Alcubierre Drive: Special Relativity Effects, Doppler Shifting, & More

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I know there are quite a few questions about this drive due to the recent news but I have a question that I haven't seen in the other threads. Say we have a working ship that has already formed the bubble of space-time distortion and accelerating towards the speed of light starting from some arbitrarily small velocity.

Do the inhabitants of the spacecraft experience any sort of special relativity effects? Since one could go faster than light, they would be able to go somewhere and land then look back at themselves traveling, well the light they emitted anyway, right? Would that light be Doppler shifted or is there something that prevents you from observing such a ship, would it not emit light outside the bubble?

Maybe these questions are nonsensical since I don't understand that much about the drive. Either way, I'd like to know more about the effects of actually traveling somewhere and then arriving there faster than the speed of light, specifically utilizing the Alcubierre drive.
 
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Strictly speaking, the Alcubierre travelers aren't moving faster than light, they're arriving at their destination before a light signal that left at the same time as them but followed a "longer" path through spacetime. You might try "www.physicsforums.com/showthread.php?t=637052" thread for some discussion of the sorts of things that could happen as a result.
 
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The problem with ftl travel is it generates paradoxes. A craft traveling at 2c would arrive at its destination before it launched [by its clock], according to the usual GR formula. Imagine launching from Earth in a vehicle traveling at 2c that has a rear oriented laser signaling device pointing at a mirror on the launch pad. Since your craft is traveling at 2c, the pulse from the rear laser can never catch up with your craft until it stops. Hence, your view of the rear pointed signal can never be obstructed by your space craft.
 
Assuming the impossible, even though the Alcubierre drive does not directly violate the laws of physics, it does have difficulties that may be impossible to overcome. One such problem states that the only way such a transport system is possible is if, like a train, it followed a pre-set path that had been laid ahead of time. To complicate matters, this "track" must also be laid at the speed of light; essentially requiring that an Alcubierre drive would have to precede itself in order to create an Alcubierre track. Since each preceding Alcubierre drive would have to precede itself, it does not seem possible that the Alcubierre track could be created. Rather it might just remain stationary like a black hole until acted upon by other gravitational influences. In other words, it could only be guided toward a direction from outside its buble of SpaceTime.
 
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