Relativistic correlation π problem and twin paradox

The first question is considered as follows: the total number or times n of rulers used by the rotary measuring instrument S' to complete the measurement is the same for both the rotary measuring instrument S' and the static measuring instrument S. Grasping this invariant, the problem becomes clear-first, starting from the static S, the circumference of the circle is C=2πR, and the moving ruler is surrounded by n or placed n times, so C=NL. Due to the shortening of the moving ruler, /c? ), so there is C=NL'√( 1-v? /c？ ); From the observation of the rotating body, if you measure n times or put n rulers on the circumference, then the circumference is C'=NL'=NL/√( 1-v? /c？ )=C/√( 1-v？ /c？ )=2πR/√( 1-v？ /c？ ), because the radial radius r is always perpendicular to the direction of motion, it can be inferred that the circumference measured by the rotating body becomes longer and the pi becomes larger.

In fact, assuming that the stationary person and the rotating person use the same ruler, the total number of rulers used by the rotating measurer S' or the number of times of placing N is greater than that used by the stationary measurer S, and the relationship is N=n/√( 1-v? /c？ ).

About the dream of five mountains: I know what you mean. You mean, at any moment, a small part of the athlete's circumference shrinks in the direction of movement. If you combine the contractions in various places, it seems that the circumference should be shortened. But the problem is that what the athlete sees is no longer a perfect circle C, but an ellipse E' that rotates with him. The diameter of the circle in the tangent direction of motion D=2R shrinks to the minor axis D'=2R√( 1-v? /c？ ); The place he measured was on the semi-axis r of the rotating ellipse e'. In one rotation, he continued to measure the exact circle C' with the diameter D=2R of the major axis of the rotating ellipse E'. When transforming to the static coordinate system, take the circle diameter D=2R perpendicular to the instantaneous motion direction as the short axis and the tangential motion direction as 2R/√( 1-v? /c？ ) is the rotating ellipse e of the long axis, not the original perfect circle c; But the static observer always puts a ruler L=L'√( 1-v? /c？ ), * * * So the measurement is completed n times, because the moving ruler L is placed on the static circle C n times, so it has a relationship that C=NL. In the eyes of a rotating body, the length of the "circle" (the big ellipse E in the static inertial system) is C'=NL'=NL/√( 1-v? /c？ )=C/√( 1-v？ /c？ )=2πR/√( 1-v？ /c？ )。

The exact circle C measured in the static inertial system is not the exact circle, but the short axis D'=2R√( 1-v? /c？ ), the long axis is an ellipse e' with D=2R, and the length of rulers varies from place to place, and L'(x, y)=√[Lx? ( 1-v？ /c？ )+Ly？ ], played dn times each, and the perimeter integral is e' = ∮ l' (x, y) dn = π r [1+√ (1-v? /c？ )]; In the static inertial system, c = nl = 2π r.

Imagine an interesting situation: first put a circle of rulers in the static inertial system, and then turn them around, you will find that the circumference becomes shorter, because the number of rulers remains the same, and each ruler becomes shorter, that is, the rotating object presents Riemann geometric characteristics with smaller pi in the static inertial system, but the static circumference that coincides with it remains the same; To the observer of synchronous rotation, the scale is not a perfect circle, but a flat ellipse. When he measured the "circumference", the pi increased, and the circumference he measured was actually a large ellipse in the static inertial system.

It can be seen that the "perimeter" of the two systems is different. The "circumference" of the rotating system is a large ellipse of the static system, and the circumference of the static system is a small ellipse of the rotating system, which leads to different measurement results. It can also be seen that the geometric characteristics such as pi are generally different in different reference systems, and the geometric measurement also changes with the reference system. In fact, the time measurement will change accordingly. In the rotating system, the spatio-temporal coordinates of the stationary inertial system based on the central fixed point no longer have practical significance to the rotating parts everywhere, and the spatio-temporal geometric characteristics such as spatio-temporal standard and pi are no longer consistent. The dynamic reason is that the acceleration is inconsistent everywhere and the geometric description is inconsistent everywhere.

