{"id":31419,"date":"2019-10-31T21:41:09","date_gmt":"2019-10-31T18:41:09","guid":{"rendered":"https:\/\/prohoster.info\/blog\/termodinamika-chernyh-dyr\/"},"modified":"2019-10-31T21:41:09","modified_gmt":"2019-10-31T18:41:09","slug":"termodinamika-chernyh-dyr","status":"publish","type":"post","link":"https:\/\/prohoster.info\/en\/blog\/news\/termodinamika-chernyh-dyr","title":{"rendered":"Thermodynamics of black holes","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

\"Thermodynamics<\/a><\/noindex>
\nHappy Cosmonautics Day! We have submitted it to the printer \"A Little Book About Black Holes\"<\/a><\/noindex>. During these days, astrophysicists showed the world what black holes look like. Coincidence? We don't think so \ud83d\ude09 So stay tuned, soon an amazing book will be released, written by Stephen Gabser and Frans Pretorius, translated by the wonderful Pulkovo astronomer aka Astrodead Kirill Maslennikov, with scientific editing by the legendary Vladimir Surdin and supported by the Trajectory Foundation.<\/p>\n

Excerpt from \"The Thermodynamics of Black Holes\" under the cut.
\n<\/a><\/noindex>
\nSo far, we have considered black holes as astrophysical objects formed in supernova explosions or lying at the centers of galaxies. We observe them indirectly by measuring the accelerations of stars near them. The famous detection of gravitational waves by the LIGO receiver on September 14, 2015, provided a more direct observation of black hole collisions. The mathematical tools we use for a better understanding of the nature of black holes include differential geometry, Einstein's equations, and powerful analytical and numerical methods applied to solve Einstein's equations and describe the geometry of the spacetime produced by black holes. Once we can provide a complete quantitative description of the spacetime generated by a black hole, the topic of black holes can be considered closed from an astrophysical perspective. However, there remains a vast range of possibilities for exploration in a broader theoretical sense. The aim of this chapter is to discuss some theoretical advancements in modern black hole physics, where concepts of thermodynamics and quantum theory merge with general relativity, giving rise to unexpected new ideas. The key idea is that black holes are not just geometric objects. They have temperature, possess immense entropy, and can exhibit signs of quantum entanglement. Our reflections on the thermodynamic and quantum aspects of black hole physics will be more fragmented and superficial than the previous chapters' analysis of the purely geometric features of spacetime in black holes. However, both these and especially the quantum aspects are essential and vital parts of ongoing theoretical investigations into black holes, and we will do our best to convey at least the spirit of these works, if not the complex details.<\/p>\n

In classical general relativity, when discussing the differential geometry of the solutions to Einstein's equations, black holes are truly black in the sense that nothing can escape from them. Stephen Hawking demonstrated that this situation completely changes when we take quantum effects into account: black holes, it turns out, emit radiation at a specific temperature known as Hawking temperature. For astrophysical-sized black holes (that is, from stellar-mass black holes to supermassive ones), Hawking temperature is negligible compared to the temperature of the cosmic microwave background \u2014 the radiation filling the entire Universe, which incidentally can itself be viewed as a variant of Hawking radiation. The calculations made by Hawking to determine the temperature of black holes are part of a broader research program in a field called black hole thermodynamics. Another significant aspect of this program is the study of black hole entropy, which characterizes the amount of information lost inside a black hole. Ordinary objects (such as a cup of water, a block of pure magnesium, or a star) also have entropy, and one of the central assertions of black hole thermodynamics is that a black hole of a given size has greater entropy than any other form of matter that can be contained within an equally sized region without forming a black hole.<\/p>\n

But before we delve deeply into the issues related to Hawking radiation and black hole entropy, let\u2019s take a quick excursion into the realms of quantum mechanics, thermodynamics, and entanglement. Quantum mechanics was developed primarily in the 1920s, and its main aim was to describe very small particles of matter, such as atoms. The development of quantum mechanics led to a blurring of such fundamental concepts of physics as the precise position of an individual particle. It turned out, for example, that the position of an electron as it moves around the atomic nucleus cannot be exactly determined. Instead, electrons are assigned so-called orbits, where their actual positions can only be defined in a probabilistic sense. For our purposes, however, it is important not to move to this \u2014 probabilistic \u2014 side of things too quickly. Let\u2019s take the simplest example: the hydrogen atom. It can exist in a specific quantum state. The simplest state of the hydrogen atom, called the ground state, is the state with the lowest energy, and this energy is known exactly. More generally, quantum mechanics allows us (in principle) to know the state of any quantum system with absolute precision.<\/p>\n

