## Wednesday, November 21, 2012

### Light Rail Transit

If I’m on a train going the speed of light and walk from the back to the front, what happens?

I recently got a fun question that highlights a seeming contradiction in what we’ve been taught in physics lessons. On the one hand we’ve been told that nothing can go faster than the speed of light, but on the other hand we’ve been told that speeds in the same direction can be added up. So what’s really going on here?

Normally if you’re riding a train going 100 km/h and you walk forward at 5 km/h along the train, those speeds can be added up, depending on the reference point you’re using. Someone sitting on the train would see you walking 5 km/h forward and the outside moving 100 km/h backwards, but someone sitting outside would see you as moving forward at 105 km/h relative to the ground. Similarly, a jet taking off from a moving aircraft carrier gets a helpful speed boost, and firing a gun from that jet massively increases the speed of the bullet.

The problem is that the speed of light is strange. In 1905 Albert Einstein suggested that the speed of light is independent of the speed of the light source and of the observer. That’s insane! No matter how quickly you’re moving towards or away from light, its speed will never change (but its colour will).
This also suggests that simply adding up speeds no longer works when we’re close to the speed of light. Fortunately a fun formula exists that relates speeds at relativity, and it looks something like this:

What’s fancy is that this equation gives the results we'd expect for speeds that are substantially slower than the speed of light, but that no matter how high the speeds are, the result of the equation is still less than light (as long as neither individually is faster than light).

Sadly, the question that’s been asked can’t be solved straight-up because nothing with mass can go faster than the speed of light (trust me, science has tried). But we can still try with something close enough.

The fastest speeds humans have achieved are at the Large Hadron Collider when a proton was accelerated to 99.9999991% the speed of light. That’s only about 10 km/h slower than the supposed universal speed limit. The fastest a human has ever run was Usain Bolt at 37 km/h. Using the equation from before, instead of topping out 27 km/h above the speed of light, Usain Bolt actually only ends up speeding up to 99.9999991000001% of the speed of light. Not really a lot of progress.

As a bonus answer, even though the speed of light isn’t exceeded in this example, other cool things happen to the train. A passenger on the train sees Bolt travelling at 37 km/h, but an observer watching outside would only see him moving one millimeter per hour faster than the speed the train is already moving. A messed-up consequence of this is that the train would appear to flatten to one ten-thousandth of its original length due to Lorentz Contraction. And if that doesn’t boggle your mind, nothing will.