In the quantum theory of gravity, the very geometry of space-time must fluctuate continuously, so that even the distinction between the past and the future can be erased. Apparently, among the fundamental forces of nature, gravity has a special status. Other forces, such as electromagnetic, operate in space-time, which serves as a simple container for physical events, a decoration against which they occur. Gravity has a completely different character. It is not a force acting against a passive background of space and time; rather, it is a distortion of spacetime itself. The gravitational field is the "curvature" of space-time. These are the concepts of gravity, established by A. Einstein as a result of the hardest, as he himself said, work in his life.
The qualitative differences between gravity and other forces become even clearer when one tries to formulate a theory of gravity that is consistent with the foundations of quantum mechanics. The quantum world is never at rest. For example, in the quantum theory of electromagnetism, the values of electromagnetic fields fluctuate continuously. In a universe obeying the laws of quantum gravity, the curvature of space-time and even its very structure will also have to fluctuate. It is possible that the sequence of some events in the world and the very meaning of the concepts of the past and the future will be subject to change.
It can be argued that if such phenomena existed, they would certainly have been discovered long ago. However, the quantum mechanical effects of gravity should manifest themselves only on extremely small scales; M. Planck was the first to draw attention to such a scale. In 1899 he introduced his famous constant called the quantum of action and denoted ħ. Planck tried to explain the radiation spectrum of a black body, i.e. light radiation emitted by a hot, closed cavity through a small hole. He noted that his constant together with the speed of light © and the Newtonian gravitational constant (G) form an absolute system of units. These units serve as natural scales for the quantum theory of gravity 1.
Planck units have nothing to do with ordinary physical representations. For example, the unit of length is 1.610–33 cm. This is 21 orders of magnitude less than the diameter of atomic nuclei. Roughly speaking, the ratio of the Planck length to the size of nuclei is the same as the ratio of the size of a person to the diameter of our Galaxy. The Planck time unit looks even more fantastic: 5.410–44 s. To study these space-time scales with the help of experimental facilities built on the basis of modern technology, an accelerator of elementary particles the size of a Galaxy is needed!
In this area of science, it is impossible to draw definitive conclusions from experiments, therefore the quantum theory of gravity has a somewhat speculative character, which is unusual for physics. However, in essence, this theory is conservative. It uses well-proven theories to draw rigorous inferences from them. If we ignore the particulars, then the main goal of quantum gravity is to combine three components into one theory: the special theory of relativity, Einstein's theory of gravity and quantum mechanics. This synthesis has not yet been fully realized, but along the way, we learned a lot. Moreover, the development of a realistic theory of quantum gravity pointed to the only way to understand the Big Bang and the ultimate fate of black holes, i.e. the beginning and the distant future of the universe.
Of all the constituents of quantum gravity, special relativity has historically emerged first. In this theory, space and time are combined on the basis of the experimentally verified postulate of the independence of the speed of light for different observers moving in empty space, free from external forces. The consequences of this postulate, introduced by Einstein in 1905, can be described using space-time diagrams, in which curved lines depict the position of objects in space as a function of time. These curves are called object worldlines.
For the sake of simplicity, I will not consider the two spatial dimensions. Then the world line can be drawn on a two-dimensional plot, where the spatial axis is directed horizontally and the temporal axis is vertical. The vertical line on such a graph represents the world line of an object that is at rest in the frame of reference selected for measurements, and the inclined line represents the world line of an object moving in this frame of reference at a constant speed. A curved world line describes the motion of an accelerated object.
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Figure: 1. The light cone, which distinguishes the regions of the Universe, reachable from a given point of space-time, in the quantum theory of gravity is difficult to define. Cone (a) is a surface in four-dimensional space-time, but here it is shown as two-dimensional: one spatial dimension is removed. If the gravitational field is quantized, then the shape of the cone can fluctuate strongly at short distances (b). In fact, the fluctuations cannot be directly distinguished; instead, the light cone will "look blurry." As a result, the question of whether any two points in space-time can be connected by a signal moving slower than light can only be given a probabilistic answer (c).
Any point on the space-time diagram determines the position of an object in space at a given moment in time; it is called an event. The spatial distance between two events depends on the selected frame of reference, the same is true for the time interval between them. The very concept of simultaneity depends on the frame of reference. If two events can be connected by a horizontal line, then they are simultaneous in this frame of reference, but not in other frames.
To establish a connection between reference frames moving relative to each other, it is necessary to introduce a common unit of measurement for spatial distances and time intervals. The multiplier for the conversion is the speed of light, which connects a given distance with the time it takes for light to cover it. I will choose meters as the unit of measure for space and time intervals. In this system of units, one meter of time equals approximately 3⅓ nanoseconds (1 ns = 10–9 s).
If space and time are measured in the same units, then the world line of the photon (quantum of light) is tilted at an angle of 45 °. The world line of any material object is deviated from the vertical by an angle less than 45 °. This is just another formulation of the statement that the speed of any object is always less than the speed of light. If the world line of some object or signal deviates from the vertical axis by more than 45 °, then from the point of view of some observers, this object or signal will move in time in the opposite direction. By creating an emitter of superluminal signals, it would be possible to transfer information to your own past, which would violate the principle of causality. Such signals are outlawed in the special theory of relativity.
Consider two events on the world line of an observer moving without acceleration. Suppose that in some frame of reference these events are separated by four meters of space and five meters of time. Then our observer moves in this frame of reference with a speed equal to 4/5 of the speed of light. In another system, its speed will be different, and the corresponding spatial and temporal distances change. There is, however, a quantity that is the same in all frames of reference. This invariant quantity is called "proper time" between two events; it is equal to the time interval measured by the watch that our observer took with him.
In the selected frame of reference, the world line between events is the hypotenuse of a right-angled triangle with a base of 4 m and a height of 5 m. The “proper time” is equal to the “length” of this hypotenuse, but calculated in an unusual way, using the “pseudo-Pythagorean” theorem. First, the legs of the triangle are squared - just as in the usual Pythagorean theorem. However, the square of the hypotenuse in special relativity is not equal to the sum, but to the difference of the squares of the legs.
