Kinematic paradoxes and the theory of relativity. Twin paradox (thought experiment): explanation

At first glance, the patent office was not the most promising
the place where the greatest revolution since Newton's time could begin.

tion in physics. But this service also had its advantages. Fast
having dealt with the patent applications cluttering his desk,
Einstein leaned back in his chair and immersed himself in childhood memories.
niya. In his youth, he read "Natural Science Books for the People"
Aaron Bernstein, "a work I read with bated breath",
Albert remembered. Bernstein invited the reader to imagine that
it follows in parallel with the electric current when it is transmitted
by wire. At the age of 16, Einstein asked himself the question: what would
looks like a ray of light if you could catch up with it? He recalled:
“This principle was born out of a paradox that I came across in
16 years old: if I chase a beam of light at a speed of c (the speed of light
in a vacuum), I must observe such a beam of light as spatially
oscillating electromagnetic field at rest. But,
it seems that such a thing cannot exist - experience says so, and
that's what Maxwell's equations say. As a child, Einstein believed that
If you move parallel to a beam of light at the speed of light, then the light
will appear frozen, like a frozen wave. However, no one
I didn't see the frozen light, so there was clearly something wrong.

At the beginning of the new century, there were two pillars in physics, on which
everything was at rest: the Newtonian theory of mechanics and gravity and
Maxwell's theory of light. In the 1860s, the Scottish physicist James
Clark Maxwell proved that light is made up of pulsating electrical
tricic and magnetic fields, constantly passing into each other.
Einstein was to discover, to his great shock, that
these two pillars contradict each other, and one of them was to
collapse.

In Maxwell's equations, he discovered the solution to a riddle that
haunted him for 10 years. Einstein found in them what
what Maxwell himself missed: the equations proved that light is transmitted
moves at a constant speed, while there was absolutely no
what matters is how fast you tried to catch up with him. speed of light
c was the same in all inertial frames of reference (i.e.
reference systems moving at a constant speed). stood
whether you were on the spot, whether you were riding a train or perched on a rushing
comet, you would definitely see a beam of light rushing ahead of you
at a constant speed. It didn't matter how fast you were moving
would have been on your own - you can’t overtake the world.

This state of affairs quickly led to the emergence of many
radoxes. Imagine for a moment an astronaut trying to catch up with the beam
Sveta. An astronaut takes off in a spaceship, and here he is rushing
head to head with beam of light. An observer on Earth who witnessed
the body of this hypothetical chase, would claim that the astronaut and beam
lights move side by side. However, the astronaut would say something different, and
namely: a ray of light was carried forward from him, as if a cosmic
the ship was at rest.

The question that confronted Einstein was:
how can two people interpret so differently
the same event? According to Newton's theory, a ray of light can always
but catch up; in Maxwell's world this was impossible. Einstein
it suddenly dawned on me that already in the fundamental foundations of physics such
there was a fundamental flaw. Einstein recalled that in the spring
1905 "a storm broke in my head." He finally found
solution: Time moves at different speeds depending on
movement speed. Basically, the faster you move, the slower you go.
time moves. Time is not absolute, as Newton once believed.
According to Newton, time is uniform throughout the universe and duration
one second on Earth would be identical to one second on Jupiter
or Mars. Clocks are absolutely synchronized with the entire universe.
However, according to Einstein, different clocks in the Universe run with different
speeds.

The main "purpose" of the set of SRT paradoxes is to show the internal contradictions of the theory. If a theory makes predictions about any phenomenon that contradict each other, then this indicates the fallacy of the theory, which requires its revision. SRT paradoxes are derived from thought experiments, that is, an imaginary experiment based on the provisions of the theory. One of these paradoxes is rightfully considered one of the oldest paradoxes - the Ehrenfest paradox of 1909, now often formulated as the "paradox of the wheel" and which, according to many authors, has not yet had a satisfactory explanation or solution.

There are several different formulations of Ehrenfest's "paradox" in the literature. Here the word paradox is put in quotation marks deliberately, since in this note it will be shown that the paradox is formulated with errors, on the basis of statements attributed to the special theory of relativity, but which it does not make. In general, these various formulations of the paradox can be reduced to three groups:

• when the wheel rotates, the spokes are deformed;
• it is impossible to spin a wheel of absolutely solid material at all;
• when spinning at the speed of light (the rim), the wheel shrinks to a point and disappears.

All these formulations are essentially close enough to each other and, under certain conditions, are combined. For example, in the work "The Theory of Relativity in an Elementary Presentation" the following formulation is given:

At first, the wheel is stationary, and then it is brought into such a rapid rotation that the linear speed of its edges approaches the speed of light. In this case, the sections of the rim ... are reduced .., while the radial "spokes" ... retain their length (after all, only longitudinal dimensions, i.e. dimensions in the direction of movement, experience relativistic shortening).

Rice. one. Illustration for the paradox of the wheel at work

And then the solution of the formulated paradox is given:

When the initially stationary wheel is brought into rapid rotation: its rim tends to shorten, and the spokes to maintain a constant length. Which of these trends will prevail depends entirely on the mechanical properties of the rim and spokes; but there will be no shortening of the rim without a proportional shortening of the spokes (unless the wheel takes the form of a spherical segment). Obviously, from a fundamental point of view, nothing will change even if the spoked wheel is replaced by a solid disk.

The essence of the solution, as we see, is that either the spokes will definitely shrink, or the rim will stretch, depending on the stiffness of the material. Apparently, if the material is homogeneous, the contraction will be mutual: both the spokes and the rim will shrink, but to a lesser extent.

The wheel paradox in Ehrenfest's version is given in Poincare's Uncorrected Error and SRT Analysis:

Consider a flat, hard disk rotating around its axis. Let the linear speed of its edge be comparable in order of magnitude to the speed of light. According to special relativity, the length of the edge of this disk must experience a Lorentz contraction...

There is no Lorentz contraction in the radial direction, so the radius of the disk must retain its length. With such a deformation, the disk technically can no longer be flat.

