Black Holes: The Key to Understanding the Universe

10

Hawking Radiation

‘Bardeen, Carter and I considered that the thermodynamical similarity was only an analogy. The present result seems to indicate, however, that there is more to it than this.’

Stephen Hawking31

Stephen Hawking’s paper triggered a revolution in theoretical physics that is still ongoing. He discovered that quantum theory predicts that a black hole will emit radiation as if it were an ordinary object with a temperature. The immediate suggestion is that the law of gravity should be regarded as a statistical law and that quantum effects lead to an elemental randomness in the geometry of space. Today, we do not know what that randomness corresponds to. It remains the holy grail of theoretical physics. But we have travelled a long way since 1974. The remainder of the book is about the quest to understand the deep origins of black hole thermodynamics; a quest that is edging us ever closer to a new theory of space and time.

The laws of black hole mechanics

In 1973, Bardeen, Carter and Hawking published a paper entitled ‘The Four Laws of Black Hole Mechanics’ in which they drew the following analogy between the laws of classical thermodynamics and the properties of black holes:32

Even if you don’t understand the symbols, the similarity is striking. One set of laws can be obtained from the other by swapping ‘temperature (T)’ with ‘surface gravity (k)’ (divided by 2π), and ‘entropy (S)’ with ‘area (A)’ (divided by 4).

Let’s start with the Zeroth Law, which as we saw in the last chapter formally anchors the concept of temperature. A system such as a box of gas is in equilibrium if everything has settled down and nothing is happening. This means that all parts of the system have the same temperature. For a black hole, the corresponding quantity is the surface gravity, k. This has the same value everywhere on the event horizon of a black hole that has settled down after swallowing a planet, for example. The surface gravity tells us how difficult it is to resist the pull of gravity just above the event horizon. Imagine that, in an admittedly surreal turn of events, an astronaut decides to conduct a spacewalk, carrying a fishing rod, close to a black hole. On the end of the fishing line is a trout of mass M. The astronaut lowers the trout down until it is dangling just above the horizon and measures the tension on the fishing line. The tension will be kM, where k is the surface gravity of the black hole.* For a perfectly spherical (Schwarzschild) black hole it is perhaps obvious that the surface gravity should not vary as one moves around the horizon. For a rotating (Kerr) black hole this is not obvious at all. A proof is provided in the Bardeen, Carter and Hawking paper.

The First Law expresses the conservation of energy. It says that if we add energy (dE) into a system at given temperature (T) we increase the entropy (dS). The corresponding law of black hole mechanics states that if we drop an amount of energy (dE) into a black hole that has a surface gravity (k) then the surface area of the horizon will increase (dA). If we are tempted to identify the surface gravity with the temperature, then we might also be tempted to identify the surface area with the entropy as Jacob Bekenstein proposed. That temptation is made all the greater by the Second Law which, for a black hole, is a statement of Hawking’s discovery that the area of the event horizon always increases. This apparent link between the purely geometric concept of area and the information content of a system is very unexpected.

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.

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.