Figure: 2. The world line represents a path through space and time. Shown here are two world lines showing one variation of Einstein's twin paradox. The "tilted" world line of the twins accelerating at the pivot point when returning from the trip appears to be longer, but this twin will register a shorter "proper time". Indeed, the straight line corresponds to the longest interval between two points on the space-time diagram. The figure shows the departure and arrival times of signals exchanged between twins.
In our example, the proper time is equal to three meters. It will remain equal to three meters in the frame of reference of any observer moving without acceleration. It is the invariance of proper time that allows you to combine space and time into one really existing space-time. The space-time geometry based on the "pseudo-Pythagorean" theorem is not Euclidean, but in many respects it is similar. In Euclidean geometry, among the many paths connecting two points, one can choose one extreme - a straight line. The same is true for the geometry of space-time. However, in Euclidean geometry, this extremum is always a minimum (a straight line is the shortest distance between points), while in space-time it is always a maximum, if two points can be connected by a world line that does not contain FTL signals.
In 1854, the German mathematician B. Riemann generalized Euclidean geometry to the case of curved spaces. Two-dimensional curved spaces have been studied since antiquity. They were called curved surfaces and were usually viewed from the perspective of the three-dimensional Euclidean space in which they were placed. Riemann showed that curved spaces can have any number of dimensions and that to study them there is no need to assume that they are in a Euclidean space of the highest dimension.
Riemann also pointed out that the physical space in which we exist can be curved. In his opinion, this question can only be solved experimentally. How is it possible, at least in principle, to perform such an experiment? They say that Euclidean space is flat. Parallel lines in flat space form a homogeneous rectangular mesh. This is the property of flat space. What happens if you try to draw the same grid on the surface of the Earth, assuming that it is flat?
The result can be seen from an airplane flying on a clear day over the cultivated fields of the Great Plains. Roads running from west to east and from north to south have divided the entire land into equal sections (say, one square mile). East-west roads are often nearly straight lines that stretch for miles. But the north-south roads look different. If you follow your gaze along such a road, every few miles you will see an unexpected bend to the east or west. These bends are due to the curvature of the earth's surface. If they are not, the roads heading north will converge and the sections they separate will be less than a square mile in area.
In the three-dimensional case, one can imagine the construction of a giant lattice structure (like a scaffold) in which the edges converge at exactly 90 ° and 180 °. If the space is flat, then the construction of such scaffolding will not cause difficulties. If the space is curved, then sooner or later you will need to use edges of different lengths, lengthening or shortening some of them in order to fit each other.
The same generalization can be applied to the geometry of special relativity that Riemann applied to Euclidean geometry; it was carried out between 1912 and 1915 by A. Einstein with the help of the mathematician M. Grossman. The result was the theory of curved spacetime. In the hands of Einstein, it became the theory of gravity. In the special theory of relativity, space-time was considered flat, i.e. the absence of gravitational fields was implied. There is a gravitational field in curved spacetime; in fact, "curvature" and "gravitational field" are just synonyms.
Since Einstein's theory of the gravitational field is a generalization of the special theory of relativity, he called it general theory of relativity. This name has been misused. General relativity is actually less "relative" than special theory. Flat space-time is devoid of characteristic features, it is homogeneous and isotropic, and this circumstance guarantees the strict relativity of positions and velocities. But as soon as “hillocks” or local areas with curvature appear in space-time, the positions and velocities acquire an absolute character: they can be determined in relation to these “bumps”. Space-time ceases to be just a passive arena of action for physics, it itself acquires physical properties.
In Einstein's theory, curvature is created by matter. In principle, the relationship between the amount of matter and the degree of curvature is simple, but the calculations are quite complex. To describe the curvature at a given point, you need to know the values at this point of twenty functions of space-time coordinates. Ten of these functions correspond to that part of the curvature that propagates freely in the form of gravitational waves, i.e. in the form of "ripple" curvature. The remaining ten functions are determined by the distribution of masses, energy, momentum, angular momentum and internal stresses in the substance, as well as the Newtonian gravitational constant G.
The constant G is very small if we take into account the values of the mass density found in terrestrial conditions. It takes a lot of masses to bend spacetime noticeably. The reciprocal of 1 / G can be regarded as a measure of the "rigidity" of space-time. From the point of view of everyday experience, spacetime is very rigid. The entire mass of the Earth creates a space-time curvature that is only one billionth of the curvature of the earth's surface.
In Einstein's theory, a body that is freely falling or freely rotating in an orbit follows in its motion along a world line called a geodesic. A geodesic connecting two space-time points is a world line of extreme length; it is a generalization of the concept of a straight line. If you mentally place a curved space-time in a flat space of the highest dimension, then the geodesic will be a curved line.
The effect of curvature on body movement is often illustrated by a model in which a ball rolls over a curved rubber surface. This model is misleading as it can only reproduce spatial curvature. In real life, we are forced to remain in the four-dimensional universe, in our ordinary space-time. Moreover, we cannot avoid motion in this Universe, as we tirelessly rush forward in time. Time is the key element. It turns out that although space is curved in the gravitational field, the curvature of time is much more important. The reason for this lies in the high value of the speed of light, which links the scales of space and time.
Near the Earth, the curvature of space is so small that it cannot be detected by static measurements. But in our unrestrained race for time, the curvature in dynamic situations becomes noticeable, just as a bump on a freeway may be invisible to a pedestrian, but becomes dangerous to a speeding car. Although near-Earth space can be considered flat with a high degree of accuracy, we are able to detect the curvature of space-time by simply throwing a ball into the air. If the ball is in flight for 2 s, then it will describe an arc with a height of 5 m. For the same 2 s, light travels a distance of 600,000 km. If we imagine that an arc with a height of 5 m is lengthened horizontally to the size of 600,000 km, then the curvature of the resulting arc will correspond to the curvature of space-time.
Figure: 3. The curvature of space-time appears as a gravitational field in the presence of masses. If you throw the ball up 5 m (left), then it will be in flight for 2 seconds. Its movement up and then down is a manifestation of the curvature of space-time near the earth's surface. The curvature of the ball's trajectory is easy to observe, but in reality it is very small if space and time are measured in the same units. For example, seconds can be converted to meters by simply multiplying by the speed of light, i.e. at 300 million meters per second. If this is done, then the trajectory becomes a very shallow arc, the height of which is only 5 m, and the length is 600 million m (right). In the figure, the height of the trajectory is increased.
