Physicist Emil Akhmedov on Newton's second law, the Minkowski metric and the nature of space-time.
You can spend years of your life defining what time is. This is a personal matter for every person who occupies his own civilizational niche. Of course, trying to answer such questions is part of human culture. But for a physicist, connections between different substances are important, and the relationships are not verbal, but formulaic. As an example of such a ratio, we can cite Newton's second law. He claims that F = ma - the force causes a body with mass m to move with acceleration a. You can spend years of your life figuring out the meaning of power. You can spend years of your life trying to determine what the substance of the mass is. But for a physicist, the formula relationship between force, mass and acceleration is important. Now I will emphasize in what sense.
It is argued that the law F = ma, Newton's second law, follows from experiment. This does not mean that there is a specific experiment in which force, mass, acceleration are measured and it is established that F = ma. There is a range of natural phenomena that are succinctly described in the form of this formula and other formulas and ratios. For a physicist, this is precisely what is important: there is a quantity measured in newtons, it is a force; there is a quantity measured in kilograms - this is mass; there is a value measured in meters per second squared - this is acceleration. From childhood, I realized for myself that physics is a science that establishes relationships between quantities that can be measured in kilograms, meters and seconds. Newton is expressed by this formula in terms of kilograms, meters and seconds.
Try to answer the question "What is the nature of time?" This question is legitimate, but for a physicist and engineer, it is not an abstract answer that is important, but a formula that connects time with something, with the left and right sides. After that, the question “What is the nature of what is on the right, and with what is the time connected?” Will become legitimate. Whoever wants, let him answer it. But physics is important about the relationship between one and the other, the cause-and-effect relationship: if I change it this way, then it changes like that. This is a fact of objective reality, no matter how we relate to it.
What is time for a physicist? There is a standard of time, which, for example, is kept in Paris. I do not know what is now taken as a standard of time, but I can take a single vibration of a molecule as a standard of time. And if a molecule made 10 billion vibrations, it used to be called a second. Previously, a second was taken as the standard, but now you can take a single vibration, then a second is 10 billion vibrations of a molecule. An atomic clock, a chronometer, is measured simply as a count of the number of oscillations between the initial moment and the final moment of a given molecule. This is how time is measured, such is its nature for a physicist.
You can also ask: what is the nature of space, how is it arranged at the microscopic level? If you get an answer to this question in the form of a formula connecting some characteristics of space with something else, then I am ready to discuss it. As a physicist, this is interesting to me. If you start to say that space as a substance is similar to clay or something else, this is not interesting to me, this statement is not informative for me.
The nature of space for a physicist is the following: in space, you can enter a coordinate grid, that is, imagine the coordinate axes in space and set the way to determine the position in these coordinates, as well as the distance between any two points in space. How to measure distance on a plane? You enter a coordinate grid - Y-axis and X-axis. You define a point, it has two coordinates. For example, you want to find the distance from this point to point Y, it also has its own coordinates. You calculate the difference in coordinates along one and the other axis, square them, add according to the Pythagorean theorem and extract the square root. This is the distance between two points - the Euclidean plane, the Euclidean two-dimensional space. This is how it is defined. For me, nothing else is needed at the moment to make predictions. Then one might ask: where does this formula come from, why is it correct? But again, the answer will be of interest to me only in the formula, and not in the verbal one.
Space in Newtonian mechanics is a three-dimensional space in which there are three axes: the vertical Z axis, the horizontal X and Y perpendicular to them. The position of a point in this space is defined as three coordinate values. I chose something outside the center of coordinates, for example, an angle in this room, directed the axes perpendicular to each other, and I say that the point is three meters from the origin in one direction, five in the other and ten in the third direction. After that, I have to set a formula that determines the distance between this point and any other. In the same way, I calculate the lengths of this segment along three axes (I have a segment connecting these points, it has three projections on three axes). I sum the squares of the projections, extract the square root, and that gives me the answer for what is the length of the line segment. As soon as I wrote this formula, I can study the movement of material points, particles under the influence of forces. For example, under the influence of some forces, my particle makes some kind of movement. I wrote this curve and, using the formula, I can determine all the characteristics of this curve and find out numerically what force and at what moment acted on the particle and gave it such and such acceleration, the particle had such and such a mass, and so on. After that, I will establish the correctness of the law - for example, F = ma. Or, using the law F = ma, I will predict how a particle will move under the influence of one force or another.
This was the case in Newtonian mechanics, where time was measured separately with the help of something. Galileo counted the oscillations of the chandeliers in the cathedral in Piazza dei Miracoli, in Pisa, he counted his own pulse: how many times his pulse ticked and how many times his chandelier swayed. For him, the unit of measurement was one sixtieth of a second. Someone else can build a Swiss chronometer, while others are not content with this and demand that there be an atomic chronometer. It all depends on the degree of accuracy of the statement he wants to get.
In Newtonian mechanics, time and distance in time were measured separately. It's just that the distance in time was measured according to the following formula: between one and the other moment there were so many oscillations, for example, of a pendulum. This means that we have measured time as the number of oscillations. In the same way, we measured distance in space - this is how Newton's mechanics worked.
