The difficulty with making remnants in classical physics

We have seen that strange objects called remnants may be possible due to the attractive nature of gravity. But how can we make such objects? In this section we note that it is not clear that we can make remnants by classical processes.

How can we make remnants ?    We have seen that the intrinsic energy \( E=mc^2 \) of a particle \( m \) is positive, but the gravitational potential energy \( PE=-\frac {GMm}{r} \) is negative. Thus by placing many such small particles \( m \) near a heavy mass \( M \) it seems that we can make an object - the 'remnant' - whose total energy is zero or close to zero.

But how do we 'place' the mass \( m \) at the desired position? Let us try a few possibilities:

(i) Free infall:   Suppose we start with \( m \) being far away from \( M \). At this stage \( m \) has only its intrinsic energy \( E=mc^2 \). Now we let \( m \) fall in towards \( M \), under the gravitational pull exerted by \( M \).

When the distance reduces, we do get the negative PE contribution. But the falling particle \( m \) also gets a kinetic energy (KE), which is positive. The total energy of \( m \) is \[ E_m=mc^2-\frac {GMm}{r}+KE \]

Since the KE is positive, it is not clear if we will be able to make the net energy \( E_m \) negative. In fact the energy \( E_m \) is conserved during the motion, so its value always remains the same as its starting value, which was \( mc^2 \). This value is positive.

So at least this way of 'placing' \( m \) close to \( M \) - letting it fall in - does not help us make a remnant.

(ii) Slow lowering:   We again start with \( m \) far away, but instead of letting it fall in freely, we lower it slowly towards \( M \) with the help of a rope

Now \( m \) will not speed up, so it will not have a large KE.

But if we study this process properly using general relativity, we find the following. As long as we lower \( m \) to a point outside the horizon, we can indeed avoid having any KE. But if we try to place \( m \) inside the horizon, then the rope cannot hold it stationary; it has to fall in.

So the process of 'slow lowering with a rope' also does not allow us to make a remnant.

Hawking's discovery :    It may appear from the above discussion that one need not worry about remnants: they are problematic objects, but if there is no way to make them, then maybe they do not matter. But Hawking showed that if we include the effects of quantum mechanics, then it is indeed possible to get remnants. In fact one cannot stop a black hole from turning into a remnant. We will discuss the Hawking process next.