# Hole-and-nail paradox in special relativity

Yesterday we started relativity on our physics class, and my professor taught us a few concepts. We did some examples on how things changed by looking them from different reference systems, and a paradox came to my mind from an example he did on the blackboard:

Suppose that we have a bug inside a hole (and suppose that the height of the bug is $$0$$). This hole is $$L$$ meters deep. There's a man outside the hole that wants to kill the bug with a nail of length $$l$$, with $$l. Obviously, it's not possible for him to kill the bug by trying to push the nail in the hole. However, if the nail moves really fast, from it's reference system the hole will have a height $$L_n, and if it moves fast enough then it will happen that $$L_n, so he finally could kill the bug.

On the other hand, the bug knows that from his reference system, the nail will have a lenght $$l_b, so it would never reach him.

Is it possible to quantify the problem? How is it possible that the nail reach the bug from it's reference system and from the bug's reference system it's even smaller than before?

Imagine a slightly different scenario: two pilots, Alice, and Bob, are in their spaceships. They move towards a tunnel of length $L$ at a velocity $v$, and remain a distance $l'$ apart. Alice is closest to the tunnel and thus enters first, approaching a wall at the end of the length of the tunnel. Just as Bob enters he decelerates, coming quickly to a halt and sending a message to Alice telling her to stop. This message takes a certain amount of time to reach her, and if she is travelling too fast, she will hit the wall before the message arrives.

This is analogous to what is happening with the nail and the rivet. The tip of the nail is Alice, and the head of the nail is Bob. This allows the bug to be crushed in its own reference frame, even if $l'< L$, and resolves the paradox.

There is an animation of this here. A more rigorous analysis can be found here. Also see Chris White's answer.

Draw a spacetime diagram. Really, there is no better way to solve relativity problems.

In the above, the nail has worldlines $1$ (back) and $2$ (front), while the hole's worldlines are $3$ (front) and $4$ (back). Let's agree that the origin $\mathcal{O}$ of the coordinates is the event of the front of the nail just entering the hole, i.e. the intersection of worldlines $2$ and $3$. Also, all my coordinates are in the order (time, space), with space on the horizontal axis, and the speed of light has been set to $1$.

In the frame of the hole (diagram on the left), the back of the nail is a distance $l/\gamma$ to the left when the front enters (event $A$). Since the worldlines of the nail are at an angle $\theta = \sin^{-1}(1/v)$, we can easily find the time at which the back of the nail reaches the hole, $l/\gamma v$ (event $B$). At this point, information that the back of the nail has been stopped travels at the speed of light ($45^\circ$ dashed line $BC$) to the front of the nail, no faster, to tell it to stop (event $C$).

Since line $\mathcal{O}C$ has formula $t = x/v$ and line $BC$ has formula $t = x + l/\gamma v$, we know the $x$-coordinate of $C$ is $$C^x = \frac{l}{\gamma(1-v)} = l \sqrt{\frac{1+v}{1-v}},$$ which for any $l$ can be made arbitrarily large by making $v$ arbitrarily close to $1$. That is, we can choose $v$ such that $C^x > L$ and in fact the collision of the front of the nail with the back of the hole (event $E$) happens before the nail stops (unlike as drawn).

A similar analysis could also have been done in the nail's frame, as shown in the diagram on the right. We know $\mathcal{O}B$ has formula $t' = -x'/v$ and worldline $1$ has formula $x' = -l$, so the front of the hole reaches the back of the nail at time $B^{t'} = l/v$. Then information propagates up the nail along $BC$, with formula $t' - l/v = x' + l$, so the front of the nail knows about the back collision at time $$C^{t'} = l (1+v)/v.$$ Meanwhile, the back of the hole reaches the front of the nail at time $$E^{t'} = L/\gamma v.$$ The bug will be crushed if and only if $E^{t'} < C^{t'}$, which turns out to be exactly equivalent to the condition $C^x > L$.

The bug will die: What does cause the nail deceleration? it's the base of the nail hitting the outside of the hole. In the best case scenario for the bug the tip of nail stops as soon as it gets the information (the shock wave from the abrupt deceleration). This will travel across the nail no faster than the speed of light. So let's assume the best case scenario again and so the tip of the nail stops as soon as a light signal is received from the base of the nail when it hits the entrance of the hole. Here you have a model that does not depend on the reference frame. It is easier to take the nail as the reference frame i think. So the hole stop moving when the light signal reach the tip of the nail. You can see that even with a Ln>L you could still crush the bug.

This is a variation of the pole and the barn paradox, and is also known as the bug and the rivet paradox, see Rod Nave's hyperphysics:

The final line of this article is "the paradox is not resolved". Some people will say the paradox is resolved via consideration of simultaneity, but I don't think it is. Another variation of the theme is when you and I are riding 2m surfboards passing each other fast with 1m butterfly nets. We can't both "scoop" each other.

All in all I think it demonstrates an interesting issue with length contraction: if you move fast past some object, you might see it as shortened. But it hasn't changed a jot. Instead you changed, along with how you see the world. And as for how the world really is, why, that's why we do physics.

PS: if somebody fired a rivet at you at 0.9c, length contraction doesn't matter. You're toast.

Edit: here's a picture of the "scoop each other" scenario. You're the guy on the left, poised to sweep your net down over the other guy, net and all. You see the other guy as length-contracted, but he sees you as length contracted too. He says he's the guy on the left, poised to sweep his net down over you and your net.

The bug lives! Since BOTH the hole and the nail will change length in the moving coordinate system and their proportions Ln/Lh will be constant, if you pick a third coordinate system that is moving at half the speed of the nail. In this system both the hole and the nail are moving at the same speed in opposite directions.