Riemann's introduction of ideas about curved spaces contributed to research in another vast area of mathematics, topology. It was known that infinite two-dimensional surfaces can exist in an infinite variety of variants that are not brought together by continuous deformation of the surface; a simple example of this is a sphere and a torus. Riemann pointed out that the same is true for curved spaces of higher dimension, and took the first steps to classify them.
Curved spacetime (more precisely, its models) can also be one of many topological types. From the point of view of the correspondence to the real Universe, some models should be rejected, since they lead to paradoxes associated with causality, or it is impossible to formulate the known laws of physics in them. But there are still a lot of possibilities.
The famous model of the Universe was proposed in 1922 by the Soviet mathematician A. A. Fridman. In the special theory of relativity, space-time is not only flat, but also infinite in both time and space. In the Friedman model, any three-dimensional spatial section of space-time has a finite volume and topology of a three-dimensional sphere. A three-dimensional sphere is a space that can be enclosed in a four-dimensional Euclidean space so that all its points will be at a given distance from a given point. Ever since E. Hubble discovered the expansion of the Universe in the 1920s, Friedman's model has become a favorite of cosmologists. Together with Einstein's theory of gravity, Friedmann's model predicts the Big Bang at the initial moment of the expansion of the universe, when the pressure was infinitely great. This is followed by an extension,whose speed is slowly decreasing due to the mutual gravitational attraction of all matter in the Universe.
In Friedmann space-time, any closed curve can be continuously contracted to a point. Such space-time is said to be simply connected. The real Universe may not have such a property. Apparently, Friedman's model describes very well the regions of space located within several billion light years from the Galaxy, but the entire Universe is inaccessible to our observation.
A simple example of a multiply connected universe is a universe whose structure in a given spatial direction is repeated ad infinitum (ad infinitum) like a wallpaper. Every galaxy in such a universe is a member of an infinite row of identical galaxies separated by some fixed (and necessarily enormous) distance. If the members of this series of galaxies are indeed absolutely identical, then the question arises whether they should be considered as different galaxies at all. It is more economical to represent the entire series as one galaxy. Then traveling from one member of the row to another means the traveler's return to the starting point. The trajectory of such a journey is a closed curve that cannot be contracted to a point. It is like a closed curve on the surface of a cylinder that encloses the cylinder once. This repeating universe is called cylindrical.
Another example of a multiply connected structure is the handle model 2 proposed in 1957 by J. Wheeler (now at the University of Texas at Austin). Here the multi-connectivity manifests itself at a much shorter distance than in the previous case. A two-dimensional "handle" can be constructed by cutting two round holes in the two-dimensional surface and smoothly joining the edges of the cuts (see Fig. 4). In three-dimensional space, the procedure remains the same, but it is more difficult to visualize it.
Figure: 4. "Handle" in space-time is a hypothetical formation that can change the topology of the Universe. You can create a "grip" on a plane by cutting two holes and extruding their edges into tubes, which are then connected. On the original plane, any closed curve can be contracted to a point (shown in color). However, the curve passing through the "handle" cannot be tightened. A "handle" in three-dimensional space does not fundamentally differ from a "handle" in four-dimensional space-time.
Since in the original space two holes can be at a great distance from each other and still connect through the “neck”, such a “handle” has become a favorite device in science fiction for moving from one place in space to another faster than light: you just need to “pierce” there are two holes in space, connect them and "crawl" through the neck. Unfortunately, even if it is possible to build such a "hole punch" (which seems very doubtful), the system will not work. If the geometry of space-time obeys Einstein's equations, then the "pen" must be a dynamic object. As it turned out, the holes that it connects must necessarily be black holes from which there is no return. What will happen to the traveler? The neck will shrink and everything inside will be compressed to an infinitely high density,before it reaches the exit.
Figure: five. Remote regions of the Universe, in principle, can be connected using a "handle". It can be assumed that this allows the exchange of signals that travel faster than light, but in reality such a scheme will not work. In the picture above on the left, the distance between the holes in the "outside world" is comparable to the distance through the "neck". For the "handle" shown at the bottom left, the outer distance is much greater. In the lower figures, to the left and in the middle, the space is represented by a curved plane, but this is its view from the perspective of an observer in a space of a higher dimension. To an observer on a plane, it will appear to be roughly flat indeed. Whatever the length of the "neck", it is impossible to pass through it. The reason is that the handle always connects two black holes. "The neck is getting thinner"as shown in the pictures in the middle, and whatever gets there will be compressed to infinite density before reaching the opposite end.
The fluctuating topology characteristic of space-time in some versions of the quantum theory of gravity leads to serious fundamental difficulties. The picture on the right shows a "handle" that gradually became thinner and finally disappeared, leaving behind two "outgrowths". If such a process is possible, then the reverse process is also possible. In other words, the “outgrowths” can merge into a new “handle”. Such an event seems likely when the “outgrowths” are close, and impossible if they are distant from each other. However, the idea of what is "near" or "far" is associated with the embedding of the surface in a space of higher dimensions. For an observer on the surface itself, both cases depicted in the figures on the right should be indistinguishable.
Quantum mechanics, the third component of the quantum theory of gravity, was created in 1925 by W. Heisenberg and E. Schrödinger, but the theory of relativity was not taken into account in its original formulation. Nevertheless, it was immediately accompanied by brilliant success, since numerous experimental observations were long awaiting their explanation, in which it was quantum effects that dominated, and relativistic effects played little or no role at all. It was known, however, that in some atoms the electrons reach speeds that cannot be neglected even in comparison with the speed of light. Therefore, the beginning of the search for relativistic quantum theory was not long in coming.
By the mid-1930s, it was fully realized that the combination of quantum mechanics with the theory of relativity led to some completely new facts. The next two are the most fundamental. First, each particle is associated with some type of field, and each field is associated with a whole class of indistinguishable particles. Electromagnetism and gravity could no longer be regarded as the only fundamental fields in nature. Second, there are two types of particles that differ in the values of the spin angular momentum. Particles with half-integer spin ½ħ, 1½ħ, etc. obey the exclusion principle (no two particles can be in the same quantum state). Particles with integer spin 0, ħ, 2ħ, etc. are more "sociable" and can gather in groups with an arbitrary number of particles.
