One wormhole with stable end points never causes time travel. You get a privileged reference frame type of FTL travel, which is completely safe. The issues arise when both spaceships have the ability to go faster than light in their own reference frame.
OK, so set up 4 starbases (called A, B, C, D), each 100 light hours from each other in a stable reference frame. They have a wormhole from A to D.
Wormhole from 1 light minute south of A, to 1 light minute south of B, travelling at 0.99c northwards according to the starbase reference frames, both ends in the same reference frame as each other, but in a different one to the starbases.
Everyone looks at A for the time, it says "time = 1200h" at A, B sees "time=1100h", C sees "time=1000h", D sees "time=900h", but they know the distance so can work out that it is currently 1200h, so all set their clocks, which run at the same time as they are in reference frames.
Event occurs at 1200h at starbase B.
Event is seen at starbase A and C at 1300h.
How can you use the wormhole to pass information back in time. Assume both ends of each wormhole are in the same reference frame, but aren't changing reference frame (no acceleration)
First off I have to admit I'm having a bit of trouble nailing down the exact math, but I'll do my best.
If we arrange the starbases on the north-south line, I believe that the roaming wormhole would allow a ship to observe the event at starbase B, then jump through and pop out next to starbase A at roughly 1115h. (If that number is wrong, then adjust the wormhole velocity I guess.)
In this scenario, you can't violate causality. You're 100 light hours from B, and the A<->D wormhole is too far away to help.
But if you changed it so that B, C, and D are only 10 light hours from each other, you could then take the wormhole to D. Now it's roughly 1116h, and you're only 20 light hours from B. You can send a signal about the event at 1200h, and it will arrive at 1136h. You sent information back in time.
starbase B looks at starbase A, sees it's 110h, but the distance is 10h, so sets their clock to 120h
starbase C sees starbase B's clock as 110h but knows it's 10 light hours away and A as 100h and 20lh away, so sets their clock to 100+20h or 120h
starbase D also sets their clock at 120h
event occurs at 200h wall clock in starbase B
event is seen at 210h wall clock in starbase A, 210h in C, 220h in D
OK, your spaceship could tell starbase D the event has occured when it pops out at t=201h, but so what. How does starbase D get a message back to starbase B back before to get them to prevent the event from occuring?
Where did you get 201h? And you completely changed the layout?? I didn't want you to change the distance between A and B.
But I can make something work with the new numbers. Here:
At 120h according to the starbases, everyone sets their clocks, just like you said.
Event occurs at 200h in starbase B.
A near-light-speed wormhole is passing by B and D in its own reference frame. As far as it can tell, the starbases have their clocks synced really badly. In this wormhole's reference frame, it's simultaneously "200h" at starbase B and "182h" at starbase D.
A messenger ship launches from B right after the event, goes through the wormhole, and lands on starbase D at 183h.
So far, we're paradox-free. We can tell D the results of the event before the light reaches them, but that's not new. People do that all the time with the normal wormhole. And since we're 20 light-hours away from B, any signal we send from here would arrive too late to affect anything.
But then the ship takes the normal wormhole, the one that's always connecting A and D. That wormhole always agrees with the clocks onboard the starbases. The ship enters at 183h, and exits at 183h, now at starbase A.
Then the ship sends a signal toward B. The signal arrives at B at 193h. This is the same location as the event, 7 hours before it happens.
