Chapter 5 Entropy and thermodynamics of black holes

  You are what you eat

  It is often said that you are what you eat. Thus if your diet is purely junk food and chocolate, then your complexion, not to mention your physical and mental well-being, will be rather different than if you subsist on a healthy diet of salad andMediterranean food.However, it seems that black holes are not fussy eaters. Whether they are hoovering up a vast expanse of interstellar dust or a cubic light-year of fried eggs, their mass will similarly increase inexorably. In fact, after a black hole has finished its sumptuous meal, you have no way of telling what it was eating, only how much it has consumed (although you could tell if what it ate had charge or angular momentum). You only know the quantity of its diet, not about the quality. The `no-hair theorem' described in Chapter 2 says that the black hole is only characterized by a very few parameters (mass, charge, and angular momentum), and thus we cannot talk about what the black hole is made of.

  This lack of knowledge about the nature of what has been sucked in by a black hole may seem like a trivial observation, but it is actually rather profound. Information about a black hole's lunch menu has been fundamentally lost. Any matter which has fallen into the black hole has surrendered its identity.We can't perform measurements on that matter, or discern any details about it.

  本章節來源於𝑏𝑎𝑛𝑥𝑖𝑎𝑏𝑎.𝑐𝑜𝑚

  Black holes and engines

  This situation is eerily familiar to those who have studied the beautiful subject of thermodynamics. In that field it is quite common to understand how information can become lost or dissipated through physical processes. Thermodynamics has a long and interesting history. The modern theory began during the industrial revolution when people were trying to work out how to make steam engines more efficient. `Energy' could be defined in such a way that it was always conserved and could be converted between different forms. This is known as the first law of thermodynamics. However, although you can make some conversions between different types of energy, there are particular conversions you are not permitted to make. For example, although you are allowed to convert mechanical work completely into heat(you do that every time you use the brakes to bring your car to a complete stop), you cannot convert heat completely into mechanical work, which unfortunately is precisely what we would like to do with a steam engine. Therefore a steam engine in a train only succeeds in making a partial conversion of heat from the furnace into mechanical work which turns the wheels. It was ultimately realized that heat is a type of energy involving the random motion of atoms, while mechanical work involves the coordinated motion of some large bit of matter, like a wheel or a piston. Therefore, a crucial component of the nature of heat is randomness: because of the jiggling of atoms in a hot body, you lose track of the motion of the individual atoms. This random motion cannot simply be unrandomized without additional cost.The randomness, or to give it the technical name, entropy, in any isolated system never decreases but must always either stay the same or increase in every physical process. (This is the second law of thermodynamics.) One way of looking at this is to say that our information about the world always decreases because we cannot keep track of the motion of all the atoms in a large system. As energy moves from macroscopic scales to microscopic scales, from a simplemoving piston to the randommotion of huge numbers of atoms, then information is lost to us. Thermodynamics allows us to make this vague-sounding notion completely quantitative.This information loss turns out to be exactly analogous to what we've been describing for matter falling into a black hole.

  Although thermodynamics was developed for steam engines, the principles are thought to apply to all processes in the Universe.One of the first people to think about this in connection with black holes was the Oxford physicist Roger Penrose. He reasoned that because a black hole has spin, it might be possible to extract energy from it and thus to use it as some kind of engine. He came up with an ingenious scheme in which matter is thrown towards a spinning black hole in such a way that some of it emerges with more energy than was thrown in. Energy is extracted from the region just outside the event horizon (in fact from the ergosphere discussed in Chapter 3). Penrose's process slows the rotation of the black hole. In principle, an enormous amount of energy can be extracted from a black hole in this way, but of course this is just a thought experiment and so doesn't seemto be at present a practical solution to planet Earth's looming energy crisis!Within a few years of Penrose's work, James Bardeen, Brandon Carter, and Stephen Hawking made a landmark advance and formulated what they called the three laws of black hole dynamics which laid the foundations for Hawking's later thinking on the thermodynamics of black holes, which required the concept of temperature for a black hole which is determined by its mass and spin.