The angular velocity of rotation decreases with increasing distance from the axis of rotation. Therefore, adjacent layers of the disk must slide relative to each other, and the disk itself will experience torsional deformations. The disk will eventually break down.

The interpretation, it should be noted, is very specific: destruction is associated not with compression of the inner layers or spokes, but with their bending, twisting. The author does not explain the cause of the difference in angular velocities, referring to Ehrenfest, and only adding:

The relativists themselves have been unable to provide any explanation of physical causes, either to explain the hypothesis or to explain the paradox.

However, this is the only description of the disc curl effect that I have come across on the Internet with a quick look.

Wikipedia describes the paradox as follows, citing a children's encyclopedia in the text:

Consider a circle (or hollow cylinder) rotating about its axis. Since the speed of each element of the circle is directed along a tangent, then it (the circle) must experience Lorentz contraction, that is, its size for an external observer must seem smaller than its own length.

An initially immobile rigid circle, after it is untwisted, must paradoxically decrease its radius in order to maintain its length.

According to Ehrenfest, an absolutely rigid body cannot be put into rotational motion, since there should be no Lorentz compression in the radial direction. Consequently, the disk, which was flat at rest, must somehow change its shape when untwisted.

Here another manifestation of the paradox is indicated with reference to Ehrenfest: it is impossible to put an absolutely rigid disk into rotation at all. A similar interpretation is given in the "Encyclopedia for Children", which, in turn, refers to Ehrenfest's author's work - a short note "Uniform rotational motion of bodies and the theory of relativity" from 1909:

The note contained a paradoxical statement: an absolutely rigid cylinder (or disk) cannot be brought into rapid rotational motion around the central axis, otherwise a contradiction of the special theory of relativity arises. Indeed, let such a disk rotate, then the length of its circumference will decrease due to the Lorentz contraction, while the radius of the disk will remain constant... In this case, the ratio of the circumference of the disk to the diameter is no longer equal to the number n. This thought experiment is the content of Ehrenfest's paradox.

Here, one can say, the main, generally accepted formulation of the Ehrenfest paradox is given, which differs from the common formulation of the paradox of the wheel. It no longer refers to the deformation of the disk or wheel spokes. Just the disk will remain motionless.

Let's experiment with a disk. We will rotate it, gradually increasing the speed. Disk sizes... will decrease; in addition, the disk will warp. When the speed of rotation reaches the speed of light, it simply disappears. And where does it go?...

The disc should deform during rotation, as shown in the figure.

That is, as above, it is concluded that the spokes are deformed, while, obviously, it is quite reasonably assumed that the hardness of the rim exceeds the flexibility of the spokes.

Finally, in order to find out which of the formulations of the paradox corresponds to the author's, we will give a description of the paradox as it is formulated in the mentioned work of Ehrenfest. The following quote is practically the entire content of that brief note:

Both definitions of non-absolute hardness are - if I understand correctly - equivalent. Therefore, it suffices to point out the simplest kind of motion, for which the given initial definition already leads to a contradiction, namely, uniform rotation about a fixed axis.

Indeed, let there be a non-absolutely rigid cylinder C with radius R and height H. Let it be gradually brought into rotation around its axis, which then occurs at a constant speed. Let's call R" the radius that characterizes this cylinder from the point of view of a stationary observer. Then the value R" must satisfy two contradictory requirements:

a) the circumference of a rotating cylinder, compared to the state of rest, should be reduced:

2πR′< 2πR,

since each element of such a circle moves in the direction of the tangent with an instantaneous speed R "ω;

b) the instantaneous speed of any radius element is perpendicular to its direction; this means that the radius elements do not undergo any contraction compared to the state of rest.

Hence it follows that

Comment. If we assume that the deformation of each radius element is determined not only by the instantaneous velocity of the center of gravity, but also by the instantaneous angular velocity of this element, then it is necessary that the function describing the deformation contain, in addition to the speed of light c, one more universal dimensional constant, or it must enter the acceleration of the center of gravity of the element .

As we can see, at least in the original author's version, the paradox directly concerns non-absolutely rigid bodies. Nothing is said about twisting the layers. Nothing about "disappearing" the disk. Perhaps all these extensions of the original idea were formulated somewhere in Ehrenfest's subsequent works, but let's leave it all to the conscience of the cited authors: they did not provide verifiable references to their statements. Thus, we can reasonably consider:

The myth of Ehrenfest's paradox

Consider, if possible, the modern versions of the paradox indicated at the beginning of the article. The simplest and, apparently, the most common version is the “paradox of the wheel”, with which, as you can see, the contradiction formulated in 1909 by Ehrenfest coincides to the greatest extent. In fact, Ehrenfest's paradox is identically the paradox of the wheel.

However, first we will consider its extreme version. This is the version where the spokes or the inside of the wheel do not rotate at all. In this case, we get rid of any doubt about whether the spokes are shortening or not. Such a "wheel", as you might guess, looks like a hollow thin-walled cylinder or a thin ring mounted on a thick axle. The solution to this "paradox" is obvious. And again, as above, the word "paradox" is put in quotation marks here solely because it is, in fact, not a paradox, but a pseudo-imaginary paradox. The special theory of relativity describes the behavior of such a wheel without any contradictions. Indeed, from the point of view of the fixed axis, the “rim” of the wheel undergoes Lorentz contraction during rotation, which leads to a decrease in its diameter. From this point of view, either the wheel will burst, or it will compress the axle, squeezing out a notch on it, or, with sufficient elasticity, the ring will stretch. In this case, an external observer will not notice any changes, even if the wheel-ring is spun up to light speed: if only the material of the wheel has enough elasticity.