Probabilities come into play when we ask certain types of questions about a quantum-mechanical system. For instance, if it is definitely known that the hydrogen atom is in its ground state, we can ask: \"Where is the electron?\" and by the laws of quantum mechanics, we will receive only a certain estimate of the probability, roughly something like: \"the electron is likely to be at a distance of up to half an angstrom from the nucleus of the hydrogen atom\" (one angstrom equals
\n10^{-10} meters). \"Thermodynamics meters). However, we have the ability, through a specific physical process, to locate an electron much more accurately than to within one angstrom. This fairly common process in physics involves sending a photon with a very short wavelength (or, as physicists say, scattering a photon on the electron) \u2014 after this, we will be able to reconstruct the position of the electron at the moment of scattering with an accuracy approximately equal to the wavelength of the photon. But this process will alter the state of the electron, meaning it will no longer be in the ground state of the hydrogen atom and will not have a precisely defined energy. However, for a brief period, its position will be almost precisely defined (to the accuracy of the wavelength of the photon used). A preliminary estimation of the electron's position can only be performed in a probabilistic sense with an accuracy of about one angstrom, but once we measure it, we know exactly what it was. In short, if we somehow measure a quantum-mechanical system, then, at least in the commonly accepted sense, we 'force' it into a state with a specific value of the quantity we are measuring.<\/p>\n

Quantum mechanics applies not only to small systems but, as we believe, to all systems. However, for large systems, quantum-mechanical rules quickly become very complex. A key concept is quantum entanglement, a simple example of which is the notion of spin (rotation). Individual electrons have spin, so in practice, a single electron can have a spin pointing either up or down relative to a chosen spatial axis. The spin of an electron is an observable quantity because the electron generates a weak magnetic field, similar to that of a magnetic bar. Therefore, an upward spin means that the north pole of the electron points downward, while a downward spin means that the north pole 'looks' upward. Two electrons can be placed in an entangled quantum state, where one has a spin pointing up and the other down, but it is impossible to say which of the electrons has which spin. Essentially, in the ground state of a helium atom, the two electrons exist in such a state, called a spin singlet, because the total spin of both electrons equals zero. If we separate these two electrons without altering their spins, we can still claim that they are collectively in a spin singlet, but we still cannot say what the spin of either one of them will be separately. If we measure one of their spins and find it pointing up, then we will be completely sure that the other is pointing down. In this situation, we say that the spins are entangled\u2014neither one has a definite value on its own, while together they are in a specific quantum state.<\/p>\n

Einstein was greatly concerned about the phenomenon of entanglement: it seemed to threaten the fundamental principles of the theory of relativity. Let's consider the case of two electrons in a spin singlet state, when they are far apart in space. For the sake of clarity, let one of them be Alice and the other Bob. Suppose Alice measured the spin of her electron and found it to be pointing up, while Bob did not make any measurements. Until Alice completed her measurement, it was impossible to say what the spin of his electron was. But as soon as she finished her measurement, she absolutely knew that Bob's electron spin was pointing down (in the direction opposite to her own electron's spin). Does this mean that her measurement instantaneously forced 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 colleagues Nathan Rosen and Boris Podolsky felt that the issue of measuring entangled systems was so serious that it threatened the very existence of quantum mechanics. The paradox they formulated, known as the Einstein-Podolsky-Rosen (EPR) paradox, uses a thought experiment similar to the one we just described to conclude: quantum mechanics cannot be a complete description of reality. Now, based on subsequent theoretical inquiries and numerous measurements, there is a general consensus that the EPR paradox contains an error, and that quantum theory is correct. Quantum mechanical entanglement is real: measurements on entangled systems will be correlated, even when these systems are far apart in spacetime.<\/p>\n

Let\u2019s return to the situation where we placed two electrons in a spin-singlet state and distributed them to Alice and Bob. What can we say about the electrons before measurements are made? Together, they are in a definite quantum state (the spin-singlet state). Alice's electron's spin is equally likely to point up or down. More precisely, the quantum state of her electron can be one (spin-up) or the other (spin-down) with equal probability. Now, the concept of probability takes on a deeper meaning than before. Previously, we considered a certain quantum state (the ground state of a hydrogen atom) and noted that there are some \u201cuncomfortable\u201d questions, such as \u201cWhere is the electron?\u201d \u2014 questions whose answers exist only in a probabilistic sense. If we asked \u201cgood\u201d questions, such as: \u201cWhat is the energy of this electron?\u201d, we would obtain definite answers. However, there are no \u201cgood\u201d questions we could ask about Alice's electron, where the answers would not depend on Bob's electron. (We are not referring to silly questions like \u201cDoes Alice's electron even have spin?\u201d \u2014 questions that have only one answer.) Therefore, to determine the parameters of one half of the entangled system, we will have to use a probabilistic language. Certainty arises only when we consider the relationship between the questions that Alice and Bob may ask about their electrons.<\/p>\n

