I Faster-than-Light Causality-Flip: Thought Experiment Reference?

[Mikeal]

TL;DR

Bullet and train thought experiment

A few years ago I was studying special relativity and came across a thought experiment that explored a faster-than-light causality-flip. It consisted of an observer on the track-side firing a bullet at a speeding train, with the bullet entering the back of a carriage and exiting the front. There was a passenger in the center of the carriage witnessing the same event. The question was what combination of bullet speed and train speed would case the carriage-bound observer to see the bullet exiting the front before they saw it entering the back. Does anyone have a link to that though experiment reference?

 

[Ibix]

Either exceeding $c$ will do it in some frame - which is impossible. The so-called "tachyonic anti-telephone" is a related thought experiment, which demonstrates the kind of "kill your own grandfather" paradoxes you could get if it were possible.

 

[PeroK]

TL;DR Summary: Bullet and train thought experiment

A few years ago I was studying special relativity and came across a thought experiment that explored a faster-than-light causality-flip. It consisted of an observer on the track-side firing a bullet at a speeding train, with the bullet entering the back of a carriage and exiting the front. There was a passenger in the center of the carriage witnessing the same event. The question was what combination of bullet speed and train speed would case the carriage-bound observer to see the bullet exiting the front before they saw it entering the back. Does anyone have a link to that though experiment reference?

Suppose we have a train traveling at speed $v$ in some inertial frame and a bullet moving at speed $u > v$ in the same direction. If we take the origin to be when the bullet enters the back of the train, then the bullet emerges from the front of the train at:$$t = \frac{L}{u-v}, \ x = \frac{Lu}{u - v}$$Where $L$ is the length of the train in this frame. We can transform that event to the rest frame of the moving train to get the time coordinate:$$t' = \frac{\gamma_v L}{u - v}\big (1 - \frac {uv}{c^2} \big )$$First, just to check SR, we can take the speed of the bullet $u \approx c$ (or consider a beam of light instead of a bullet and have $u =c$). In which case, we get:$$t' = \frac{\gamma_vL}{c} > 0$$And we see that the order of events is preserved across both frames (and, indeed, any other frame) for all valid speeds.

Suppose, however, we naively allow $u > c$. If $u$ is large enough, then we can make $\frac{uv}{c^2} > 1$ and get $t' < 0$. Thereby reversing the order of the events in the train frame.

This shows that FTL speeds and the Lorentz Transformation do not in general preserve causality. E.g. if in this case we consider the bullet fired from the back of the train, then we have a physical contradiction between the two frames.

 

[Mikeal]

Suppose we have a train traveling at speed $v$ in some inertial frame and a bullet moving at speed $u > v$ in the same direction. If we take the origin to be when the bullet enters the back of the train, then the bullet emerges from the front of the train at:$$t = \frac{L}{u-v}, \ x = \frac{Lu}{u - v}$$Where $L$ is the length of the train in this frame. We can transform that event to the rest frame of the moving train to get the time coordinate:$$t' = \frac{\gamma_v L}{u - v}\big (1 - \frac {uv}{c^2} \big )$$First, just to check SR, we can take the speed of the bullet $u \approx c$ (or consider a beam of light instead of a bullet and have $u =c$). In which case, we get:$$t' = \frac{\gamma_vL}{c} > 0$$And we see that the order of events is preserved across both frames (and, indeed, any other frame) for all valid speeds.

Suppose, however, we naively allow $u > c$. If $u$ is large enough, then we can make $\frac{uv}{c^2} > 1$ and get $t' < 0$. Thereby reversing the order of the events in the train frame.

This shows that FTL speeds and the Lorentz Transformation do not in general preserve causality. E.g. if in this case we consider the bullet fired from the back of the train, then we have a physical contradiction between the two frames.

The above derivation confirms the result I came up with a while ago and explains the steps involved. Thanks.