An interesting question about the effects of time dilation

Let's say we build a train track around the earth running from the Arctic to the Antarctic and back again. A train is going to run on this track at a speed of almost the speed of light lets say $$299,792,457 \;\text{m/s}$$ ($$1 \;\text{m/s}$$ less than the speed of light). We have observers on the ground observing the train from outside and passengers inside the train acting as observers inside and also looking outside through the windows. Let the train start moving from today. What does special relativity say about the observations of the observers and the clock changes involved? Also, let's say a girl starts running inside the train with $$3 \;\text{m/s}$$ speed. Now it would be impossible for her to break the light speed barrier since special relativity says that the speed of light is the same in all frames. So how would be the observations of all observers regarding her?

• Regarding the girl's speed, see here & here.
– J.G.
Jun 19 '21 at 7:10
• You need to be very careful about what you are asking here . Are you asking how fast are the clocks in one frame ticking , in the frame of the other clock ? Or are you asking how fast do the clocks in one frame "APPEAR" to tick to the observers in the other frame. These are different because of the travel delay of light Jun 19 '21 at 8:26
• FWIW, at 299,792,457 m/s the Lorentz gamma factor is ~12243.211323 Jun 19 '21 at 10:06

What physics says is that what you have proposed is strictly impossible. The energy required to accelerate the train could not be delivered to it. Gravity would not be sufficiently strong to constrain the train to the track. The train would burn-up with friction long before it reached a tiny fraction of light speed. The passengers would die as a result of the acceleration required. Nobody would 'see' anything because the speeds involved would cheat the human eye. And so on and so on.

However, to simplify matters, let us suppose the train was passing a long platform in space, and let us ignore all the practical difficulties.

Let us also assume there are clocks all along the platform and clocks all along the train.

To the passengers on the train, the clocks on the platform would all appear to be running slow and out of synchronisation. The people on the platform would have exactly the same opinion of the clocks on the train.

The girl walking on the train at 3m/s would be appear to be almost stationary (forgive the pun) in the train by the people on the platform, so she would not be breaking the speed of light from their perspective.

• What about a maglev train Jun 19 '21 at 6:58
• I am talking about a thought experiment. the train may not be a conventional train. it is a hypothetical train. Instead of train we can also think of a space ship over the planet. Jun 19 '21 at 6:59
• Sagnik, have you considered the g-forces that would be involved if you had a spaceship circling the planet at nearly the speed of light? It is utterly utterly utterly unrealistic. Stick with a spaceship travelling in a straight line to begin with. Jun 19 '21 at 9:55
• @SAGNIKUPADHYAY How does maglev help? The centripetal acceleration is a little over 1.4385 billion g. Jun 19 '21 at 10:12
• You see? Even with gritting their teeth and gripping the arms of their chairs as tightly as they could, the passengers would struggle to withstand 1,438,500,000g. The coffee cups and newspapers would be all over the place. Jun 19 '21 at 10:26

SITTING PASSENGERS:

What they see outside:

• looking back, the light they observe is so much redshifted (lowered in frequency) that their eyes see nothing. Not even expensive radio equipment could detect those such long waves.
• looking forward, the light they observe is so much blue shifted, that it's in the gamma ray range, destroying their DNA and probably destroying much more than that.
• looking sideways, they finally observe visible light, just that it's not distinguishable - it changes so fast that they see constant gray. This gray is just a thin line. As soon as the eyes move a bit backwards from the line - huge redshift; forwards from the line - huge blueshift.

How they are seen from the outside: The reverse applies. They are observed as very thin gray lines passing at such high speed that they are blending into the background, so fast that it's not visible with the eye. But many observers die because of the high gamma rays from the train.

GIRL MOVING INSIDE THE TRAIN: She is not observed from outside as surpassing the speed of light, because at such high speeds, her 3 m/s don't just add up with the + sign to the train speed. The formula is more complicated and makes her speed gain insignificant.

CLOCKS: From the outside, people wonder why they keep detecting those gamma rays from the train, even after 1 billion years. That's an amazing lifetime for the conductor and the passengers. From the inside, ironically, there's not much time for anything. The train has barely reached its cruising speed and suddenly the sun grows into a red giant melting the tracks and everything and bye-bye.

Edit: Good point from Marco regarding acceleration (at least centripetal kind) - if the train goes around the Earth, it would have to bind itself to Earth with so much energy, to keep it from escaping into space, that the huge energy would add a huge mass to the Earth turning it into a black hole. And so on, an uncountable number of strange effects occur when you go to the ultimate extreme.

• thats one hell of an answer. Jun 19 '21 at 7:05