In the special theory of relativity, a new statement has appeared that there is no separate way to measure distances along time and there is no separate way to measure distances along space, but there is a single way to measure distances in space-time. This method is given by the Minkowski formula, which states that the distance between two points in space-time is calculated by the following formula: multiply c 2 by the length of time, by delta t 2, minus the square of the length in space - c 2 * Δt 2 - (x 1 -x 0) 2 - (y 1 -y 0) 2 - (z 1 -z 0) 2. The same square of length, that is, minus the X distance squared minus the Y squared distance minus the Z squared distance.
Minkowski's formula follows from the same place as F = ma - from the description of the set of experimental data. If you accept this formula, then you are capaciously describing a certain range of experimental data. There is no need to say anything more about this formula at this stage.
When they talk about the space-time continuum or space-time, they actually mean the way of specifying coordinates in space and the way of specifying the distance by the Minkowski metric. This is the nature of space-time for a physicist.
Minkowski's formula is very different from Euclid's formula, from the way of specifying distances in Euclidean space. Because of this, from the point of view of a person familiar with Euclid's geometry, most of the statements of Minkowski's geometry look very paradoxical, so most of the statements of the special theory of relativity look paradoxical. But at the same time, a person does not realize that we are talking about very subtle phenomena.
Any physical law, no matter how fundamental it may be, has limits of applicability. He is not completely correct. Unlike the laws of logic, a physical law has limits of applicability. For example, Newtonian mechanics is applicable if we are moving at speeds that are much less than the speed of light, we are dealing with accelerations that are small enough, and the gravitational field is weak enough. If we start moving at high speeds, we are dealing with very strong gravitational fields, Newtonian mechanics is replaced by the special and general theory of relativity. They do not refute it, but include it as a component. It's just that if we take the mechanics of special and general relativity and go to small gravitational fields and low speeds, we get the same laws as in Newtonian mechanics.
The Minkowski formula is applicable only in the approximation when we neglect the curvature of space-time and when we are talking about inertial frames of reference. If we are talking about non-inertial frames of reference, then the formula is no longer applicable. And if space-time is curved, then this formula is not applicable. Most of the paradoxes of special relativity arise from the fact that people forget about the limits of applicability of this formula.
Emil Akhmedov, Doctor of Physical and Mathematical Sciences, Leading Researcher at the Alikhanov Institute for Theoretical and Experimental Physics, Professor of the Department of Theoretical Physics at the Moscow Institute of Physics and Technology.
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Emil Akhmedov
I will tell you what supersymmetry is. Supersymmetry is not yet an experimentally discovered phenomenon, but, firstly, great hopes were pinned on it at the moment when it was born, and secondly, it is an important part of mathematical physics and mathematics. Despite the fact that it has not been experimentally discovered at the moment, no one said that it will not be found in the future, it is an important part of modern science. So, in order to tell what supersymmetry is, I need to say what supersymmetric coordinates are or, in general, what coordinates are.
Emil Akhmedov
What is an elementary particle? The word "particle" comes from the word "part", so it usually seems that it is some kind of brick from which we build the whole. A brick is associated with something solid, solid, compact, small, and a particle - with some kind of ball (this is the first thing that comes to mind of a layman when they say "elementary particle"). Physicist Emil Akhmedov on Thomson's model of the atom, alpha particles and Rutherford's scattering.
Emil Akhmedov
It is known that the speed of light does not depend on the frame of reference. This statement is true only in flat space-time, and not curved, and, moreover, only in the transition from an inertial frame of reference to an inertial one. If you have passed in flat space-time from an inertial frame of reference to an inertial one, then the speed of light does not depend on the speed of movement of one frame relative to another. But if you switch to a non-inertial frame of reference, then the speed of light is not such a holy cow, it can even depend on coordinates, if you understand it as dividing a spatial increment by a temporal increment. Physicist Emil Akhmedov on Fermat's principle, Newtonian gravity and the effects of general relativity.
Emil Akhmedov
In the modern understanding, it turns out that the law of conservation of energy and the law of conservation of momentum follow from a more fundamental principle, which is the so-called translational invariance in space and time. What does it mean? What does translational invariance mean in general?
Emil Akhmedov
My story will be more historical: I will talk about how Maxwell's theory and the concept of electromagnetic waves originated. Coulomb's laws, Bio-Savard's law, various Faraday's laws of induction and others were known. This set of experimental data Maxwell tried to describe theoretically. As far as I know, his work is about six hundred pages long. He tried to explain Faraday's laws purely mechanically, describing the electromagnetic field as a set of gears with different types of engagement. In the 19th century, the mechanical description of nature was very popular. Most of those 600 pages are missing because they did not contain any constructive statements. Maybe I'm exaggerating a little, but the only constructive thing that was in this work of Maxwell is his equations, formulas.
Emil Akhmedov
Physicist Emil Akhmedov on determining the position on the plane and in space, the required coordinates and atomic clock. I will talk about the general principles of GPS and GLONASS operation. Then I'll explain how this has to do with special and general relativity. I'll start from afar. A triangle is a rigid figure on a plane in the sense that if you take three hinges and connect them with three rigid sticks, then these hinges cannot be displaced, cannot be moved. If you take four or more hinges and connect them with the appropriate number of sticks to make a polygon, then that polygon can shake. The quadrilateral can be deformed, therefore, if there are more than three angles, the figure on the plane is already non-rigid.