Now let's move on to the wheel-rim reference system. Obviously, it is impossible to tie the rest frame to the entire wheel, since the velocity vectors of the points are directed in different directions. At rest, there can be only one point at a time touching a fixed surface. It is clear that such a "stationary" wheel is just a wheel rolling on a stationary surface. We can only say about it that the speed of its center is equal to half the speed of the element on the top. But this remark suddenly unexpectedly reminds us of the already considered paradox - the paradox of the conveyor. Indeed, in that paradox there are also three points: fixed; the top one, moving at some speed, and the middle one, moving at half the top speed. What can be common between a wheel and a conveyor?

However, let's take a closer look. Let's look at the wheel at an angle to its axis. The larger this angle, the more the wheel “flattens”, taking the form of an elongated ellipse, which quite noticeably resembles a conveyor.

Rice. 2. When viewed from a large angle, the wheel looks like an ellipse. The thickened circle is the outer surface of the wheel axle. Thin line circle - rotating rim (wheel)

Although on the resulting conveyor the belt - the wheel rim moves along an elliptical path, we can well consider the "projection" of this rim onto the horizontal axis. In this case, we get a completely acceptable analogy of the conveyor belt problem and its obvious solution:

In both cases, both from the point of view of the beam (frame) and from the point of view of ... the belt, the result will be a tension on the belt, leading either to deformation ... of the frame, or to deformation ... of the belt. Depending on the initial conditions: what will be given more durable. The paradox of the transporter turned out to be an imaginary, seeming paradox.

The wheel rim, seen as a conveyor belt, as in the conveyor problem, will shrink, which will inevitably lead either to its rupture or to deformation of the axle, which at the selected angle looks like a conveyor bed. It is clear that the axle can be segmented, that is, consist of spokes, which, like a solid axle, will be deformed if the rim is stronger.

Thus, the version of the "paradox" of a wheel with a thin rim and a fixed axle is not a paradox, since the theory of relativity makes consistent predictions about it.

Now let's move on to a solid disk. Moreover, we will consider it absolutely solid, that is, consider the variant of Ehrenfest's paradox about the impossibility of spinning up such a disk.

Let's imagine a disc as concentric circles placed on top of each other - rims of rather small thickness and rigidly fastened to each other. Let us denote the radius of each such rim as Ri. The circumference of each rim, respectively, is 2πRi. Let's say we managed to spin up the disk. The angular velocity of the disc ω is the same for each point of the disc and determines the linear velocity of each particular rim of the disc. Here we strongly reject the idea of ​​twisting as unfounded. Tangential speed of each point of the rim vi = ωRi. The reduced circumference of each rim is determined by the Lorentz equations:

L i= 2π R i1 − ω 2R 2 i−−−−−−−−√ Li=2πRi1−ω2Ri2

Here we consider the problem in the system of units, in which the speed of light c = 1. Consider two rims: the outer one with R0 and one of the inner ones - R1, let R1 = kR0, where k = 0...1. From equation (1) we get:

L1= 2 π k R01 − ω 2k2R20−−−−−−−−−√ L0= 2π R01 − ω 2R20−−−−−−−−√ L1=2πkR01−ω2k2R02L0=2πR01−ω2R02

When the disc was "untwisted", these two rims reduced their length. Therefore, the radii of their new circles will be:

lR 1 w= L12 π= k R01 − ω 2k2R20−−−−−−−−−√ R 0 ω = L02 π= R01 − ω 2R20−−−−−−−−√ lR1ω=L12π=kR01−ω2k2R02R0ω=L02π=R01−ω2R02

The ratio of the rim radii after spinning is:

R 1 wR 0 ω = k R01 − ω 2k2R20−−−−−−−−−√ R01 − ω 2R20−−−−−−−−√ = k 1 − ω 2k2R201 − ω 2R20−−−−−−−−−−√ R1ωR0ω=kR01−ω2k2R02R01−ω2R02=k1−ω2k2R021−ω2R02

This expression shows that the ratio of the radii of adjacent layers depends on the rotation speed. We should be interested in what the rotation speed can be so that the radii that differ by k times in a stationary state become equal after spin-up. Apparently, this will be the maximum speed, after which the layers will "crawl" on top of each other. Let's calculate this ratio for the specified condition:

R 1 wR 0 ω = k 1 − ω 2k2R201 − ω 2R20−−−−−−−−−−√ = 1 R1ωR0ω=k1−ω2k2R021−ω2R02=1

For clarity, we discard the left equality:

k 1 − ω 2k2R201 − ω 2R20−−−−−−−−−−√ = 1 k1−ω2k2R021−ω2R02=1

We divide everything by k

1 − ω 2k2R201 − ω 2R20−−−−−−−−−−√ = 1k 1−ω2k2R021−ω2R02=1k

Squaring both sides of the equation

1 − ω 2k2R201 − ω 2R20= 1 k2 1−ω2k2R021−ω2R02=1k2

Getting rid of the fractional form

k2− ω 2k4R20= 1 − ω 2R20 k2−ω2k4R02=1−ω2R02

Moving members with radii to the left and members without radii to the right

ω 2R20− k4ω 2R20= 1 − k2ω2R02−k4ω2R02=1−k2

Gathering like members

ω 2R20(1 − k4) = 1 − k2ω2R02(1−k4)=1−k2

Rewriting the equation as a solution for the term with radius

ω 2R20= 1 − k21 − k4ω2R02=1−k21−k4

We see that on the right in the equality there are reducible terms

ω 2R20= 1 − k2(1 − k2) (1 + k2) ω2R02=1−k2(1−k2)(1+k2)

We reduce

ω 2R20= 1 1 + k2ω2R02=11+k2

Change angular velocity to linear velocity

v 2 0= 1 1 + k2 v02=11+k2

We extract the root and find the value of the speed

v0= 1 1 + k2−−−−−√ v0=11+k2

The intersection can start between adjacent layers, for which almost k = 1. The actual intersection occurs at the speed of the outer rim:

v0= 1 1 + 1 −−−−√ = 1 2 –√ = 2 –√ 2 ≈ 0 , 7 v0=11+1=12=22≈0.7

Firstly, this means that our assumption about the possibility of spinning up the disk turned out to be justified. Secondly, we find that two adjacent infinitely thin rims will only push against each other at speeds greater than 0.7 times the speed of light. And this, in turn, means that when untwisted, each rim reduces both the length of its circumference and the radius corresponding to it. Thus, here we discover a delusion regarding the reduction of the spokes of a rotating wheel. All authors, when formulating the paradox, explicitly state that the rim contracts, but the spokes do not. We have found that, on the contrary, each rim, each thin layer of the wheel shrinks and reduces its own radius. Therefore, it does not prevent the reduction of the layer, the rim, which is above it. In the same way, the layer, the rim, below it does not interfere with its own compression. Since the considered rims together form a solid disc of the wheel, this wheel as a whole does not experience any internal deformations that prevent its compression. The statements of all authors, including the author of the paradox - Ehrenfest - are erroneous: the radius of the wheel will decrease without any obstacles:

Radius elements do not undergo any reduction compared to the rest state.