These amazing consequences of the combination of special relativity and quantum mechanics have been repeatedly confirmed over the past 50 years. Quantum theory combined with relativism gave birth to a theory that is grander than the simple sum of its parts. The synergistic, mutually reinforcing effect is even more pronounced when gravity is included in the theory.
In classical physics, empty flat spacetime is called a vacuum. The classical vacuum has no physical properties. In quantum physics, the name "vacuum" is given to a much more complex object with a complex structure. This structure is a consequence of the existence of non-vanishing free fields, i.e. fields far from their sources.
A free electromagnetic field is mathematically equivalent to an infinite set of harmonic oscillators, which can be thought of as springs with masses at their ends. In a vacuum, each oscillator is in the ground state (the state with minimum energy). A classical (non-quantum mechanical) oscillator in its ground state is at rest at a certain definite point corresponding to the minimum of potential energy. But this is impossible for a quantum oscillator. If the quantum oscillator were at a certain point, then its position would be known with infinite precision. According to the uncertainty principle, the oscillator would then have to have an infinitely large momentum and infinite energy, which is impossible. In the ground state of a quantum oscillator, neither its position nor momentum is precisely determined. Both are subject to random fluctuations. In a quantum vacuum, an electromagnetic field (and any other fields) fluctuates.
Despite the fact that field fluctuations in vacuum are random, they belong to a special class of fluctuations. Namely, they obey the principle of relativity in the sense that they "look" the same for any observer moving at an arbitrary speed, but without acceleration. As can be shown, it follows from this property that the mean value of the field is zero and that the magnitudes of fluctuations increase with decreasing wavelength. The final result is that the observer will not be able to use quantum fluctuations to determine his velocity relative to vacuum.
However, fluctuations can be used to determine acceleration. This was shown in 1976 by W. Unruh of the University of British Columbia (Vancouver, Canada). Unruh's result was that a hypothetical particle detector undergoing constant acceleration should respond to vacuum fluctuations as if it were at rest in a gaseous medium (hence not in a vacuum) with a temperature proportional to the acceleration. An unaccelerated detector should not react at all to quantum fluctuations.
The possibility of such a connection between temperature and acceleration has led to a rethinking of the term "vacuum" and an understanding of the fact that there are different types of vacuum. One of the simplest non-standard vacuums can be generated by repetition in a quantum mechanical version of a thought experiment first proposed by Einstein. Imagine a closed elevator car drifting freely in empty space. A certain "playful spirit" begins to "pull" the cabin so that it comes into a state of motion with constant acceleration towards its ceiling. Let us also assume that the walls of the cabin are made of a perfect conductor, impervious to electromagnetic radiation, and that the cabin is completely evacuated so that it does not contain any particles. Einstein came up with this imaginary setting to illustrate the equivalence of gravity and acceleration,however, an analysis of the thought experiment from a modern standpoint shows that some purely quantum effects are to be expected here.
Let's start with the fact that at the moment when acceleration occurs, the floor of the car emits an electromagnetic wave, which propagates to the ceiling and then, reflected, rushes back and forth. (To show why the wave is emitted, a detailed mathematical analysis of the properties of the accelerated electrical conductor is required.) The effect is similar to the creation of an acoustic pressure wave in an air-filled cabin. If we allow for some time the possibility of dissipation of radiation in the walls of the cabin, then the electromagnetic wave will turn into a gas of photons with a thermal energy spectrum, i.e. there will be radiation of an absolutely black body, characteristic of a certain temperature.
The cabin now contains a rarefied gas of photons. To get rid of them, you can use a refrigerator with a radiator outside. This will require a certain amount of energy, which is supplied from an external source. As a result, after all the photons have been pumped out, a new vacuum is formed. The new vacuum is slightly different from the standard vacuum outside the cab. The difference is as follows. First, the Unruh detector, which, together with the elevator car, participates in accelerated motion, would have to react to field fluctuations in a standard vacuum outside; however, he will not find any reaction to the new vacuum inside. Secondly, the two vacuums differ in energy content.
In order to calculate the energy of a vacuum, one must first resolve some fundamental questions of quantum field theory. I noted above that a free field is equivalent to a set of harmonic oscillators. Fluctuations in the ground state create some residual energy near the vacuum field, known as zero point energy. Since an infinite number of field oscillators are concentrated in a unit volume, the vacuum energy density, apparently, should also be infinite.
The infinite value of the vacuum energy density presents a serious problem. However, theorists have succeeded in inventing a number of technical means to eliminate it. These tools are part of a general program called renormalization theory, which provides a recipe for dealing with the various infinities that arise in quantum field theory. Whatever means are used, they should be universal in the sense that they should not be specially created for a specific physical situation, but can be used in all cases. They should also lead to a vanishing energy density for a standard vacuum. The latter requirement is essential for consistency with Einstein's theory, since the standard vacuum is the quantum equivalent of empty flat spacetime. If some energy is concentrated in it,then spacetime will not be flat.
As a rule, different approaches to renormalization theory give the same results for the same problems. This instills faith in their justice. When these approaches are applied to the vacuums inside and outside the elevator car, they will result in zero energy density outside and negative energy density inside the elevator. The surprise is the negative energy of the vacuum. What can be less than nothing? However, with a little thought, the reasonableness of the negative value becomes clear. Thermal photons must be placed inside the cabin so that the Unruh detector responds in the same way as the detector in a standard vacuum outside. The addition of photons will lead to the fact that together with their energy, the total energy inside the cabin becomes equal to zero, i.e. the same as for the vacuum outside.
It should be emphasized that such bizarre effects are quite difficult to detect. Accelerations encountered in everyday life, and even in the case of high-speed mechanisms, are too small for negative vacuum energy to be registered in experiments. However, there is one case where negative vacuum energy was observed, albeit indirectly. We are talking about the effect predicted in 1948 by H. Casimir from the Philips research laboratory in the Netherlands. Two parallel, polished, uncharged metal plates are placed very close together in a vacuum. It was found that they are weakly attracted due to the force, the origin of which is associated with the negative energy density of the vacuum between the plates.