Emil Akhmedov
The 18th – 19th centuries were marked by the success of Newton's mechanics, which showed amazing efficiency in describing the motion of the planets of the solar system. But science began to move forward when it abandoned this mechanistic approach. Under the sign of all this happening, such a Laplace paradox arose, which says that there is no will everywhere. That is, a person cannot act of his own free will, everything is predetermined and predictable. Physicist Emil Akhmedov on differential equations, ideal lines and points and the solution of the Laplace paradox.
Emil Akhmedov
Almost everyone knows the relationship E0 = mc ^ 2. Any educated person knows that E = mc ^ 2. At the same time, they forget that if you look more closely and look at it non-colloquially, then the ratio looks like E0 = mc ^ 2, E has an index of 0, and it connects the rest energy with the mass and speed of light. It should be remembered that energy is a key concept here. So, colloquially speaking, this ratio says that any mass is energy, but not any energy is mass. We must not forget about this, that not all energy is mass! Any mass is energy, but the opposite is not true. And not for any energy, but only for the rest energy, it is true that it is equal to mc ^ 2. Where does this ratio come from? Physicist Emil Akhmedov on the ratio of mass and energy, Minkowski space-time and 4-vector coordinates.
Emil Akhmedov
What observations underlie the special theory of relativity? How was the postulate that the speed of light does not depend on the frame of reference was derived? What is Noether's theorem about? And are there any phenomena that contradict SRT? Emil Akhmedov, Doctor of Physical and Mathematical Sciences, talks about this.
Emil Akhmedov
Emil Akhmedov, Doctor of Physical and Mathematical Sciences, talks about the Lorentz transformations, the special theory of relativity, the paradox of twins and the paradox of the rod and the barn.
Cultural and educational center "Arche" invites to the course Doctor of Physical and Mathematical Sciences Emil Akhmedov "Fundamental Physics".
Topic of the second lecture: "Quantum field theory".
The lecture will talk about what a field is and how to quantize it. Then let's talk about what new phenomena arise when the field is quantized. At the end of the lecture, we will talk about the Hawking effect and fundamental problems of quantum field theory.
About the lecturer:
- Doctor of Physical and Mathematical Sciences, Leading Researcher at the Alikhanov Institute for Theoretical and Experimental Physics, Associate Professor at the Department of Theoretical Physics at MIPT, Associate Professor at the Faculty of Mathematics at the Higher School of Economics.
About the course of lectures "Fundamental Physics":
The lectures are designed for high school students, junior students and people interested in mathematics and physics. Using simple and clear examples, using elementary formulas, the lecturer will try to answer the following questions:
- What is Special Relativity, and how does the geometry of space-time work? Why is the speed of light independent of the frame of reference and why it cannot be exceeded? Several examples of paradoxes (eg the twins paradox) will be discussed and how they are resolved.
- How does curved spacetime work, and what is General Relativity? Some simple and visual examples of curved spaces will be analyzed.
- What are black holes? How to present them? What happens to objects near black holes?
- What is Quantum Mechanics? How does the transition from particle mechanics to wave mechanics take place? Where does the probabilistic interpretation of quantum mechanics come from? Several paradoxes will be sorted out. For example, the Schrödinger's Cat paradox and the Einstein-Podolsky-Rosen paradox.
- What is a field? What is quantum field theory? What is the Higgs field and how does it work?
- What is Hawking radiation?
- What is string theory? Why is it needed?
Physicist Emil Akhmedov on differential equations, ideal lines and points and the solution of the Laplace paradox.
The 18th – 19th centuries were marked by the success of Newton's mechanics, which showed amazing efficiency in describing the motion of the planets of the solar system. In addition, it undoubtedly led to successes in other, more mundane areas, and turned out to be effective in describing the nature of heat and thermodynamics. That is, the thermodynamics of gases was described in the form of the motion of atoms in it purely mechanistically. And Maxwell, when writing his equations of electrodynamics, tried mechanically, with the help of gears and gears, to describe even electromagnetic fields. But in fact, this has nothing to do with the nature of electromagnetic fields, and science began to move when it abandoned this mechanistic approach.
Under the sign of all this happening, such a Laplace paradox arose, which says that there is no will everywhere. That is, a person cannot act of his own free will, everything is predetermined and predictable.
If we believe in both the mechanistic and the field approach, then all natural phenomena are described in the form of some functions and differential equations on them. We will now discuss what functions and differential equations are. For example, the simplest function is the position of a particle. These are three functions, that is, a coordinate in three directions. There is the position of the particle at a given moment of time t in this position, at the next moment in a different position, and so on.
The result is a function - the time dependence of the position of the particle. This function is described by the well-known differential equation, called Newton's second law. It is differential because it contains two derivatives of this function. This is acceleration multiplied by mass, and all this is determined by the force acting on this particle. Here's a differential equation. If you specify the initial position of the particle and its initial velocity, then the solution to this equation is uniquely determined.
In thermodynamics, everything is also described unambiguously. You only have not one particle, but a lot of them. The Avogadro number gives an idea of how many particles are in the gas. There are a huge number of particles in a certain volume of gas. These particles move, collide with each other, collide with walls, and this leads to thermodynamic phenomena. It turns out that if you have a sufficiently powerful computer that can operate with such a huge amount of data, then, knowing the initial position of all particles and their initial velocities, you can unambiguously determine their subsequent evolution and gas behavior, fully predict all the details of the gas behavior, and its constituent particles and so on.
