Look at the different solutions for the 'Twin Paradox'.
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html
Its best to work out the problems. There are no paradox in relativity.
As for time travel into the future. This is definitely possible. Instead of using a rocket speeding away at some fraction of the speed of light I will use Einsteinian orbits to give an example. From the Schwarzchild metric ( a solution on Einstein's equation) you can derive an expression for dt/dt with the numerator being the wristwatch time of the frame of the orbiter and the denominator being the far-away time lapse recorded by the Schwarzchild bookkeeper. The ISS orbit Earth frame dt is a very close approximation of the Schwarzchild bookkeeper dt so we can say the orbiting ISS will be dt in our example.
Lets say that gravitational physicists have solved all the problems related to developing a Warp drive and we have a prototype at rest with respect to the inertial frame of the ISS orbit around Earth. The Warp 1 frame (inside the bubble) remains at rest with respect to the ISS during its journey to a black hole at Cygnus X-1 where the Warp 1 drops out of warp and free falls to establish a knife edge orbit slightly greater than the photon sphere at r = 3M (M is in meters, geometric units).
set r* = r/M
dt/dt = (1-3M/r)1/2 = (1-3/r*)1/2
Our knife edge orbit will be at r* = 3.00000001
and the total wristwatch time that Warp 1 remains in that knife edge orbit will be 172,800 seconds (2 Earth days).
So
dtISS Earth orbit = dtWarp 1 during knife edge orbit/(1-3/r*)1/2
= 172,800 seconds / [1-(3/3.00000001)]1/2 = 2,993,133,456 seconds.
There are 3.156x107 seconds per Earth year. Dividing this into the result for dt gives the result that approximately 95 Earths years of ISS wristwatch time have elapsed while Warp 1 was in the knife edge orbit (2 Earth days wristwatch time) slightly greater than the photonsphere. So when Warp 1 returns to Earth they will have traveled ~ 95 years into Earths future. This example is based on Einsteinian orbits and both the SR and GR effects are accounted for. |