As you know, photons, particles of light, of which it consists, move at the speed of light. The special theory of relativity will help us in this matter.

In science fiction films, interstellar spaceships fly almost at the speed of light without exception. Usually this is the so-called hyperspeed by science fiction writers. Both writers and film directors describe and show it to us with almost the same artistic device. Most often, in order for the ship to make a rapid dash, the heroes pull or press the control button, and the vehicle instantly accelerates, accelerating to almost the speed of light with a deafening pop. The stars that the viewer sees over the side of the ship first flicker, and then completely stretch out in lines. But is this what the stars really look like in the windows of a spaceship at hyperspeed? Researchers say no. In reality, instead of the stars stretched out in a line, the passengers of the ship would see only a bright disk.

If the object moves at almost the speed of light, then it can see the Doppler effect in action. In physics, this is the name given to the change in frequency and wavelength due to the rapid movement of the receiver. The frequency of the light of stars flashing in front of the viewer from the ship will increase so much that it will shift from the visible range to the X-ray part of the spectrum. The stars seem to disappear! At the same time, the length of the relic electromagnetic radiation remaining after the Big Bang will decrease. The background radiation will become visible and appear as a bright disk, fading at the edges.

But what does the world look like from the side of an object that reaches the speed of light? As you know, photons, particles of light, of which it consists, move at such speeds. The special theory of relativity will help us in this matter. According to it, when an object moves at the speed of light for an arbitrarily long time, the time spent on the movement of this object becomes equal to zero. In simple terms, if you move at the speed of light, then it is impossible to perform any action, such as observing, seeing, seeing, and so on. An object traveling at the speed of light will actually see nothing.

Photons always travel at the speed of light. They do not waste time accelerating and decelerating, so their whole life for them lasts zero time. If we were photons, then our moments of birth and death would coincide, that is, we would simply not realize that the world exists at all. It is worth noting that if an object accelerates to the speed of light, then its speed in all frames of reference becomes equal to the speed of light. Here is such a photo physics. Applying the special theory of relativity, we can conclude that for an object moving at the speed of light, the entire surrounding world will appear infinitely flattened, and all events occurring in it will take place at one moment in time.

We often talk about maximum speed of light in our universe, and that there is nothing that can move faster than the speed of light in a vacuum. And even more so - we. Approaching the near-light speed, the object acquires mass and energy, which either destroys it or contradicts Einstein's general theory of relativity. Suppose we believe in this and look for workarounds (like or we will figure it out) in order to fly to the nearest star not for 75,000 years, but for a couple of weeks. But since few of us have a higher physical education, it is not clear why they say on the streets that the speed of light is maximum, constant and equal to 300,000 km/s?

There are many simple and intuitive explanations for why this is so, but you can start to hate them. An Internet search will lead you to the concept of "relativistic mass" and that it requires more force to accelerate an object that is already moving at high speed. This is the usual way of interpreting the mathematical apparatus of special relativity, but it misleads many, and especially you, our dear readers. Because many of you (and us too) taste high physics, as if dipping one finger into its salty water before entering for a swim. As a result, it becomes much more complex and less beautiful than it really is.

Let's discuss this issue in terms of a geometric interpretation that is consistent with general relativity. It's less obvious, but a little more complicated than drawing arrows on paper, so many of you will immediately understand the theory behind abstractions like "force" and outright lies like "relativistic mass."

First, let's define what a direction is in order to clearly mark your place. "Down" is the direction. It is defined as the direction in which things fall when you let them go. "Up" is the opposite direction of "down". Pick up a compass and determine additional directions: north, south, west and east. All these directions are defined by serious uncles as "an orthonormal (or orthogonal) basis", but it's better not to think about it now. Let's assume that these six directions are absolute, since they will exist where we will deal with our complex issue.

Now let's add two more directions: to the future and to the past. You cannot easily move in these directions of your own free will, but it should be easy enough for you to imagine them. The future is the direction where tomorrow comes; the past is the direction where yesterday is.

These eight basic directions - up, down, north, south, west, east, past and future - describe the fundamental geometry of the universe. We can call each pair of these directions a "dimension", so we live in a four-dimensional universe. Another term for this 4D understanding would be "space-time", but we will try to avoid using that term. Just remember that in our context "space-time" will be equivalent to the concept of "universe".

Welcome to the stage. Let's look at the actors.

Sitting in front of the computer now, you are on the move. You don't feel it. You feel like you are at rest. But this is only because everything around you also moves relative to you. No, do not think that we are talking about the fact that the Earth is spinning around the Sun or the Sun is moving through the galaxy and pulling us along. This, of course, is true, but we are not talking about that now. By movement, we mean movement in the direction of the "future".

Imagine that you are in a train car with the windows closed. You can't see the street, and let's say the rails are so perfect that you don't know if the train is moving or not. Therefore, just sitting inside the train, you cannot say whether you are actually traveling or not. Look out into the street - and realize that the landscape is rushing past. But the windows are closed.

There is only one way to know if you are moving or not. Just sit and wait. If the train stops at the station, nothing will happen. But if the train is moving, sooner or later you will arrive at a new station.

In this metaphor, the car represents everything that we can see in the world around us - a house, Vaska the cat, stars in the sky, etc. "The next station is Tomorrow."

If you sit motionless, and the cat Vaska peacefully sleeps his hours put in the day, you will not feel movement. But tomorrow will surely come.

This is what it means to move towards the future. Only time will tell which is true: movement or parking.

So far, it should have been pretty easy for you to imagine all this. It may be difficult to think of time as a direction, and even more so of yourself as an object passing through time. But you will understand. Now turn on your imagination.

Imagine that while you are driving in your car, something terrible happens: the brakes fail. By a strange coincidence, at the same moment, the gas and gearbox are jammed. You can neither accelerate nor stop. The only thing you have is a steering wheel. You can change the direction of the movement, but not its speed.

Of course, the first thing you will do is try to drive into a soft bush and somehow gently stop the car. But let's not use this technique for now. Let's just focus on the features of your broken car: you can change direction, but not speed.

This is how we move through the universe. You have a steering wheel but no pedal. Sitting and reading this article, you are rolling into a bright future at maximum speed. And when you get up to make yourself a seagull, you change the direction of movement in space-time, but not its speed. If you move very quickly through space, time will flow a little slower.

This is easy to imagine by drawing a couple of axes on paper. The axis that will go up and down is the axis of time, up means the future. The horizontal axis represents space. We can only draw one dimension of space, since a sheet of paper is two-dimensional, but let's just imagine that this concept applies to all three dimensions of space.