The second question is a good one, because the usual explanation only emphasizes the general relativity effect of acceleration and ignores the special relativity effect of relative motion. In fact, once the acceleration process is fixed, the time delay caused by acceleration is determined, and the time delay caused by relative motion becomes the key.

Lz consideration ("If the time of the second and fifth stages is long enough! It will lead B to think that A's clock is slow enough, so that the other four acceleration and deceleration stages will cause B's clock to slow down and shift, thus making the most important A on the earth younger than B. The comparison has some truth, but the defect is that the time change of B coordinate system is not considered.

As the speed of B changes, its frame of reference changes, and the time coordinate changes accordingly. When B moves from uniform to static, its inherent time, that is, its own age, deviates from the time coordinate of the reference system, and it will be found that the coordinate time is rapidly advanced due to the coordinate transformation, and the final coordinate time will be equal to A's age, which means that B will measure that A's age will increase rapidly, which not only delays the time of B's own speed change, but also changes the coordinate system where B is located and finally transforms it to A's coordinate system.

Therefore, even if the time delay caused by the acceleration of B is not considered, due to the deviation between the coordinate system time and its own age caused by the speed change of B, when B finally returns to the A coordinate system, A will be found to be older. This can be achieved by Shi Kongtu analysis, or by referring to the following small example:

Both Party A and Party B are zero years old. Party A is on the earth, and now Party B has left the earth at high speed, and then Party B feels that it will come back one year later (Party B uses its own clock to calculate the time), but during this period, Party A's observation of Party B concludes that it took Party B a hundred years to fly back (Party A also uses its own clock to calculate the time). The question is: Who is younger now? How old is A and how old is B?

The problem can be simplified and solved within the framework of special relativity, that is, imagine such an ideal situation:

B accelerates from the earth to V instantly, turns around instantly after leaving for a period of time, returns to the earth at the original speed, and then lands instantly. Because of the instantaneous change of speed, it is not necessary to consider the general relativity effect of acceleration process, but only the special relativity effect of uniform linear motion process. The space-time coordinates at the beginning of a are the same origin of a and B.

Before B left for half a year and turned around, B measured that it had passed 0.5 years, while A only passed 0.005 years, which was 0.495 years younger than himself, because the space-time coordinate of A in the B frame of reference is (x', t')=(-0.5v, 0.5), which is equivalent to (x, t) = (0. In the A frame of reference, the space-time coordinate of A is (x, t) = (0,50), which is equivalent to (x', t') = (-5000 V, 5000) in the B frame of reference. Similarly, in the frame of reference B, the space-time coordinate of B is (x', t') = (0,0.5), which is equivalent to (x, t') = (0 50V,50) in the frame of reference A; At this time, if B suddenly stands still relative to A, its space-time coordinate system will suddenly become a reference system, and it is found that its time coordinate is actually 50 years and its age is only 0.5 years. At this time, the measured age of A is 50 years old, which is equal to the time of a reference system.

Similarly, when B left for half a year and suddenly turned around, V became -v, and the space-time coordinates of B changed through reflection. In B's frame of reference, the space-time coordinate of A becomes (x', t')=(0.5v, 0.5), which is equivalent to the instantaneous displacement of A, which is far from V at -0.5v and close to -v at 0.5v, which is equivalent to the sudden reversal of the space-time coordinate in B's frame of reference to 650. But as far as age measurement is concerned, the time-space coordinate of A's age is (x, t) = (0,50) in A's frame of reference, and suddenly changes from (x', t')=(-5000v, 5000) to (x', t')=(5000v, 5000) in B's frame of reference. That is to say, the time origin in frame B suddenly returns to 10000, and the time coordinate of A in frame B is known to return to 1 year, then the relative time coordinate of A in frame B jumps from 0.5 years to 0.5+10000-1= 9999.5v years. 9999.5), which is equivalent to (0,99.995) in the A frame of reference. The effect is that the age of A measured in the B frame of reference suddenly jumps from 0.005 to 99.995, which can be seen from their Shi Kongtu projections. So when B returns to Earth after half a year, it will be found that A's age has increased by 0.005, reaching 100 year, which is 0.5+0.5= 1 year.

