energy) has the opposite sign to the square of the spatial components (called the relativistic momentum). So, the magnitude of the vector above works out to
Crucially, this quantity is frame-independent. As such it is often referred to as the “invariant” mass (also “rest” mass). Okun’s argument (one I happen to agree with) is that by merely saying mass is the same as energy, implies that momentum is the same as energy (given the above equation), and thus the difference in the sign of the signature is “brushed under a rug.” This difference is very important since it points out the fact that time and space, though similar, are not quite the same exact thing (that’s a discussion for another time). Note that the four-momentum is conserved which means that it is conserved by component. We’ll see what that means in a moment.
To illustrate some of these concepts – and raise some interesting questions – let’s consider the following thought experiment. Imagine a very thin, spherical optical cavity the size of, say, a star, i.e. a very thin, hollow sphere with perfectly (or near perfectly) reflective material coating the inside, that is at “rest” (roughly inertial). Now suppose that the cavity is filled with neutral pions. Neutral pions decay to photons. Now at some point prior to any pion decay, the total four-momentum of the system will be
where we assume there are n pions. The system has a total mass equal to the magnitude of the system’s four-momentum vector (large enough for stuff to orbit it in this example).
For the sake of simplicity, let us assume that any pion decays to two photons (this is the primary decay mode with probability 0.98798) that, in order to conserve four-momentum, must move off in opposite directions. After some time, t, all the pions have decayed and the total four-momentum is
where we now have 2n photons rattling around inside the cavity. Note that, in order to conserve four-momentum, each decay produces two photons that move off in opposite directions. Given the large, randomly oriented quantity of pions, on average we expect the total relativistic momentum of the photons to equal zero. This is true even though the photons from a single decay may hit the insides of the cavity at different times because we assume that there is a comparable decay on the other side the balances it out (note that it might be possible for the cavity to “jiggle” a bit if these decays are slightly uneven temporally, but in the end it’s center-of-mass should not have budged an inch).
Naively, we might be tempted to say that the system has lost a great deal of mass. But it clearly hasn’t since, assuming four-momentum is conserved, it must be that
where the system’s mass is the magnitude of each of these vectors. So, while the pions’ mass has seemingly disappeared – turning into a bunch of massless photons – the system’s mass remains unchanged. In fact, if we suppose that a planet orbits our contrived “star” (cavity), assuming the cavity is internally perfectly reflecting, the planet should experience no change in its orbit. Thus, in this broad sense, mass is conserved (invariant). It’s crucial for the definition, however, that we’re dealing with four-vectors here. If it were an ordinary vector, none of this would work. That’s one reason the metric is so important.
Now, if we view this from the standpoint of information, is information created, destroyed, or conserved in this process? Traditionally, despite being spin-1 particles (which can serve as a representation of a qutrit), photons only have two degrees of freedom and so they are typically treated as a representation of qubits. Either way, it’s fairly easy to see how we can encode information in the photons. Pions, on the other hand, are spin-0. How could we possibly encode information in plain, old neutral pions? I suppose there might be a way, but it still begs the question of whether information gets created in this process or whether it’s conserved.
So now let’s ask ourselves what happens if the inside of the cavity is not a perfect reflector. In this case, let us assume that a certain fraction of all photon collisions with the cavity result in a photon being absorbed by the cavity. Given enough absorptions, the cavity will begin to heat up, re-radiating thermal photons, some outward and some inward. Eventually, the cavity should radiate enough heat outward that the total mass of the cavity plus all the photons inside it is now lower. In other words, in a process that’s somewhat analogous to Hawking radiation and the evaporation of black holes, our “cavity” radiates away its mass. Does this have any effect informationally?
The next question that should be asked, of course, is what would be different for this system if we treated it quantum mechanically? In that case, all the photons (and pions, for that matter) can be assigned wavefunctions and we can do a modified “particle in a box” model where the box is now a sphere. I haven’t yet given this enough thought. We’ll have to wait and see. If anyone out there has thoughts on this, please do share. 😉