Draw an arrow from the origin of the coordinate axis where they converge and point it up along the vertical axis. It doesn't matter how long it is, just keep in mind that it will only have one length. This arrow, now pointing into the future, is what physicists call "four-velocity." This is the speed of your movement through space-time. Right now you are in a stationary state, so the arrow is directed only to the future.

If you want to move through space - to the right on the coordinate axis - you need to change your four-velocity and turn on the horizontal component. It turns out that you need to rotate the arrow. But once you do that, you'll notice that the arrow isn't as confidently pointing up into the future as it was before. You are now moving through space, but you have to sacrifice future motion as the four-speed needle can only rotate, never expand or contract.

This is where the famous “time slowdown” effect begins, which is talked about by everyone even a little initiated into the special theory of relativity. If you are moving through space, you are not moving through time as fast as you could if you were sitting still. Your clock will keep time slower than the clock of a person who is not moving.

And now we come to the resolution of the question why the phrase "faster than light" does not make sense in our universe. See what happens if you want to move through space as quickly as possible. You turn the four-speed needle all the way until it points along the horizontal axis. We remember that the arrow cannot stretch. She can only rotate. So, you have increased the speed in space as much as possible. But it became impossible to move even faster. The arrow has nowhere to turn, otherwise it will become "straighter than straight" or "more horizontal than horizontal". To this concept and equate "faster than light." It is simply impossible how to feed a huge people with three fish and seven loaves of bread.

This is why nothing in our universe can move faster than light. Because the phrase "faster than light" in our universe is equivalent to the phrase "straighter than straight" or "more horizontal than horizontal."

Yes, you have a few questions. Why can four-velocity vectors only rotate but not expand? There is an answer to this question, but it is related to the invariance of the speed of light, and we will leave it for later. And if you just believe it, you will be a little less informed on this subject than the most brilliant physicists who have ever existed on our planet.

Skeptics may question why we use a simplified model of the geometry of space when talking about Euclidean rotations and circles. In the real world, the space-time geometry obeys the Minkowski geometry, and the rotations are hyperbolic. But a simple version of the explanation has the right to life.

As well as a simple explanation for that, .

Shadows can travel faster than light, but cannot carry matter or information

Is superluminal flight possible?

Sections in this article have subheadings and you can refer to each section separately.

Simple examples of FTL travel

1. Cherenkov effect

When we talk about superluminal motion, we mean the speed of light in a vacuum. c(299 792 458 m/s). Therefore, the Cherenkov effect cannot be considered as an example of superluminal motion.

2. Third observer

If the rocket A flies away from me with speed 0.6c to the west, and the rocket B flies away from me with speed 0.6c east, then I see that the distance between A And B increases with speed 1.2c. Watching the missiles fly A And B from the outside, the third observer sees that the total removal velocity of the missiles is greater than c .

but relative speed is not equal to the sum of the speeds. rocket speed A regarding the rocket B is the rate at which the distance to the rocket increases A, which is seen by an observer flying on a rocket B. Relative velocity must be calculated using the relativistic velocity addition formula. (See How do You Add Velocities in Special Relativity?) In this example, the relative velocity is approximately 0.88c. So in this example we didn't get FTL.

3. Light and shadow

Think about how fast the shadow can move. If the lamp is close, then the shadow of your finger on the far wall moves much faster than the finger moves. When moving the finger parallel to the wall, the speed of the shadow in D/d times greater than the speed of a finger. Here d is the distance from the lamp to the finger, and D- from the lamp to the wall. The speed will be even greater if the wall is at an angle. If the wall is very far away, then the movement of the shadow will lag behind the movement of the finger, since the light takes time to reach the wall, but the speed of the shadow moving along the wall will increase even more. The speed of a shadow is not limited by the speed of light.

Another object that can travel faster than light is a spot of light from a laser aimed at the moon. The distance to the Moon is 385,000 km. You can calculate the speed of movement of the light spot on the surface of the Moon by yourself with small fluctuations of the laser pointer in your hand. You might also like the example of a wave hitting a straight line of beach at a slight angle. With what speed can the point of intersection of the wave and the shore move along the beach?

All these things can happen in nature. For example, a beam of light from a pulsar can run along a dust cloud. A powerful explosion can create spherical waves of light or radiation. When these waves intersect with a surface, circles of light appear on that surface and expand faster than light. Such a phenomenon is observed, for example, when an electromagnetic pulse from a lightning flash passes through the upper atmosphere.

4. Solid body

If you have a long, rigid rod and you hit one end of the rod, doesn't the other end immediately move? Is this not a way of superluminal transmission of information?

That would be right if there were perfectly rigid bodies. In practice, the impact is transmitted along the rod at the speed of sound, which depends on the elasticity and density of the rod material. In addition, the theory of relativity limits the possible speeds of sound in a material by the value c .

The same principle applies if you hold a string or rod vertically, release it, and it begins to fall under the influence of gravity. The top end you let go starts to fall immediately, but the bottom end will only start moving after a while, as the loss of the holding force is transmitted down the rod at the speed of sound in the material.

The formulation of the relativistic theory of elasticity is rather complicated, but the general idea can be illustrated using Newtonian mechanics. The equation of longitudinal motion of an ideally elastic body can be derived from Hooke's law. Denote the linear density of the rod ρ , Young's modulus Y. Longitudinal offset X satisfies the wave equation

ρ d 2 X/dt 2 - Y d 2 X/dx 2 = 0

Plane wave solution travels at the speed of sound s, which is determined from the formula s 2 = Y/ρ. The wave equation does not allow the perturbations of the medium to move faster than with the speed s. In addition, the theory of relativity gives a limit to the amount of elasticity: Y< ρc 2 . In practice, no known material approaches this limit. Note also that even if the speed of sound is close to c, then the matter itself does not necessarily move with relativistic speed.

Although there are no solid bodies in nature, there is motion of rigid bodies, which can be used to overcome the speed of light. This topic belongs to the already described section of shadows and light spots. (See The Superluminal Scissors, The Rigid Rotating Disk in Relativity).

5. Phase velocity

wave equation
d 2 u/dt 2 - c 2 d 2 u/dx 2 + w 2 u = 0

has a solution in the form
u \u003d A cos (ax - bt), c 2 a 2 - b 2 + w 2 \u003d 0

These are sinusoidal waves propagating at a speed v
v = b/a = sqrt(c 2 + w 2 /a 2)

But it's more than c. Maybe this is the equation for tachyons? (see section below). No, this is the usual relativistic equation for a particle with mass.