This idea can be continued further. We also consist of molecules, atoms that interact with each other, act on each other with some kind of forces. And if we set the initial positions and the initial speed of all these particles of which we are composed, then our behavior is completely predetermined, because our consciousness and everything else, if you believe in this mechanistic model, is completely determined by those chemical reactions taking place inside our brain and body and so on. Accordingly, there is no will. Any of my subsequent actions are predetermined by everything that happens around. This means that this is the Laplace paradox that everything is predetermined.
It was believed that the Laplace paradox is solved by quantum mechanics, because there is a probabilistic interpretation. However, a probabilistic interpretation of quantum mechanics arises when the system is opened. That is, if you act on a small quantum system with a large classical system, this is called a measurement, the state of the quantum mechanical system is measured, and at this moment a probabilistic interpretation appears. And if the quantum mechanical system is closed, then it is completely described by the so-called wave function. Because of its probabilistic interpretation, it is called a probability wave, but that doesn't matter.
Whatever it is called, a closed quantum mechanical system is described by a wave function, which also obeys a differential equation called the Schrödinger equation. The following is important: if you know the initial conditions for this differential equation, that is, the initial values of the wave function, its derivatives, then after that you unambiguously restore the wave function at all times. And a quantum mechanical system, if it is closed, is described uniquely using this wave function. And no probabilistic interpretation is needed because you are not opening the system.
We can say that everything is predetermined again. One can argue with this, but whatever theory we are dealing with - with the theory of relativity, with the general theory of relativity, with the equation of gravity, with Maxwell's equations, equations describing weak and strong interactions - all these forces are described by differential equations of the second order ... These equations contain fields that are functions of coordinates, that is, of the position in space and in time, the value of a field. Its changes in space and time are described by a differential equation. That is, everything seems to be predetermined again.
Where do paradoxes come from? Let's digress for a second and try to explain what's going on in general. An essential part of the paradoxes arises when we try to extrapolate some kind of natural law for all occasions. For example, the well-known paradox: which came before - the chicken or the egg? A philosophical problem that suggests that in the entire history of the universe, there were chickens that laid eggs, chickens hatched from eggs, and so on. It is clear that this was not always the case. As a result of evolution, there were intermediate states that gave birth to something like an egg, closer and closer and more like an egg, and from these eggs, or the like of eggs, birds or animals hatched, which were more and more close to what we now call a chicken. The chicken-and-egg paradox is solved in this way.
Returning to Laplace's paradox, we natural scientists always use some approximation. Any natural-scientific law, no matter how fundamental it may be, is always true in some approximation. Newton's second law is true if we are dealing with sufficiently large objects - from a grain or more - moving at speeds that are much less than the speed of light, with accelerations close to what we experience on Earth and in the solar system, in gravitational fields that create something like the sun, stars like the sun, or planets like the earth. If we start discussing objects moving at very high speeds, we have to deal with the special theory of relativity. If we are discussing very strong gravitational fields, we have to deal with general relativity. If we have to deal with very small objects, we have to deal with quantum mechanics. If we have to deal with very high speeds for very small objects, we have to deal with quantum field theory. In the next step, if we want to deal with quantum field theory in very strong gravitational fields, we will probably have to deal with something like quantum gravity, which is still in the creation stage, and the rest of the theories have been developed.
Where does this approximation come from? Mathematics, as they like to say with great pathos, is what allows us to find some kind of order in the chaos that surrounds us. That is, we always use mathematical formulas to describe something mathematically idealized, which approximately describes what actually happens in nature. And we can even determine in what approximation, and even improve this approximation, approaching the real situation. For example, there are no ideal, infinitely thin straight lines, there are no ideal points and sizeless objects, there are no ideal inertial frames of reference.
But in reality, what is happening? We can calculate the yield from a given area, describing it with a rectangle or polygon, the edges of which are composed of straight line segments, considering them infinitely thin. This allows us to estimate the area of this flat figure and the harvest that we will collect, often neglecting the fact that this surface is not flat, but inside this polygon there are hills, depressions, and so on. The question is, in what approximation we are working.
Likewise, using perfect thin lines, dots, and so on, we can calculate at home. For accuracy, a few millimeters are enough for calculating houses so that we do not have gaps in the windows. On the other hand, with what accuracy do we need to calculate an object like a detector in an accelerator (and this is something comparable to a three-, four-, or five-story building)? There, its various parts are adjusted to one another with micron precision. There, the accuracy is needed higher, because it is necessary to determine the tracks of particles and the peaks of reactions with such accuracy. The question is how precisely what we want to describe. Therefore, we always make some kind of approximation, limiting ourselves to some precision with which we want to describe something, and from this everything flows.
Therefore, the differential equations that describe the laws of nature are actually some kind of approximation to what really happens in nature. Nobody said that if we go to even smaller sizes, we will see a fine structure in space and time, some kind of granular structure, the behavior of which will no longer be described by differential equations, but by finite-difference equations. Yes, in such equations again there will be a problem with the fact that everything is predictable. But what if these are not finite-difference equations? The fact is that, most likely, the Laplace paradox is explained by the fact that it is not necessary to extrapolate the laws of nature, applicable to a given situation, to all cases in life and nature.
Emil Akhmedov, Doctor of Physical and Mathematical Sciences, Leading Researcher at the Alikhanov Institute for Theoretical and Experimental Physics, Professor of the Department of Theoretical Physics at the Moscow Institute of Physics and Technology.
