relativity - How do tachyons violate causality?

Something moving faster than light should have imaginary mass, like photons have zero mass and thus travel at $c$.

I have read this article of physicist E. C. George Sudarshan.

He said taking mass to be imaginary we get real energy and momentum. (I think I have understood that.) However, if something moves at a speed greater than $c$, its proper time $\tau=t\sqrt{1-v^2/c^2}$ becomes imaginary (where $t$ is coordinate time).

Does this imply causality violation? Does imaginary time mean time is going backward?

Answer

The conclusion derives from looking at how tachyon signals would behave as seen in slower-than-light inertial frames, not from trying to consider a tachyon's own "time" (if you can call it that, since a tachyon's worldline would have to be space-like, not time-like)--basically, it's a consequence of the relativity of simultaneity. It can be shown that any signal that moves even slightly faster than light in one frame would move instantaneously in some other inertial frame, and if the first postulate of SR applies to tachyons, then if it's possible to send a message instantaneously in one frame, this must be possible in all inertial frames. Likewise, if a signal travels instantaneously in one frame, there must be some other inertial frame where it actually travels backwards in time (i.e. the event of the signal being received occurs before the event of it being sent), and if that's possible in any frame, it must be possible in all frames too.

If you're familiar with the basics of SR spacetime diagrams and how a surface of simultaneity in one frame looks tilted in other frames, then you could take a look at the helpful explanation with diagrams on this page, which shows how the ability of two different slower-than-light observers to send signals that move instantaneously as seen in their own frames implies that they can bounce a message back and forth, and in each one's frame the other one's signal is going backwards in time, so that the message gets returned to the original sender at an earlier point on his worldline than the point where he sent the message, a clear causality violation in all frames. You could also take a look at the tachyonic antitelephone article which goes into more detail, with equations and a numerical example.

Note that this answer is really about why the ability to send FTL signals would violate causality--as other answers have noted, in quantum field theories there could be tachyons that would be impossible to use for transmitting information, in which case no causality violation would occur (the last section of this article has a helpful discussion about why tachyons in QFT wouldn't be usable for information transmission).