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FTL and Time Travel Paradoxes |
Here's a discussion of (instantaneous) FTL and causality that shows how to make a time machine (and hence results in time-travel paradoxes).
I need the help of my friends from the Canadian Space Corps to help me violate causality. I arrange for Alice to move out into space, and then assume a position at rest relative to me. Bob has to fly past me at event C, and Carol has to fly past Alice at event B. Things are arranged so that Bob and Carol are moving with the same speed, and both are moving in the direction from me towards Alice. At event A I use my ansible to transmit a message instantaneously to Alice at event B. Alice gives a copy of the message to Carol, who is flying past at the time that Alice receives my transmission. Now, Carol and Bob are in a different inertial frame to Alice and me, so they have a different slice through spacetime that they consider the present. This means that if Carol transmits the message to Bob with her ansible, then it will arrive at event C, just as Bob's ship if flying past me. Bob can then give me a copy of my message. The net result is that I have transmitted a message into my own past.
For any faster-than-light system there will be some inertial frame in which it appears to act instantaneously, and so the same argument may easily be applied to any such system.
Notice that you have to use the ansible (note) twice to make a causality violation. You might think that things would be okay if your system only worked once in the entire history of the universe, but this won't work - in general there is no way in which to decide which use of the system was the first. Another possible way to avoid causality violations is to say that all the ansibles only work if they are at rest relative to each other, so that they share the same simultaneities. This, however, defines a preferred frame, so you'd have to throw away the principle of relativity.
I should note that this violation of causality does not involve a closed timelike curve (parts of the message's trajectory are spacelike).
Related links:
Author's Afterword:
I've just posted to my weblog the article on FTL and causality that I've been threatening people with for a while. Here's the URL:
http://www.theculture.org/rich/sharpblue/archives/000089.html
I think it's worth the effort to install the SVG plugin for IE if you don't already have it, because I'm proud of the diagrams. (Unfortunately, Mozilla seems to have broken its plugin interface recently...)