But the discovered contraction, contraction of the radii, has a rather strange feature: this contraction is possible only up to the tangential velocity of the outer rim, which does not exceed 0.7 of the speed of light. Why exactly 0.7? From where, from what physical features of the wheel does this number arise? And what happens if the wheel spins even faster?

However, why do we claim that the spokes will shrink, because in our model there are no spokes, the wheel is solid. And in a wheel with spokes there are no “thin rims”, there is an empty space between adjacent spokes.

As correctly stated in the work, there is no difference between a solid disk and a disk with spokes. All elements that are the same distance from the center are subject to Lorentz contraction. That is, in this case, the “thin layer” is a sequence of “lobes” of spokes and an empty space between them. Here a bewildered objection may arise: how is it, why is it that each “slice” of the spoke is compressed along the circumference? After all, they have empty space next to them! Yes, empty. But all elements without exception are subject to Lorentz contraction, this is not a real physical contraction, it is a contraction visible to an external observer. As a rule, when describing the Lorentz contraction, it is always emphasized that the object from the point of view of an external observer has reduced its size, although nothing has happened to it from the point of view of the object itself.

To explain this tangential contraction, the thinning of the spokes, let us imagine a moving platform on which, for example, bricks are laid at intervals. To an outside observer it will appear that the platform has shrunk. And what will happen to the intervals between the bricks? The bricks, of course, will shrink, but if the interval between them remains unchanged, they will simply push each other off the platform. However, in reality the bricks and the spaces between them shrink as one single object. Any observer moving past the platform will see its reduced length, depending on the relative speed, and the reduced length of the "bricks at intervals" object. With the platform itself, the bricks and the intervals between them, as you know, nothing will happen.

So it is with the spoked wheel. Each individual radial layer of the wheel - the rim will be a "layer cake", consisting of successive pieces of spokes and the space between them. Reducing in length, such a "puff" rim will simultaneously reduce its radius of curvature. In this sense, it is useful to imagine that the wheel is first spinning, then slowed down to a stop. What will happen to him? It will return to its original state. The decrease in its size has nothing to do with its physical deformation, these are dimensions visible to an external, motionless observer. Nothing happens to the wheel itself.

From this, by the way, it directly follows that the wheel can be absolutely solid. No deformation forces are applied to it, changing its diameter does not require direct physical compression of the wheel material. You can spin the wheel, then slow it down as many times as you like: for the observer, the wheel will reduce its size and restore them again. But on one condition: the tangential speed of the outer rim of the wheel should not exceed the mysterious value - 0.7 of the speed of light.

It is obvious that when this speed is reached by the outer rim of the wheel, the speeds of all underlying ones will certainly be less. Therefore, the “wave” of overlap will start from the outside and will gradually move inside the wheel, towards its axis. In this case, if the outer rim is spun up to the speed of light, the overlap of the layers will be only up to the layer having 0.7 of the initial radius of the wheel. All layers closer to the axis will not overlap each other. It is clear that this is a hypothetical model, since it is not yet clear what will happen to layers located further from the axis than 0.7 of the original radius. Recall the exact value of this quantity: √2/2.

The diagram shows the process of reducing the radii of the layers and the point at which they begin to intersect:

Rice. 3. The degree of compression of the rim radii depending on their distance from the center and the tangential velocity of the outer rim

With an increase in the tangential speed of the outer edge of the disk, its layers - rims reduce their own radii to a different extent. The radius of the outer edge decreases the most, down to zero. We see that the rim, the radius of which is equal to a tenth of the radius of the outer edge of the disk, practically does not change its radius. This means that with a strong spin-up, the outer rim will shrink to a radius smaller than the inner one, but how this will look in reality is still unclear. So far, it is only obvious that deformation occurs only when the speed of the outer rim exceeds √2/2 of the speed of light (approx. 0.71 s). Up to this speed, all rims are compressed without crossing each other, without deformation of the disk plane, the outer radius of which will then decrease to 0.7 of the initial value. To visualize this point, the diagram shows two adjacent outer layers of the rim, which have almost the same radii. These are the first "candidates" for mutual intersection during unwinding.

If uniformly concentric circles are applied to the disk, at regular intervals, then in the process of its unwinding for an external observer, these circles will be located at intervals uniformly decreasing from the center (almost the initial value of the interval) to the periphery (decreasing down to zero).

In order to find out what happens to the wheel after the outer rim exceeds the speed of 0.7 of the speed of light, we change the shape of the wheel so that the layers do not interfere with each other. Let's move the layers of the wheel along the axis, turning the wheel into a thin-walled cone, a funnel. Now, when compressing each layer, there are no other layers below it, and nothing prevents it from compressing as much as it likes. Let's start spinning the cone from rest to a speed of 0.7 of the speed of light and then to the speed of light, after which we will decrease the speed in the reverse order. Let's depict this process as an animation:

Rice. 4. Lorentz deformation of a cone during unwinding. On the left is a view along the axis of the cone - funnels, on the right - a side view, perpendicular to the axis. The red thin line on the cone shows its contour.

