05 Bell's spaceship paradox

    Hello. Today we will look at Bell's space paradox. Relativity has a lot to do with common sense. Changes in length, time, and mass are also exciting, but there are many paradoxes that are hard to understand logically. I think the most disturbing thing is the "Bell's spaceship paradox".

In general, the paradox of relativity can have various opinions. Until someone has experimentally verified it, it is difficult to judge who is right. Experimentation of relativity theory must be done in extreme environments, so it is difficult to see even if there are various opinions. But today's talk about Bell's space paradox is a bit different. In this paradoxical story, even if only the experiment of thought is carried out, the obvious logical flaw is revealed.

Bell's paradox is this. There are two spaceships on the Earth's hangar. The specifications of these two spaceships are the same, and the pilots are twins, so all conditions are the same. But the two spaceships are connected to each other by a thin string. If the distance between the two ships is a little further away, this string will be cut off.

Figure 1. Two spaceships connected by a thin string

Now these spaceships have flown away from Earth with exactly the same propulsion. Gravity conditions are ignored. What if an observer on earth observes two spaceships? Answering this question is the heart of this paradox. Here are some answers we can answer. Let's try it on a multiple choice basis.

1. There is no change.
2. There is no change in the length of the string, and the length of the spaceships has been reduced due to Lorentz contraction.
3. Both string and spaceship are shrinking in length due to Lorentz contraction.
4. The spaceship shrinks in length by the Lorentz contraction, but the length of the string increases.

Figure 2. Four possibilities

Some people think that there will be no change, as in the first interpretation (1), because of the assumption that sometimes the two spaceships move at a certain distance. If relativity is not applied here, it will be correct, but if relativity is applied, the length will change, so the first interpretation (1) should be excluded.

Then, it is situation (2). The spaceships, which are rigid objects, were reduced in length by the Lorentz contraction, but some might think that the distance between the two spaceships would be constant. However, this is not supported by the current theory of relativity. In this Bell's spaceship paradox, many scholars have a heated debate, but no one claims to be like (2). Above all, if you look at the spacetime diagram, you cannot make the same conclusion as (2).

If you accept the fact that it is purely Lorentz contraction, it would be natural to think that it would be like (3). If the spaceships are shrinking and the string is reduced, everything is logically correct. However, there is one problem that can be interpreted in this way. This logic conflicts with the interpretation of the Minkowski spacetime diagram. The picture below is the world line of two spaceships.

Figure 3. The distance between two spaceships when stationary
and the distance between two spaceships when moving

When stationary, the distance between the two spaceships is AB, but if the spaceships are moving it will be the distance PQ. The problem is this distance PQ. According to the correct spacetime diagram interpretation method, the length of PQ is always greater than the distance of AB. I will not explain it here. This is about the spacetime diagram itself, so the explanation would be too long. 

Most scholars agree that distance PQ is greater than distance AB. Most scholars such as Dewan, Beran, Bell, Petkov and Franklin agree here. It is not my claim that PQ's distance is greater than AB. Clearly, many scholars argue. Jerrold Franklin has kindly induced the equation. 3)

Jerrold Franklin's "space expansion"

Looking closely at the equation derived here, the length we already know is different from the contraction. Compared to the length contraction we knew, it is rather the opposite. The equation is derived in the form of an expanding length. It's the first time you see an increase in length or distance. Let's name it something here. Let's call it "space expansion." Dewan and Beran, who first proposed this paradox, divided the length into two parts as follows. 4)

[1] The distance between the two ends of a connected rod

[2] The distance between two objects which are not connected but each of which independently and simultaneously moves with the same velocity with respect to an inertial frame.

So, let's call [1] the general 'Lorentz contraction' and [2] the 'space expansion'. For this reason, most of the scholars agree that the two fast-moving spaceships are like the fourth interpretation (4). If you plot this opinion, contraction and expansion must be applied differently. Since contraction and expansion are applied differently, the string is broken as shown below.

Figure 4. General interpretation of Bell's spaceship paradox

What do you think of this? This is a question of relativity, but this leads to a philosophical question of existence. It also raises the topic of relativity and causality. 

I will focus on solving this paradox today. According to scholars claiming that the lines are broken, the contents are as follows. In the case of a rigid body, Lorentz shrinkage is applied, and the space between the two points is subjected to "space expansion". Let's draw this picture.

Figure 5. Applying different laws of physics to an object

These claims are not justified on two points. First, 'There is no ideal rigid-body in this world.' As you may well know, it is the empty space of most of the atoms. And the atomic nucleus, which occupies most of the mass of the atom, is actually almost empty space. So, where is the object that can be applied to the Lorentz contraction in the world?

Second, how can three objects (two spaceships and one string) move together with different relativistic effects? Are the particles that make up atoms and strings like intelligence as humans? How do they separate themselves and apply the laws of physics? It can never be. Rather, it is a violation of 'the impossibility of division of electrons in quantum mechanics'. (In quantum mechanics, it is assumed that all electrons cannot be distinguished from each other. Essentially, atomic bonding is also explained by this principle.) Do the atomic nuclei discriminate themselves? Can quarks be different by themselves? Is there any difference between atoms forming string and atoms forming spaceships? Why do different laws of physics have to be applied? If so, what is the basis for that?

Now let's look at the right interpretation. The solution to this paradox is simple enough. The truth is simple. The logic is simple, so there is justification. Only one condition is required to solve this paradox. It is not just a length contraction, but a length expansion. 

Figure 6. Bell's paradox as interpreted as length expansion: There is no tension on the string

Figure 7. Length expansion applied to spacetime diagram

This is shown in the spacetime diagram as follows. The spaceships also expand and the string also expands. As everything expands, the string does not get tension. So, the string does not break. In addition, the spaceships and thin string are subject to the same laws of physics. Although not listed here, this paradox leaves a heavy theme: 'Coordinate transformation and reality', 'Relativity theory and causality'. However, these things can be interpreted simply by interpreting the expansion of the length. I will post on this issue later. Thank you for reading this post.

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