Exploring the Paradox of Faster-Than-C Travel

[ehrenfest]

Even if someone could travel faster than c, could he return before somewhere before he set out? Could she arrive at some intermediate destination before she set out? Draw a spacetime diagram representing this trip?

OK. Obviously the first question is no because in a spacetime diagram your wordline would just get closer and closer to the x-axis the faster you travel.

However, I am confused about the point of second and third questions. The second question is obviously no for the same reason, but what is it getting at? And how can you draw a spacetime diagram since it is impossible?

 

[Dick]

It's not so obvious. If you can travel faster than c, then your origin and destination define a spacelike interval. That means that there is another inertial frame in which the origin is in the future of the destination. So the traveler could boost into that frame and then by traveling faster than c, could return to the origin before he left.

 

[ehrenfest]

Wow. That's true. So my argument holds only when you stay in a single inertial reference frame. I am still confused about what the question wants me to draw. How do you represent that in a spacetime diagram?

 

[Dick]

Don't really know what they want you to draw, but from the viewpoint of the stay at home observer, it's pretty simple. He travels out on a spacelike line, then adjusts his frame and returns on another spacelike line joining him with a point t<0 on the observer's timeline. You can see that the restriction that he can only reach spacelike points with t>0 is artificial, by adjusting his frame he can reach any spacelike point.

 

[ehrenfest]

Don't really know what they want you to draw, but from the viewpoint of the stay at home observer, it's pretty simple. He travels out on a spacelike line, then adjusts his frame and returns on another spacelike line joining him with a point t<0 on the observer's timeline. You can see that the restriction that he can only reach spacelike points with t>0 is artificial, by adjusting his frame he can reach any spacelike point.

Okay. He starts at rest in S. Let's say his (maximum) velocity is 3/2c. Then he travels 3/2c meters in a second and then stops. t is now 1 in S and some very small number for the traveller. How can you be sure that there are points with t < 0 that are accessibly to him? Is there a maximum distance he can travel before points with t<0 become inaccessible to him?

 

[Dick]

Because he's now spacelike separated from his origin. I thought we agreed there were frames in which his origin is in the future of his current location. Try a Lorentz transformation at his current position and see what's accessible at the origin. Though I'm not sure it's worth messing around with the numbers too much.