Because (at least, according to special and general relativity) when two events U,V are simultaneous (in the sense of “the time coordinate is the same) in one frame, then from another frame U is before V (in the sense of the time coordinate in that frame’s coordinate system) , and in a third frame V is before U.

Hm, that’s probably not a very clear explanation..

Ok, suppose you depict time and one dimension of space on a grid, where traveling at c is depicted with a line that makes a 45 degree angle with the space axis. Draw two such lines crossing at the t=0, x=0 point. If you apply a “hyperbolic rotation” aka a “boost”, keeping the origin fixed, these lines will stay fixed, and each point in one of the 4 quadrants made by these lines will remain in that quadrant.

The upper quadrant consists of all events in the absolute future of the event at t=0, x=0 . The bottom quadrant consists of all events in the absolute past of the event at t=0, x=0 .

These boosts are what happens when one changes one’s reference frame by changing velocity. Of course, changing one’s velocity cannot change what events can have a causal effect on what other events, as it only changes one’s perspective.

Any line segment that you draw that would be faster than light speed can be hyperbolically rotated so that the endpoint has an earlier time coordinate in the boosted frame than the start point.

Whereas, the points in the upper quadrant always have positive time coordinates.

Now, if you only allowed one instance of FTL ever, that might not cause any cycles in causation? Maybe even if you only allow it in one direction? But those don’t seem particularly plausible. If you have it in both directions, and they aren’t separated enough I guess, then one could combine those line segments with ones that go at 45 degrees (I.e. light speed) in the forward-in-time direction, to make a loop.

Uh.

Ok so specifically these hyperbolic rotations, in units where c=1 , are t’ = t cosh(theta) - x sinh(theta) and x’ = x cosh(theta) - t sinh(theta)

Uhh…

Ok I think if FTL was only possible if the signals are still forward in time with respect to one fixed preferred reference frame, then there’s no issue. It is just when it is possible in both directions in a variety of reference frames that it results in a loop.

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