In the figure, the cone (funnel) is shown in two views: along the axis, as the paradox of the wheel is always depicted, and perpendicular to the axis, a side view, on which the “profile” of the cone is visible. In the side view, we clearly see the behavior of each layer-rim of the cone, the former wheel. Each of these layers is represented by a colored line. These lines repeat the corresponding circles, rims, for which the graph in the previous figure is built. This allows you to see each rim independently and see how the outer rim reduces its radius more than the inner rims.

The following obvious circumstances should be especially noted. According to the theory of relativity, there is no deformation of the disk or the cone shown as such. All changes in its shape are visibility for an external observer, nothing happens to the disk itself and the cone. Therefore, it may well be from an absolutely solid material. Products made from such material do not shrink, do not stretch, do not bend or twist - they are not subject to any geometric deformation. Therefore, the appearance of deformation quite admits the unwinding of this disk up to the speed of light. An external observer will see, as shown in the animation, a completely logical, albeit rather strange picture. The outer rim of the cone decreases to a speed of 0.7 s, after which it continues to shrink further. In this case, the inner rim, which had a smaller radius, is on the outside. However, this is quite an obvious phenomenon. The painted rims in the animation show how the outer rims approach the center of the disc, turning the cone into a kind of closed vessel, an amphora. But you need to understand that in this case the cone itself remains the same as it was originally. If you reduce the speed of its rotation, then all the layers will return to their places and the amphora for a stationary observer will again turn into a cone. This apparent movement of layers, rims due to compression towards the center of the disk from the point of view of an external observer is in no way connected with the real geometric deformation of the disk itself. That is why there are no physical obstacles for the cone to be made of absolutely solid material.

But this applies to the cone. And how will a flat wheel behave, in which all the layers are still one above the other? In this case, a stationary observer will see a very strange picture. After the outer rim of the disk decreases at a speed of 0.7 s, it will attempt further compression. In this case, the inner rim, which had a smaller radius, will resist this. Here we recall the obvious condition - at any speed, the disk must remain flat.

With all the strangeness of the picture, you can quite easily guess what will happen next. You just need to remember the picture discussed above with a thin-walled wheel mounted on a fixed axle. The only difference is that in the considered case, the fixed axis does not experience Lorentz contraction. Here, the layers, from zero to 0.7 of the radius of the wheel, experienced compression themselves and somewhat reduced their size. Despite this, the outer layers still "caught up" with them. Now the Lorentz compression of the inner layers is not enough, they do not allow the outer layers to continue their own compression. As options, we can identify three scenarios for the further development of events, not taking into account the action of centrifugal forces and the fact that an infinitely powerful engine is required for such a spin-up.

For a conventional material, when the rim layers interact, the inner layers experience compressive deformation, while the outer layers experience tension. Therefore, a rupture of the outer rims is more likely than an elastic decrease in the volume of the inner ones. This is obvious, since their material is the same.

Rice. five. Lorentz deformation of a disk made of ordinary solid material

Here and on subsequent animations, the coloring of the stripes is made like a "vest" - lighter colors alternate with darker ones. In this case, when the disk is compressed, it is better seen on its section that they do not intersect each other, but, as it were, fold in the form of an "accordion". In the compression animation of a conventional hard (brittle) disk, layers (rims) are repainted in red, which come into close contact, press each other with force. In this case, their material experiences both compressive force (inner layers) and tensile force (outer layers). With some effort, the outer layers, which is more likely, will simply be torn apart and scatter in different directions. As you can see in the animation, the conditions for the break come after reaching the limit speed of 0.7 s.

For a perfectly elastic material, the picture is slightly different. Breaking the layers is impossible, but their infinite compression is possible. Consequently, when the speed of the outer rim is close to the speed of light, for an external observer, the wheel can turn into an infinitely small point.

Rice. 6. Lorentz deformation of an elastic disk

This is the case if less force is needed for compression than for tension. Otherwise, the shape of the wheel with the equality of these forces will remain unchanged. After the rotation stops, the wheel will return to its original dimensions without any damage. In the animation, as above, you can see that the rim layers are folded in the form of an "accordion" without crossing each other. True, here one should show the thickening of the disk in the gap between the outer rim and the axle. The disk, obviously, should take the form of a donut when compressed. Upon reaching the speed of the outer rim, equal to the speed of light, the disk will shrink into a point (or rather, into a thin tube put on the axle).

For a completely rigid wheel material that does not compress, stretch or bend, the picture will also be different from the previous ones.

Rice. 7. Lorentz deformation of a disk made of absolutely rigid material

The outer rims cannot break, and the inner rims cannot shrink. Therefore, there will be no destruction of either one or the other, but the force of their pressure on each other will rapidly increase after the maximum rotation speed is reached. What are the sources of this power? Obviously, due to the forces that cause the wheel to rotate. Therefore, the external source will have to apply more and more force up to infinity. It is clear that this is impossible, and we come to the conclusion: when the outer rim of an absolutely rigid wheel reaches a speed of √2/2 of the speed of light, there will be no further increase in this speed. The drive motor seems to rest against the wall. This is about the same as running, for example, behind a tractor cart, trailer. You can run at any speed, but when you reach the cart, the speed will be immediately limited by its speed, the speed of the tractor.

So, let's sum up. As we can see, the behavior of the spinning wheel has strictly consistent and consistent predictions in the special theory of relativity for all variants of the wheel paradox.

