Happy Cosmonautics Day! We handed over to the printing house . It was during these days that astrophysicists showed the whole world what black holes look like. Coincidence? We don’t think 😉 So wait, an amazing book will soon appear, written by Steven Gabser and Frans Pretorius, translated by the wonderful Pulkovo astronomer aka Astroded Kirill Maslennikov, scientific editing by the legendary Vladimir Surdin and supported by the Trajectory Foundation.
An excerpt from "Thermodynamics of black holes" under the cut.
Until now, we have considered black holes as astrophysical objects that were formed during supernova explosions or lie at the centers of galaxies. We observe them indirectly by measuring the accelerations of nearby stars. The famous detection of gravitational waves by the LIGO receiver on September 14, 2015 was an example of more direct observations of the collision of black holes. The mathematical tools we use to gain a better understanding of the nature of black holes are differential geometry, Einstein's equations, and powerful analytical and numerical methods used to solve Einstein's equations and describe the geometry of the space-time that black holes generate. And as soon as we can give a complete quantitative description of the space-time generated by a black hole, from an astrophysical point of view, the topic of black holes can be considered closed. In a broader theoretical perspective, there is still a lot of room for research. The purpose of this chapter is to describe some of the theoretical developments in modern black hole physics, in which the ideas of thermodynamics and quantum theory are combined with general relativity, giving rise to unexpected new concepts. The basic idea is that black holes are not just geometric objects. They have a temperature, they have huge entropy, and they can exhibit manifestations of quantum entanglement. Our discussion of the thermodynamic and quantum aspects of black hole physics will be more sketchy and superficial than the analysis presented in previous chapters of the purely geometric features of space-time in black holes. But these aspects, and especially the quantum aspects, are an essential and vital part of the ongoing theoretical research on black holes, and we will try very hard to convey, if not the complex details, then at least the spirit of these works.
In classical general relativity—in terms of the differential geometry of solutions to Einstein's equations—black holes are truly black in the sense that nothing can escape from them. Stephen Hawking has shown that this situation reverses when we take quantum effects into account: black holes appear to emit radiation at a certain temperature, known as the Hawking temperature. For black holes of astrophysical dimensions (that is, from black holes of stellar masses to supermassive ones), the Hawking temperature is negligible compared to the temperature of the cosmic microwave background - radiation that fills the entire Universe, which, by the way, can itself be considered as a variant of Hawking radiation. Hawking's calculations to determine the temperature of black holes are part of a larger research program in a field called black hole thermodynamics. The other big part of this program is the study of the entropy of black holes, which characterizes the amount of information lost inside a black hole. Ordinary objects (such as a cup of water, a bar of pure magnesium, or a star) also have entropy, and one of the central claims of black hole thermodynamics is that a black hole of a given size has more entropy than any other form of matter that can be contained in a region of the same size, but without the formation of a black hole.
But before we dive deep into the problems associated with Hawking radiation and black hole entropy, let's take a quick detour into quantum mechanics, thermodynamics, and entanglement. Quantum mechanics was developed mainly in the 1920s and its main purpose was to describe very small particles of matter, such as atoms. The development of quantum mechanics led to the blurring of such basic concepts of physics as the exact position of an individual particle: it turned out, for example, that the position of an electron as it moves around an atomic nucleus cannot be accurately determined. Instead, so-called orbits were assigned to the electrons, in which their actual positions can only be determined in a probabilistic sense. For our purposes, however, it is important not to jump to this probabilistic side of the matter too quickly. Let's take the simplest example: the hydrogen atom. It can be in a certain quantum state. The simplest state of the hydrogen atom, called the ground state, is the state with the lowest energy, and this energy is precisely known. More generally, quantum mechanics allows us (in principle) to know the state of any quantum system with absolute precision.
Probabilities come into play when we ask certain kinds of questions about a quantum mechanical system. For example, if we know for sure that the hydrogen atom is in the ground state, we can ask, "Where is the electron?" and according to the laws of quantum
mechanics, we get only some probability estimate for this question, approximately something like: “probably the electron is at a distance of up to half an angstrom from the nucleus of a hydrogen atom” (one angstrom is equal to 
meters). But we have the possibility, through a certain physical process, to find the position of an electron much more accurately than to one angstrom. This fairly common process in physics is to launch a photon with a very short wavelength into an electron (or, as physicists say, scatter a photon on an electron) - after that we can reconstruct the location of the electron at the moment of scattering with an accuracy approximately equal to the wavelength photon. But this process will change the state of the electron, so that after that it will no longer be in the ground state of the hydrogen atom and will not have a precisely defined energy. But for a while, its position will be almost exactly determined (up to the wavelength of the photon used for this). A preliminary estimate of the position of an electron can only be made in a probabilistic sense with an accuracy of about one angstrom, but once we have measured it, we know exactly what it was. In short, if we measure a quantum mechanical system in some way, then, at least in the conventional sense, we “force” it into a state with a certain value of the quantity we are measuring.