To eliminate the paradox, you need to distinguish between "phase velocity" v ph , and "group velocity" v gr , and
v ph v gr = c 2

The solution in the form of a wave may have dispersion in frequency. In this case, the wave packet moves with a group velocity that is less than c. Using a wave packet, information can only be transmitted at the group velocity. Waves in a wave packet move with phase velocity. Phase velocity is another example of FTL motion that cannot be used to communicate.

6. Superluminal galaxies

7. Relativistic rocket

Let an observer on Earth see a spacecraft moving away at a speed 0.8c According to the theory of relativity, he will see that the clock on the spacecraft is running 5/3 times slower. If we divide the distance to the ship by the time of flight according to the onboard clock, we get the speed 4/3c. The observer concludes that, using his on-board clock, the pilot of the ship will also determine that he is flying at a superluminal speed. From the pilot's point of view, his clock is running normally, and interstellar space has shrunk by a factor of 5/3. Therefore, it flies the known distances between the stars faster, at a speed 4/3c .

Time dilation is a real effect that could in principle be used in space travel to cover great distances in a short amount of time from the point of view of astronauts. At a constant acceleration of 1g, astronauts will not only have a comfortable artificial gravity, but will also be able to traverse the galaxy in just 12 years of proper time. During the journey, they will age by 12 years.

But it's still not superluminal flight. You can't calculate speed using distance and time defined in different frames of reference.

8. Gravity speed

Some insist that the speed of gravity is much faster c or even infinite. See Does Gravity Travel at the Speed ​​of Light? and What is Gravitational Radiation? Gravitational perturbations and gravitational waves propagate at a speed c .

9. EPR paradox

10. Virtual photons

11. Quantum tunnel effect

In quantum mechanics, the tunnel effect allows a particle to overcome a barrier, even if its energy is not enough for this. It is possible to calculate the tunneling time through such a barrier. And it may turn out to be less than what is required for light to overcome the same distance at a speed c. Can it be used to send messages faster than light?

Quantum electrodynamics says "No!" Nevertheless, an experiment was carried out that demonstrated the superluminal transmission of information using the tunnel effect. Through a barrier 11.4 cm wide at a speed of 4.7 c Mozart's Fortieth Symphony was presented. The explanation for this experiment is very controversial. Most physicists believe that with the help of the tunnel effect it is impossible to transmit information faster than light. If it were possible, then why not send a signal to the past by placing the equipment in a rapidly moving frame of reference.

17. Quantum field theory

With the exception of gravity, all observed physical phenomena correspond to the "Standard Model". The Standard Model is a relativistic quantum field theory that explains the electromagnetic and nuclear forces and all known particles. In this theory, any pair of operators corresponding to physical observables separated by a spacelike interval of events "commutes" (that is, one can change the order of these operators). In principle, this implies that in the Standard Model the force cannot travel faster than light, and this can be considered the quantum field equivalent of the infinite energy argument.

However, there are no impeccably rigorous proofs in the quantum field theory of the Standard Model. No one has yet even proven that this theory is internally consistent. Most likely, it is not. In any case, there is no guarantee that there are no yet undiscovered particles or forces that do not obey the ban on superluminal movement. There is also no generalization of this theory, including gravity and general relativity. Many physicists working in the field of quantum gravity doubt that the simple concepts of causality and locality will be generalized. There is no guarantee that in a future more complete theory the speed of light will retain the meaning of the limiting speed.

18. Grandpa Paradox

In special relativity, a particle traveling faster than light in one frame of reference moves back in time in another frame of reference. FTL travel or information transmission would make it possible to travel or send a message to the past. If such time travel were possible, then you could go back in time and change the course of history by killing your grandfather.

This is a very strong argument against the possibility of FTL travel. True, there remains an almost improbable possibility that some limited superluminal travel is possible, which does not allow a return to the past. Or maybe time travel is possible, but causality is violated in some consistent way. All this is very implausible, but if we are discussing FTL, it is better to be ready for new ideas.

The reverse is also true. If we could travel back in time, we could overcome the speed of light. It is possible to go back in time, fly somewhere at low speed, and arrive there before the light sent in the usual way arrives. See Time Travel for details on this topic.

Open questions of FTL travel

In this last section, I will describe some serious ideas about possible faster-than-light travel. These topics are not often included in the FAQ, because they are more like a lot of new questions than answers. They are included here to show that serious research is being done in this direction. Only a short introduction to the topic is given. Details can be found on the Internet. As with everything on the Internet, be critical of them.

19. Tachyons

Tachyons are hypothetical particles that travel faster than light locally. To do this, they must have an imaginary mass value. In this case, the energy and momentum of the tachyon are real quantities. There is no reason to believe that superluminal particles cannot be detected. Shadows and highlights can travel faster than light and can be detected.

So far, tachyons have not been found, and physicists doubt their existence. There were claims that in experiments to measure the mass of neutrinos produced by the beta decay of tritium, neutrinos were tachyons. This is doubtful, but has not yet been definitively refuted.

There are problems in the theory of tachyons. In addition to possibly violating causality, tachyons also make the vacuum unstable. It may be possible to circumvent these difficulties, but even then we will not be able to use tachyons for superluminal transmission of messages.

Most physicists believe that the appearance of tachyons in a theory is a sign of some problems with this theory. The idea of ​​tachyons is so popular with the public simply because they are often mentioned in fantasy literature. See Tachyons.

20. Wormholes

The most famous method of global FTL travel is the use of "wormholes". A wormhole is a slit in space-time from one point in the universe to another, which allows you to get from one end of the hole to the other faster than the usual path. Wormholes are described by the general theory of relativity. To create them, you need to change the topology of space-time. Maybe this will become possible within the framework of the quantum theory of gravity.

To keep a wormhole open, you need areas of space with negative energies. C.W.Misner and K.S.Thorne proposed to use the Casimir effect on a large scale to create negative energy. Visser suggested using cosmic strings for this. These are very speculative ideas and may not be possible. Maybe the required form of exotic matter with negative energy does not exist.