An erroneous version of the Ehrenfest paradox is the impossibility of spinning an absolutely rigid body:

Ehrenfest's reasoning shows the impossibility of bringing an absolutely rigid body (initially at rest) into rotation

These are erroneous conclusions that do not correspond to the predictions of the special theory of relativity. In addition, in the work of Ehrenfest, which should be considered the first formulation of the paradox, there is no such reasoning. It is believed that the absolutely rigid body itself is, by definition, impossible in special relativity, since it allows superluminal signal transmission. Therefore, SRT mathematics is initially inapplicable to such bodies. However, such a body, as we have shown, can be spun up to more than two-thirds the speed of light. In this case, no SRT paradoxes arise, since for an external observer, a relativistic contraction of the entire circle, including its spokes, occurs. The assertion of Ehrenfest and other authors that the spokes do not shrink longitudinally is erroneous. Indeed, since the rims move without slipping relative to each other, we can glue them together, treating them as one solid disk. If now on such a solid disk we “draw” the spokes, then obviously they will reduce their length, following the decrease in the diameters of the rims. Also, the spokes can be made as a corrugation on the surface of the disk and even by making radial (or at an angle) cuts inside it. The resulting spokes and the empty intervals (space) between them move like parts of rims connected to each other, that is, they are objects that contract as a whole. Both the material of the spokes and the spacing between them experience tangential Lorentz contraction equally, which, accordingly, leads to the same radial contraction.

The original, widespread in the literature, author's version of Ehrenfest's paradox is also erroneous - the untwisting of an ordinary body: the radius of the wheel is simultaneously equal to the original and shortened value.

The error lies in the statement on behalf of the theory of relativity that the radius (spokes) of the wheel does not experience Lorentz contraction. But special relativity makes no such prediction. According to her predictions, the spokes experience the same Lorentz contraction as the wheel rim. At the same time, depending on the material of the wheel, its part, which exceeds 0.7 of the radius when the rim unwinds to light speed, will either be destroyed, torn, if the material is not elastic enough, or the entire wheel will experience Lorentz compression to an infinitely small radius from the point of view of an external observer . If the wheel is stopped before it is destroyed and before reaching a speed of 0.7 of the speed of light, then it will take its original shape for an external observer without any damage. An elastic body, when reaching a speed above 0.7 of the speed of light, may experience some deformations. For example, if there were inclusions of brittle material in it, then they will be destroyed. After stopping the wheel, destruction will not be restored.

Thus, it should be recognized that none of the considered formulations allows us to speak of a paradox. All kinds of wheel paradox, Ehrenfest are imaginary, pseudo-paradoxes. Correct and consistent application of SRT mathematics allows for each situation described to make consistent predictions. By paradox, we mean correct predictions that contradict each other, but this is not the case here.

After reviewing a number of sources (which, of course, cannot be called exhaustive), it turned out the following. The stated solution of Ehrenfest's paradox (the paradox of the wheel) is, apparently, the first since its formulation by Ehrenfest in 1909, the correct solution of the paradox within the framework of the special theory of relativity. The considered solution was first discovered in October 2015 and on 10/18/2015 this article was sent for publication on the website of the International Association of Scientists, Teachers and Specialists (Russian Academy of Natural Sciences) in the Correspondence Electronic Conferences section.

The main purpose of the thought experiment called "Twin Paradox" was to refute the logic and validity of the special theory of relativity (SRT). It is worth mentioning right away that there is actually no question of any paradox, and the word itself appears in this topic because the essence of the thought experiment was initially misunderstood.

The main idea of ​​STO

The paradox (twin paradox) says that a "stationary" observer perceives the processes of moving objects as slowing down. In accordance with the same theory, inertial frames of reference (frames in which the motion of free bodies occurs in a straight line and uniformly, or they are at rest) are equal relative to each other.

The twin paradox in brief

Taking into account the second postulate, an assumption about inconsistency arises. To solve this problem visually, it was proposed to consider the situation with two twin brothers. One (conditionally - a traveler) is sent on a space flight, and the other (a homebody) is left on planet Earth.

The formulation of the twin paradox under such conditions usually sounds like this: according to the homebody, the time on the clock that the traveler has is moving more slowly, which means that when he returns, his (the traveler's) clock will lag behind. The traveler, on the contrary, sees that the Earth is moving relative to him (on which there is a homebody with his watch), and, from his point of view, it is his brother who will go more slowly.

In reality, both brothers are on an equal footing, which means that when they are together, the time on their clocks will be the same. At the same time, according to the theory of relativity, it is the brother-traveler's watch that should fall behind. Such a violation of the apparent symmetry was considered as an inconsistency in the provisions of the theory.

Twin paradox from Einstein's theory of relativity

In 1905, Albert Einstein derived a theorem that states that when a pair of clocks synchronized with each other is at point A, one of them can be moved along a curved closed trajectory at a constant speed until they again reach point A (and on this will be spent, for example, t seconds), but at the time of arrival they will show less time than the clock that remained motionless.

Six years later, Paul Langevin gave this theory the status of a paradox. "Wrapped" in a visual story, it soon gained popularity even among people far from science. According to Langevin himself, the inconsistencies in the theory were due to the fact that, returning to Earth, the traveler moved at an accelerated rate.

Two years later, Max von Laue put forward a version that it is not the acceleration moments of an object that are significant, but the fact that it falls into a different inertial frame of reference when it finds itself on Earth.

Finally, in 1918, Einstein himself was able to explain the paradox of two twins through the influence of the gravitational field on the passage of time.

Explanation of the paradox

The twin paradox has a rather simple explanation: the initial assumption of equality between the two frames of reference is incorrect. The traveler did not stay in the inertial frame of reference all the time (the same applies to the story with the clock).

As a consequence, many felt that special relativity could not be used to correctly formulate the twin paradox, otherwise incompatible predictions would result.

Everything was resolved when it was created. It gave the exact solution for the existing problem and was able to confirm that out of a pair of synchronized clocks, it was those that were in motion that would lag behind. So the initially paradoxical task received the status of an ordinary one.

controversial points

There are assumptions that the moment of acceleration is significant enough to change the speed of the clock. But in the course of numerous experimental tests, it was proved that under the influence of acceleration, the movement of time does not accelerate or slow down.

As a result, the segment of the trajectory, on which one of the brothers accelerated, demonstrates only some asymmetry that occurs between the traveler and the homebody.

But this statement cannot explain why time slows down for a moving object, and not for something that remains at rest.

Verification by practice

The formulas and theorems describe the twin paradox accurately, but this is quite difficult for an incompetent person. For those who are more inclined to trust practice, rather than theoretical calculations, numerous experiments have been carried out, the purpose of which was to prove or disprove the theory of relativity.