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Free will is an old philosophical problem, but there have been many interesting advances in this area in recent decades. The participants in the conversation will discuss these innovations. Speech, in particular, will go about "Examples of Frankfurt", "Manipulative Argument" by D. Pereboom and "The Principle of Ultimate Responsibility" by R. Kane. Consideration of these and other conceptual tools will allow the participants in the conversation to assess the real progress in understanding the problem of free will.
Determinism is a general scientific concept and philosophical doctrine of causality, regularity, genetic connection, interaction and conditionality of all phenomena and processes occurring in the world.
With the help of an anesthesiologist, Jennifer Aniston, and a scientist armed with a sledgehammer, Professor Marcus du Sautoy seeks the answer to the question of what "I" is. To do this, he subjects himself to several interesting and unusual experiments. Marcus learns at what age our self-awareness appears and whether other living beings possess it. He puts his mind to sleep in an anesthetic experience to better understand it, then has an out-of-body experience to localize his self. Marcus then travels to Hollywood to understand how celebrities are helping to better understand the microscopic activity of our brains. He then takes part in a mind-reading experiment that radically changes his understanding of what "I" is.
If the initial conditions of the system are known, it is possible, using the laws of nature, to predict its final state.
Free will is an important part of thinking about free will in general. Religions differ greatly in how they respond to the main argument against free will, and thus can provide different responses to the paradox of free will - the assertion that omniscience is incompatible with free will.
“Modern events have a connection with previous events, based on the obvious principle that no object can begin to be without the cause that produced it ... The will, no matter how free, cannot, without a certain motive, give rise to actions, even those that are considered neutral ... We must consider the present state of the Universe as a result of its previous state and the cause of the subsequent one. The mind, which for any given moment would know all the forces acting in nature, and the relative location of its constituent parts, if it were, moreover, vast enough to subject these data to analysis, would embrace in a single formula the movements of the most enormous bodies in the Universe and the lightest atom; for him there would be nothing obscure, and the future, like the past, would be in front of his eyes ... what is imposed by our ignorance "
There is a problem in the compatibility of our concept of freedom and how the world works. On the one hand, we know that every event has its own cause. The chain of reasons goes very far. And it seems that what is happening today is predetermined by the events of the past. On the other hand, there is the idea that we are able to initiate action ourselves; we can really change the future. The metaphysical problem of free will is the problem of the relationship between the causal order, between the fact that all events are deterministic, and the fact that we make a free choice or free action. But this is not an abstract problem. The concept of personality and responsibility is built on the concept of freedom. Can we perform free actions, what is moral and legal responsibility based on, and will a robot become a person? In this issue, we invite you to discuss the Derk Pereboom manipulation argument.
Ilya Shchurov
When was the concept of "function" introduced into scientific circulation? What were the proposed solutions to the problem of string vibration? What were the approaches to understanding the function? And how did the string controversy develop? Mathematician Ilya Shchurov talks about this.
In fundamental physics, unlike mathematics, there are only three main unsolved problems that virtually all scientists from this field of science are engaged in - the problem of the cosmological constant, the problem of quark confinement, and the problem of quantum gravity.
The cosmological constant problem
Imagine a hole containing a ball. If you move it, it will begin to oscillate and without friction it will oscillate forever - you get a classic oscillator. But if the ball is not touched, then it will simply lie at the bottom.
However, a quantum particle is not a ball, but a wave. Therefore, the ground state of a quantum oscillator has nonzero energy. This is a wave with a single crest inside the hole. That is, a quantum particle vibrates even in the ground state. These are the so-called zero fluctuations. They occur in any quantum system, including quantum field theory.
In quantum field theory, a vacuum is not a void. It consists of zero-point fluctuations. If there is no gravity, then the energy is calculated from the total energy of these zero-point vibrations. They seem to be thrown away. And all particles in quantum field theory are excitation over zero-point vibrations.
However, in the presence of gravity, zero-point oscillations cannot be discarded. After all, they "weigh" something, that is, they distort space-time. Therefore, a problem arises.
It is theoretically predicted that zero-point vibrations constitute enormous vacuum energy. However, observations show that the vacuum energy in our Universe is very small. This is what is now called dark energy in space. It leads to an accelerated expansion of the Universe, as something "weighs". This is precisely the problem of the cosmological constant: on the one hand, quantum field theory predicts that it is huge, and on the other hand, we observe it very small. Where does the enormous vacuum energy predicted by quantum field theory go? And what is then the nature of dark energy?
The quark confinement problem
It is known that the nucleus consists of protons and neutrons. They interact with each other using nuclear forces. If protons collide, increasing energy, we will see the birth of a huge variety of new particles - hadrons.
All hadrons are described in one way: they are made up of quarks. This is observed by scattering an electron on a proton at very high energies. It turns out that in this case, the electron is scattered in much the same way as alpha particles on atoms. The latter was studied by Rutherford at the beginning of the twentieth century: he saw that an alpha particle is scattered at a very concentrated center of the nucleus, which is very small in size. It turns out that in the same way an electron is scattered by a proton, but with one caveat: the proton seems to have three centers with corresponding charges.
