If moving object decelerated into your frame, will it lengthen?

[Aziza]

OK, so let us see what we can make of this in GR context, which is what your professor seems to be indicating, making some assumptions.

We are talking about a 'hole'. Presumably, it is a hole on Earth (or some alien planet), and a T stick is moving towards it at high velocity. For an observer sitting beside and and at rest w.r.t. the hole, the stick is length contracted, while the length of the hole is unchanged.

Therefore, A & B will connect before the C & D do. Presumably, the signal from A & B traveling at the speed of light MAY reach D before C has had a chance to lengthen itself to reach D (since it must do so at less than speed of light). This is still not a definite answer, as C has a smaller distance to cover to D, than the signal from B does! Anyway, the explosion MAY not happen, as the signal to disarm could have reached D before C can. It depends on the velocity v.

If this is what your professor meant, well and good. Otherwise:
- See if anyone else in this forum can come up with a definitely better answer
- If not, ask your professor to explain in detail what he meant, and post the answer in this forum
- If he refuses to do so, ask him to show you his ivy league credentials

ahah now i am confused beyond confusion!

now i won't be able to sleep until monday when i can finally ask my professor

i am scared he's going to just confuse me more thus causing my brain to explode haha

I propose we just go back to Newtonian mechanics and throw SR/GR out the window:D

 

[Fredrik]

OK, so let us see what we can make of this in GR context,

What you said after this doesn't involve a spacetime other than Minkowski spacetime, so it's just SR.

Another question just occurred to me related to my initial question of length decontraction. Fredrik and harrylin, I understand why the longitudinal wave from the top of the T will not reach the bottom, and so i guess the T should get torn apart because the bottom doesn't know that the top stopped moving, but are you saying that the T gets torn apart in the hole's reference frame? Because if you consider the T frame or the frame in which both the T and the hole are the same length, then surely in these frames the T can't get actually torn apart...? But then if you consider "T gets torn apart" as an event, then different observers will not agree on what events actually occurred? But I don't think this is possible...?

Good point. Unfortunately, I don't have time to think this through and write a good answer right now. I will return to this later today.

 

[Dale]

Sorrys! I think its because I sent the pic from my phone to email and used the url of the location of the picture but it was https instead of just http. so it was displaying for me but i guess not for you guys. ok so i attached the picture to this post

This is essentially the same as the bug-rivet paradox, which is one of my favorites.

http://math.ucr.edu/~jdp/Relativity/Bug_Rivet.html
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/bugrivet.html

EDIT: note that the hyperphysics link incorrectly analyzes the scenario and believes that the paradox is unresolved

 

[harrylin]

Another question just occurred to me related to my initial question of length decontraction. Fredrik and harrylin, I understand why the longitudinal wave from the top of the T will not reach the bottom, and so i guess the T should get torn apart because the bottom doesn't know that the top stopped moving, but are you saying that the T gets torn apart in the hole's reference frame? Because if you consider the T frame or the frame in which both the T and the hole are the same length, then surely in these frames the T can't get actually torn apart...? But then if you consider "T gets torn apart" as an event, then different observers will not agree on what events actually occurred? But I don't think this is possible...?

Hmm, did I write that? Oops, you're right:
"ripped apart at one end, and smashed together at the other end."
That was a bit too hasty, but probably correct for most materials at high speed.

My answer in a nutshell: also the top doesn't immediately know that the bottom stopped moving.

This kind of problem requires from us to develop an intuition for extraordinary situations. Strictly speaking, "torn apart" requires material strength analysis. The most "sure" reference systems to use for material strength analysis are those in which the structure is stationary at the start of an event (event in its true meaning of happening over a short duration of time). A very brittle T will crack (be torn apart) if locally put under too much stress. That stress builds up by stretching (strain). For this local stress not to build up too much, the pulling force must not be allowed to develop too much - for example thanks to counter force from the other end. However, in this particular case the support coming from the "bottom" is in first instance just as ineffective as the trigger signal coming from the "top". For a quantitative conclusion you'd probably have to do dynamic FEA.

A safe way to explain what will happen without bringing up such complications is that at one end there will be stretching, and at the other end compression. OTOH, I guess that at γ=2, "ripped apart at one end, and smashed together at the other end" may be more appropriate and even valid for rubber.

like I said my professor is saying exact opposite..and seeing as he is Yale and Harvard and whatnot graduate, I feel like he can't possible be wrong haha

The paper that I linked illustrates that even "big shots" can make mistakes (if you did not notice this, ask me by private message or email). If your university trains you correctly, you are expected - even demanded - not to trust on opinions by others.

[EDIT]: It's a riddle to me how your professor can explain away the middle figure in the drawing. Out of curiosity, can you reproduce his reasoning?

 

[Dale]

OK, so let us see what we can make of this in GR context, which is what your professor seems to be indicating, making some assumptions.

GR is irrelevant. Gravity is not significant in this problem, so it is SR only. Please stop confusing students with unnecessary and irrelevant complications.

 

[harrylin]

This is essentially the same as the bug-rivet paradox, which is one of my favorites.
[..] http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/bugrivet.html

That's a very nice variant.

A bit shocking though, is the last sentence: after at a first glance nicely explaining the paradox, the author states: "The paradox is not resolved." Perhaps the author did not read recent publications...

Aziza, I give you an "A" for this (although that won't count for your uni) if you now explain what hyperphysics overlooks.

 

[Dale]

A bit shocking though, is the last sentence: after at a first glance nicely explaining the paradox, the author states: "The paradox is not resolved." Perhaps the author did not read recent publications...

