## Physics Friday 138

Here is a classic special relativity exercise: the “detonator paradox.” Suppose we have two objects, one shaped like a “T” and the other resembling a “U”

The T fits into the U, and comes just short of touching the “inside” of the U, where we have a pressure switch; this pressure switch is connected to the detonator for some explosives.

Now suppose we start with the T some distance away from the U, and send the T toward the U at a relativistic speed , so that γ=2. In the lab frame, the U frame, the U is at rest, and the T, moving at v, undergoes length contraction to half it’s rest length, leaving it clearly too short to reach the detonator.

However, in the T’s frame of reference, it is at rest and the U approaches at speed v. In this frame, the U is length contracted to half, and so the T can easily reach the switch, and trigger the explosion.

So, is there an explosion, or not?

Let us label the ends of the U as “A” and the switch as “B”, and label the arms of the T as “P” and the end of the T as “Q”:

Let us now draw spacetime diagrams for our frames. First, in the U frame,

we see that A meets P, the “stopping” event where the arms of the T meets the arms of the U, before Q can reach B and trigger the explosion.
Now, draw the T frame diagram:

Here, Q meets B, and the explosion is triggered, before A can meet P.
So we still have the paradox. What is wrong?

The answer is that we are treating our T and our U as classical rigid bodies, so that the entire body comes to a stop when one portion of it is stopped by collision with another. However, one can see that this requires each portion of the object to instantaneously “know” that another portion has experienced a stopping force; this violates special relativity’s restriction that information can only travel at the speed of light. In fact, the “stopping force” propagates through the object at the speed of sound in the material, which cannot exceed the speed of light. So we revisit the U frame. Here, although the arms of the T stop when they meet the ends of the U (A meets P), Q continues on; until a speed of light signal from AP can reach Q, the Q will still travel at velocity v. From the time of AP in the U frame, it only takes Q a time of  to reach B, while a speed of light signal from AP takes a time of  to reach B

So an explosion is definitely triggered. This is an example of what people mean when they say that “there are no rigid bodies in special relativity.” (See here for a little history on this point.)