If you’re a listener of this podcast, you’re probably aware of Einstein’s Theory of Relativity, and its strange consequences for objects traveling close to the speed of light. In particular, such an object will appear to have its length shortened in the direction of motion, as measured from its rest frame. It’s not a huge factor— where v is the object’s velocity and c is the speed of light, it’s the square root of 1 minus v squared over c squared. At ordinary speeds we observe while traveling on Earth, the effect is so close to zero as to be invisible. But for objects near the speed of light, it can get significant.

A question we might ask is: if some object traveling close to the speed of light passed you by, what would it look like? To make this more concrete, let’s assume you’re standing at the side of the Autobahn with a souped-up camera that can take an instantaneous photo, and a Nissan Cube rushing down the road at .99c, 99% of the speed of light, is approaching from your left. You take a photo as it passes by. What would you see in the photo? Due to length contraction, you might predict a side view of a somewhat shortened Cube. But surprisingly, that expectation is wrong— what you would actually see is weirder than you think. The length would be shorter, but the Cube would also appear to have rotated, as if it has started to turn left.

This is actually an optical illusion: the Cube is still facing forward and traveling in its original direction. The reason for this skewed appearance is a phenomenon known as Terrelll Rotation. To understand this, we need to think carefully about the path a beam of light would take from each part of the Cube to the observer. For example, let’s look at the left rear tail light. At ordinary non-relativistic speeds, we wouldn’t be able to see this until the car had passed us, since the light would be physically blocked by the car— at such speeds, we can think of the speed of light as effectively infinite. Thus we would capture our usual side view in our photo. But when the speed gets close to that of light, the time it takes for the light from each part to travel to the observer is significant compared to the speed of the car. This means that when the car is a bit to your left, the contracted car will have moved just enough out of the way to actually let the light from the left rear tail light reach you. This will arrive at the same time as light more recently emitted from the right rear tail light, and light from other parts of the back of the car that are in between. In other words, due to the light coming from different parts of the car having started traveling at different times, you will be able to see an angled view of the entire rear of the car when you take your photo, and the car will appear to have rotated overall. This is the Terrell Rotation.

I won’t go into the actual equations in this podcast, since they can be a bit hard to follow verbally, but there is a nice derivation & some illustrations linked in the show notes. But I think the most fun fact about the Terrell Rotation is that physicists totally missed the concept for decades. For half a century after Einstein published his theory, papers and texts claimed that if you photographed a cube passing by at relativistic speeds, you would simply see a contracted cube. Nobody had bothered carefully thinking it through, and each author just repeated the examples they were used to. This included some of the most brilliant physicists in our planet’s history! There were some lesser-known physicists such as Anton Lampa who had figured it out, but they did not widely publicize their results. It was not until 1959 that physicists James Terrell and Roger Penrose independently made the detailed calculation, and published widely-read papers on this rotation effect. This is one of many examples showing the dangers of blindly repeating results from authoritative figures, rather than carefully thinking them through yourself.

And this has been your math mutation for today.

References:

- https://en.wikipedia.org/wiki/Terrell_rotation
- https://en.wikipedia.org/wiki/Length_contraction
- https://www.math.ubc.ca/~cass/courses/m309-01a/cook/terrell1.html
- http://cosmic-horizons.blogspot.com/2013/11/seeing-length-contraction.html
- https://www.revolvy.com/main/index.php?s=Terrell%20rotation&item_type=topic