There are actually three quarks inside the proton. But for some unknown reason, we cannot separately obtain these quarks, we always see them only in the composition of hadrons. We know the theory of quarks, and this is quantum chromodynamics, which describes quarks and gluons. The latter transfer interactions between quarks, just like photons between electric charges. We understand quantum chromodynamics well at high energies. Then it really describes the physics of hadrons. But at low energies, the electron is scattered by hadrons as a whole. How does one description, with the help of practically free quarks, go to another - in the form of hadrons as bound states from quarks? And why don't quarks exist individually? These questions are the essence of the confinement problem.
The problem of quantum gravity
Quantum field theory has problems with the existence of infinite frequencies. Roughly speaking, the field can be bent as desired, with an arbitrarily high accuracy. Because of this, so-called divergences arise, namely: when calculating various physical quantities in quantum field theory, we get infinite contributions. In all currently developed quantum field theories that we deal with, these divergences can be eliminated by redefining several coupling constants, such as the charges and masses of particles, for example.
Moreover, in order to eliminate a similar problem when quantizing gravity, it is necessary to redefine an infinite number of coupling constants. As the energy rises, the theory needs to be complicated more and more. This suggests that the theory of gravity is applicable only at low energies, and should be based on a more fundamental (high-energy) theory that we do not yet know.
In September 2015, Stephen Hawking on a new idea that, according to the physicist, will help solve the 40-year-old paradox of information loss in black holes. This paradox is formulated at the interface between quantum field theory and general relativity, so its resolution can help in formulating the theory of quantum gravity.
The scientist referred in his message to some special properties of space, using correctly which it is possible to indicate how and in what form the information leaves the black hole. "Hot on the heels" after the announcement, we have already figured out Hawking's proposal, but all the details of the hypothesis were still awaiting publication.
Three months later, almost immediately after the New Year, an article appeared on the arXiv.org electronic preprint service, in which the physicist, together with colleagues - Andrew Strominger and Malcolm Perry - revealed in more detail the essence of his proposal. Simultaneously with the publication of the preprint, Hawking sent the article for publication to one of the most respected physics journals - Physical Review Letters... Five months later, the work was reviewed and on June 6 appeared on the journal's website.
This caused an unexpected surge in publications about portals to other universes located in black holes and other strange phenomena. Their source is a popular science lecture that Hawking gave back in August 2015. In the published work, there is not a word about alternative universes, but there are the very details explaining how to cope with the information paradox.
Today we return to the discussion of the information paradox and again turned for a comment to Emil Akhmedov, Doctor of Physical and Mathematical Sciences and a leading researcher at the Institute of Theoretical and Experimental Physics.
Before you start
In order to formulate the information paradox, it is necessary to recall several important properties of black holes. The most famous of them is that a black hole has a certain surface called the event horizon, after finding itself behind which even light cannot leave the vicinity of the object. The second important property is the so-called “black hole no hair theorem”. According to her, any fields created by a resting black hole are stationary, that is, they do not change in time. This property of a black hole follows from the properties of the event horizon.
An important step in the emergence of the information paradox was the prediction of Hawking radiation, due to which the black hole slowly evaporates. This is a quantum effect associated with the amplification (amplification) of zero point oscillations as a result of collapse - the process of formation of a black hole.
The energy spectrum of this radiation is thermal, and the smaller the black hole, the higher the temperature that corresponds to this radiation. This is due to the fact that a black hole cannot contain quantum excitations with a wavelength longer than its size. Therefore, from general considerations, it will emit with a characteristic wavelength of the order of the size of its horizon. And the radius of the horizon of a black hole is proportional to its mass. Accordingly, the characteristic radiation energy, being proportional to the frequency, should be inversely proportional to the mass of the black hole. But the characteristic energy of radiation quanta is its temperature. This heuristic reasoning, which belongs to Vladimir Naumovich Gribov, is confirmed by a detailed calculation.
Hawking's temperature is very low - for a black hole with the mass of the Sun, it will be ten millionths of a kelvin. And a black hole of even greater mass will, accordingly, have an even lower temperature. Therefore, it is most likely impossible to see Hawking radiation in practice in the foreseeable future. Unless it will be possible to detect the decay of the so-called primordial black holes, which were formed in the early stages of the development of the universe. Indeed, then the density of matter should have been very high and, therefore, black holes of very small mass could be formed. Such holes would have very high temperatures. One can hope to see the results of their decay by Hawking radiation, if we look at the most distant, that is, the earliest, parts of the visible part of the universe. But so far such phenomena have not been detected.
Hawking radiation does not depend on what material the black hole was formed from as a result of the collapse. In it, for a given energy, different particles can appear with equal probability - say photons and neutral pi-mesons. As a result, a situation that is unacceptable for physics is obtained - the fundamental possibility of restoring the "fate" of a single atom that has fallen into a black hole is lost. In the language of mathematics, this means that the transformation matrix that transfers the system from the state before the formation of the black hole to the state after its evaporation turns out to be non-unitary (we are talking about the S-matrix, one of the central objects in quantum field theory). This means, for example, that the probabilities of some processes may turn out to be more than one.
This is the paradox of information loss - based on general relativity and quantum field theory, Hawking managed to get a situation that simply should not exist in physics. You can treat the formulation of this paradox in different ways, but its clear and precise solvability is one of the properties of the "real" theory of quantum gravity.