Oh, I am so embarassed to have posted the link! Oops.

I guess I should work it out completely, but I am not sure if I will have time today.

 

[harrylin]

Oh, I am so embarassed to have posted the link! Oops.

I guess I should work it out completely, but I am not sure if I will have time today.

I hope that you won't, for it would be great to leave the opportunity to Aziza!

 

[arindamsinha]

GR is irrelevant. Gravity is not significant in this problem, so it is SR only. Please stop confusing students with unnecessary and irrelevant complications.

Yes DaleSpam, I see what you mean. My mistake.

 

[PAllen]

So I assume everyone here is agreed the explosion does happen for the written problem, and we would very much like to here how the professor could possibly argue otherwise. I will give a brief recap using the wording and hints in the problem. Then I will answer one of loose end mysteries on this thread.

1) The special reference frame is obviously the one where T and the hole are moving towards each other at the same speed. In this frame, event 1 (AB collision), and event 2 (CD collision) are simultaneous. The invariant fact is that no matter what is assumed about material behavior, events 1 and 2 are causally disconnected (spacelike separation) in every frame, even though their order must change for different frames. However, the wording of the problem requires that to avoid an explosion, event 1 must have a point on worldline of D before event 2 in its future light cone. However, since 1 and 2 are causally disconnected, all points on D's world line that are in the future light cone of event 1 are in the future of event 2 on D's world line. Thus explosion must occur.

2) One loose end that I don't see picked up is how to understand the breaking of the T (C leg pulling away (towards C) from the A bar of the T, in the reference frame of the T. I presume it is clear why the T breaks in the hole reference frame, and also pretty clear why it happens in the reference frame I describe in (1) - T leg doesn't know yet about the CD collision and moves away from the A bar that has hit at B. So on to the T reference frame. It is best to think of a piece of the hole near B. When CD collision occurs (first event in this frame), matter at B can't be influenced by it, so it continues to move left at near c. Because of the head start along T leg that B has, even assuming matter influence could travel at c, the influence of the CD collision cannot catch up to B before B hits A and breaks the T.

 

[harrylin]

[..] One loose end that I don't see picked up is how to understand the breaking of the T (C leg pulling away (towards C) from the A bar of the T, in the reference frame of the T. [..]

I discussed that in my post #34, but it's always useful to have it presented in different ways. As you made use of the picture, your explanation is likely easier to understand.

Note however that your discussion suffers from the same non-generality as my remark in post #12: it's not true that a T of any material that hits such a hole at any speed, will break.

 

[Demystifier]

Aziza, I would suggest you to show your tutor this paper
http://xxx.lanl.gov/abs/physics/9810017
Relativistic contraction of an accelerated rod, Am. J. Phys. 67 (1999) 1007.

 

[Fredrik]

...i guess the T should get torn apart because the bottom doesn't know that the top stopped moving, but are you saying that the T gets torn apart in the hole's reference frame?

I don't have time to think this through and write a good answer right now. I will return to this later today.

Sorry about not getting back to this the same day. I have thought about it now. I was a bit confused at first, because it seemed to me that in the T's frame, the T will be crushed from the front, and in the hole's frame, it will be pulled apart from behind. This would have been a contradiction. After some thought, I think all frames will agree that the front will be crushed from the front by the collision with the bottom of the hole, and that the top of the T will be torn off from the rest of it by the collision with the top of the hole. There's no contradiction here, since these two statements are referring to different parts of the T.

 

[A.T.]

I think all frames will agree that the front will be crushed from the front by the collision with the bottom of the hole, and that the top of the T will be torn off from the rest of it by the collision with the top of the hole. There's no contradiction here, since these two statements are referring to different parts of the T.

That's why it is confusing to speak about "T's frame". There is no frame where the whole T will remain at rest, during the entire scenario.

In the rest frame of the T-bottom, the T-bottom hits the bottom of the hole. Then the frame becomes non-inertial: The rest of the T suddenly accelerates down and contracts, while the previously contracted hole expands, hitting the T-top with the edges and tearing it off.

In the rest frame of the T-top, the T-bottom hits the bottom of the hole. The frame is still inertial here. The rest of the T suddenly accelerates up and contracts, while the hole depth remains constant. At some point the rest of the T has contracted below the constant hole depth, and the T-top hits the hole edges.

 

[PAllen]

I discussed that in my post #34, but it's always useful to have it presented in different ways. As you made use of the picture, your explanation is likely easier to understand.

Note however that your discussion suffers from the same non-generality as my remark in post #12: it's not true that a T of any material that hits such a hole at any speed, will break.

I allowed for speed of sound up to c. Really, I assume only that 'all causal influence is limited by c'. Then, if speed and material are such that breakage occurs in any frame, my discussion shows why it would happen in the T frame. It is obviously true that if all components are made of idealized chewing gum (for example) no breakage will occur in any frame at any speed.

 

[harrylin]

I allowed for speed of sound up to c. Really, I assume only that 'all causal influence is limited by c'. Then, if speed and material are such that breakage occurs in any frame, my discussion shows why it would happen in the T frame. [..]

Yes the speed is not given in the problem as presented in the OP. However, I still think that it's reasonable practical assumption that it will break, the way it is presented.

It is obviously true that if all components are made of idealized chewing gum (for example) no breakage will occur in any frame at any speed.

That's a guess, and your guess differs from mine (perhaps because I don't consider anything "idealized"). However, with that the discussion is drifting off-topic, and I don't want to spend time now on such theoretical material research.