Figure: 6. An accelerated elevator car is a thought experiment that explains the nature of vacuum in quantum mechanics and the effect of acceleration or gravitational field on it. The cabin is assumed to be empty and insulated so that there is initially an absolute vacuum inside and outside. As the car accelerates, its floor emits an electromagnetic wave and the elevator is filled with a rarefied gas of quanta of electromagnetic radiation - photons (left). The refrigerator, connected to some kind of external energy source, "pumps out" the photons (in the middle). When all photons are removed, photon detectors measure the energy of the vacuum inside and outside (right). As the outdoor device accelerates through the vacuum, it responds to quantum mechanical field fluctuations that permeate space even in the absence of particles. The internal detector is at rest in relation to the elevator and does not “feel” fluctuations. It follows from this that the vacuums inside and outside the cabin are not equivalent. If we assume that the energy of the "standard" vacuum outside is zero, then the vacuum inside the cabin must have negative energy. To restore the zero value of the vacuum energy inside the elevator, it is necessary to return the removed photons. The gravitational field can also create a negative energy vacuum.
If space-time is curved, then the vacuum becomes even more complex. Curvature affects the spatial distribution of fluctuations of quantum fields and, like acceleration, is capable of inducing negative vacuum energy. Since the curvature can change from point to point, the vacuum energy can also change, being positive in some places and negative in others.
In any self-consistent theory, energy must be conserved. Suppose that an increase in curvature leads to an increase in the energy density of the vacuum. The very existence of fluctuations of quantum fields means then that energy is needed to bend space-time. Thus, spacetime resists curvature in exactly the same way as in Einstein's theory.
In 1967 AD Sakharov suggested that gravity could be a purely quantum phenomenon arising from the energy of the vacuum. He also suggested that the Newtonian constant G, or, equivalently, the rigidity of space-time can be calculated from the first principles of the theory. This proposal met with a number of difficulties. First, it was required that gravity, as a fundamental field, was replaced by some kind of "gauge field of the great unification" generated by the known elementary particles. In order to still obtain the absolute scale of units, it is necessary to introduce some fundamental mass. Hence, one fundamental constant will simply be replaced by another.
Second, and apparently more important, the calculated dependence of the vacuum energy on curvature, as it turned out, leads to a more complex theory of gravity than Einstein's. The vacuum energy depends on the number and type of the selected elementary fields and the renormalization method: it turned out that the energy can even decrease with increasing curvature. Such feedback would mean that flat spacetime is unstable and should tend to wrinkle, like a plum on drying. In what follows, we will consider the gravitational field as fundamental.
True vacuum is defined as a state of thermal equilibrium at a temperature equal to absolute zero. In quantum gravity, such a vacuum can exist only when the curvature is independent of time. If this is not the case, then particles can spontaneously appear in the vacuum (as a result of which the vacuum, of course, ceases to be a vacuum).
The particle production mechanism can again be explained in terms of the harmonic oscillator model. When the curvature of space-time changes, the physical properties of the field oscillators also change. Suppose that a conventional oscillator is initially in the ground state and is subject to zero fluctuations. If you change one of its characteristics, for example, the value of the mass or the stiffness of the spring, then the zero oscillations must adapt themselves to these changes. After that, there is a finite probability of detecting the oscillator not in the ground, but in an excited state. This phenomenon is analogous to the increase in vibration of a piano string as its tension increases; the effect is known as parametric excitation. In quantum field theory, the analogue of parametric excitation is the production of particles.
Particles generated by changes in curvature over time appear randomly. It is impossible to predict exactly when and where a given particle will be born. However, the statistical distribution of the energy and momentum of the particles can be calculated. Particle production is most abundant where the curvature is greater and where it changes most rapidly. Perhaps the most abundant particle production occurred during the Big Bang, when it could be the main effect that determines the dynamics of the universe in the early stages of its development. And it does not seem at all improbable that the then-born particles are responsible for all the matter existing in the Universe!
Attempts to calculate the production of particles in the Big Bang were first undertaken about 10 years ago independently by the Soviet academician Ya. B. Zeldovich and L. Parker from the University of Wisconsin in Milwaukee. Since then, many scientists have dealt with these issues. While some of the results look promising, none of them are accurate. Moreover, the main question remains unresolved: what is chosen as the initial quantum state at the moment of the Big Bang? Here physicists can take on the role of god. None of the suggestions made so far appear to be perfect.
Another phenomenon in the Universe where curvature can change rapidly is the collapse of a star into a black hole. Here, quantum mechanical calculations, regardless of the initial conditions, led to a real surprise. In 1974, S. Hawking of the University of Cambridge showed that a change in curvature near a collapsing black hole generates a stream of emitted particles. This flow is uniform and continues long after the black hole becomes geometrically stationary. It can continue due to time dilation in a huge gravitational field near the surface of the black hole horizon, when it seems to an external observer that all processes freeze. Particles born near the horizon delay their journey to the outside world.
Although the delay in emission means that there are a huge number of particles "hovering" near the horizon and waiting for their "turn" before departure, the total energy density in this region is still negative and rather small. The positive energy of the particles is mostly compensated by the huge negative energy of the vacuum, which would exist in the absence of these particles (for example, if a black hole has always existed and was never born in gravitational collapse).
It can be shown that the emission of particles is not statistically correlated and that their energy spectrum has a thermal character. Hawking radiation is similar to black body radiation, which, perhaps, is its main property. This allows us to attribute both temperature and entropy to the black hole. Entropy, which is a measure of thermodynamic disorder in the system, turns out to be proportional to the surface area of the horizon. It is huge for a black hole with a mass on the order of the mass of stars: 19 orders of magnitude more than the entropy of the star from which the black hole arose. On the other hand, the temperature is inversely proportional to the mass and in our example should be 11 orders of magnitude less than the temperature of the progenitor star.
Since the amount of radiation emitted by an object depends on its temperature, Hawking radiation from astrophysical black holes is completely negligible. It becomes important only for black "mini-holes" with a mass less than 1010 grams. The only conceivable reason for the formation of mini black holes is the enormous pressure during the Big Bang. It is possible that then there was their multiple birth. In this case, they must make a significant contribution to the entropy of the universe.
