This was the view of many theoretical physicists until very recently. In the last few years, however, there has been a realisation that the clues to quantum gravity may not only reside near the singularity. They may also be found in the physics of the horizon. This came as a very welcome surprise, because it was long assumed that whatever happens close to the event horizon of a black hole, the physical processes should have nothing at all to do with the extreme conditions at the singularity where we expect quantum gravitational effects to be important. The horizon, after all, is a place in space through which an astronaut can happily fall and suffer no ill effects. This assumption now appears to have been too pessimistic. The study of the thermodynamics of black holes, which began in the 1970s as an investigation into quantum mechanical effects in the vicinity of the horizon, has opened an entirely unexpected window into the deep mystery of quantum gravity. And it is to black hole thermodynamics that we now turn.
* Strictly speaking, black holes can also carry electric charge but astrophysical black holes are electrically neutral.
† The waters were muddied by a Lifshitz and Khalatnikov paper of 1963 which claimed to prove no singularity would occur. Following work with Belinskii, Lifshitz and Khalatnikov withdrew their claim in 1970.
‡ On the diagram, the Schwarzschild radius is at r = 2m where m is the mass of the star because Penrose is working in units where G = c = 1.
§ Strictly speaking this logic works only for the region outside of the horizon and we are guessing somewhat as to what happens inside the horizon.
9
Black Hole Thermodynamics
‘Black holes ain’t so black’
Stephen Hawking
Up to this point, we’ve thought about black holes as objects that, broadly speaking, mind their own business. Stuff can fall in, causing the black hole to grow, but nothing that crosses the horizon can come out. All traces of anything and everything that falls into a black hole would seem to be erased from the Universe forever. This is the description of a black hole according to general relativity. In 1972, John Wheeler and his graduate student Jacob Bekenstein realised that this raises a deep question. Wheeler tells the story of how he, in a ‘joking mood one day’, told Bekenstein that he always felt like a criminal when he put a cup of hot tea next to a glass of iced tea and let them come to the same temperature. The energy of the world doesn’t change, but the disorder of the Universe would be increased, and that crime ‘echoes down to the end of time’.25 Wheeler was referring to the Second Law of Thermodynamics, which roughly speaking says that when any change occurs in the world, the world becomes more disordered as a result. ‘But let a black hole swim by and let me drop the hot tea and the cold tea into it. Then is not all evidence of my crime erased forever?’ Jacob took the remark seriously, recounts Wheeler, and went away to think about it.
The physical quantity that measures disorder is known as entropy. In Wheeler’s words, ‘Whatever is composed of the fewest number of units arranged in the most orderly way (a single, cold molecule for instance) has the least entropy. Something large, complex and disorderly (a child’s bedroom perhaps) has a large entropy.’ The Second Law of Thermodynamics, phrased in terms of entropy, states that in any physical process, entropy always increases. Wheeler was concerned that he had increased the entropy of the Universe by placing his two cups of tea in contact, and then decreased it again by throwing them into a black hole.
As is so often the case, there is a lyrical Eddington quote that reflects the importance of the Second Law: ‘The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.’
Bekenstein returned a few months later with an answer: the black hole does not conceal the crime. His answer was inspired by Hawking’s prior observation that the area of a black hole’s horizon always increases, no matter what. To Bekenstein, this ‘area always increases’ law reminded him of the ‘entropy always increases’ law. He therefore made the bold claim that throwing objects into a black hole causes an increase in the area of the event horizon, which in turn signals a corresponding increase in entropy. In other words, when something falls into a black hole, a record is kept, the Second Law is obeyed, and deepest humiliation is avoided. Looking back at Wheeler’s description of entropy in terms of molecules and messy bedrooms, however, the assignment of an entropy to a black hole would seem to be questionable, as Bekenstein and Wheeler were well aware. According to general relativity, black holes are simple things: a Schwarzschild black hole can be described by a single number; its mass. But Wheeler begins his description of entropy with the phrase ‘Whatever is composed of the fewest number of units arranged in the most orderly way … has the least entropy.’ With no apparent units to re-arrange, what, then, is the meaning of the entropy of a black hole?
Entropy was introduced in the nineteenth century as one of the fundamental quantities in the newly emerging science of thermodynamics, alongside the more familiar notions of heat, energy and temperature. Perhaps because of its historical origins in the industrial revolution, thermodynamics has the reputation of being closer to engineering than black holes, but this is emphatically not the case. Thermodynamics is connected at a deep level to quantum mechanics and the structure of matter. When applied to black holes, we will see that thermodynamics is connected at a deep level to quantum gravity and the structure of spacetime. Before we get to the thermodynamics of black holes, let’s go back in time to the nineteenth century for a tour of the origins of the subject and to introduce the important concepts of heat, energy, temperature and entropy.
The fundamental physics of the fridge
The foundations of thermodynamics were laid by practical people doing practical things: people like Salford brewer James Prescott Joule who were interested in building better steam engines, developing more efficient industrial processes, and beer.
In the early 1840s, Joule performed a range of experiments to demonstrate that heat and work are different but interchangeable forms of energy. Figure 9.1 illustrates his most famous experiment. A weight falling under gravity turns a paddle that stirs some water, causing the temperature of the water to rise. In thermodynamical jargon, the falling weight does work on the water. Joule’s skill was in being able to make very precise measurements of the temperature increase, which he demonstrated to be proportional to the amount of work done by the falling weight. Initially, his findings were not met with enthusiasm. They went against the thinking of the day, which held that heat is an ethereal fluid (‘caloric’) that flows from hot to cold objects. Joule submitted his findings to the Royal Society in 1844, but his paper was rejected, partly because it was not believed that he could measure temperature increases to the claimed 1/200th of a degree Fahrenheit. The Royal Society, then as now, was not awash with experts in the brewing of fine ales, the demands of which meant that Joule had access to instruments capable of the required precision. A notable exception is the former President of the Royal Society and Nobel Laureate Sir Paul Nurse, who began his distinguished career as a technician in a brewery because he wasn’t accepted onto a university degree course due to his lack of a modern language qualification. Sir Paul was subsequently awarded the 2001 Nobel Prize for his work on yeast. Joule remained undaunted, and by the mid-1850s his work had become widely accepted after a fruitful collaboration with William Thomson (later Lord Kelvin).