The Third Law is interesting too. In classical thermodynamics, it states that it is not possible to cool something down to zero temperature in a finite series of steps. One way to see this is to think about a fridge again. As the temperature inside the fridge gets closer and closer to zero, the efficiency of the fridge also gets closer and closer to zero. That’s because removing energy from something at very low temperature involves a huge entropy change, which has to be accounted for by sending an appropriately huge amount of energy out into the environment. Ultimately, the fridge would have to do an infinite amount of work to transfer the last drips of energy from inside to outside and cool the interior to absolute zero. For a Schwarzschild black hole, we could make the surface gravity go to zero by making its mass infinite, which obviously requires an infinite amount of energy. For a Kerr black hole, the situation is different. It is possible to reduce the surface gravity by throwing matter into the hole, if the matter is rotating. At first sight, it looks as if this could be used to dodge the Third Law and get the surface gravity down to zero, but that isn’t the case. Remarkably, it turns out that as the surface gravity gets smaller, it becomes harder to throw stuff into the hole. The matter either misses the hole or gets repelled by it.
Bardeen, Carter and Hawking conclude their 1973 discussion with the following comments: ‘It can be seen that k is analogous to temperature in the same way that A is analogous to entropy. It should be emphasised, however, that k and A are distinct from the temperature and entropy of the black hole. In fact, the effective temperature of a black hole is absolute zero.’
A few months later in his 1974 paper Hawking disagrees with himself,33 which is one of a scientist’s most important abilities: ‘… it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate one would expect if the black hole was a body with temperature …’ His more detailed 1975 follow-up, ‘Particle Creation by Black Holes’, carefully derives the following equation for the temperature of a black hole:14
which states that black holes have a temperature equal to their surface gravity divided by 2π. Hawking has now realised that the similarity between the laws of thermodynamics and the laws of black hole mechanics is not just an analogy. Rather, it appears to be an exact correspondence and black holes are thermodynamic objects. As Hawking writes, ‘if one accepts that black holes do emit particles at a steady rate, the identification of k/2π with the temperature and A/4 with the entropy is established and a Generalised Second Law confirmed.’
Hawking’s discovery that black holes emit particles is of profound importance, not least because it suggests that the origin of the law of gravity is statistical. This is the shock wave that reverberated through the theoretical physics community in the early 1970s. Just as the concepts of temperature and entropy for a box of gas emerge from a hidden microscopic world composed of lots of little things jiggling and reconfiguring, so it seems do the laws of gravitation. But how is it possible for a thing from which nothing should escape to glow like a hot coal? To understand Hawking’s discovery we need to turn to quantum theory, and to the physics of nothing.
Hawking radiation
We haven’t met quantum theory in any detail yet because we’ve been discussing general relativity, which is a classical theory. Classical theories describe a reality that fits nicely with our intuitive mental picture of the world. The Universe is composed of particles, fields, and forces. At any moment there is a single configuration of the Universe, and this evolves in a predictable way into a new configuration as things interact with each other in the arena of spacetime. General relativity tells us how spacetime reacts to the particles and fields and how the particles and fields react to spacetime.
Quantum mechanics is different. It describes a world of probabilities and of multiple possibilities. For example, when a particle moves from A to B, quantum mechanics says we must take all possible paths into account if we are to make predictions that agree with experimental observations. In classical physics, the particle follows a single path, but this is not so in quantum physics.
A central difference between classical theory and quantum theory is the unavoidable appearance of probabilities in the description of Nature. This is encapsulated by the famous Heisenberg Uncertainty Principle, which states that we cannot simultaneously know the precise position and momentum of a particle. If we know with high precision where a particle is, we know with less precision how fast it is moving. The consequence is that we cannot predict with certainty where a particle will be in the future, even if we know everything it is possible to know about its current state. Rather the theory gives us a list of probabilities for possible future locations. This is not because of a lack of knowledge or skill on our part. It’s the way Nature is. Very importantly, though, we can still predict how the so-called quantum state of the particle changes over time. Precise knowledge of the quantum state provides us with a list of probabilities to find the particle in some region of space, and we can predict with precision how this list of probabilities changes over time, though we can never say for sure where the particle will be. We are therefore not able to know precisely where an electron will be at some moment, or precisely how much energy the electromagnetic field will carry in some region of space. We can only know the probability that a particle will be somewhere, or the probability that a field will be in a certain configuration. It is this inherent uncertainty in the configuration of particles and fields that ultimately leads to Hawking radiation.
We should emphasise that, as far as we can tell, all of Nature is quantum mechanical. Quantum theory is as venerable as general relativity and underpins not only our understanding of atoms and molecules and all of chemistry and nuclear physics, but also modern-day electronics. For example, the semiconductor transistor used in their billions in modern electronic devices is an inherently quantum mechanical device. We live in a quantum universe.
The quantum vacuum
There are certain words in colloquial usage that mean something very different in physics. Vacuum is one such word. The important feature of our quantum universe that leads to Hawking radiation is that the vacuum of empty space is not empty. It’s natural to picture a vacuum as being empty – devoid of all particles and fields – but this is not correct. The vacuum can’t be empty, because ‘empty’ is a precise statement about the energy and configuration of the fields, and quantum theory does not allow that. The vacuum is therefore an active place with a complex structure. There is no way to isolate a region of space and suck all the particles out of it to leave it perfectly empty. Roughly speaking, the vacuum is to humans as water is to a fish: it is an ever-present backdrop to our everyday experience. Particles can be thought of as excitations of the vacuum – ripples in the vacuum sea – and quantum theory describes a sea that always ripples.