Relativity & Surroundings: Train within a Train Theory

[Morkike]

Hi been curious about a question. If a train were to be designed within a train within a train within a train... etc... then couldn't the speed of light be reached and surpassed? could this work in theory - what would the main engineering problems which would have to be overcome? and also what affects would this have on the person in central train after let's say a 1 day trip? thanks

 

[DrGreg]

No, it wouldn't work because the velocity relative to the ground of a train within a train is not obtained by adding the velocities, but by the formula$$w = \frac{u + v}{1 + \frac{uv}{c^2}}$$

u = velocity of outer train relative to ground
v = velocity of inner train relative to outer train
w = velocity of inner train relative to ground
c = velocity of light in vacuum​

If u and v are both less than c then so is w.

 

[mfb]

Related: xkcd #675

And it does not matter how fast the trains run and how many you stack, the formula given by DrGreg will always give a speed below the speed of light (you can apply it multiple times for stacks).

what would the main engineering problems which would have to be overcome?

Air resistance, the absurd centripetal force required to keep the train near the surface of the Earth as trains need the speed of light go around the Earth 7 times per second.
A 40000km vacuum tunnel... and more energy than all power plants on Earth produce in a year, if the train has at least ~1000kg of mass and should travel at something like 90% of the speed of light.

and also what affects would this have on the person in central train after let's say a 1 day trip?

No person could ever survive this due to the involved accelerations.