In one case, they were used. They are extremely accurate, and for a minimum desynchronization they will need more than one million years. Placed in a passenger plane, they circled the Earth several times and then showed quite a noticeable lag behind those watches that did not fly anywhere. And this despite the fact that the speed of movement of the first sample of the watch was far from light.

Another example: the life of muons (heavy electrons) is longer. These elementary particles are several hundred times heavier than ordinary particles, have a negative charge and are formed in the upper layer of the earth's atmosphere due to the action of cosmic rays. The speed of their movement towards the Earth is only slightly inferior to the speed of light. With their true lifespan (2 microseconds), they would have decayed before they touched the surface of the planet. But during the flight, they live 15 times longer (30 microseconds) and still reach the goal.

The physical cause of the paradox and the exchange of signals

Physics also explains the twin paradox in a more accessible language. During the flight, both twin brothers are out of range for each other and cannot practically make sure that their clocks move in sync. It is possible to determine exactly how much the movement of the traveler’s clocks slows down if we analyze the signals that they will send to each other. These are conventional signals of "exact time", expressed as light pulses or video transmission of the clock face.

You need to understand that the signal will not be transmitted in the present time, but already in the past, since the signal propagates at a certain speed and it takes a certain time to pass from the source to the receiver.

It is possible to correctly evaluate the result of the signal dialogue only taking into account the Doppler effect: when the source moves away from the receiver, the signal frequency will decrease, and when approached, it will increase.

Formulation of an explanation in paradoxical situations

There are two main ways to explain the paradoxes of these twin stories:

1. Careful consideration of existing logical constructions for contradictions and identification of logical errors in the chain of reasoning.
2. Implementation of detailed calculations in order to assess the fact of time deceleration from the point of view of each of the brothers.

The first group includes computational expressions based on SRT and inscribed in Here it is understood that the moments associated with the acceleration of motion are so small in relation to the total flight length that they can be neglected. In some cases, they can introduce a third inertial frame of reference, which moves in the opposite direction in relation to the traveler and is used to transmit data from his watch to the Earth.

The second group includes calculations built taking into account the fact that moments of accelerated motion are still present. This group itself is also divided into two subgroups: one uses the gravitational theory (GR), and the other does not. If general relativity is involved, then it is understood that the gravitational field appears in the equation, which corresponds to the acceleration of the system, and the change in the speed of time is taken into account.

Conclusion

All discussions connected with an imaginary paradox are due only to an apparent logical error. No matter how the conditions of the problem are formulated, it is impossible to ensure that the brothers find themselves in completely symmetrical conditions. It is important to consider that time slows down precisely on moving clocks, which had to go through a change in reference systems, because the simultaneity of events is relative.

There are two ways to calculate how much time has slowed down from the point of view of each of the brothers: using the simplest actions within the framework of the special theory of relativity or focusing on non-inertial frames of reference. The results of both chains of calculation can be mutually consistent and equally serve to confirm that time passes more slowly on a moving clock.

On this basis, it can be assumed that when the thought experiment is transferred to reality, the one who takes the place of a homebody will indeed grow old faster than the traveler.

Do you want to surprise everyone with your youth? Embark on a long space flight! Although, when you return, there will most likely be no one to be surprised ...

Let's analyze history two twin brothers.
One of them - the "traveler" goes on a space flight (where the speed of the rockets is near the light), the second - the "homebody" remains on Earth. And what is the question? - at the age of brothers!
Will they remain the same age after space travel, or will one of them (and who exactly) become older?

Back in 1905, Albert Einstein in the Special Theory of Relativity (SRT) formulated relativistic time dilation effect, according to which clocks moving relative to an inertial frame of reference run slower than stationary clocks and show a shorter time interval between events. Moreover, this slowdown is noticeable at near-light speeds.

It was after the nomination of SRT by Einstein that the French physicist Paul Langevin formulated "twin paradox" (or otherwise "clock paradox"). The twin paradox (otherwise "clock paradox") is a thought experiment with which they tried to explain the contradictions that have arisen in SRT.

So, back to the twin brothers!

It should seem to the homebody that the clock of the moving traveler has a slow pace of time, so when returning, it should lag behind the clock of the homebody.
And on the other hand, the Earth is moving relative to the traveler, so he believes that the homebody's clock should fall behind.

But, both brothers cannot be at the same time one older than the other!
This is where the paradox lies...

From the point of view of the “twin paradox” that existed at the time of the emergence, a contradiction arose in this situation.

However, the paradox, as such, does not really exist, since we must remember that SRT is a theory for inertial frames of reference! Ah, the frame of reference for at least one of the twins was not inertial!

At the stages of acceleration, deceleration or turnaround, the traveler experienced accelerations, and therefore, at these moments, the provisions of SRT are not applicable.

Here you have to use General Theory of Relativity, where it is proved by calculations that:

Let's get back, to the question of slowing down time in flight!
If light travels any path in time t.
Then the duration of the flight of the ship for the "homebody" will be T = 2vt / s

And for the “traveler” on the spaceship, his clock (based on the Lorentz transformation) will take only To=T times the square root of (1-v2/c2)
As a result, calculations (in general relativity) of the magnitude of time dilation from the position of each brother will show that brother-traveler will be younger than his brother-homebody.

For example, you can mentally calculate the flight to the star system Alpha Centauri, which is 4.3 light years away from Earth (a light year is the distance that light travels in a year). Let time be measured in years and distances in light years.

Let the spacecraft move half of the way with an acceleration close to the free fall acceleration, and slow down the other half with the same acceleration. Making the way back, the ship repeats the stages of acceleration and deceleration.

In this situation the flight time in the earth's reference system will be approximately 12 years, while according to the clock on the ship, 7.3 years will pass. The maximum speed of the ship will reach 0.95 of the speed of light.