But it turned out that it is possible; now it is believed that we will never be able to travel faster than light ... ". But in fact it is not true that someone once believed that it was impossible to travel faster than sound. Long before supersonic aircraft appeared, it was already known that bullets fly faster than sound. managed supersonic flight, and that was the mistake. SS movement is a completely different matter. It was clear from the start that supersonic flight was hampered by technical problems that simply had to be solved. But it is completely unclear whether the problems that hinder the SS movement can ever be solved. The theory of relativity has a lot to say about this. If SS travel or even signal transmission is possible, then causality will be violated, and absolutely incredible conclusions will follow from this.

We will first discuss simple cases of CC motion. We mention them not because they are interesting, but because they resurface again and again in discussions of the STS movement and therefore have to be dealt with. Then we will discuss what we consider to be difficult cases of STS movement or communication and consider some of the arguments against them. Finally, we will consider the most serious assumptions about the real STS movement.

Simple SS move

1. The phenomenon of Cherenkov radiation

One way to move faster than light is to first slow down the light itself! :-) In a vacuum, light travels at a speed c, and this value is a world constant (see the question Is the speed of light constant), and in a denser medium like water or glass, it slows down to the speed c/n, where n is the refractive index of the medium (1.0003 for air; 1.4 for water). Therefore, particles can move faster in water or air than light travels there. As a result, Vavilov-Cherenkov radiation appears (see question ).

But when we talk about SS motion, we, of course, mean exceeding the speed of light in a vacuum c(299 792 458 m/s). Therefore, the Cherenkov phenomenon cannot be considered an example of SS motion.

2.Third party

If the rocket BUT flies away from me at a speed 0.6c west and the other B- from me with speed 0.6c east, then the total distance between BUT And B in my frame of reference increases with speed 1.2c. Thus, an apparent relative velocity greater than c can be observed "from a third party".

However, this speed is not what we usually understand by relative speed. Real rocket speed BUT regarding the rocket B- this is the rate of increase in the distance between the rockets, which is observed by the observer in the rocket B. Two velocities must be added according to the relativistic formula for adding velocities (see the question How to add velocities in particular relativity). In this case, the relative speed is approximately 0.88c, that is, is not superluminal.

3. Shadows and bunnies

Think about how fast the shadow can move? If you create a shadow on a distant wall from your finger from a nearby lamp, and then move your finger, then the shadow moves much faster than your finger. If the finger moves parallel to the wall, then the speed of the shadow will be D/d times the speed of the finger, where d is the distance from the finger to the lamp, and D- distance from the lamp to the wall. And you can get even more speed if the wall is located at an angle. If the wall is very far away, then the movement of the shadow will lag behind the movement of the finger, since the light will still have to fly from the finger to the wall, but still the speed of the shadow will be as many times greater. That is, the speed of the shadow is not limited by the speed of light.

In addition to shadows, bunnies can also move faster than light, for example, a speck from a laser beam directed at the moon. Knowing that the distance to the Moon is 385,000 km, try to calculate the speed of the bunny if you move the laser slightly. You can also think of a sea wave hitting the shore obliquely. With what speed can the point at which the wave breaks move?

Similar things can happen in nature. For example, a light beam from a pulsar can comb through a cloud of dust. A bright flash generates an expanding shell of light or other radiation. When it crosses the surface, it creates a ring of light that grows faster than the speed of light. In nature, this occurs when an electromagnetic pulse from lightning reaches the upper atmosphere.

All these were examples of things moving faster than light, but which were not physical bodies. With the help of a shadow or a bunny, you cannot transmit a CC message, so communication faster than light is not possible. And again, this is apparently not what we want to understand by CC motion, although it becomes clear how difficult it is to determine what exactly we need (see the question FTL Shears).

4. Rigid bodies

If you take a long hard stick and push one end of it, does the other end move immediately or not? Is it possible to carry out the SS transmission of the message in this way?

Yes it was would could be done if such solid bodies existed. In reality, the influence of a blow to the end of a stick propagates along it at the speed of sound in a given substance, and the speed of sound depends on the elasticity and density of the material. Relativity imposes an absolute limit on the possible hardness of any bodies so that the speed of sound in them cannot exceed c.

The same thing happens if you are in the field of attraction, and first hold the string or pole vertically by the upper end, and then release it. The point that you let go will start moving immediately, and the lower end will not be able to start falling until the influence of letting go reaches it at the speed of sound.

It is difficult to formulate a general theory of elastic materials in terms of relativity, but the basic idea can also be shown using the example of Newtonian mechanics. The equation for the longitudinal motion of a perfectly elastic body can be obtained from Hooke's law. In variables mass per unit length p and Young's modulus Y, longitudinal displacement X satisfies the wave equation.

Plane wave solution moves at the speed of sound s, and s 2 = Y/p. This equation does not imply the possibility of a causal influence propagating faster s. Thus, relativity imposes a theoretical limit on the amount of elasticity: Y < pc2. Practically, there are no materials even close to it. By the way, even if the speed of sound in the material is close to c, matter in itself is not required to move with relativistic velocity. But how do we know that, in principle, there can be no substance that overcomes this limit? The answer is that all substances are made up of particles, the interaction between which obeys the standard model of elementary particles, and in this model no interaction can propagate faster than light (see below about quantum field theory).

5. Phase velocity

Look at this wave equation:

It has solutions like:

These solutions are sine waves moving at a speed

But this is faster than light, so we have the equation of the tachyon field in our hands? No, this is just the usual relativistic equation of a massive scalar particle!

The paradox will be resolved if we understand the difference between this speed, also called the phase speed vph from another speed, called the group speed vgr which is given by the formula,

If the wave solution has a frequency spread, then it will take the form of a wave packet , which moves with a group velocity not exceeding c. Only wave crests move with phase velocity. It is possible to transmit information using such a wave only with a group velocity, so the phase velocity gives us another example of superluminal speed, which cannot carry information.

7. Relativistic rocket

A controller on Earth watches a spacecraft leaving at a speed of 0.8 c. According to the theory of relativity, even after taking into account the Doppler shift of the signals from the ship, he will see that the time on the ship is slowed down and the clocks there go slower by a factor of 0.6. If he calculates the quotient of the distance traveled by the ship divided by the elapsed time measured by the ship's clock, he will get 4/3 c. This means that the ship's passengers travel through interstellar space at an effective speed greater than the speed of light they would have if measured. From the perspective of the ship's passengers, interstellar distances are subject to Lorentzian contraction by the same factor of 0.6, which means they too must admit that they cover known interstellar distances at a rate of 4/3 c.