Quantum mechanics applies not only to small systems, but (we believe) to all systems, but for large systems, the rules of quantum mechanics quickly become very complex. The key concept is quantum entanglement, a simple example of which is the notion of spin. Individual electrons have a spin, so in practice a single electron can have a spin up or down with respect to the chosen spatial axis. The electron spin is an observable quantity because the electron generates a weak magnetic field, similar to the field of a magnetic bar. Then spin up means the north pole of the electron is pointing down, and spin down means the north pole is "looking" up. Two electrons can be placed in a conjugated quantum state, in which one of them has a spin up and the other down, but it is impossible to tell which of the electrons has which spin. In fact, in the ground state of a helium atom, two electrons are in just such a state, called spin-singlet, since the total spin of both electrons is zero. If we separate these two electrons without changing their spins, we can continue to say that together they are spin singlets, but we still cannot say what the spin of either of them will be individually. Now, if we measure one of their spins and establish that it is pointing up, then we will be completely sure that the second is pointing down. In this situation, we say that the spins are entangled—none of them have a definite meaning on their own, while together they are in a definite quantum state.
Einstein was very worried about the phenomenon of entanglement: it seemed to threaten the basic principles of the theory of relativity. Let us consider the case of two electrons in the spin-singlet state, when they are far apart in space. For definiteness, let Alice take one of them and Bob take the other. Let's say that Alice measured the spin of her electron and found that it was pointing up, but Bob did not measure anything. Until Alice made her measurement, it was impossible to tell what the spin of his electron was. But once she completed her measurement, she absolutely knew that Bob's electron was spinning down (in the opposite direction of her own electron's spin). Does this mean that her measurement instantly put Bob's electron into a state where its spin is pointing down? How could this happen if the electrons are spatially separated? Einstein and his collaborators Nathan Rosen and Boris Podolsky felt that the story of measuring entangled systems was so serious that it threatened the very existence of quantum mechanics. The Einstein-Podolsky-Rosen (EPR) paradox they formulated uses a thought experiment similar to the one we have just described to conclude that quantum mechanics cannot be a complete description of reality. Now, based on the theoretical research that followed and many measurements, there is a general opinion that the EPR paradox contains an error, and the quantum theory is correct. Quantum mechanical entanglement is real: measurements of entangled systems will correlate even if the systems are far apart in spacetime.
Let's go back to the situation where we put two electrons in a spin-singlet state and distributed them to Alice and Bob. What can we say about electrons before measurements are made? That both together they are in a certain quantum state (spin-singlet). The spin of Alice's electron is equally likely to be directed up or down. More precisely, the quantum state of its electron is equally likely to be one (spin up) or the other (spin down). Now for us the concept of probability takes on a deeper meaning than before. We have previously looked at a certain quantum state (the ground state of the hydrogen atom) and have seen that there are some "uncomfortable" questions, such as "Where is the electron?", questions to which there are only probabilistic answers. If we were to ask “good” questions, such as “What is the energy of this electron?”, we would get certain answers to them. Now, there are no “good” questions we can ask about Alice's electron that don't depend on Bob's electron. (We're not talking about stupid questions like "Does Alice's electron even have a spin?"—questions to which there is only one answer.) Thus, to determine the parameters of one of the halves of an intricate system, we will have to use probabilistic language. Certainty only arises when we consider the relationship between the questions that Alice and Bob might ask about their electrons.
We purposely started with one of the simplest quantum mechanical systems that we know of: systems of spins of individual electrons. It is hoped that quantum computers will be built on the basis of such simple systems. The system of spins of individual electrons or other equivalent quantum systems are now called qubits (short for "quantum bits"), emphasizing their role in quantum computers, similar to the role played by ordinary bits in digital computers.
