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Fuck it, let's assume we can build jump gates.
Let's say they're just big enough to send a tiny unmanned drone through.
I hop into my space ship and accelerate with a conventional engine to 86% of light speed.
No violation of physics needed, just shitloads of energy.
I fly to another star, which takes 10 years from earth's point of view.
Due to time dilation at 86% light speed, time in my space ship passes half as fast as on earth.
If someone on earth had a strong enough telescope, they could look at a clock on my ship and see that it ticks half as fast as the clocks on earth.
But in my frame of reference, earth moves away from me at 86% light speed.
So if I look at earth through a telescope, I see that the clocks on earth tick half as fast as mine.
There isn't a universal time. Time is always relative to speed and this is no problem when the reference frames are separated.
I arrive at the star, after 5 years have passed on earth.
I activate a jump gate and send the drone through with a message. It arrives on earth instantly, 5 years after I left.
But from their reference frame, they could see my clock ticking only half as fast as theirs.
After earth's 5 years, only 2.5 years have passed for the space ship they see.
They activate their jump gate and send the drone back with a reply.
It arrives instantly at the star, 7.5 years before my space ship gets there.
This is why FTL travel isn't and will never be possible. Even with tricks like jump gates or wormholes, it creates time paradoxes.
Assuming a mechanism exists that changes the universe from being singly connected to multiply connected (i.e., wormholes exist), it is possible to have wormholes permitting faster-than-light travel without time paradoxes, though some additional restrictions may apply.
source: Galactic Library