The energy of a particle, born as a result of a change in curvature in time, is not drawn from nothing. It is taken from space-time itself. In turn, the particle acts on space-time. Various attempts have been made to calculate this "back reaction" in the case of the Big Bang in order to determine its impact on the dynamics of the early universe. In particular, can the back reaction suppress (compensate) the infinitely high initial density of matter required by the classical theory of Einstein. Infinite density is a barrier to all further research. If it were possible to replace it simply by a huge density, then the question would arise: what happened in the Universe before the Big Bang?
In the 60s, R. Penrose of Oxford University and S. Hawking showed that Einstein's classical theory is incomplete. It predicts the appearance in the past or future of infinite densities and infinite curvatures under a number of physically acceptable conditions. A theory that leads to infinite values of physically observable quantities is not able to predict their behavior beyond these points. Because physicists believe in the knowability of nature, they believe that such a theory should be modified to include a broader class of phenomena. Currently, the conservative view is that the inclusion of quantum effects is the only acceptable means that can save Einstein's theory from some limitations.
Calculations of the inverse effect of the produced particles on the Big Bang process were carried out by methods of numerical simulation on computers. So far they have given uncertain results. One of the difficulties consisted in the problem of choosing (as the initial data for the computer) a reliable value of the total energy density of the generated particles and the quantum vacuum in which they are placed.
The reverse effect is especially important for black holes. Hawking radiation "steals" both temperature and entropy from a black hole. Accordingly, the mass of the black hole decreases. The rate of decrease in mass is initially small, but increases sharply with increasing temperature. Eventually, the rate of change becomes so great that the approximations used to calculate the Hawking radiation are violated. What happens next is unknown. Hawking thinks that his approximation will remain qualitatively correct, so that the black hole will cease to exist in a spectacular outburst, after which a "naked singularity" will remain in the causal structure of space-time.
Any singularity (whether naked or not) means the theory is inconsistent. If Hawking is right, then not only is Einstein's theory incomplete, but also quantum theory. The fact is that any particle born outside the surface of the horizon corresponds to another particle born inside. These two particles are correlated in the sense that the observer could detect "interference of probability" if he were able to communicate with both particles simultaneously. Hawking suggested that internal particles are compressed to infinite density and cease to exist. At this point, the standard probabilistic interpretation of quantum mechanics is violated: probability disappears in a collision with infinity.
An alternative and equally plausible assumption is that the very framework of quantum field theory that is erected around Einstein's theory does not allow probability and information to get lost in the collapse. It is possible that the backlash effect becomes so great that it can prevent infinities from arising. The horizon is more of a mathematical construction than a physical one. It may or may not exist at all as an absolute one-sided barrier. Matter that collapses to form a black hole can eventually be fully accounted for, particle by particle. There is no doubt that there must be huge densities within the black hole and a final burst of Hawking radiation. However, the pressure to which nuclear particles are subjected can turn them into photons and other massless particles that can escape,taking away the little remaining energy and all the quantum correlations. These end products should not carry with them the original entropy of the black hole, since all of it has already been "abducted" by Hawking radiation.
Now I come to the more difficult parts of the quantum theory of gravity. When quantum effects, such as particle creation or vacuum energy, reverse the curvature of spacetime, the curvature itself becomes a quantum object. Self-consistency of the theory requires quantizing the gravitational field. For wavelengths longer than the Planck length, the fluctuations of the quantized gravitational field are small. They can be carefully taken into account by considering them as small perturbations against a classical background. Perturbations can be analyzed as if they were independent fields. They contribute their share both to the energy of the vacuum and to the creation of particles.
At Planck wavelengths and energies, the situation becomes incredibly complicated. The particles associated with a weak gravitational field are called gravitons; they have no mass, and their spin angular momentum is 2ħ. It is unlikely that a single graviton will ever be directly detected. Ordinary matter, even if you take an entire galaxy, is almost completely transparent to gravitons. Only at Planck energies can they noticeably interact with matter. But at such energies, gravitons are capable of generating a Planck curvature in the background geometry. Then the field with which gravitons are associated cannot be considered weak, and under such conditions the very concept of "particle" is poorly defined.
At long wavelengths, the energy carried by the graviton distorts the geometry of the background. At shorter wavelengths, it distorts the waves associated with the graviton itself. This is a consequence of the nonlinearity of Einstein's theory: when two gravitational fields are superimposed, the resulting field is not the sum of its components. All non-trivial field theories are nonlinear. To combat nonlinearities in some of them, it is possible to apply methods of successive approximations, called perturbation theory (this name comes from celestial mechanics). The essence of the method is to refine the initial approximation by building a sequence of progressively decreasing corrections. The application of perturbation theory to quantized fields leads to the appearance of infinities, which can be eliminated by renormalization.
In the case of quantum gravity, perturbation theory does not work, and for two reasons. First, at Planck energies, the successive terms of the perturbation theory series (i.e., successive corrections) are all comparable in magnitude. Breaking off a series at some finite number of terms does not mean getting a good approximation here; instead, the entire infinite series must be summed. Second, individual members of the series cannot be renormalized consistently. In each approximation, new types of infinities appear, which have no analogues in ordinary quantum field theory. They arise because when the gravitational field is quantized, space-time itself is quantized. In conventional quantum field theory, spacetime is a fixed background. In quantum gravity, this background not only influences quantum fluctuations, but also participates in them.
A narrowly technical answer to these difficulties has been some attempts to summarize certain infinite subsets of the perturbation theory series. The results, especially the complete reduction of all infinities, are encouraging and at the same time questionable. These results must be treated with caution, since various approximations were introduced in obtaining them, and the perturbation theory series was never completely summed up. Nevertheless, these results were used to calculate improved estimates of the kickback effect on the Big Bang.
In a more general case, one should expect the appearance of other problems that cannot be solved even by summing the series as a whole. The causal structure of quantized spacetime is undefined and subject to fluctuations. At Planck distances, the very distinction between the past and the future is erased. It can be expected that processes that are forbidden in classical Einstein's theory, including travel to Planck distances at superluminal speed, will become possible. This can be a phenomenon similar to tunneling in atomic systems, when an electron leaks through an energy barrier that it cannot "climb". It is completely unknown how to calculate the probability of such processes in quantum gravity. In many cases, it is not even clear how to correctly pose the questions and which ones. There are no experimentswhich would point us in the right direction. Therefore, you can still afford to indulge in flights of fantasy.