  Black holes and entropy

  Penrose's insight was a significant impetus and got others thinking about the thermodynamics of black holes. Together with R. M.Floyd, he showed that in his imagined process the area of the black hole's event horizon would tend to increase. Stephen Hawking started working on Penrose's clever scheme. The area depends on the mass and spin (and charge) in a rather complicated way, but Hawking was able to prove that in any physical process this area always increases or remains the same.One of the consequences of this intriguing result is that if two black holes coalesce then the area of the black hole event horizon of the merged black holes is larger than the sum of the areas of the two original black hole event horizons. (This is intuitively reassuring because the radius of the event horizon scales with mass, and surface area has a well-known dependence on radius.)This is the same sort of behaviour that we see with entropy in thermodynamics and therefore people began to wonder whether the entropy of a black hole and its area were somehow connected.Is this more than just an interesting analogy? One of John Wheeler's students, Jacob Bekenstein, went ahead and proposed a direct connection in his PhD thesis. Bekenstein used the ideas from the information theory of thermodynamics to argue that the area of a black hole event horizon is proportional to its entropy.(The choice he made means that you take the area of the event horizon and divide by one of physicists' fundamental constants,the Planck area, which is roughly 10-70 square metres, and within a numerical factor you get the entropy. This choice of unitsmakes the entropy of a black hole absolutely enormous.)

  Initially Hawking didn't believe Bekenstein's results, but on further examination he was able not only to confirm the approach but deepen our understanding of how black hole thermodynamics works. It is perhaps worth understanding how these analyses are done so one can appreciate both their power but also their limitations. The ideal way forward in this field would be to use a combination of quantum mechanics and general relativity, called quantum gravity, to study systems which are both very small, like a singularity in a black hole, but in which gravity plays a big role.Unfortunately we do not have a good theory of quantum gravity at present. A good approach is to use general relativity to model how spacetime curves and then use this together with quantum mechanics to understand the behaviour of particles in the curved spacetime. This was the approach that Hawking took to attempt to understand the thermodynamics of black holes.

  Is empty space empty?

  The concept of the vacuum (a region where there is `nothing' there) has had a long and tortuous history.Most of the ancient Greek philosophers hated the idea, on grounds that today seem extraordinarily arcane, but there were a small band of atomists who included the vacuum in their description of the world. Until the scientific renaissance, the idea of the vacuum was therefore very much out of fashion. However, following the invention of the air pump in 1650, the vacuum was something that you could experimentally demonstrate. Even though the amount of air that you could pump out of a vessel in the seventeenth century still gave you a rather poor vacuum by modern standards, the idea of nothingness had become substantially more believable. Once the existence of atoms had been demonstrated beyond all reasonable doubt in the early 20th century, the idea of a region of space with no atoms in it became not only uncontroversial, but inevitable.

  No sooner had atoms been demonstrated than a new theory of physics arose: quantum mechanics. One of the surprising consequences of this new theory was that there were fleeting moments when it seems like energy needn't be conserved. The first law of thermodynamics, the grand and seemingly unbreakable principle of physics, insisted that at every moment and at every place there had to be a strict accountancy between energy debits and energy credits. `Energy must always balance!' thunders the Cosmic Accountant. In fact, it seems that the Universal accountancy rules are more lenient and it is possible to obtain credit. It is perfectly acceptable to borrow energy for a short period of time as long as you pay it back quickly afterwards. The amount you can borrow depends on the duration of the loan, by an amount described by the Heisenberg Uncertainty principle. For example, even in the supposedly-empty vacuum it is possible to borrow enough energy to make a particle and anti-particle pair.These two objects can wink into existence and then after an extremely short period annihilate each other, thereby paying the energy back within the maximum allowed time limit (a time interval which is shorter the more energy is borrowed). Such a process goes on everywhere, all the time. It can even be measured!We now understand that the vacuum is actually not empty, but is a soup of these pairs of so-called virtual particles winking in and out of existence. Thus, the vacuum is not sterile and unoccupied, but is teeming with quantum activity.

  Black hole evaporation and Hawking radiation

  Hawking used the modern theory of the vacuum, quantum field theory, to study its behaviour close to the event horizon of a black hole. His analysis was mathematical but we can picture it in quite a simpleway. The essence is that a pair of `virtual' particles, a particle and its antiparticle (opposite in charge, identical in mass),created close to the event horizon of a black hole may end up becoming torn apart from one another. If one of that pair, either the particle or the anti-particle, falls into the event horizon it will plunge into the singularity and can be never recovered. However,its partner may remain outside the black hole. This particle has lost its virtual partner but it is now nonetheless a real particle and has the possibility of escape. If the particle does escape, rather than falling back in, it forms part of something called Hawking radiation. As far as a distant observer is concerned, the black hole has lost mass because a particle has been emitted.What had been realized is that, taking account of quantum field theory, black holes are not completely black, but they can actually emit particles. This argument also applies to photons, and so very weak light (also known as electromagnetic radiation) emerges from a black hole if Hawking's argument is correct.