This is a real phenomenon and in principle it can be used by space travelers to overcome huge distances during their lifetime. If they accelerate at a constant acceleration equal to the acceleration of free fall on Earth, then not only will they have perfect artificial gravity on the ship, but they will still have time to cross the Galaxy in just 12 of their years! (See the question What are the equations of a relativistic rocket?)

However, this is not a real SS movement. The effective speed is calculated from distance in one frame of reference and time in another. This is not real speed. Only the ship's passengers benefit from this speed. The dispatcher, for example, will not have time in his life to see how they fly a gigantic distance.

Difficult cases of SS movement

9. Paradox of Einstein, Podolsky, Rosen (EPR)

10. Virtual photons

11. Quantum tunneling

Real Candidates for the SS Travelers

This section contains speculative but serious assumptions about the possibility of FTL travel. These will not be the kind of things that are usually put in a FAQ, as they raise more questions than they answer. They are presented here mainly to show that serious research is being carried out in this direction. Only a brief introduction is given in each direction. More detailed information can be found on the Internet.

19. Tachyons

Tachyons are hypothetical particles that locally travel faster than light. To do this, they must have an imaginary mass, but their energy and momentum must be positive. It is sometimes thought that such CC particles should be impossible to detect, but in fact, there is no reason to believe so. Shadows and bunnies tell us that stealth does not follow from the CC of the movement.

Tachyons have never been observed and most physicists doubt their existence. It was once stated that experiments were carried out to measure the mass of neutrinos emitted during the decay of Tritium, and that these neutrinos were tachyon. This is highly doubtful, but still not excluded. There are problems with tachyon theories, because in terms of possible violations of causality, they destabilize the vacuum. It may be possible to get around these problems, but then it will be impossible to use tachyons in the SS message we need.

The truth is that most physicists consider tachyons to be a sign of an error in their field theories, and interest in them from the general public is fueled mainly by science fiction (see Tachyons article).

20. Wormholes

The most well-known supposed possibility of STS travel is the use of wormholes. Wormholes are tunnels in space-time that connect one place in the universe to another. They can move between these points faster than light would take its usual path. Wormholes are a phenomenon of classical general relativity, but in order to create them, you need to change the topology of space-time. The possibility of this may be contained in the theory of quantum gravity.

Huge amounts of negative energy are needed to keep wormholes open. Misner And Thorn suggested that the large-scale Casimir effect can be used to generate negative energy and Visser proposed a solution using cosmic strings. All these ideas are highly speculative and may simply be unrealistic. An unusual substance with negative energy may not exist in the form necessary for the phenomenon.

Thorne found that if wormholes could be created, they could create closed time loops that would make time travel possible. It has also been suggested that the multivariate interpretation of quantum mechanics suggests that time travel will not cause any paradoxes, and that events will simply unfold differently when you get into the past. Hawking says that wormholes may simply be unstable and therefore unusable in practice. But the topic itself remains a fruitful area for thought experiments, allowing you to figure out what is possible and what is not possible based on both known and assumed laws of physics.
refs:
W. G. Morris and K. S. Thorne, American Journal of Physics 56 , 395-412 (1988)
W. G. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. letters 61 , 1446-9 (1988)
Matt Visser, Physical Review D39, 3182-4 (1989)
see also "Black Holes and Time Warps" Kip Thorn, Norton & co. (1994)
For an explanation of the multiverse see, "The Fabric of Reality" David Deutsch, Penguin Press.

21. Deformer motors

[I have no idea how to translate this! The original warp drive. - approx. translator
translated by analogy with the article on Membrane]

The warp could be a mechanism for twisting space-time so that an object can travel faster than light. Miguel Alcabière became famous for having developed the geometry that describes such a deformer. Space-time distortion makes it possible for an object to travel faster than light while remaining on a time-like curve. The obstacles are the same as when creating wormholes. To create a deformer, you need a substance with a negative energy density u. Even if such a substance is possible, it is still not clear how it can be obtained and how to use it to make the deformer work.
ref M. Alcubierre, Classical and Quantum Gravity, 11 , L73-L77, (1994)

Conclusion

First, it was not easy to define in general what an SS travel and SS message means. Many things, like shadows, make CC move, but in such a way that it cannot be used, for example, to transmit information. But there are also serious possibilities of real SS movement, which are proposed in the scientific literature, but their implementation is still technically impossible. The Heisenberg uncertainty principle makes it impossible to use apparent CC motion in quantum mechanics. In general relativity there are potential means of SS propulsion, but it may not be possible to use them. It seems extremely unlikely that in the foreseeable future, or at all, technology will be able to create spacecraft with SS engines, but it is curious that theoretical physics, as we now know it, does not close the door to SS propulsion for good. SS movement in the style of science fiction novels is apparently completely impossible. For physicists, the question is interesting: "why, in fact, is this impossible, and what can be learned from this?"

Doctor of Technical Sciences A. GOLUBEV.

In the middle of last year, a sensational report appeared in the magazines. A group of American researchers have discovered that a very short laser pulse travels hundreds of times faster in a specially selected medium than in a vacuum. This phenomenon seemed absolutely incredible (the speed of light in a medium is always less than in a vacuum) and even gave rise to doubts about the validity of the special theory of relativity. Meanwhile, a superluminal physical object - a laser pulse in an amplifying medium - was first discovered not in 2000, but 35 years earlier, in 1965, and the possibility of superluminal motion was widely discussed until the early 70s. Today, the discussion around this strange phenomenon has flared up with renewed vigor.

Examples of "superluminal" motion.

In the early 1960s, high-power short light pulses began to be obtained by passing a laser flash through a quantum amplifier (a medium with an inverse population).

In an amplifying medium, the initial region of a light pulse causes stimulated emission of atoms in the amplifier medium, and its final region causes energy absorption by them. As a result, it will appear to the observer that the pulse is moving faster than light.

Lijun Wong experiment.

A beam of light passing through a prism of a transparent material (such as glass) is refracted, that is, it experiences dispersion.

A light pulse is a set of oscillations of different frequencies.

Probably everyone - even people far from physics - knows that the maximum possible speed of movement of material objects or the propagation of any signals is the speed of light in vacuum. It is marked with the letter from and is almost 300 thousand kilometers per second; exact value from= 299 792 458 m/s. The speed of light in vacuum is one of the fundamental physical constants. The impossibility of achieving speeds exceeding from, follows from the special theory of relativity (SRT) of Einstein. If it were possible to prove that the transmission of signals with superluminal speed is possible, the theory of relativity would fall. So far, this has not happened, despite numerous attempts to refute the ban on the existence of speeds greater than from. However, recent experimental studies have revealed some very interesting phenomena, indicating that under specially created conditions it is possible to observe superluminal velocities without violating the principles of the theory of relativity.