One of the favorite fantasies, which is repeatedly referred to in the literature on quantum gravity, is the fluctuating topology. The basic idea, proposed by Wheeler in 1957, is as follows. Vacuum fluctuations of the gravitational field, as well as fluctuations of all other fields, increase in magnitude at shorter wavelengths. If we extrapolate the results obtained in the weak-field approximation to the region of Planck dimensions, then the curvature fluctuations will become so intense that they can, it seems, "cut" holes in space-time and change its topology. The vacuum, according to Wheeler, is in a state of endless disorder, when "handles" and more complex topological formations are constantly being born and disappeared. The sizes of these formations are of the order of the Planck ones,so that disorder can only be "seen" at the Planck level. At coarser resolution, spacetime appears smooth.
Figure: 7. The quantum vacuum, as it was presented in 1957 by J. Wheeler, becomes more and more chaotic, if we consider it at increasingly small distances in space. At the scale of atomic nuclei (top), space looks very smooth. At distances of about 10-30 cm, some irregularities begin to appear (in the middle). At distances that are about 1000 times smaller, i.e. on the Planck length scale (bottom), the curvature and topology of space fluctuate strongly.
However, an objection can be raised: any topological change is necessarily accompanied by the appearance of a singularity in the causal structure of spacetime, so that such an approach faces the same difficulty that follows from Hawking's views on black hole decay. Suppose, however, that Wheeler's point of view is correct. Here is one of the first questions that should then be asked: what is the contribution of topological fluctuations to the energy of the vacuum and how do they affect the resistance of space-time to curvature (at least in a rough approximation)? Until now, no one has answered this question convincingly, primarily due to the fact that no consistent picture of the topological transition process itself has been built.
To be able to assess what kind of obstacles stand in the way of constructing such a picture, consider the process presented in Fig. 5. On the left and in the middle of the figure, there are two representations of the same event: the “handle” became so thin that two “outgrowths” remained from it in a simply connected space. In one image the space is shown flat, in the other it is curved.
Now let's look at the reverse process: the formation of a "handle". If there is a finite probability that the "pen" becomes thinner and finally simply disappears, then there is a finite probability of its formation. A new difficulty arises here. If we look at our illustration in the reverse direction in time, we see that it depicts two "outgrowths" that spontaneously formed in a quantum vacuum. For one of the views, it seems acceptable to be able to connect the two “outgrowths” into a “handle”. For another, this seems to be incredible. However, the physical situation is the same in both cases. The formation of a "handle" in one of the cases seems quite probable, since the "outgrowths" are close to each other. However, "proximity" is not an intrinsic property of a given location in space, as follows from the two cases considered. The concept of "proximity" requires the existence of a space of higher dimension, in which space-time is embedded. Moreover, the space of the highest dimension must have the appropriate physical properties so that the “outgrowths” can convey to each other a “sense of closeness”. But then spacetime is no longer the universe. The universe is now something more. If we remain true to the notions that the properties of space-time should be its internal characteristics, and not the result of something from outside, then a consistent picture of topological transitions, apparently, cannot be built.so that the "outgrowths" could convey to each other a "sense of closeness." But then spacetime is no longer the universe. The universe is now something more. If we remain true to the notions that the properties of space-time should be its internal characteristics, and not the result of something from outside, then a consistent picture of topological transitions, apparently, cannot be built.so that the "outgrowths" could convey to each other a "sense of closeness." But then spacetime is no longer the universe. The universe is now something more. If we remain true to the notions that the properties of space-time should be its internal characteristics, and not the result of something from outside, then a consistent picture of topological transitions, apparently, cannot be built.
Another difficulty in considering topological fluctuations is that they can violate the macroscopic dimension of space. If "handles" are able to form spontaneously, then they themselves can give rise to other "handles" and so on ad infinitum. Space can unfold into a structure that remains three-dimensional at the Planck level, but has four or more dimensions on a large scale. A familiar example of such a process is the formation of foam, which is built entirely from two-dimensional surfaces, but has a three-dimensional structure (see Figure 8).
Figure: 8. The dimension of space is questionable due to the fact that space-time can have a complex topology. The depicted surface is two-dimensional, but its topological connections are such that it appears to be a three-dimensional object. It is possible that three-dimensional space, when viewed on a microscopic level, actually has fewer dimensions, but is topologically composed of interweaving.
Because of these difficulties, some physicists have suggested that the generally accepted description of spacetime as a smooth continuum ceases to be correct at the Planck level and must be replaced by something else. What constitutes this “other” has never been clear enough. Taking into account the success of the generally accepted description at distances extending more than 40 orders of magnitude (or even 60 orders of magnitude, if we assume that such a description becomes incorrect only at Planck distances), it can be assumed that it is valid at all scales and that topological transitions are simple does not exist. This would be an equally reasonable assumption.
Even if the topology of space does not change, it does not have to be simple, even at a microscopic level. It is possible that from the very beginning the space has a "foamy" structure. In this case, its apparent dimension may differ from the true dimension - be more or less than it.
The latter possibility was proposed in the theory put forward by T. Kaluza in 1921 and O. Klein in 1926. In Kaluza – Klein theory, space is four-dimensional and space-time is five-dimensional. The reason space appears to be three-dimensional is because one of its dimensions is cylindrical, as in the universe discussed above. There is, however, a significant difference from the previous case: the circumference of the universe in the cylindrical direction is now not billions of light years, but several (perhaps 10 or 100) Planck units of length. As a result, the observer who is trying to penetrate the fourth spatial dimension will almost instantly return back to the starting point. In fact, it doesn't even make sense to talk about such an attempt, since the atoms from which the observer is created are much larger than the circumference of a cylinder. The fourth dimension, as such, is simply unobservable.
Nevertheless, it can manifest itself in another way: as light! Kaluza and Klein showed that if the five-dimensional space-time is described using exactly the same mathematical methods that describe the four-dimensional space-time in Einstein's theory, then their theory is equivalent to Maxwell's theory of electromagnetism and Einstein's theory of gravity. The components of the electromagnetic field are implicitly contained in the equation for the curvature of space-time. Thus, Kaluza and Klein invented the first successful unified field theory; in their theory, a geometric explanation of electromagnetic radiation is given.
