Detonator Paradox" Re-Cast: Can a Thin Wire Prevent a Bomb?

[jonspalding]

TL;DR Summary

This is Taylor and Wheeler's problem 6-5, but reconsidered in a different form.

In Taylor and Wheeler, problem 6-5 "Detonator Paradox," a T-shaped plunger is not long enough to strike a detonator inside a U-shaped container. However, if the T-shaped plunger is moving relativistically, then in the reference frame of the T, the U shape is shorter and so the T is able to strike the detonator. The paradox is that in the U-shape reference frame, the T is too short to strike the detonator.

The answer in the back of the book says that the T does strike the detonator and explodes a bomb, even though this only happens in the T-reference frame.

Suppose this is re-cast as a problem in which the T is replaced by a thin metal wire, 1 m - 1 micron in length, that needs to pass through two lasers, separated by 1 m, that cross its' path; suppose that if the two beams are simultaneously blocked, a bomb is detonated. The wire needs to go about 300 km/s for the frame of the laser beams to be shortened enough to be blocked by the wire.

Does the bomb detonate?

Note that this is the exact same problem, but in a different form in which the structural properties of the T (the wire) have no implications in the scenario. As well, perhaps this could be realized experimentally.

 

[Dale]

I am not following the setup. A diagram would help a lot.

 

[jonspalding]

I just realized the solution -- in the laser version, there is no detonation because the two laser crossings are not simultaneous in the frame of the lasers.

This is the same as the train paradox with simultaneous lightning strikes -- the lightning strikes are not simultaneous in both frames.

 

[Nugatory]

suppose that if the two beams are simultaneously blocked,

You have overlooked the relativity of simultaneity, which makes it impossible to build such a device (rather like the impossibility of finding the integral square root of a prime number).

What we can do is build a device that sends a light signal to some central point when one of the lasers is blocked, and if both signals arrive at that point together the bomb detonates. When we allow for length contraction, time dilation, and relativity of simultaneity (don’t leave any of them out, they all matter) we will find that no matter which frame we use to analyze the problem, they all agree about whether the bomb explodes or not. They won’t necessarily agree about the distance the signals travel, how long they are in flight, or crucially when the beam interruptions happened and whether they happened at the same time but all of these effects will cancel so that we end up with a consistent result.

 

[Ibix]

Summary:: This is Taylor and Wheeler's problem 6-5, but reconsidered in a different form.

The answer in the back of the book says that the T does strike the detonator and explodes a bomb, even though this only happens in the T-reference frame.

If the T hits the bottom in one frame it hits the bottom in all frames.

I think that the general point being made here is that there may be a frame in which it is obvious that detonation happens or not, and since detonation or not is an invariant fact, you can assert that other frames' analyses must lead to detonation even if such analysis is complicated.

In Taylor and Wheeler's problem, the simple case is the T-piece rest frame. The first point of contact in that frame is the detonator, so boom. Careful analysis in other frames will show the same result (edit: by "result" here I mean the "boom", not the "first point of contact is the detonator"), but you have to worry about light cones and the implications thereof to show it.

Your laser device isn't properly specified because you haven't said what frame's simultaneity is used for the detonation condition. You do imply it's the lasers' frame in your next post, so I shall assume that. In this case, as you seem to have realized, the simple frame to analyse is the lasers' rest frame and there is clearly no boom. Careful analysis in other frames will show that your simultaneity detector doesn't function as a simultaneity detector in those frames, and whatever condition it does detect is never satisfied.

 

[A.T.]

In Taylor and Wheeler, problem 6-5 "Detonator Paradox," a T-shaped plunger is not long enough to strike a detonator inside a U-shaped container.

I know this as the bug-rivet paradox. I guess a bug's life wasn't high enough stakes?

 

[PeroK]

The answer in the back of the book says that the T does strike the detonator and explodes a bomb, even though this only happens in the T-reference frame.

I'm sure the answer says that the bomb explodes. Full stop. Events don't "happen in reference frames". Events happen and each reference frame specifies the coordinates (time and space) of the event.

 

[Ibix]

I'm sure the answer says that the bomb explodes. Full stop.

For the record, the exact words are "Yes, explosion. (Sorry!)"

 

[Nugatory]

Note that this is the exact same problem, but in a different form in which the structural properties of the T (the wire) have no implications in the scenario.

google for “pole barn paradox, that‘s exactly what this is.

 

[PAllen]

A few general concepts I find important for problems (not paradoxes) like this are:

1) What happens (collisions, explosions, signal receptions, etc.) are frame independent. Descriptions of where and when things happen are frame dependent.

2) A unique, frame independent, Newtonian description is typically neither unique, nor frame independent in relativity. Specifically, relative distances, relative velocities, angles, time differences, and the property of simultaneity are all frame independent for Galilean frames. None of these are frame independent in special relativity, so the frame of specification must be indicated for each problem element. This can lead to multiple different physical scenarios in relativity, depending on which frame is used to specify initial conditions. However, (1) remains true: for each different specification of initial conditions, the result is frame independent, with different descriptions in different frames (for each of the distinct initial setups).

3) Material bodies are best thought of as molded jello floating in space, for near c relative velocities. This is actually more fundamental than that bonding is based on EM forces - the causal structure of SR forces that spacelike separated events cannot influence each other, so one part of a body cannot immediatly be affected by the displacement of a spacelike separated part of the body. Thinking about the problem as floating jello molds, with high speeds being like bullet velocity, using only Newtonian physics, quickly gets you to a correct perception of what happens to different parts of the system. Then, relativity only modifies the description of when things happen, with the sequence description being frame dependent. Note, that thinking in terms of jello molds helps avoid physically unrealizable problem setups, as well as understanding the dynamics.

4) (highly optional): You can try to recover some properties of rigid bodies using the notion of Born rigidity. However, this is a topic that is both technically sophisticated and physically unrealizable. The key concept is that for a body to change its motion rigidly, you must think in terms of each body element being independently programmed to undergo accelerations such as to avoid changes to its relative distance from any nearby elements, in its local rest frame of the moment (MCIF - momentarily comoving inertial frame). This is systematized by the notion of a timelike congruence, with definition of expasions scalar, shear tensor, and vorticity tensor. Born rigidity is the absence of expansion and shear. There are surprising restrictions on what is possible, even mathematically, encapsulated in the Herglotz-Noether Theorem(s).

Note, for example, for this problem, specifying that all elements are born rigid throughout, completely changes the problem result - the bomb is not detonated, but the description in some frames is really weird. For example, assuming the cavity undergoes no proper accelerations, then, in the initial rest frame of the T, the base of the T starts accelerating first, contracting the T base length, then the T arms start accelerating to match the cavity velocity as the cavity reaches them.