Does this break relativity? A humming bird is flying 30 km/h inside of a train

Question

If I am in a high speed train, and I am sitting there, and I see a hummingbird flying $ 30 km/h $ relative to me towards the head of the train.

We all know, the scenario above is 100% possible by physics. I can measure it flying away at $ 30 km/h $.

However, by Galileo, we can't really tell what speed the train is going. No matter what speed the train is going, we can say it is going at $ 200 km/h $, or $ 600 km/h $, or close to the speed of light. It just depends on what the train is moving relative to.

So if I consider relative to a certain point in the universe, the train is going near the speed of light, let's say $ c - 20 km / h $ where $ c $ is the speed of light, and by the fact that the humming bird is moving faster than the train, then the hummingbird is going at a speed of, as @Dan suggested, using the velocity addition formula, $ v $, where $ v $ is getting closer to $ c $, but cannot exceed $ c $. For simplicity, let's say $ v $ is something like $ c - 2km/h $ or $ c - 3km/h $, just a speed that is greater than $ c - 20 km / h $ but less than $ c $.

So from my perspective, the hummingbird is $ 30 km/h $ faster than me, but by relativity, the hummingbird is no more than $ 20 km / h $ faster than me.

So doesn't this have a contradiction? What should be adjusted in the above statements?

Answer 1

Relativity is fine! You have stumbled across the fact that velocities in special relativity don't add as in the Galilean case. Instead, we must use the velocity addition formula, which guarantees that velocities indeed never exceed $c$.

Answer 2

You and the train cannot actually move at the speed of light. If you and the train are slightly slower than speed of light as viewed from the ground, the the entire train is slightly larger than zero length. The front of the train, the back of the train, you, and the hummingbird are almost at the same location (along the axis of the train's motion). The hummingbird moves quickly past you in your reference frame because the start and end positions are far apart. Because the train never gets this close to the speed of light, the above situation is a limit. When perhaps half the speed of light, the distances, lengths, and times as viewed from the ground compress so that your speed and the hummingbird's speed are much closer together from the ground. This is what leads to the velocity addition formula. Calculate $\frac{\Delta x'}{\Delta t'}=u'$ and $\frac{\Delta x}{\Delta t}=u$ with $v$ as the velocity of the primed system relative to the unprimed system.