Supplementary information about the landlord:

1. Gemini paradox. The landlord's understanding is basically correct, but he misunderstood my delay effect and didn't consider the acceleration-I mean, because of the existence of acceleration, there is also the delay effect caused by acceleration besides the relative motion, so the calculation result is that when A passes through 100 years, B returns less than 1 year; However, if the problem is confined to the scope of special relativity and the time delay effect of acceleration is ignored, for example, it is idealized as "instantaneous speed change", then B will come back just at 1 year. But who receives the acceleration is not uncertain because of the relative acceleration between them, because one of them can feel the change of its own motion state, that is, the existence of acceleration, while the other can't feel the change of its own motion state.

2. CD-ROM problem. How to understand that in our world, the circumference of this ruler is shorter, but the diameter is the same? It can be considered that the space between the moving ruler and us is curved, deviating from the plane space which is relatively static with us. Imagine that the plane of the turntable in our eyes is physically bent into a sphere. Although the radius from the center of the circle to the circumference (similar to the length of the spherical meridian from the north pole to the equator of the earth) remains the same, the circumference becomes shorter (similar to the equatorial circumference of a sphere being shorter than a plane with the same radius). So we can think that the rotating object is in a curved space, which can't be reasonably explained by our usual plane space.

How to understand the influence of general relativity on gravitational field, such as the clock slowing down and the scale shrinking? The best way to understand it is to analyze it with the help of general relativity equation. It can be seen that the space-time scale deviates greatly from the straight space-time because of the strength of the gravitational field. When the inherent time or distance there is converted into generalized space-time coordinates, you will see that the clock slows down. You can also use simple examples to help you understand, such as the turntable. We see that the inherent circumference of the turntable becomes smaller in the inertial coordinate system established by the center of the circle due to the scale contraction effect, which can be obtained by combining the instantaneous special relativity analysis with the same time delay. But the general understanding only stays on the simple relationship obtained by special relativity, so it is not clear whether it is the effect of special relativity or general relativity.

In fact, the analysis of special relativity is only applicable to an instantaneous point, and it is relative, that is, the special relativity relationship is applied to each other between points that coincide instantaneously in static and dynamic coordinate systems; However, when analyzing the period of a rotating body, we can't simply generalize the analysis principle of special relativity, otherwise we will come to contradictory conclusions. At this time, due to the existence of acceleration, the instantaneous local mutual symmetry at the coincidence point of static and dynamic systems fails in a large range, and only the local special relativity relationship between dynamic systems and static systems works as a whole, just like the relationship between variable speed and constant speed in Gemini paradox. Just like this, the overall general relativity effect reflects the quantitative relationship consistent with the special relativity effect to some extent, but the simple special relativity effect can not lead to the asymmetric clock slow-scale effect. Therefore, the reasoning mentioned by the landlord that "the outer edge is fast, so according to the special theory of relativity, the clock is slow, and the acceleration here is large, which is equivalent to the strength of the gravitational field and the local clock is slow" is a simple statement of the image. In fact, this is not the case, because strict analysis is abstract, so many books are loosely presented. Strictly speaking, this kind of reasoning is incorrect. It is ok to help you understand it simply, but it is misleading if it is used for logical analysis. We can see that the landlord loves to think deeply, which is very beneficial, but to understand deeply, we can't be limited to such a simple example. It is best to have a certain mathematical analysis basis of non-Euclidean geometry and general relativity, and mathematics and physics in high school and even university are difficult to understand. Therefore, only by encouraging the landlord to continue his studies can we explore it better.

In addition, the time delay of a moving object in the gravitational field has both gravitational effect and motion effect, but only gravitational effect from its subsequent generalized coordinate system and only motion effect from its local static inertial system; The special relativity effect is included in the general relativity effect as a local effect.