We deliberately started with one of the simplest quantum-mechanical systems known to us: the spin systems of individual electrons. There is hope that quantum computers will be built on the basis of such simple systems. The spin systems of individual electrons or other equivalent quantum systems are currently referred to as qubits (short for \u201cquantum bits\u201d), highlighting their role in quantum computers, similar to the role conventional bits play in digital computers.<\/p>\n

Now, let's imagine that we have replaced each electron with a much more complex quantum system with multiple, rather than just two, quantum states. For example, let's give Alice and Bob bars of pure magnesium. Before Alice and Bob go their separate ways, their bars can interact, and we agree that during this interaction, they acquire a certain shared quantum state. Once Alice and Bob part ways, their magnesium bars stop interacting. Similar to electrons, each bar is in an indeterminate quantum state, although together, as we assume, they form a distinctly defined state. (In this discussion, we assume that Alice and Bob can move their magnesium bars without altering their internal state, just as we previously assumed that Alice and Bob could share their entangled electrons without changing their spins.) However, the difference between this thought experiment and the one with electrons is that the uncertainty of the quantum state of each bar is enormous. A bar can potentially acquire more quantum states than there are atoms in the universe. Here is where thermodynamics comes into play. Very poorly defined systems can still have some well-defined macroscopic characteristics. One such characteristic is temperature. Temperature is a measure of how likely any part of the system has a certain average energy, with a higher temperature corresponding to a greater likelihood of having higher energy. Another thermodynamic parameter is entropy, which is essentially the logarithm of the number of states that the system can take. Another thermodynamic characteristic that would be significant for a magnesium bar is its total magnetization, which essentially indicates how many more electrons in the bar can have their spins pointing up than those pointing down.<\/p>\n

We have introduced thermodynamics in our discussion as a means to describe systems whose quantum states are not precisely known due to their entanglement with other systems. Thermodynamics is a powerful analytical tool for such systems, but its creators did not foresee this application. Sadi Carnot, James Joule, and Rudolf Clausius were figures of the 19th-century industrial revolution, concerned mainly with the most practical of questions: how do engines work? Pressure, volume, temperature, and heat are the essence of engines. Carnot established that energy in the form of heat can never be completely converted into useful work, such as lifting loads. Part of the energy will always be wasted. Clausius made a fundamental contribution by formulating the idea of entropy as a universal tool for determining energy losses in any heat-related process. His main achievement was recognizing that entropy never decreases\u2014almost in all processes, it increases. Processes in which entropy increases are called irreversible, precisely because they cannot reverse without reducing entropy. The next step in the development of statistical mechanics was made by Clausius, Maxwell, and Ludwig Boltzmann (among many others)\u2014they showed that entropy is a measure of disorder. Usually, the more you act on something, the more disorder you introduce into it. And even if you have designed a process aimed at creating order, it will inevitably generate more entropy than it will destroy\u2014such as during heat emission. A crane that organizes steel beams in perfect order creates order in terms of beam arrangement, but during its operation, it will emit so much heat that the overall entropy will still increase.<\/p>\n

However, the difference in the perspective on thermodynamics from 19th-century physicists compared to that associated with quantum entanglement is not as significant as it seems. Each time a system interacts with an external agent, its quantum state becomes entangled with the quantum state of the agent. Usually, this entanglement leads to an increase in the uncertainty of the quantum state of the system; in other words, it results in a greater number of quantum states in which the system can exist. As a result of interactions with other systems, entropy, defined in terms of the number of accessible quantum states available to the system, generally increases.<\/p>\n

In general, quantum mechanics provides a new way to characterize physical systems in which some parameters (for example, position in space) become uncertain, while others (such as energy) are often known precisely. In the case of quantum entanglement, two fundamentally separate parts of a system share a known common quantum state, while each part individually has an uncertain state. A standard example of entanglement is a pair of spins in a singlet state, where it is impossible to determine which spin is pointing up and which is pointing down. The uncertainty of the quantum state in a large system requires a thermodynamic approach, wherein macroscopic parameters like temperature and entropy are known with great accuracy, even though the system has many possible microscopic quantum states.<\/p>\n

