Length Contraction and the Bomb Paradox

[HungryChemist]

Question "Length Contraction"

Hello, I just finished reading special theory of relativity written by Albert Einstein himself. Even though I understand the book is only written to convey general idea it still left me with a lot of question. Einstein said that any event that occurs in one reference frame should occur in another reference frame whether simultaneously or not.(may be he didn't phrase this way) While I was thinking a lot about what he said, I came up with one situation and it starts to puzzle me so much. Please, consider the question and tell me what I have misunderstood. Sorry if I speak bad English.

Question

There is two people namely A and B. A will be riding a train that has length L. B is standing on the ground next to the train rail devising a bomb to kill A for some reason. B makes the bomb finally and this is how the bomb works. The bomb has two switch connected to the main body. Only when the two switchs are in physical contact with the train's end point the bomb will be activated. (Bomb will not work if only one of the switch is being in contact to the one end of the train. Bomb will not work if two switchs are being contact to the train where it is not the end point of the train).

So, the person B places the two switches on the rail track L meters apart knowing the train has length of L. But then he quickly realizes the length contraction of the train(Since the train will be moving with certain velocity v when it passes by him). So he calculates the new length L' that is shorter than L using special relativity and places the two swtches L' apart from each other and waits for the train to come.

Now, in the view of B, the bomb should work and blow the train. But let's look at this whole set up in the view of A. For A, he will see the bomb approaching to his train with certain speed. A knows how the bomb works. He quickly sees the two switches of the bomb on the rainway and realize the length of the distances bewteen the two switchs are shorter than L(since now the length contraction will be observed by A). So A will conclude with absolute certainty that he is going to survive.

So, will A survive? or will A die? is my question. Remember, the train is never accerlates respect to ground, so I don't think it is similar to that of twin paradox. And I feel somehow that this is little different than the barn paradox but can't tell how exactly they are different. Feel free to reply to this post. I am only to learn.

 

[jcsd]

Someone asked a very simlair question:

https://www.physicsforums.com/showthread.php?t=38116

You have to realize that the condition: "Only when the two switchs are in physical contact with the train's end point the bomb will be activated" will also transform between frames i.e. in one frame it will be as stated above but in another frame it will be: "Only when switch 1 is in physical contact with the train's end point and then time t later switch 2 is in physical contanct with the train's endpoint will the bomb will be activated".

This may not be that obvious, but it becomes more obvious when you look at how such a device as the triggering mechanism will be activated, foir example in order for the 2 switches to 'know' that they are both depressed, they must communicate with each other across space (so lemngth contraction plays a part) which will take time (so time dialtion also plays a part).

 

[zefram_c]

This is a simultaneity measurement - are the contacts at the two ends triggered simultaneously or not? This, of course, depends on who makes the measurement. So if the bomb is placed on the ground and waits for the incoming train, it will explode. If the bomb is traveling on the train, it won't. A and B will agree on what the bomb, wherever it is, will do.

 

[zefram_c]

Nice! According to PF, our posts were almost simultaneous!

 

[jcsd]

Depends which frame of refrence your posting from

 

[Zanket]

A different way of putting jcsd’s answer:

Presumably when you say, “Only when the two switchs are in physical contact with the train's end point the bomb will be activated” you mean, “Only when the two switches are in physical contact with the train's end points simultaneously in B’s frame the bomb will be activated.”

Let’s say that in B’s frame the end points of the train simultaneously hit the switches at precisely noon. Then we know that from A’s frame the end points of the train simultaneously hit the switches at precisely noon on B’s clock. From either perspective (frame) the bomb explodes.

Now it is true that in A’s frame the end points of the train do not hit the switches simultaneously on A’s clock. But the bomb isn’t rigged to care about that.

Affix clocks all along the track; these are B’s clocks, at rest with respect to B. A observes that these track clocks, at any given moment for A (simultaneously for A), all show different times (after adjusting for the time it takes the image of the clock to reach A—in relativity discussions the word observe implies that you’ve made such adjustment). The track clock with the latest time is the one furthest away from A in the direction A is headed.

Affix clocks all along the train; these are A’s clocks, at rest with respect to A. B observes that these train clocks, at any given moment for B (simultaneously for B), all show different times. The train clock with the latest time is the one furthest away from B in the direction A came from, which is the clock at the back of the train.

For further study check out the ladder paradox (also called the pole in the barn paradox) http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node80.html and elsewhere on the web. This paradox is similar to your thought experiment.

 

[Zanket]

Also if you want to learn relativity then I suggest you put Einstein’s book aside in favor of contemporary layman’s versions. An excellent one (if you get only one, get this one) is Relativity Visualized. Another is Understanding Einstein's Theories of Relativity: Man's New Perspective on the Cosmos. Both can be found at Barnes & Noble.

 

[HungryChemist]

Thank you for the posts. I've come to understand still with many questions though. A friend of mine who is majoring physics just agree to give me his book called Intro. Relativity written by French. I think I will stick to that one for now. ^^
Thank you people!