Quantum mechanics: the notion of vacuum fluctuations

We have seen that it is not clear if we can make remnants by the processes allowed in classical physics. But Hawking showed that if we consider the effects of quantum mechanics, then remnants appear to form quite naturally. We explain the relevant process, which arises from quantum fluctuations of the vacuum.

Let us now see the physics underlying Hawking's computation.

Vacuum fluctuations :    We normally think of the vacuum as 'empty'. This is indeed true in classical physics. The vacuum has no matter, so no mass. Using the relation \( E=mc^2 \), we see that the energy \( E \) of the vacuum is zero.

But things are more complicated when we take into account quantum mechanics. The energy can now fluctuate, in a way governed by the uncertainty principle \[ \Delta E~ \Delta t \sim h \] where \( h \) is the Planck constant. This relation means the following. Even if we are in the vacuum, the energy can fluctuate up to a value \( \Delta E \), as long as the fluctuation lasts for a time that is no longer than \[ \Delta t \sim \frac { h} {\Delta E} \]

If we do have an energy fluctuation \( \Delta E \), what does it mean? Suppose at a given time the vacuum is empty of all particles. If there is an energy fluctuation \( \Delta E \), then this energy can take the form of new particles. The figure (a) below shows an electron and a positron that have appeared in the vacuum. If each of these particles has a mass \( m \), then we need an energy fluctuation \( \Delta E = 2 m c^2 \).

In figure (b), we are at a time \( \Delta t \) later, and the particles have disappeared. This creation and subsequent annihilation of a particle pair is called a 'vacuum fluctuation'.

Vacuum fluctuations near the horizon:    Something novel happens if we consider such vacuum fluctuations near a horizon. One member of the pair can end up inside the horizon, where it has negative energy. The other can be outside, where it has positive energy. The total energy of such a pair is \( \Delta E =0 \).

But because \( \Delta E=0 \), we find that such a fluctuation can live for a time \( \Delta t \rightarrow \infty \). In other words, the particle pair does not ever vanish. The particle outside (the blue circle) drifts away from the hole. The particle inside the hole (the orange particle) stays in the hole as a negative energy particle, just as was required for making a remnant !

We will next put this understanding into an overall picture of the Hawking radiation process.