Figure 10.1. A vacuum fluctuation. A pair of particles emerges (at A) from the vacuum for a fleeting instant of time before recombining at B. One of the particles can be pictured as having negative energy such that the sum-total of the two particle energies is zero.
One way to picture the quantum vacuum is to imagine particles constantly popping into it and disappearing again a fleeting instant later. These ghosts – the momentary flickers of particles – are known as vacuum fluctuations or ‘virtual’ particles. If we could somehow freeze time and peer with high-resolution vision deep into any region of space, we’d see these particles, not as fleeting ghosts but as real particles. But that’s what happens to time and space close to the horizon of a black hole, as viewed from the outside. The black hole behaves like a magnifying glass, freezing time and changing the way we view the vacuum fluctuations. Virtual particles from one point of view can be as real as the particles that make up our bodies from another.
Virtual particles emerge from the vacuum in pairs, and if you could watch them flickering in and out of existence in front of your nose you’d observe that one of the ghostly particles would have positive energy and the other negative energy. In the normal scheme of things, the particles come back together again in a very short time and the energy is repaid, so that the total energy of the vacuum remains unchanged on the average. This process, however fantastical it may sound, is familiar. When you switch on a fluorescent light, atoms of the vapour inside the tube are supplied with energy and jump into an excited state. This means that electrons inside the atoms now occupy energy levels above the ground state. We met these energy levels in our discussion of temperature and entropy in the previous chapter: one can think of a single atom rather like a box, containing electrons that are distributed among the available energy levels. The electrons occupy the higher energy levels for a while before dropping down to lower levels. In doing so, they emit photons which carry the energy away and cause the fluorescent tube to glow. For a time, the reason for the electrons dropping back down to lower energy levels in the atom was not understood, and physicists called it ‘spontaneous emission’. It is now understood that vacuum fluctuations cause the electrons to fall back down to the lower energy levels inside the atom. They ‘tickle’ the atom and trigger the emission of light. Hawking radiation has the same origin. Vacuum fluctuations tickle the black hole, causing it to lose energy by emitting particles.
In his 1975 paper, Hawking gives a heuristic explanation for the origin of black hole radiation. As he is careful to note, these physical pictures are not meant to be a rigorous argument and ‘the real justification of the thermal emission is the mathematical derivation’. Nevertheless, as Hawking appreciated, pictures are very useful for developing understanding. Here’s Hawking’s picture.
Let’s station ourselves outside the event horizon of a black hole and focus on the vacuum fluctuations close to the horizon. From this perspective, the vacuum fluctuations can be disrupted such that one of the particles escapes recombination with its partner. The reason is that the negative energy particle in the fluctuating pair can be inside the horizon, where it can exist until it reaches the singularity. The possibility of negative energy particles existing inside a black hole is something we encountered in the Penrose process for extracting energy from a black hole. In that case, the fact that space and time ‘switch roles’ inside the ergosphere was responsible. That same reversal of the roles of space and time is why the negative energy particle inside the horizon will reduce the mass of the black hole while its partner can head outwards into the Universe and appear as Hawking radiation.†
For a Schwarzschild black hole, it is possible to express the equation for the temperature of a black hole in terms of the mass M of the black hole, rather than the surface gravity. The result is that‡
This wonderful equation reveals the marriage of quantum theory and general relativity that is present in Hawking’s calculation, and it establishes that the laws of black hole mechanics are the fundamental laws of thermodynamics in disguise. This is what convinced physicists that it is correct to treat black holes as thermodynamic objects that can store information and exchange energy with the Universe beyond the horizon. Stephen Hawking’s calculation of the temperature of a black hole is so important to our understanding of the Universe that it is now literally written in stone on the floor of Westminster Abbey.
Figure 10.2. The temperature of a Schwarzschild black hole, written on Stephen Hawking’s memorial stone in Westminster Abbey. (Copyright © Dean and Chapter of Westminster)
* The surface gravity is inversely proportional to the mass of the black hole.
† Not all the positive energy particles will travel away from the hole. Some will fall into it and end up in the singularity. The point is that some of the particles can escape.
‡ ħ is Planck’s constant, h, divided by 2π.
11
Spaghettified and Vaporised
‘In microphysics, however, the information does not sit out there. Instead, Nature in the small confronts us with a revolutionary pistol, “No question, no answer.” Complementarity rules.’
John Archibald Wheeler34
The ‘black holes are a bit like atoms in the way they emit radiation as a consequence of being tickled by the vacuum’ picture is a good one, but it disguises a major difference between the emission of light from everyday hot objects and the emission of Hawking radiation. The difference can be traced to the fact that it is gravitational effects that make the vacuum fluctuations real. This unique production mechanism gives rise to three properties of Hawking radiation that, taken together, appear nothing short of bamboozling.
Someone falling freely near to the horizon of a large black hole will not encounter any radiation.
Someone accelerating, so that they hover just above the horizon of a large black hole, will be vaporised by a flux of very hot radiation.
Someone far away from the black hole will experience a flux of cool radiation which appears to have been emitted by a glowing object at the Hawking temperature.
Let’s deal with each of these in turn.
That someone falling freely near the horizon of the large* black hole should not encounter any radiation is not hard to understand. This is Einstein’s Equivalence Principle. A freely falling observer feels as if they are at rest in plain old flat spacetime. From their perspective therefore, they experience the vacuum fluctuations (those particle–antiparticle pairs) in the same benign way as someone floating far away from the black hole. As a result, they float onwards unawares until, inexorably, they are spaghettified as they approach the singularity.
In contrast, someone hovering just above the horizon will encounter the positive energy part of the vacuum fluctuations. From their perspective they will be bombarded by a flux of real particles, separated from their partners by the geometry of spacetime. A very similar effect occurs even in flat spacetime, as was appreciated by Paul Davies and Bill Unruh in the mid-1970s.†35 In the Davies–Unruh effect, an accelerating rocket ship far from any gravitating objects will experience a thermal bath of particles and the temperature of the bath is proportional to the acceleration. The Equivalence Principle can be invoked to say that the same thing will happen to a rocket that accelerates to maintain a fixed position close to the horizon of a black hole. In that case, because spacetime in the immediate vicinity of the rocket is approximately flat, the rocket’s experience is just like that of a rocket accelerating in flat spacetime. And therefore, as a result of being immersed in a hot bath of particles, the rocket heats up. If it is close enough to the horizon, it will be vaporised.