Imagine now that we have replaced each electron with a much more complex quantum system with many, not just two, quantum states. For example, they gave Alice and Bob bars of pure magnesium. Before Alice and Bob go about their business in different directions, their bars can interact, and we will agree that in doing so they acquire a certain common quantum state. As soon as Alice and Bob separate, their magnesium bars stop interacting. As in the case of electrons, each bar is in an indeterminate quantum state, although together, we believe, they form a well-defined state. (In this discussion, we assume that Alice and Bob are able to move their magnesium bars without disturbing their internal state in any way, just as we previously assumed that Alice and Bob could share their entangled electrons without changing their spins.) But the difference is between this thought experiment and the experiment with electrons is that the uncertainty in the quantum state of each bar is enormous. The bar may well acquire more quantum states than the number of atoms in the universe. This is where thermodynamics comes into play. Very imprecisely defined systems may, however, have some well-defined macroscopic characteristics. Such a characteristic is, for example, temperature. Temperature is a measure of how likely any part of a system is to have a certain average energy, with higher temperature being more likely to have more energy. Another thermodynamic parameter is entropy, which is essentially equal to the logarithm of the number of states a system can take. Another thermodynamic characteristic that would be significant for a bar of magnesium is its total magnetization, that is, in essence, a parameter showing how much more electrons can be in the bar with an upward spin than with a downward spin.
We have drawn thermodynamics into our story as a way of describing systems whose quantum states are not exactly known due to their entanglement with other systems. Thermodynamics is a powerful tool for analyzing such systems, but its creators did not at all envision such an application. Sadi Carnot, James Joule, Rudolf Clausius were the industrial revolutionaries of the XNUMXth century, and they were interested in the most practical of all questions: how do engines work? Pressure, volume, temperature and heat are the lifeblood of engines. Carnot found that energy in the form of heat can never be fully converted into useful work like lifting loads. Part of the energy will always be wasted. Clausius made the main contribution to the creation of the idea of entropy as a universal tool for determining energy losses during any heat-related process. His main achievement was the realization that entropy never decreases - in almost all processes it increases. Processes in which entropy increases are called irreversible, precisely because they cannot be reversed without a decrease in entropy. The next step in the development of statistical mechanics was taken by Clausius, Maxwell and Ludwig Boltzmann (among many others) - they showed that entropy is a measure of disorder. Usually, the more you act on something, the more you mess it up. And even if you have designed a process whose purpose is to restore order, it inevitably creates more entropy than it destroys, for example, when heat is released. A crane that lays down steel beams in perfect order creates order in terms of the arrangement of beams, but so much heat is released in the course of its operation that the total entropy still increases.
But still, the difference between the view of thermodynamics of nineteenth-century physicists and the view associated with quantum entanglement is not as great as it seems. Every time the system interacts with an external agent, its quantum state becomes entangled with the agent's quantum state. Usually, this entanglement leads to an increase in the uncertainty of the quantum state of the system, in other words, to an increase in the number of quantum states in which the system can be. As a result of interactions with other systems, entropy, defined in terms of the number of quantum states available to the system, tends to increase.
In general, quantum mechanics provides a new way to characterize physical systems in which some parameters (such as position in space) become uncertain, while others (such as energy) are often known exactly. In the case of quantum entanglement, two fundamentally separate parts of the system have a known common quantum state, and each part separately has an indeterminate state. A standard example of entanglement is a pair of spins in a singlet state, in which it is impossible to tell which spin is up and which is down. The uncertainty of a quantum state in a large system requires a thermodynamic approach in which macroscopic parameters such as temperature and entropy are known with great precision, even though the system has many possible microscopic quantum states.
