Is this the best SR paradox?

By Brian Clegg

I’ve just read Jim Al-Khalili’s excellent new book On Time (review out next week) in which he spends a little time on the special theory of relativity.* He brings up a couple of the well-known paradoxes that arise, specifically the twins paradox and the barn doors paradox – but it reminded me of my absolute favourite of the genre: the rivet and the beetle.

I’ve seen paradoxes defined as contradictions that can’t be true, but I think a much more appropriate definition for physics is situations that appear to involve a mind-boggling contradiction, that the physics tells us really is the way things are. And special relativity is full of them. I got this one from Andrew Stearne.

Here’s the scenario. We’ve got a table with a 10mm deep hole in it. At the bottom of the hole a beetle is happily beetling about, unaware that we are about to fire a rivet into the hole. The good news is that the shank of the rivet, the bit that will go into the hole, is only 8mm long, leaving room for our (rather small) beetle to feel safe and snug.

Unfortunately, though, the rivet is fired towards the table at a fair percentage of the speed of light. Stearne tells us the rivet’s speed by saying that γ is 2, which doesn’t really give you an idea of how fast its travelling, but if my back of an envelope calculations are right, this is around 0.87 times the speed of light. Quite a fast rivet, then.

Now one of the weird effects of special relativity is that an object moving at high speed is squashed up in the direction of travel as seen by an observer. So from the bug’s viewpoint the rivet won’t by 8mm long, it will be just 5mm long. ‘Wow,’ thinks the bug, ‘what was I worried about?’ But before it sits back and starts reading the newspaper, there’s something it needs to consider. Relativity works both ways. From the rivet’s viewpoint, it’s not the rivet that’s moving, it’s the table. This means from the rivet’s view it remains 8mm long – but the hole is contracted and is now only 5mm deep. Squish goes bug.

So what really happens? (I use ‘really’ loosely. Let’s face it, Wickes does not sell a 0.87 times the speed of light rivet gun.) Is the beetle somehow both live and dead, in the manner of the famous quantum cat? Sadly no. It’s squish all the way.

Let’s follow the event from the beetle’s viewpoint. Down comes the rivet and slams into the table. At the moment before the impact the rivet is still just 5mm long as far as the bug is concerned. But here’s the thing. Just because the head of the rivet has come to a sudden stop doesn’t mean the whole rivet does. A wave has to pass along the rivet to its end saying ‘Stop!’ The end of the rivet will just keep on going until this wave, typically travelling at the speed of sound, reaches it. That fast-moving end will crash into the beetle long before the wave arrives. It will then send a counter wave back up the rivet and after a degree of shuddering will eventually settle down as an 8 mm rivet in a 10 mm hole. Too late, though, for that bug.

If you think this remarkable, check out Jim’s book for plenty more time-based mind boggling.

* Jim calls the special theory ‘special relativity’, which I have no problem with, but I know it drives John Gribbin round the bend, because he says that it’s the theory that’s special, not the relativity.

Image from Unsplash+ by Elantro

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Now Appearing is the blog of science writer Brian Clegg (www.brianclegg.net), author of Inflight Science, Before the Big Bang and The God Effect.