Lately faster-than-light travel seems to be cropping up in physics, in at least three places; FTL neutrinos, maybe; and nonlocal entanglement; and closed timelike loops in general relativity. FTL plus the Lorentz transformation implies time travel, which is notoriously paradoxical. In this post I consider these paradoxes, formalize some solutions, and propose experiments to test the theories.
Time travel has these problems; time paradoxes and time loops. Take for instance the so-called Grandfather Paradox. What if a time-traveler kills his own grandfather, does he exist or not? If yes, then no; if no then yes.
Well, I dislike this formulation of the paradox, partly for its sensationalism. Brutality in tale-telling is counter-educational; it evokes anxiety not thought. Also missed in this gratuitously-violent story is the logical exit; the killer grandchild killed the wrong man. A more rigorous version of the tale would be a Time Suicide, who kills himself in the past; does he exist or not? The Time Suicide Paradox, though still too crazy, is closer to pure paradox than the flawed Grandfather.
There are two main theories of time paradoxes; I call them the "odds-bending hypothesis" and the "multiple-worlds hypothesis". In the odds-bending hypothesis, there is one consistent world-line, and improbable events prevent time paradoxes. So the Time Suicide's gun always jams. In the multiple-worlds hypothesis, there are alternate time-lines, and meddling in the past merely meddles with an alternate past. So the Time Suicide kills an alternate self from an alternate world-line, and thus prevents that other self from later going back to kill him. (Arguably a case of self-defense.) You could also have a compromise between the two theories; miracles happen, or the universe splits, whichever is more probable.
To test these theories, you'd need time-phones. Consider this thought experiment:
Program a time phone to receive a bit from the future, invert that bit (1 to 0 and 0 to 1), and store the result as bit B; but also program the time-phone to later transmit bit B into the past. And let the entire apparatus be connected to the power supply through a single, but highly reliable, circuit-breaker. When the device is activated, what output do you measure?
If the odds-bending hypothesis is true, then you measure nothing because the circuit breaker _always_ trips, shutting down the apparatus and preventing the experiment from running at all. If the multiple-worlds hypothesis is true, then you measure a 0 or a 1 because the alternate you in the other time track measures a 1 or a 0; which one is indeterminate in principle because both happen. Or perhaps sometimes the circuit breaker trips, and sometimes the apparatus gives a seemingly-random output; whichever is more probable.
So either odds bend or output seems random, or a mix. Odds-bending has obvious technological uses; just substitute any desired improbable result for the circuit-breaker. For instance, the time-phone paradoxes itself unless an ordinary rock emits heat energy from spontaneous fission. Multiple worlds and random-seeming output are hallmarks of quantum mechanics; so that is the result I tend to expect; but on the other hand, I suspect that universe-splitting is a rare process, governed perhaps by the density of dark energy. Since that density is off by a factor of 10^100, that's my guess for the splitting probability. So you can bend odds down to one in a googol, but no more; that's my theory; and it's testable, given time-phones.
That's how to test time-paradox physics. You can also test time-loop physics. Time-loops are self-consistent cycles of causation; they are allowed in general relativity, and hinted at by nonlocal entanglement. There are many examples in SF of time loops; for instance "By His Bootstraps" by Heinlein. Time-loops in SF usually involve data from the future; where did it come from?
So consider this experiment; two time phones, A and B; A transmits to B the negation of the bit it had received from B; B transmits to A the negation of the bit that it had received from A. AB transmits either 01 or 10, but there's no prior reason to predict either. Which do you measure? Is it a 50-50 distribution? And does the circuit-breaker trip more often than usual?
If information can come from a time-loop, seemingly spontaneously, then you can use this effect to compute and to decode. So consider this experiment: wire a network of time-phones into a time circuit that is programmed to be self-consistent only if the phones transmit the password to DSK's bank account. When you turn on the phones, do you always break in? Or does the circuit breaker always go off? Or sometimes one, sometimes the other; and if so then with what probability distribution? What are the odds that the raider gets raided?