To begin with, let us recall the main aspects related to the problem of the speed of light. First of all: why is it impossible (under normal conditions) to exceed the light limit? Because then the fundamental law of our world is violated - the law of causality, according to which the effect cannot outstrip the cause. No one has ever observed that, for example, a bear first fell dead, and then a hunter shot. At speeds exceeding from, the sequence of events becomes reversed, the time tape rewinds. This can be easily seen from the following simple reasoning.

Let's assume that we are on a certain cosmic miracle ship moving faster than light. Then we would gradually catch up with the light emitted by the source at earlier and earlier points in time. First, we would catch up with photons emitted, say, yesterday, then - emitted the day before yesterday, then - a week, a month, a year ago, and so on. If the light source were a mirror reflecting life, then we would first see the events of yesterday, then the day before yesterday, and so on. We could see, say, an old man who gradually turns into a middle-aged man, then into a young man, into a youth, into a child ... That is, time would turn back, we would move from the present to the past. Cause and effect would then be reversed.

Although this argument completely ignores the technical details of the process of observing light, from a fundamental point of view, it clearly demonstrates that the movement at a superluminal speed leads to a situation that is impossible in our world. However, nature has set even more stringent conditions: movement is unattainable not only at superluminal speed, but also at a speed equal to the speed of light - you can only approach it. It follows from the theory of relativity that with an increase in the speed of movement, three circumstances arise: the mass of a moving object increases, its size decreases in the direction of movement, and the passage of time on this object slows down (from the point of view of an external "resting" observer). At ordinary speeds, these changes are negligible, but as we approach the speed of light, they become more and more noticeable, and in the limit - at a speed equal to from, - the mass becomes infinitely large, the object completely loses its size in the direction of motion and time stops on it. Therefore, no material body can reach the speed of light. Only light itself has such a speed! (And also the "all-penetrating" particle - the neutrino, which, like the photon, cannot move at a speed less than from.)

Now about the signal transmission speed. Here it is appropriate to use the representation of light in the form of electromagnetic waves. What is a signal? This is some information to be transmitted. An ideal electromagnetic wave is an infinite sinusoid of strictly one frequency, and it cannot carry any information, because each period of such a sinusoid exactly repeats the previous one. The speed at which the phase of the sine wave moves - the so-called phase speed - can exceed the speed of light in a vacuum under certain conditions. There are no restrictions here, since the phase speed is not the speed of the signal - it does not exist yet. To create a signal, you need to make some kind of "mark" on the wave. Such a mark can be, for example, a change in any of the wave parameters - amplitude, frequency or initial phase. But as soon as the mark is made, the wave loses its sinusoidality. It becomes modulated, consisting of a set of simple sinusoidal waves with different amplitudes, frequencies and initial phases - a group of waves. The speed of movement of the mark in the modulated wave is the speed of the signal. When propagating in a medium, this velocity usually coincides with the group velocity characterizing the propagation of the above group of waves as a whole (see "Science and Life" No. 2, 2000). Under normal conditions, the group velocity, and hence the speed of the signal, is less than the speed of light in vacuum. It is no coincidence that the expression "under normal conditions" is used here, because in some cases the group velocity can also exceed from or even lose meaning, but then it does not apply to signal propagation. It is established in the SRT that it is impossible to transmit a signal at a speed greater than from.

Why is it so? Because the obstacle to the transmission of any signal at a speed greater than from the same law of causality applies. Let's imagine such a situation. At some point A, a light flash (event 1) turns on a device that sends a certain radio signal, and at a remote point B, under the action of this radio signal, an explosion occurs (event 2). It is clear that event 1 (flash) is the cause, and event 2 (explosion) is the effect that occurs later than the cause. But if the radio signal propagated at a superluminal speed, an observer near point B would first see an explosion, and only then - that reached him with a speed from flash of light, the cause of the explosion. In other words, for this observer, event 2 would have happened before event 1, that is, the effect would have preceded the cause.

It is appropriate to emphasize that the "superluminal prohibition" of the theory of relativity is imposed only on the movement of material bodies and the transmission of signals. In many situations it is possible to move at any speed, but it will be the movement of non-material objects and signals. For example, imagine two rather long rulers lying in the same plane, one of which is located horizontally, and the other intersects it at a small angle. If the first line is moved down (in the direction indicated by the arrow) at high speed, the intersection point of the lines can be made to run arbitrarily fast, but this point is not a material body. Another example: if you take a flashlight (or, say, a laser that gives a narrow beam) and quickly describe an arc in the air, then the linear speed of the light spot will increase with distance and, at a sufficiently large distance, will exceed from. The spot of light will move between points A and B at superluminal speed, but this will not be a signal transmission from A to B, since such a spot of light does not carry any information about point A.

It would seem that the question of superluminal speeds has been resolved. But in the 60s of the twentieth century, theoretical physicists put forward the hypothesis of the existence of superluminal particles, called tachyons. These are very strange particles: they are theoretically possible, but in order to avoid contradictions with the theory of relativity, they had to be assigned an imaginary rest mass. Physically imaginary mass does not exist, it is a purely mathematical abstraction. However, this did not cause much concern, since tachyons cannot be at rest - they exist (if they exist!) only at speeds exceeding the speed of light in vacuum, and in this case the mass of the tachyon turns out to be real. There is some analogy with photons here: a photon has zero rest mass, but that simply means that the photon cannot be at rest - light cannot be stopped.

The most difficult thing was, as expected, to reconcile the tachyon hypothesis with the law of causality. Attempts made in this direction, although they were quite ingenious, did not lead to obvious success. No one has been able to experimentally register tachyons either. As a result, interest in tachyons as superluminal elementary particles gradually faded away.

However, in the 60s, a phenomenon was experimentally discovered, which at first led physicists into confusion. This is described in detail in the article by A. N. Oraevsky "Superluminal waves in amplifying media" (UFN No. 12, 1998). Here we briefly summarize the essence of the matter, referring the reader interested in the details to the said article.