In 64 years of proper time, the spacecraft with a similar acceleration can travel to the Andromeda galaxy (back and forth). On Earth, during such a flight, about 5 million years will pass.

The reasoning behind the story of the twins only leads to an apparent logical contradiction. With any formulation of the “paradox”, there is no complete symmetry between the brothers.

The relativity of the simultaneity of events plays an important role in understanding why time slows down precisely for a traveler who has changed his frame of reference.

Already conducted experiments on lengthening the lifetime of elementary particles and slowing down the clock during their movement confirm the theory of relativity.

This gives grounds to assert that the time dilation described in the story of the twins will also occur in the actual implementation of this thought experiment.

PARADOXES OF THE SPECIAL THEORY OF RELATIVITY. The word "paradoxes" in this case means those conclusions from SRT, which, although they are absolutely correct in essence and are confirmed by experiments, nevertheless contradict intuitive ideas based on classical physics.

Two conclusions from the postulates of SRT (by the way, experimentally confirmed) have always been of particular interest, although in practice they almost never come across explicitly (these effects are implicitly contained in any relativistic formula).

The thing is that these conclusions, at first glance, cannot correspond to reality at all.

1. The most famous - the twin paradox is usually formulated as follows. Let twin brother A go on a space flight to a star X, located at a distance of, say, 20 light years from us. The speed of a starship is close to the speed of light: v = 0,9from. Having reached the star in about 22.3 years (according to its clock), the ship turns around and flies back. Thus, according to the clock of brother A, who made this flight, about T= 44.6 years. The second twin brother B was waiting for the return of brother A on Earth. At the gangway of the spaceship, brother A was met by a decrepit old man who had to wait for a meeting for more than 100 years.

Actually, there is no paradox here yet. Indeed, when moving at a speed v = 0,9c the Lorentz factor is equal to g » 2.3, and due to the effect of time dilation according to the clock of an earthly observer, a time equal to g T» 103 years.

A paradox arises when trying to reverse the argument. After all, from the point of view of brother A (a stationary observer), brother B is moving, and more time passes by his watch. But from the point of view of brother B, brother A moves, and according to his watch more time must pass. Thus brother A must return older. It would seem that the SRT formulas are symmetrical with respect to the replacement v on the - v. What's the matter?

This paradox is resolved as follows. The fact is that the world lines of brothers A and B are different. One of them (B) is at rest, the other (A) moves at a constant speed, which at a certain moment changes to the opposite, which is possible only when the spacecraft decelerates and then accelerates (which corresponds to motion in a non-inertial frame of reference). Thus, brother A moves from the Earth and to the Earth, being at rest first with respect to one inertial frame, and then with respect to the other, and on the way passes for a short time into a non-inertial frame. At the same time brother B is at rest relative to the same inertial frame. It can be seen that A and B are in different physical conditions, and this resolves the paradox. An exact calculation shows that from the point of view of any of the brothers, the one that is stationary relative to the Earth will age more.

In accelerators, short-lived particles moving at speeds close to the speed of light “live” much longer than “resting” particles.

2. Another effect is the Lorentz contraction of length and related paradoxes.

Let there be two inertial frames of reference - S" And S. In system S"rigid rod of length D x" rests along the axis x and you need to determine its length in the system S, relative to which the rod moves with a speed v. To measure the length of a rod in any inertial frame relative to which the rod moves along the longitudinal axis, one must simultaneously observe its ends. This is the key point, the misunderstanding of which sometimes leads to paradoxes.

In SRT it is necessary to distinguish what the observer sees from what he knows, as it were, after the fact. What the observer sees or photographs at any fixed point in time is called the picture of the world at that moment. This concept is practically not very important, and theoretically very difficult, because. what the observer sees at the moment is a mixture of events that took place farther in the past and farther in space. If you look at the night sky full of stars, then the distances to these stars range from a few to hundreds of thousands of ly. years, therefore, the observer sees the light from these stars, emitted at different times and simultaneously reaching his eye, i.e. he. sees different events.

The concept of a world map is more useful. It can be represented as a map of events in a section of the 4-dimensional Minkowski space by a plane of constant time t = t 0. A map of the world is, as it were, a three-dimensional instant photograph in full size, taken simultaneously everywhere, a frozen moment in the observer's spatial frame of reference. Such a map of the world can be realized by joint photographs taken by auxiliary observers located at the nodes of the spatial lattice in a given inertial frame, and each photographs his surroundings at a predetermined moment of time t = t 0, and then the pictures are glued together.

When they say that the length of the body in the system S is equal to such and such a value, we are talking about a map of the world, i.e. about the simultaneous fixation of the positions of the ends of the rod at a given moment in time. What the eye actually sees when observing a moving body is a completely different and not very important question.

To derive a formula for reducing the length of the Lorentz transformation from the system S to the system S" are written for coordinate increments:

D xў0 = g(D x 0 – v D x 1), D xў1 = g(D x 1 – v D x 0).

In the second formula, you need to put D x 0 = 0 (simultaneous fixation of the ends of the rod in the system S!). Then D xў1 = gD x 1. If we denote D xў1 = L 0 and D x 1 = L, then

L = L 0/g

(g is the Lorentz factor).

All paradoxes of length contraction are connected, of course, with the symmetry of the effect: if the observer is in S sees a contraction of length, then the observer in S" must see the same thing. From the "paradoxes" of SRT, one can draw an important conclusion: whatever result is obtained by correct reasoning in some inertial frame of reference, it is true in any other inertial frame of reference.

When used correctly, SRT does not allow any "paradoxes".

Some seemingly obvious things turn out to be not at all so obvious within SRT. For example, it would seem that if along the axis x If a cube of a given size flies, then, due to the Lorentz contraction, it should look flattened in the direction of motion in the laboratory frame, turning into a parallelepiped. A detailed calculation shows, however, that this is not the case: the visible cube does not change its dimensions and only rotates through a certain angle relative to the axis x. This result (“the invisibility of the Lorentz contraction”) was obtained only fifty years after the creation of SRT.

Alexander Berkov