# Length Contraction Paradox Scenario

1. May 1, 2014

### bmbuncher

Hi! I have been pondering a scenario involving a paradox with length contraction. I brought it up with my physics professor, and I somewhat understand what is supposed to happen, but I'm still somewhat confused, so I was wondering if you could help me figure out what is going on.

In this scenario, a meter stick is travelling at relativistic speeds above the flat ground. On the ground, there is a setup where two laser tripwires (consisting of a laser emitter and receiver), each located one meter apart, are lying in the path of the meter stick. When both lasers are tripped simultaneously (as in the receivers simultaneously detect that there is no light entering it), an electrical signal is transmitted that turns on a light. So, from what I understand, when the meter stick is travelling extremely fast, it experiences length contraction (say, it is travelling fast enough to contract to 0.9 meters), and so it will never obstruct the path of both lasers simultaneously; therefore, in the frame of reference where the lasers are still and the meter stick is moving, no electronic pulse will be created, and so the light will not turn on. However, in the frame of reference that the lasers are moving and the meter stick is still, the distance between the lasers will contract (say, to 0.9 meters), so both lasers will be obstructed simultaneously, and so the light will turn on.

When I posed this to the professor, he said that the observation in both frames of reference should be analogous because causality must be preserved; therefore, the light will either turn on in both scenarios, or will not remain in both scenarios. He said that, while the length contraction will occur in both scenarios, the time dilation caused by special relativity should "balance this out" so that the same result occurs regardless of the frame of reference. However, I am still somewhat confused. In this scenario, how does time dilation cause the light to behave similarly in both frames of reference? Can anyone explain what is going on in this scenario that prevents causality from being violated? Thank you!

2. May 1, 2014

### phyzguy

This is the well-known "Pole in the Barn Paradox". If you google search on this phrase you will find many explanations for how it is resolved by correctly considering the relativity of simultaneity. Here is one of them.

3. May 1, 2014

### Staff: Mentor

This is one element of your scenario which is not really present in the standard "pole and barn" paradox. To help resolve your scenario, in addition to all the good info you will get by considering the pole and barn paradox, you might want to think carefully about exactly how the setup you describe here, that turns on a light if both lasers are tripped "simultaneously", will be realized physically. (Hint: "simultaneously" is frame-dependent; which frame is meant? And how do you physically detect "simultaneity" in that frame?)

4. May 1, 2014

### bmbuncher

So if I'm understanding this correctly, simultaneous events are not necessarily simultaneous because the Lorentz transformations cause different locations experience different effects (I know that is somewhat unscientific, but I haven't spent a huge amount of time investigating the mathematics behind special relativity). Therefore, when both lasers would be covered because of the length contraction of the distance between the lasers, the event still may not occur because of a new requirement for simultaneity due to time dilation. Is this correct? Thank you!

5. May 1, 2014

### Staff: Mentor

The correct statement is: pairs of events that are simultaneous in one frame are not simultaneous in other frames.

Not really.

Let me repeat the suggestion I made in my previous post: consider the frame in which the laser triggers are at rest. You have said that when the two lasers are triggered "simultaneously", an electrical signal is sent that turns on a light. How, specifically, do you propose to do this? How is the signal sent? From where to where? And since there must actually be at least two signals (one from each laser), how do you determine that both are sent "simultaneously"? Think carefully about how you would actually design the apparatus to do all this.

6. May 1, 2014

### bmbuncher

I see. My thought is that a receiver would trigger when the light is interrupted like a normal laser tripwire would activate. Both of these would feed into an AND gate, which would send an electrical pulse when both were active simultaneously; however, I do not know enough about electronics and logic gates to really understand what is going on in this scenario.

7. May 1, 2014

### Staff: Mentor

Okay, but the light interruption is happening at two spatially separated locations, one meter apart. How can that trigger a single receiver? Wouldn't you need two of them?

Where is the AND gate located in space? How does the information about the light interruption get to where the AND gate is? How does the gate detect when both lights are interrupted "simultaneously"?

You don't need to; I'm not trying to get into the details of logic gates. An AND gate for combining the two signals is fine, once both signals are at the same spatial location; the question is how to get to that point. I'm trying to get you to think specifically about how each piece of the apparatus is laid out in space and how the signals required to make it work have to travel in space and time.

8. May 1, 2014

### bmbuncher

I'm sorry, it would be two receivers. Each of these receivers would send an electrical signal into the AND gate at a separate location when the individual laser is covered. When both are covered, an electrical pulse would leave the AND gate and turn on the light. The AND gate and wires leading from the lasers are still in relation to the lasers. The AND gate would be created using electronic components, as detailed in the link below. Does that help at all?

http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/and.html

9. May 1, 2014

### Staff: Mentor

Where is the separate location? How much time will the signals from each receiver take to travel from the receivers to the separate location?

