We have seen that we can make objects with very small mass (or perhaps even zero mass) by using the attractive nature of gravity. Now we note that the number of such objects is potentially infinite. This observation is important, because it will be a key element in the black hole information paradox.
We have seen that we can make objects with very small mass as follows. We start with a big mass \( M \). We place several small masses \( m \) near \( M \), all within the critical radius \( r_0\). The overall energy of the configuration gets reduced, and we may even be able to bring it down to zero. This entire configuration we call our 'object'. Since \( E=mc^2 \) for any object, the small energy of the object implies that the object has a small mass.
Let us now ask: if we can indeed make such objects, how many will there be? There are three reasons why the number will be very large:
(a) The nature of the starting mass \( M \) can be chosen in many different ways. For example, it could be made of iron, or copper, or gold, or any mixture of different elements. We depict this by different colors of the central mass \( M \) in the future below
(b) We can have many possible ways of choosing the small masses \( m \). Suppose for simplicity that any of the masses \( m \) could be of two kinds: an electron or a positron. We depict these by the orange and green dots respectively in the figure below. Then we can arrange the small masses in many different ways, as shown in the figure.
Suppose there are \( N \) small masses \( m \). We have assumed that each such mass can be of two kinds (electron or positron). Then the number of possible configurations of these small masses is \( 2^N \). Typically \( N \) will be a large number, and then \( 2^N \) will be very large indeed.
(c) The third reason is actually the most important, because it shows that the number of objects is not just large, but actually infinite. As shown in the figure below, we can start with a bigger value of the mass \( M \). We will need to add a correspondingly large number of small masses \( m \) to lower the energy down to a small value. But at the end of this process we will again get an object of small mass.
But there is no limit to how large a mass \( M \) we start with. Thus the number of small mass objects we can make by the above method would be infinite.
The problem : We can now state how the central problem of the information paradox will arise. While classical physics is adequate for most of our everyday observations, we know that our world is really described by quantum theory. And it turns out that in quantum theory, there is a serious difficulty if there are an infinite number of objects with a small mass.
We will explain the reason for this difficulty later. But for now we note that in our usual world, we have only a few objects of small mass. Suppose we count all particles that have mass less than, say, \( 10^{-25} \) grams. We have the photon, which is massless. We have the electron, with mass \( 10^{-27} \) grams. There are also particles called pi mesons, that are less familiar, but whose mass is around \( 10^{-24} \) grams. But there are no other objects within our chosen mass bound, so the number of such objects is finite.
But when we use the negative potential energy of gravity, it appears that we can make an infinite number of light objects. The natural mass scale for such objects, if they exist, would be planck mass, which is \( 10^{-5} \) grams. This is large by the standards of particle physics, where elementary particles are typically much lighter. But the problem is that the number of objects with mass smaller than \( 10^{-5} \) grams would be infinite. And normal quantum theory would fail if we had an infinite number of such objects.