Having concluded our brief excursion into the realms of quantum mechanics, entanglement, and thermodynamics, let\u2019s now try to understand how all of this leads to the realization that black holes possess temperature. The first step in this direction was made by Bill Unruh \u2014 he demonstrated that an accelerating observer in flat space will have a temperature equal to their acceleration divided by 2\u03c0. The key to Unruh's calculations lies in the fact that an observer moving with constant acceleration in a specific direction can only see half of flat spacetime. The other half essentially lies beyond a horizon, similar to the horizon of a black hole. At first, this seems impossible: how can flat spacetime behave like the horizon of a black hole? To understand how this works, let\u2019s invoke our trusty observers Alice, Bob, and Bill. At our request, they line up, with Alice positioned between Bob and Bill, and the distance between each pair of observers is exactly 6 kilometers. It has been agreed that at time zero, Alice will jump into a rocket and fly towards Bill (and thus away from Bob) with constant acceleration. Her rocket is very good, capable of achieving an acceleration 1.5 trillion times greater than the gravitational acceleration experienced by objects near the Earth's surface. Of course, sustaining such acceleration is not easy for Alice, but, as we will see, these figures have been chosen for a specific purpose; ultimately, we are merely discussing potential possibilities, that\u2019s all. Exactly at the moment Alice jumps into her rocket, Bob and Bill wave to her. (We are allowed to use the phrase \"exactly at the moment when...\" because while Alice has not yet begun her flight, she is in the same frame of reference as Bob and Bill, so they can all synchronize their clocks.) Alice can certainly see Bill waving to her: however, while in the rocket, she will see him earlier than she would if she had remained where she was, as her rocket is flying directly towards him. On the other hand, she is moving away from Bob, so we can reasonably suppose that she will see him waving to her somewhat later than she would have if she had stayed in her original position. But the truth is even more astonishing: she will not see Bob at all! In other words, the photons traveling from waving Bob to Alice will never catch up with her, even considering that she can never reach the speed of light. If Bob began waving while being slightly closer to Alice, then the photons that left him at the moment of her launch would have caught up with her; conversely, if he were slightly further away, they certainly would not have. In this sense, we say that Alice can see only half of spacetime. At the moment Alice begins her movement, Bob is just a bit beyond the horizon that Alice can observe.<\/p>\n

In our discussion of quantum entanglement, we have already become accustomed to the idea that even if a quantum-mechanical system as a whole has a definite quantum state, some of its parts may not possess it. In fact, when we discuss a complex quantum system, some part of it might be best characterized within the framework of thermodynamics: a specific temperature can be assigned to it, despite the highly uncertain quantum state of the entire system. Our latest story involving Alice, Bob, and Bill is somewhat similar to this situation, but the quantum system we are discussing here is empty spacetime, and Alice only sees half of it. Let us specify that the spacetime as a whole is in its ground state, meaning there are no particles in it (of course, excluding Alice, Bob, Bill, and the rocket). But the part of spacetime that Alice sees will not be in the ground state, but in a state entangled with the part she cannot see. The spacetime perceived by Alice exists in a complex uncertain quantum state characterized by a finite temperature. Unruh's calculations show that this temperature is approximately 60 nanokelvins. In short, as she accelerates, Alice effectively immerses herself in a warm bath of radiation with a temperature equal to (in the appropriate units) the acceleration divided by. \"Thermodynamics
\n
\n\"Thermodynamics<\/p>\n

Fig. 7.1. Alice is accelerating from a state of rest, while Bob and Bill remain stationary. Alice's acceleration is exactly such that she will never see the photons that Bob sends her way at time t = 0. However, she does receive the photons that Bill sent her at time t = 0. As a result, Alice is only able to observe one half of spacetime.<\/p>\n

The peculiarity of Unruh's calculations lies in the fact that while they pertain entirely to empty space from start to finish, they contradict the well-known words of King Lear: \"nothing comes from nothing.\" How can empty space be so complex? Where could particles possibly arise from? The truth is that according to quantum theory, empty space is anything but empty. Short-lived excitations, known as virtual particles, constantly emerge and disappear here and there within it, with energies that can be both positive and negative. An observer from a distant future\u2014let's call her Carol\u2014who is able to see almost all of empty space can confirm that there are no long-lasting particles present. At the same time, the presence of particles with positive energy in the part of spacetime that Alice can observe is correlated, through quantum entanglement, with excitations of equal and opposite energy in the part of spacetime not observable by Alice. The full truth about empty spacetime as a whole is revealed to Carol, and that truth is that there are no particles there. However, Alice's experience tells her that there are particles present!<\/p>\n

However, it turns out that the calculated Unruh temperature seems to be just a fiction \u2014 it is not so much a property of flat space as it is a property of the observer experiencing constant acceleration in flat space. However, gravity itself is a similarly 'fictitious' force in the sense that the 'acceleration' caused by it is nothing more than motion along a geodesic in curved metric. As we explained in Chapter 2, Einstein's principle of equivalence states that acceleration and gravity are essentially equivalent. From this perspective, there is nothing particularly shocking about the black hole horizon having a temperature equal to the calculated Unruh temperature of the accelerating observer. But we may 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 this mean that to determine the effective temperature we measure of the black hole, we should use the corresponding small value of acceleration? This question turns out to be quite tricky because, as we believe, the temperature of an object cannot arbitrarily decrease. It is assumed to have some fixed finite value that could be measured even by a very distant observer.
\n
Source: habr.com<\/a><\/p>","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

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