And, once you've got all this clear in the frame in which the receivers (and, presumably, the AND gate as well) are at rest, transform to the frame in which the moving meter stick is at rest. What do things look like there? How does the transformation affect the time it takes for each signal to reach the AND gate, now that both receivers and the gate are moving at a relativistic speed?

10. May 1, 2014

### bmbuncher

Oh, I see! It never occurred to me that the electronics would be affected, but that does make sense. I'll do some research on Lorentz contractions to figure out how to apply the equations (I've never used them before), and if it still doesn't make sense, I'll ask my questions here. Thank you!

Last edited: May 1, 2014
11. May 1, 2014

### Staff: Mentor

The electronics are affected, but that's not really the key point. The key point is the signals that have to travel from the receivers to the AND gate.

12. May 1, 2014

### Bill_K

Peter, I dissent. I think you've misunderstood the problem. It's just Pole in the Barn.

13. May 1, 2014

### Staff: Mentor

I agree the Pole and Barn scenario captures most of it; but in the Pole and Barn scenario, the key question is how the pole can fit in the barn even though, in the pole's rest frame, the barn is shorter. Here the key question is how the two signals from the laser triggers can *fail* to light up the light even though, in the meter stick's rest frame, the distance between the triggers is shorter than the stick, so in that frame, both triggers are triggered simultaneously. To resolve that question, at least if you want to actually fully understand *how* things work in the meter stick's rest frame, just considering relativity of simultaneity is not enough; you also have to consider how the signals from the triggers actually travel to a common location, which is something that isn't necessary in the standard Pole and Barn scenario.

To put it another way, the standard Pole and Barn paradox can be resolved just by pointing out relativity of simultaneity; but this "paradox" additionally requires you to think about how "simultaneity", in any frame, is actually *tested for*, physically.

14. May 1, 2014

### bmbuncher

I see, that makes sense. I think Peter is right, in that the main question here is not how the meter stick will behave when experiencing length contraction; rather, it is how to resolve the appearance that the light will be on or off depending on which reference frame is used, which, with my limited knowledge of causality and how that affects the scenario, should be impossible. Is that what you're saying?

15. May 1, 2014

### nearlynothing

Say you can determine to infinite accuracy if the meter stick blocks the lights simultaneously.
Still, this refers to simultaneity on one frame of reference, when you go to the rest frame of the meter stick, you can see the lights are blocked simultaneously "in the rest frame of the meter stick", but things are arranged so that the light turns on only when the meter stick blocks both lights simultaneously on the rest frame of the detectors.

16. May 1, 2014

### Staff: Mentor

I'm saying that the light must be either on or off in all reference frames, yes. (In fact I telegraphed the answer in my response to Bill_K; the light stays off.) The light being on or off is a direct observable, and direct observables must be frame-independent. The question is how to explain why the light stays off from the viewpoint of the meter stick frame, even though in that frame, both lasers are triggered simultaneously.

17. May 1, 2014

### Staff: Mentor

And *how* are things arranged so they work that way? That's the question.

18. May 1, 2014

### nearlynothing

I'm having problems understanding why the "how" you determine this simultaneity is the key question here.
When you think about the pole in the barn paradox, how do you then determine when both ends of the pole are inside the barn? what is the detection apparatus used there?
The detection is just a means to make it apparent to us that something is somewhere at a given time on a given reference frame, but wether you detect this or not doesnt change the physical reality.

19. May 1, 2014

### Staff: Mentor

There doesn't have to be one, because the positions of the pole don't cause anything else in the scenario to happen. In the scenario under discussion here, that's not the case: the positions of the ends of the meter stick directly cause the light to turn on (or not).

It doesn't change the physical reality of the paths through spacetime of each end of the pole (or meter stick). But if the scenario is set up so the detection of the positions of the ends of the meter stick has other consequences (like whether or not a light turns on), then how the detection is done *does* make a difference for physical reality.

20. May 1, 2014

### PAllen

Well, in my favorite formulation of the Barn Pole paradox, there is detection. In the barn frame, the doors close momentarily, enclosing the pole. The fact they are both closed, and the fact that they were timed to close at the same time per the Barn frame, is then what must explained in the pole frame. You then get into the question of how the doors were preprogrammed to close and open, specifically, how the clocks controlling them were synchronized.

This example is very similar, but the difference is a different mechanism for probing simultaneity. Instead of pre-programmed closing and opening of doors, you have communication of beam interruption to something that determines simultaneity. And the thing to model is how the beams are both interrupted in both frames, but if the simultaneity detector is set up to work in the laser beam/detector frame, it finds the interruptions non-simultaneous. Meanwhile, in the rod frame, the interruptions are simultaneous, but the moving detector (having been set up per the laser frame), determines otherwise. All facts are readily explained in both frames, but the details are interestingly related to how you propose to detect the simultaneity of the interruption.

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