Essentially yes. Faster than Light travel is essentially equivalent to traveling to the pass changing the reference system. That's why I'm very skeptic.
But you need a more complex setup. If the two stars are static (or the relative velocity is 0) then in both trips you arrive later than the departure time, so there is no easy to spot paradox.
But let's suppose that the Sun is static and the other star is moving very fast (with a speed <c). In the first part of the travel, form the Sun point of view you are traveling forward in time with a speed than is greater than c (in spite allegedly you are in a "bubble" so you locally are traveling wit a speed smaller than c. The bubble and everything that is in it is traveling with speed >c.). From the other moving star point of view you (or the bubble) are traveling backwards in time. You reach there and then they see you approaching there.
We can compare this to sound of a supersonic plane, where the plane arrives first and then the sound arrives, but if you do the calculations the sound was emitted before the plane arrival time. But if you travel fast enough and the star is moving fast enough then the effect is different, with the correct standard calculations the light arrives later than the space ship, but the light "was" emitted after the ship arrival time. From their point of view the space ship was traveling backwards in time.
Once you are there, you only have to recalibrate all the systems to use the other star reference frame. All the inertial reference frames are equivalent, so this is easy.
And now you start the engines and travel forward in time with a speed > c in the other star reference frame. They don't see anything too special, only that you go too fast but they are unable to tell you that. But form the Sun's point of view you are now traveling backwards in time, so you arrive before you departure the other star!
So, let's peek the Sun's reference frame. You travel first forward in time and then backwards in time. Is the sum positive? If the other star is static then the sum is always positive an there are no surprises. But if the other star moves fast enough then the sum is negative and you arrive before you departure.
If you don't find a fast enough star then you can use a fast space station or a fast space ship as the other extreme in your journey. Or even use an abstract point traveling fast enough in the middle of the space.
But you need a more complex setup. If the two stars are static (or the relative velocity is 0) then in both trips you arrive later than the departure time, so there is no easy to spot paradox.
But let's suppose that the Sun is static and the other star is moving very fast (with a speed <c). In the first part of the travel, form the Sun point of view you are traveling forward in time with a speed than is greater than c (in spite allegedly you are in a "bubble" so you locally are traveling wit a speed smaller than c. The bubble and everything that is in it is traveling with speed >c.). From the other moving star point of view you (or the bubble) are traveling backwards in time. You reach there and then they see you approaching there.
We can compare this to sound of a supersonic plane, where the plane arrives first and then the sound arrives, but if you do the calculations the sound was emitted before the plane arrival time. But if you travel fast enough and the star is moving fast enough then the effect is different, with the correct standard calculations the light arrives later than the space ship, but the light "was" emitted after the ship arrival time. From their point of view the space ship was traveling backwards in time.
Once you are there, you only have to recalibrate all the systems to use the other star reference frame. All the inertial reference frames are equivalent, so this is easy.
And now you start the engines and travel forward in time with a speed > c in the other star reference frame. They don't see anything too special, only that you go too fast but they are unable to tell you that. But form the Sun's point of view you are now traveling backwards in time, so you arrive before you departure the other star!
So, let's peek the Sun's reference frame. You travel first forward in time and then backwards in time. Is the sum positive? If the other star is static then the sum is always positive an there are no surprises. But if the other star moves fast enough then the sum is negative and you arrive before you departure.
If you don't find a fast enough star then you can use a fast space station or a fast space ship as the other extreme in your journey. Or even use an abstract point traveling fast enough in the middle of the space.