  All bodies at non-zero temperature emit thermal radiation as photons. You do this yourself, which is why you would show up on an infra-red camera even in the dark (and this is why the police and the military use such cameras). The hotter the body, the higher the frequency of the radiation.We emit infra-red radiation,but a red-hot poker is hot enough to emit visible light. Because a black hole emits Hawking radiation, it has a temperature (known as the Hawking temperature) as we have seen earlier, although this is normally incredibly low. A black hole with a mass of one hundred times that of the Sun has a Hawking temperature less than a billionth of a degree above absolute zero (which is273 degrees below the freezing point of water)! This is one reason why Hawking radiation has not yet been detected: it is incredibly weak. But it is believed to be there.

  Hawking radiation does however have an interesting consequence on the evolution of black holes: it is ultimately responsible for a black hole's eventual death. Think again about the two virtual particles. The energy of the real particle which escapes from the black hole has to be positive, but since the virtual particle pair appeared spontaneously from the vacuum, then the virtual particle sucked into the black hole must have negative energy to compensate. Because energy and mass are connected, the net effect of this process is that the black hole has had negative mass added to it, and therefore its mass will have decreased due to the emission of Hawking radiation.

  Hawking had therefore discovered a mechanism by which a black hole can evaporate. Slowly, over time, the black hole will emit radiation and lose mass. This process is initially incredibly slow. It turns out that the larger a black hole is the smaller is its `surface gravity'. This is because even though the surface gravity depends on mass, which is larger for a big black hole, gravitational attraction follows an inverse-square law and more massive black holes are larger. The net result is that large black holes have very little surface gravity and this equates to a very low temperature.A large black hole therefore emits lessHawking radiation than a small black hole.

  However, as a black hole evaporates and loses mass, the amount of Hawking radiation goes up as the surface gravity and hence temperature increases. Assuming the black hole isn't receiving any other energy, this makes the rate of mass loss faster and faster until, at the end of its life, the black hole simply pops out of existence. Thus the life of a black hole ends not with a bang but with that quiet pop. This evaporation process is only possible for black holes whose temperatures are higher than their surroundings. At the current epoch in cosmic history, the temperature of the Universe, measured from the spectral shape of the CosmicMicrowave Background radiation, is 2.7 degrees above absolute zero. Black holes with masses greater than a hundred million million kilos will not evaporate at the current epoch because their temperatures are lower than that of their surroundings. These black holes which have a slender fraction of the mass of the Sun, however, will be able to evaporate when the Universe has cooled more following further expansion. Up to this point in cosmic time, all black holes whose masses were less than one per cent of this slender value would have evaporated away by now.

  The black hole information paradox

  One question which arises from all of this is what happens to the information stored in the matter that fell into the black hole? One school of thought holds that this information is lost for ever, even if the black hole subsequently evaporates. Another point of view claims that that information is not lost. Because black holes evaporate, the argument goes, the information contained within the original matter that fell into the black hole must somehow be stored in the radiation from the black hole. Thus if you could analyse all the Hawking radiation from a black hole and understand it completely you would be able to reconstruct the details of all the matter that had originally fallen into the black hole. There was a famous bet between, on the one hand Stephen Hawking and Kip Thorne, and John Preskill on the other, about this very matter. Thorne and Hawking took the former position,while Preskill took the latter. The agreement was that the loser would reward the winner with an encyclopaedia of the winner's choice. In 2004, Hawking was sufficiently persuaded by the idea that information could indeed be encoded in the radiation from a black hole that he conceded the bet, supplying Preskill with an encyclopaedia about baseball (whether that constitutes a repository of meaningful information depends on your opinion of baseball); however, the matter is still debated.

  Despite all these ingenious theoretical speculations, it is worth saying again that even ordinary Hawking radiation from a black hole has not yet been observed. The history of physics is littered with the relics of old, ingenious but ultimately wrong, theories.Experiments and observation have frequently been surprisingly effective at bringing forth unexpected results. Indeed,observations of spectacular phenomena have emerged that probably no-one at all would have predicted from first principles for black holes. One of the reasons that the faint Hawking radiation has not been observed is that many black holes we know about are at the centres of some of the brightest objects in the Universe, and these black holes are way too massive, and hence way too cold, to evaporate via Hawking radiation. These objects are extraordinarily bright for a completely different reason, which is examined in Chapter 6 and in Chapter 8.