Having finished our brief digression into the field of quantum mechanics, entanglement and thermodynamics, we now try to understand how all this leads to an understanding of the fact that black holes have a temperature. The first step towards this was taken by Bill Unruh - he showed that an accelerating observer in flat space would have a temperature equal to his acceleration divided by 2π. The key to Unruh's calculations is that an observer moving with constant acceleration in a certain direction can only see half of flat space-time. The other half, in fact, lies beyond a horizon similar to that of a black hole. At first this looks impossible: how can flat spacetime behave like the horizon of a black hole? To understand how this works out, let's call on the help of our faithful observers Alice, Bob and Bill. At our request, they line up, with Alice between Bob and Bill, and between the observers in each pair, the distance is exactly 6 kilometers. We agreed that at the zero moment of time, Alice will jump into the rocket and fly towards Bill (and hence away from Bob) with constant acceleration. Her rocket is very good, capable of developing an acceleration of 1,5 trillion times the gravitational acceleration with which objects move near the surface of the Earth. Of course, it is not easy for Alice to withstand such an acceleration, but, as we will see in a moment, these numbers are chosen for a specific purpose; after all, we're just discussing potentialities, that's all. Exactly at the moment when Alice jumps into her rocket, Bob and Bill wave to her. (We are right to use the expression "exactly at the moment when ...", because until Alice has started flying, she is in the same frame of reference as Bob and Bill, so they can all synchronize their clocks.) Waving Alice, of course, sees Bill to her: however, being in a rocket, she will see him earlier than it would have happened if she had remained where she was, because her rocket, along with her, flies exactly to him. She, on the contrary, moves away from Bob, so we can reasonably assume that she will see him waving to her a little later than she would have seen if she had stayed in the same place. But the truth is even more amazing: she won't see Bob at all! In other words, the photons that fly from the waving Bob to Alice will never catch up with her, even though she can never reach the speed of light. If Bob started waving, being a little closer to Alice, then the photons that flew away from him at the moment of her departure would overtake her, and if he were a little further away, they would not overtake her. It is in this sense that we say that only half of space-time is visible to Alice. At the moment when Alice starts moving, Bob is slightly further than the horizon that Alice is observing.
In our discussion of quantum entanglement, we have become accustomed to the idea that even if a quantum mechanical system as a whole has a certain quantum state, some parts of it may not. In fact, when we discuss a complex quantum system, some part of it can be best characterized precisely in terms of thermodynamics: a well-defined temperature can be assigned to it, despite the highly uncertain quantum state of the whole system. Our latest story involving Alice, Bob and Bill is a bit similar to this situation, but the quantum system we're talking about here is empty spacetime, and Alice only sees half of it. Let's make a reservation that space-time as a whole is in its ground state, which means the absence of particles in it (of course, not counting Alice, Bob, Bill and the rocket). But that part of space-time that Alice sees will not be in the ground state, but in a state entangled with that part of it that she does not see. The space-time perceived by Alice is in a complex indefinite quantum state characterized by a finite temperature. Unruh's calculations show that this temperature is about 60 nanokelvins. In short, as she accelerates, Alice seems to plunge into a warm bath of radiation with a temperature equal (in appropriate units) to the acceleration divided by
Rice. 7.1. Alice accelerates from rest while Bob and Bill remain stationary. Alice's acceleration is just such that she never sees the photons that Bob sends in her direction at time t = 0. However, she receives photons that Bill sent her at time t = 0. The result is that Alice can only observe one half of space-time.
The oddity of Unruh's calculations is that although they refer from beginning to end to empty space, they contradict King Lear's well-known saying "nothing will come of nothing." How can empty space be so complicated? Where can the particles come from? The fact is that, according to quantum theory, empty space is by no means empty. Short-lived excitations, called virtual particles, whose energy can be both positive and negative, constantly appear and disappear in it here and there. An observer from the distant future — let's call her Carol — who can see almost all of empty space can confirm that there are no particles that persist for a long time. At the same time, the presence of particles with positive energy in that part of space-time that Alice can observe, due to quantum entanglement, is associated with excitations of equal and opposite energy in the part of space-time unobservable for Alice. The whole truth about empty space-time as a whole is open to Carol, and that truth is that there are no particles. However, Alice's experience tells her that the particles are there!
But then it turns out that the temperature calculated by Unruh seems to be just a fiction - it is not so much a property of flat space as such, but rather a property of an observer experiencing constant acceleration in flat space. However, gravity itself is the same "fictitious" force in the sense that the "acceleration" that it causes is nothing but movement along a geodesic in a curved metric. As we explained in Chapter 2, Einstein's principle of equivalence is that acceleration and gravity are essentially equivalent. From this point of view, there is nothing particularly shocking about the fact that the horizon of a black hole has a temperature equal to Unruh's calculated temperature of an accelerating observer. But, may we ask, what value of acceleration should we use to determine the temperature? By moving far enough away from the black hole, we can make its gravitational pull arbitrarily weak. Does it follow from this that in order to determine the effective temperature of the black hole that we measure, we need to use the corresponding small value of the acceleration? This question turns out to be rather insidious, because, as we believe, the temperature of an object cannot arbitrarily decrease. It is assumed that it has some fixed final value, which even a very distant observer can measure.
Source: habr.com