Emil Akhmedov belongs to the group of physicists who believe that there is no paradox with the loss of information. The ambiguity associated with the paradox is due to the large number of crude assumptions Hawking made in his formulation. Among others, these are:
1) The energy of the black hole radiation particles is small enough compared to the total energy or mass of the black hole.
2) The event horizon is far enough from the singularity and the general theory of relativity is applicable to it.
3) Quantum corrections make a small contribution to the Hawking radiation spectrum.
However, Emil believes that it is very important to understand in detail how black holes decay, and how decay products carry information about the initial state of the collapsing matter.
New job and its prerequisites
A new article by Hawking, Strominger and Perry is titled Soft Hair on Black Holes. According to its popular exposition by Gary Horowitz, Distinguished Professor of Physics at the University of California, the article revises the fundamental facts behind the paradox's formulation, such as the validity of the "black hole no hair theorem."
N+1: As I understand it, during the time that has passed since the release of the preprint, there have probably already been several seminars that analyze Hawking's work in detail?
Emil Akhmedov: We even hosted an undergraduate and graduate school conference in April. We called in the students of Malcolm Perry and Hawking. They lectured and we more or less figured out what was stated in the work. We can say they have reached a deep misunderstanding.
N+1: Strominger, Perry, and Hawking examined two assumptions that were made by Stephen Hawking in the original 1975 paper. Looks like they said that it really isn't. Where did this statement come from?
E.A .: Remember, I explained to you last time that there is a so-called "no hair theorem"? Spacetime in the presence of a black hole at a given moment in time, at any distance from her, characterized by three numbers: mass, moment of rotation and charge. Accordingly, the ground state of quantum field theory against the background of a black hole should be characterized by these parameters. And since Hawking radiation does not carry any information, it means that almost everything that was before the collapse is gone.
Now Strominger, Perry, and Hawking have reconsidered this claim. To begin with, they say that if you leave the black hole for long distances not at a given moment in time, but in the direction of light infinity - that is, moving with the light - then the characteristics of this radiation contain much more parameters, more precisely - infinitely many parameters.
N+1: That is, they are not limited to the angular momentum, charge and mass of the black hole?
E.A .: Yes. I can even give an analogue from electromagnetism, which will probably be more understandable.
Let's look at the electromagnetic field of a group of charges. If we take a given moment in time and just look at this group from a very long distance, then we will see just a Coulomb field. Corrections can appear to it - dipole moment, quadrupole moment, but the dominant quantity at large distances will be the Coulomb field.
Moreover, there is an analogue of the “no hair theorem” - the solution of Maxwell's equations, which does not change when turning around the center and drops to zero at large distances - the only one, and this is the Coulomb field. Its only characteristic is charge. In this sense, the situation is similar to the “no hair theorem”. If there is no rotation invariance, then there can be corrections in the form of dipole, quadrupole, and higher moments.
All of the above is true if we look at the charges at a given time and forget about their movement. If the charges make some kind of movement, then they emit something. Then, in addition to the above characteristics, you will also have radiation characteristics. And even at a great distance, in addition to the Coulomb field, there will also be a radiation field carrying infinitely many characteristics. A similar situation exists in the presence of gravitational fields and radiation. I would like to emphasize that so far all this is not directly related to the resolution of the information paradox.
This is what was known before the article by Hawking, Strominger and Perry - back in the 60s and 70s. New interest in this issue has arisen thanks to the work of Strominger with his students and co-authors. The point is that this infinite number of characteristics of radiation at a great distance is associated with the existence of some very large asymptotic symmetry in this part of space-time. Strominger studied it, trying to generalize the principle of the AdS / CFT correspondence to the case of flat space. [a little more about this in the previous interview]
What's new suggested by Hawking, Perry and Strominger
E.A .: Everything I said about the infinitely large number of characteristics of radiation true when you are very far away from all kinds of sources of gravitational and electromagnetic fields. Namely, this is true in a rough approximation of the leading order at light infinity, that is, without any corrections. Hawking, Perry and Strominger now say that a similar situation should be not only at an infinite distance from radiation sources, but also near the black hole event horizon.
N+1: It's definitely not an infinitely far distance.
E.A .: Yes, this is definitely not an infinitely far distance, but Hawking and co-authors claim that they managed to describe how the symmetries described above can be continued from infinity to the black hole horizon. Moreover, not for the most general case of fields, but so far only for electromagnetic radiation.
There are many questions to this statement. They say literally that there is literally the same symmetry at the horizon of a black hole as at infinity. Where this statement follows, I could not understand in detail. If you look at the article by Hawking, Strominger and Perry, there are not many formulas, there are more words. And from these words I could not extract the tested formula.
N+1: Where did this statement come from then?
E.A .: Hawking was interested in the fact that the space-time metric at large distances from a black hole can be described by more parameters than just charge, mass and angular momentum. This is an obvious violation of the no hair theorem. He thought the same could be generalized to the characteristics of the spacetime metric near the black hole's event horizon.
Indeed, from general considerations, it is clear that if we take into account the influence on the black hole of particles / waves incident or emitted by Hawking from the event horizon, then this horizon will somehow be deformed. These deformations can be characterized by an infinitely large number of parameters, since they can occur locally in any part of it. And this picture is similar to how the deformation of space-time at light infinity occurs as a result of radiation going there. That is, the analogy between the event horizon and light infinity is obvious.