Shortly after the discovery of lasers, in the early 1960s, the problem arose of obtaining short (with a duration of the order of 1 ns = 10 -9 s) high-power light pulses. To do this, a short laser pulse was passed through an optical quantum amplifier. The pulse was split by a beam-splitting mirror into two parts. One of them, more powerful, was sent to the amplifier, and the other propagated in the air and served as a reference pulse, with which it was possible to compare the pulse that passed through the amplifier. Both pulses were fed to photodetectors, and their output signals could be visually observed on the oscilloscope screen. It was expected that the light pulse passing through the amplifier would experience some delay in it compared to the reference pulse, that is, the speed of light propagation in the amplifier would be less than in air. What was the amazement of the researchers when they discovered that the pulse propagated through the amplifier at a speed not only greater than in air, but also several times greater than the speed of light in vacuum!

After recovering from the first shock, physicists began to look for the reason for such an unexpected result. No one had even the slightest doubt about the principles of the special theory of relativity, and this is precisely what helped to find the correct explanation: if the principles of SRT are preserved, then the answer should be sought in the properties of the amplifying medium.

Without going into details here, we only point out that a detailed analysis of the mechanism of action of the amplifying medium has completely clarified the situation. The point was a change in the concentration of photons during the propagation of the pulse - a change due to a change in the gain of the medium up to a negative value during the passage of the rear part of the pulse, when the medium is already absorbing energy, because its own reserve has already been used up due to its transfer to the light pulse. Absorption does not cause an increase, but a decrease in the impulse, and thus the impulse is strengthened in the front and weakened in the back of it. Let us imagine that we observe the pulse with the help of an instrument moving at the speed of light in the medium of an amplifier. If the medium were transparent, we would see an impulse frozen in immobility. In the medium in which the process mentioned above takes place, the strengthening of the leading edge and the weakening of the trailing edge of the pulse will appear to the observer in such a way that the medium, as it were, has moved the pulse forward. But since the device (observer) moves at the speed of light, and the impulse overtakes it, then the speed of the impulse exceeds the speed of light! It is this effect that was registered by the experimenters. And here there really is no contradiction with the theory of relativity: it's just that the amplification process is such that the concentration of photons that came out earlier turns out to be greater than those that came out later. It is not photons that move with superluminal speed, but the envelope of the pulse, in particular its maximum, which is observed on the oscilloscope.

Thus, while in ordinary media there is always a weakening of light and a decrease in its speed, determined by the refractive index, in active laser media, not only amplification of light is observed, but also propagation of a pulse with superluminal speed.

Some physicists have tried to experimentally prove the presence of superluminal motion in the tunnel effect, one of the most amazing phenomena in quantum mechanics. This effect consists in the fact that a microparticle (more precisely, a microobject that exhibits both the properties of a particle and the properties of a wave under different conditions) is able to penetrate the so-called potential barrier - a phenomenon that is completely impossible in classical mechanics (in which such a situation would be analogous : a ball thrown at a wall would end up on the other side of the wall, or the undulating motion imparted to a rope tied to the wall would be transmitted to a rope tied to the wall on the other side). The essence of the tunnel effect in quantum mechanics is as follows. If a micro-object with a certain energy encounters on its way an area with a potential energy exceeding the energy of the micro-object, this area is a barrier for it, the height of which is determined by the energy difference. But the micro-object "leaks" through the barrier! This possibility is given to him by the well-known Heisenberg uncertainty relation, written for the energy and interaction time. If the interaction of the microobject with the barrier occurs for a sufficiently definite time, then the energy of the microobject, on the contrary, will be characterized by uncertainty, and if this uncertainty is of the order of the barrier height, then the latter ceases to be an insurmountable obstacle for the microobject. It is the rate of penetration through the potential barrier that has become the subject of research by a number of physicists who believe that it can exceed from.

In June 1998, an international symposium on the problems of superluminal motions was held in Cologne, where the results obtained in four laboratories - in Berkeley, Vienna, Cologne and Florence were discussed.

And finally, in 2000, two new experiments were reported in which the effects of superluminal propagation appeared. One of them was carried out by Lijun Wong and co-workers at a research institute in Princeton (USA). His result is that a light pulse entering a chamber filled with cesium vapor increases its speed by a factor of 300. It turned out that the main part of the pulse leaves the far wall of the chamber even before the pulse enters the chamber through the front wall. Such a situation contradicts not only common sense, but, in essence, the theory of relativity as well.

L. Wong's report provoked intense discussion among physicists, most of whom are not inclined to see in the results obtained a violation of the principles of relativity. The challenge, they believe, is to correctly explain this experiment.

In the experiment of L. Wong, the light pulse entering the chamber with cesium vapor had a duration of about 3 μs. Cesium atoms can be in sixteen possible quantum mechanical states, called "ground state hyperfine magnetic sublevels". With the help of optical laser pumping, almost all atoms were brought to only one of these sixteen states, corresponding to almost absolute zero temperature on the Kelvin scale (-273.15 o C). The length of the cesium chamber was 6 centimeters. In a vacuum, light travels 6 centimeters in 0.2 ns. As the measurements showed, the light pulse passed through the chamber with cesium in a time 62 ns shorter than in vacuum. In other words, the transit time of a pulse through a cesium medium has a "minus" sign! Indeed, if we subtract 62 ns from 0.2 ns, we get a "negative" time. This "negative delay" in the medium - an incomprehensible time jump - is equal to the time during which the pulse would make 310 passes through the chamber in vacuum. The consequence of this "time reversal" was that the impulse leaving the chamber managed to move away from it by 19 meters before the incoming impulse reached the near wall of the chamber. How can such an incredible situation be explained (unless, of course, there is no doubt about the purity of the experiment)?

Judging by the discussion that has unfolded, an exact explanation has not yet been found, but there is no doubt that the unusual dispersion properties of the medium play a role here: cesium vapor, consisting of atoms excited by laser light, is a medium with anomalous dispersion. Let us briefly recall what it is.

The dispersion of a substance is the dependence of the phase (ordinary) refractive index n on the wavelength of light l. With normal dispersion, the refractive index increases with decreasing wavelength, and this is the case in glass, water, air, and all other substances transparent to light. In substances that strongly absorb light, the course of the refractive index reverses with a change in wavelength and becomes much steeper: with a decrease in l (increase in frequency w), the refractive index sharply decreases and in a certain range of wavelengths becomes less than unity (phase velocity V f > from). This is the anomalous dispersion, in which the pattern of light propagation in a substance changes radically. group speed V cp becomes greater than the phase velocity of the waves and can exceed the speed of light in vacuum (and also become negative). L. Wong points to this circumstance as the reason underlying the possibility of explaining the results of his experiment. However, it should be noted that the condition V gr > from is purely formal, since the concept of group velocity was introduced for the case of small (normal) dispersion, for transparent media, when a group of waves almost does not change its shape during propagation. In regions of anomalous dispersion, however, the light pulse is rapidly deformed and the concept of group velocity loses its meaning; in this case, the concepts of signal velocity and energy propagation velocity are introduced, which in transparent media coincide with the group velocity, while in media with absorption they remain less than the speed of light in vacuum. But here's what's interesting about Wong's experiment: a light pulse, passing through a medium with anomalous dispersion, does not deform - it retains its shape exactly! And this corresponds to the assumption that the impulse propagates with the group velocity. But if so, then it turns out that there is no absorption in the medium, although the anomalous dispersion of the medium is due precisely to absorption! Wong himself, recognizing that much remains unclear, believes that what is happening in his experimental setup can be clearly explained as a first approximation as follows.