In a sense, the Kaluza-Klein theory was too successful. Although she combined the theories of Maxwell and Einstein, she did not predict anything new and therefore could not be tested along with other theories. The reason was that Kaluza and Klein imposed restrictions on the way that spacetime is allowed to bend in the extra dimension. If these limitations were removed, the theory should have predicted new effects, but these effects did not appear to correspond to reality. Therefore, this theory was viewed simply as a beautiful curiosity and was shelved for many years.
She was remembered in the 60s. It became clear that new gauge theories, whose popularity was growing, can be reformulated in the style of the Kaluza – Klein theory, when space has not one but several additional microscopic dimensions at once. The impression was that all physics could be explained in terms of geometry. As a result, the question arose: what happens if the restrictions on curvature in closed dimensions are removed.
One possible consequence is the prediction of curvature fluctuations in extra dimensions; these fluctuations appear as massive particles. If the circumference in additional closed dimensions is of the order of 10 Planck units, then the masses of these particles have a value, roughly speaking, of the order of one tenth of the Planck mass, i.e. about 10–6 g. Since the creation of such heavy particles requires enormous energy, they are almost never born. Therefore, for everyday practice, it does not really matter whether restrictions on curvature fluctuations are imposed or not. Problems remain. The main one is that large values of curvature in extra dimensions lead to very high energy density in classical vacuum. Observations exclude large values of the vacuum energy.
Figure: nine. Additional spatial dimensions, in addition to the known three, can exist if they have a "closed" character (compactified). For example, the fourth spatial dimension can be rolled up into a cylinder with a circumference of the order of 10–32 cm. In the figure, a hypothetical "closed" dimension is "not rolled up" and is represented by the vertical axis on the space-time diagram. Therefore, the path of a particle has a cyclic component: every time a particle reaches the maximum coordinate value in a closed dimension, it is again at a point with an initial coordinate in this dimension. The observed path is the projection of the true path onto the spacetime of macroscopic measurements. If the path is geodesic, it may look like the path of a charged particle moving in an electric field. A theory of this type was proposed in the 1920s by T. Kaluza and O. Klein, who showed that it can explain both gravity and electromagnetism. Recently, there has been a resurgence of interest in such theories.
Kaluza-Klein models have never received much attention and their role in physics is still unclear. However, in the past two or three years, they have been scrutinized again, this time in connection with a remarkable generalization of Einstein's theory known as supergravity. Supergravity was invented in 1976 by D. Friedman, P. van Nieuwenhuisen and S. Ferrara and (in an improved version) by S. Deser and B. Zumino.
One of the inconsistencies of the Kaluza – Klein models with reality is that they predict the existence of particles only with integer spin 0, ħ and 2ħ, and even these particles must be either massless or superheavy. There was no room in it for particles of ordinary matter, most of which have a spin angular momentum of ½ħ. It turned out that if Einstein's theory is replaced by supergravity, and space-time is considered similar to the Kaluza – Klein model, then a true unification of all spins is achieved.
In the Kaluza-Klein "supermodel", which is now the most popular, seven extra dimensions are added to the dimension of space-time. These measurements have the topology of a seven-dimensional sphere, i.e. space, which in itself has very interesting properties. The resulting theory is unusually complex and rich in content; it establishes the existence of huge particle multiplets. The masses of these particles are still either zero or extremely large. It is possible that "breaking" the symmetry of the seven-dimensional sphere will lead to the appearance of more realistic mass values for some particles. The great energy of the classical vacuum also survived, but it can be reduced with the negative energy of the quantum vacuum. It remains to be seen whether this strategy of correcting theory will succeed. In reality it will take a lot of workto find out exactly all the consequences of the theory.
If Einstein could see what happened to his theory, he would certainly be surprised and, I suppose, delighted. He would be pleased that, after so many years of doubt, physicists have finally come to his point of view that mathematically beautiful theories deserve study, even if it is not yet clear at the moment whether they have anything to do with physical reality. He would be glad if physicists would dare to hope that unified field theories would be achievable. And he would be especially pleased to find that his old dream - to explain all physics in terms of geometry seems to come true.
But mostly he would be surprised. I am surprised that quantum theory still stands at the basis of everything intact and unshakable, enriching field theory and in turn enriching it. Einstein never believed that quantum theory expresses ultimate truth. He himself did not come to terms with the indeterminism introduced by the theory of quanta, and believed that some day some nonlinear field theory would replace it. The opposite happened. Quantum theory absorbed and changed Einstein's theory.
Translator's Notes:
1.
$ / hbar ~ $ - Dirac constant (Planck constant divided by $ 2 / pi ~ $)
$ / c ~ $ - speed of light
$ / G ~ $ - gravitational constant
$ / k ~ $ - Boltzmann constant
$ / frac 1 {4 / pi / varepsilon_0} ~ $ is the coefficient of proportionality in the Coulomb's law, where $ / varepsilon_0 ~ $ is the electric constant.
All other Planck units are derived from them, for example:
Planck mass $ M_ {Pl} = / sqrt { frac { hbar c} G} cong 2 {,} 17644 (11) times 10 ^ {- 8} ~ $ kilogram;
Planck length $ l_ {Pl} = / frac / hbar {M_ {Pl} c} = / sqrt { frac { hbar G} {c ^ 3}} cong 1 {,} 616252 (81) times 10 ^ {-35} ~ $ meters;
Planck time $ t_ {Pl} = / frac {l_ {Pl}} c = / sqrt { frac { hbar G} {c ^ 5}} cong 5 {,} 39124 (27) times 10 ^ {- 44} ~ $ seconds;
Planck temperature $ T_ {Pl} = / frac {M_ {Pl} c ^ 2} k = / sqrt { frac { hbar c ^ 5} {k ^ 2 G}} cong 1 {,} 416785 (71) times 10 ^ {32} ~ $ Kelvin
Planck charge $ q_ {Pl} = / sqrt {4 / pi / varepsilon_0 / hbar c} = / sqrt {2 ch / varepsilon_0} = / frac {e} { sqrt { alpha}} cong 1 {,} 8755459 / times 10 ^ {- 18} ~ $ Pendant;
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2.
The term "pen" used in Russian scientific literature is borrowed from topology.
By Bryce S. De Witt