History repeats itself
There is a parallel between Bekenstein’s proposal for the entropy of a black hole and the development of nineteenth-century statistical mechanics. When Boltzmann died in 1906, aged 62 years, his atomistic explanation of the Second Law was still not universally accepted. Led by the greatly influential Austrian physicist Ernst Mach, many scientists still doubted the very existence of atoms. Mach’s objections were initially of a philosophical nature, but they developed a momentum in part because Boltzmann’s work led to a good deal of confusion that Boltzmann himself struggled to dispel. The argument became centred around the statistical nature of the Second Law. According to Boltzmann, if matter is made of atoms all moving around then it is overwhelmingly likely that entropy will increase, but it is not guaranteed. For example, there is a near-vanishingly small probability that all the atoms in a room will end up clustered in one corner. Mach and his followers felt that a fundamental law of Nature should not be statistical. ‘Entropy almost always increases’ didn’t sound authoritative enough, especially since Clausius’s formulation of the Second Law has no ‘almost’ about it. Today, we know Boltzmann was right – the Second Law does involve an element of probability.
A similar debate now centres around the physical significance of the thermodynamic behaviour of black holes. If we accept the idea that the entropy of a black hole is signalling the presence of ‘moving parts’ of some sort, the astonishing implication is that general relativity is underpinned by a statistical theory just as classical thermodynamics is underpinned by statistical mechanics. This means that we should regard spacetime as an approximation; an averaged-out description of the world akin to the description of a box of gas in terms of temperature, volume and weight. In 1902, the pioneer of statistical mechanics Josiah Willard Gibbs wrote: ‘The laws of thermodynamics … express the approximate and probable behaviour of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles …’ Could it be that, in the first decades of the twenty-first century, we find ourselves in a similar position, as beings who have not the fineness of perception to appreciate the underlying structure of space and time?
When Bekenstein made his suggestion that black holes have an entropy, however, there was one huge fly in the ointment. As we’ve seen, entropy and temperature go hand in hand, and the assignment of a thermodynamic entropy to a black hole requires it to have a temperature. But for an object to have a temperature it must be able to emit things as well as to absorb them. Temperature, after all, can be defined in terms of the net transfer of energy between objects – in Feynman’s analogy, the ‘ease of removing water’. If it is not possible to extract anything from a black hole then the temperature must be zero. And, as everyone knew in 1972 and as general relativity makes abundantly clear, nothing can escape from a black hole.
Then, in 1974, everything changed. Along came Stephen Hawking with a short paper entitled ‘Black Hole Explosions?’
* Taken from Clausius’s excellently titled 1865 paper, ‘The Main Equations of the Mechanical Heat Theory in Various Forms that are Convenient for Use’.
† Specifically, the change in entropy dS = dQ/T, where dQ is the amount of heat energy transferred to the cup of tea and T is its temperature.
‡ A technical note. The entropy of a system at zero temperature is zero if the ground state is not degenerate, which means that there are not multiple ground states of the same energy. Solid carbon monoxide and ice are two examples of solids that have degenerate ground states and therefore have a ‘residual entropy’ at zero temperature because there will still be uncertainty about which state a randomly selected molecule came from.
§ log 1 = 0. We use ‘log W’ to indicate the natural logarithm of W.
¶ log 2 = 1.4427.
** We explain why a black hole has the largest possible entropy shortly.
†† A more careful calculation, which properly accounts for the quantum physics, gives an entropy equal to the horizon area divided by (4 x Planck length squared), which differs from our estimate by a numerical factor.
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.