A light pulse consists of many components with different wavelengths (frequencies). The figure shows three of these components (waves 1-3). At some point, all three waves are in phase (their maxima coincide); here they, adding up, reinforce each other and form an impulse. As the waves propagate further in space, they are out of phase and thus "extinguish" each other.

In the region of anomalous dispersion (inside the cesium cell), the wave that was shorter (wave 1) becomes longer. Conversely, the wave that was the longest of the three (wave 3) becomes the shortest.

Consequently, the phases of the waves also change accordingly. When the waves have passed through the cesium cell, their wavefronts are restored. Having undergone an unusual phase modulation in a substance with anomalous dispersion, the three considered waves again find themselves in phase at some point. Here they add up again and form a pulse of exactly the same shape as that entering the cesium medium.

Typically in air, and indeed in any normally dispersive transparent medium, a light pulse cannot accurately maintain its shape when propagating over a remote distance, that is, all of its components cannot be in phase at any remote point along the propagation path. And under normal conditions, a light pulse at such a remote point appears after some time. However, due to the anomalous properties of the medium used in the experiment, the pulse at the remote point turned out to be phased in the same way as when entering this medium. Thus, the light pulse behaves as if it had a negative time delay on its way to a remote point, that is, it would have arrived at it not later, but earlier than it passed the medium!

Most physicists are inclined to associate this result with the appearance of a low-intensity precursor in the dispersive medium of the chamber. The fact is that in the spectral decomposition of the pulse, the spectrum contains components of arbitrarily high frequencies with negligible amplitude, the so-called precursor, which goes ahead of the "main part" of the pulse. The nature of the establishment and the form of the precursor depend on the dispersion law in the medium. With this in mind, the sequence of events in Wong's experiment is proposed to be interpreted as follows. The incoming wave, "stretching" the harbinger in front of itself, approaches the camera. Before the peak of the incoming wave hits the near wall of the chamber, the precursor initiates the appearance of a pulse in the chamber, which reaches the far wall and is reflected from it, forming a "reverse wave". This wave, propagating 300 times faster from, reaches the near wall and meets the incoming wave. The peaks of one wave meet the troughs of another so that they cancel each other out and nothing remains. It turns out that the incoming wave "returns the debt" to the cesium atoms, which "borrowed" energy to it at the other end of the chamber. Someone who watched only the beginning and end of the experiment would see only a pulse of light that "jumped" forward in time, moving faster from.

L. Wong believes that his experiment is not consistent with the theory of relativity. The statement about the unattainability of superluminal speed, he believes, is applicable only to objects with a rest mass. Light can be represented either in the form of waves, to which the concept of mass is generally inapplicable, or in the form of photons with a rest mass, as is known, equal to zero. Therefore, the speed of light in a vacuum, according to Wong, is not the limit. Nevertheless, Wong admits that the effect he discovered does not make it possible to transmit information at a speed greater than from.

"The information here is already contained in the leading edge of the impulse," says P. Milonni, a physicist at the Los Alamos National Laboratory in the United States.

Most physicists believe that the new work does not deal a crushing blow to fundamental principles. But not all physicists believe that the problem is settled. Professor A. Ranfagni, of the Italian research team that carried out another interesting experiment in 2000, says the question is still open. This experiment, carried out by Daniel Mugnai, Anedio Ranfagni and Rocco Ruggeri, found that centimeter-wave radio waves propagate in ordinary air at a speed exceeding from by 25%.

Summarizing, we can say the following. The works of recent years show that under certain conditions, superluminal speed can indeed take place. But what exactly is moving at superluminal speed? The theory of relativity, as already mentioned, forbids such a speed for material bodies and for signals carrying information. Nevertheless, some researchers are very persistent in their attempts to demonstrate the overcoming of the light barrier specifically for signals. The reason for this lies in the fact that in the special theory of relativity there is no rigorous mathematical justification (based, say, on Maxwell's equations for an electromagnetic field) for the impossibility of transmitting signals at a speed greater than from. Such an impossibility in SRT is established, one might say, purely arithmetically, based on Einstein's formula for adding velocities, but in a fundamental way this is confirmed by the principle of causality. Einstein himself, considering the question of superluminal signal transmission, wrote that in this case "... we are forced to consider a signal transmission mechanism possible, when using which the achieved action precedes the cause. But, although this result from a purely logical point of view does not contain itself, in my opinion, no contradictions, it nevertheless contradicts the character of all our experience so much that the impossibility of supposing V > c appears to be sufficiently proven." The principle of causality is the cornerstone that underlies the impossibility of superluminal signal transmission. And this stone, apparently, will stumble all searches for superluminal signals, without exception, no matter how much experimenters would like to detect such signals because that is the nature of our world.

In conclusion, it should be emphasized that all of the above applies specifically to our world, to our Universe. Such a reservation was made because recently new hypotheses have appeared in astrophysics and cosmology that allow the existence of many Universes hidden from us, connected by topological tunnels - jumpers. This point of view is shared, for example, by the well-known astrophysicist N. S. Kardashev. For an outside observer, the entrances to these tunnels are marked by anomalous gravitational fields, similar to black holes. Movements in such tunnels, as suggested by the authors of the hypotheses, will make it possible to circumvent the limitation of the speed of movement imposed in ordinary space by the speed of light, and, consequently, to realize the idea of ​​creating a time machine... things. And although so far such hypotheses are too reminiscent of plots from science fiction, one should hardly categorically reject the fundamental possibility of a multi-element model of the structure of the material world. Another thing is that all these other Universes, most likely, will remain purely mathematical constructions of theoretical physicists living in our Universe and trying to find the worlds closed to us with the power of their thoughts ...

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