N+1: That is, I understand correctly that the work claims that Hawking radiation will have infinitely many characteristics, and not just a temperature distribution that depends on mass, charge and angular momentum?
E.A .: Yes. And, accordingly, with the help of these characteristics, you can fully characterize the state of a black hole. In words, all this has long been clear to me and to many of my colleagues, but I have not seen clear and easily testable formulas on this topic. Moreover, even from people who understand this issue and discussed it with Hawking, Strominger and Perry.
N+1: So this is more of a philosophical work?
E.A .: It looks more like formulating an idea. As an idea, I like it. I repeat, it was initially understood by both me and many of my colleagues. That is, for me this is nothing new, except that such famous people spoke on this topic in the same vein in which other, less famous people spoke out.
N+1: There was one more little thing besides the "hair". Hawking, Strominger and Perry say that the vacuum state is somehow not unique?
E.A .: The characteristics of a black hole are the same as the characteristics of the vacuum (ground state) in quantum field theory against the background of a black hole. The fact is that even in the presence of Hawking radiation, we are dealing with the ground state of quantum field theory, because Hawking radiation is the amplification of zero-point vibrations that are present in a vacuum, that is, in the ground state. Previously, they thought that there were only three of these characteristics, but now they saw that there should be infinitely many such characteristics. It has long been known that there are infinitely many such characteristics at infinity, but now they claim that in the region of a black hole everything is exactly the same. Thus, the ground state of quantum field theory in the presence of a black hole has an infinitely large degeneracy, and different ground states differ by means of the aforementioned characteristics and they are translated into each other by means of transformations of infinite symmetry.
Hawking, Perry and Strominger even claim that they have rigorously proven it. That is, if you ask Malcolm Perry directly, he will say that they proved this statement. And he is a man who does not throw words to the wind. It's just that I have not yet fully understood these statements.
N+1: During the last interview, you mentioned another factor that Hawking did not take into account. I wonder if it was corrected when "patching holes" in the description of the paradox?
E.A .: I said the following - quantum field theory against the background of a black hole is in a non-stationary state. I may have phrased it a little differently, but that's what I meant. Hawking, Strominger and Perry talk about vacuum and its characteristics. For me, this is not enough - due to the fact that quantum field theory is in a non-stationary state against the background of a black hole, it does not remain in a vacuum state, but passes into some kind of excited state. Namely, the internal degrees of freedom of the field theory are excited. That is, in addition to zero-point oscillations, the excited states of quantum field theory will also contribute to the radiation of the black hole. And this, too, certainly characterizes the state of quantum field theory against the background of a black hole and complements the picture.
But what I have just said is by no means a generally accepted point of view. It is shared, perhaps, by about five people in the world. However, this point of view can be supported by detailed calculations [Emil T. Akhmedov et al. / PRD, 2016], and the formula is objective. Anyone can check it and make sure it is correct or incorrect.
Thought experiment
N+1: And if you dream up - can you still imagine some kind of experiment that can test the theory? After all, each theory gives its own predictions, which are the criteria for correctness.E.A .: Of course, all these effects are weak and at the moment they are of only academic interest. Unfortunately, it is hopeless to check the existence of Hawking radiation and see its characteristics near those objects in the sky that we consider to be black holes.
N+1: And if we imagine that we can send the device?
E.A .: Even if we imagine that we can send the apparatus, these effects are still very weak. The temperature of a black hole with the mass of the Sun will amount to some ten-millionths of a Kelvin - a negligible value even against the background of cosmological relic radiation.
The only thing scientists hope for is to see phenomena from microscopic black holes. When we look at the sky, we are looking not only into the distance, but also into the past. In the early stages of the development of the Universe, when it was very dense, small primordial black holes could form. If we take a black hole with a mass equal to the mass of Mount Everest, then it rather not slowly radiates, but explodes, because its temperature is enormous.
N+1: It is clear - the smaller the black hole, the higher the temperature of the radiation. But if we can catch at least one quantum emitted by a black hole from afar?
E.A .: In order to confirm the observations of Hawking, Strominger and Perry experimentally, one quantum from the flux emitted by a black hole is not enough. If we look at a black hole from afar, then the entire stream is given by an infinite number of characteristics.
N+1: That is, if we could catch the entire flux of radiation from a black hole, then we could get an answer as to whether the theory is correct or not.
E.A .: Well, theoretically, if we surround the black hole with a box and collect everything that it radiated, then we can determine the value of an infinite number of charges. Let me emphasize that some of them will be equal to zero, and some will not. All this would fully characterize the state of the black hole.
But I will clarify again, this should not be done at infinity, because a black hole may not be alone, it may be surrounded by something. These bodies can also emit gravitational and electromagnetic radiation. To get the characteristics of a particular black hole, we need to catch radiation near its horizon.
N+1: It turns out that we just need to build a huge detector around the black hole - a kind of Dyson sphere.
E.A .: N no. Of course, I am not saying that confirmation of the above observations is necessary to set up just such a complex and even impossible experiment. If we saw that some microscopic (for example, primary) black hole emits and its characteristics change, and the radiation carries away exactly those characteristics that have changed, then that would be enough.
Interviewed by Vladimir Korolev
Chris Friel is a British photographer, the author of the illustrations used in the material. He has spent the last 10 years trying to get a photo he likes. He has worked in 150 countries and would like to have time to visit the remaining 46 until he becomes a couch potato.
