Rocket faster than light? In this article, wikipedia describes a constantly accelerated rocket, assuming special relativity :
$$ x(\tau) \;=\; \frac{c^2}{a} \left(\cosh \frac{a \ \tau}{c} -1 \right) $$
The proper time $\tau$ is less than $\frac{d_0}{c}$, where $d_0$ is the distance to the foreign star. For example, Alpha centauri is 4.37 light-years away from Earth, and the constantly accelerated rocket arrives there in 3.6 years (including the deceleration on the second half of the trip).
Doesn't this mean that, from the rocket's perspective, Alpha centauri moves faster than light ?
Let $d(\tau)$ be the distance from the rocket to Alpha centauri, as perceived by the rocket. That's the length of the spacelike geodesic orthogonal to the rocket's 4-speed, joining the rocket and the star. $d(0)$ is 4.37 light-years, when the rocket leaves Earth and $d(3.6)$ is zero, when the rocket reaches the star. By the mean value theorem, at some proper time $\tau$, $|\frac{dd}{d\tau}|>c$.
 A: While $dx/dt$ is subluminal, $dx/d\tau$ can be superluminal. This is due to time dilation. 
To take a simpler example, suppose a ship travelled to a distant star at a constant speed satisfying  $\beta \gamma >1$.  Then  $c\beta \gamma $  is the ratio of the traversed distance from the perspective of people on Earth to the journey's duration from the perspective of people on the ship. Those watching may die of old age even if the crew doesn't.
A: No.
Just as in classical mechanics, in relativistic mechanics objects other than you move with respect to you with the same speed as you are seen moving with respect to them.
The way in which the "you" onboard the rocket reconciles this is the phenomenon of length contraction: both the distance between the objects as well as the objects themselves are shrunken in the direction of your motion by the Lorentz factor $\gamma$. Thus while they are not moving toward you any faster than $c$, you have less distance to cover to get to them from your point of view. You cannot see "external" objects as moving faster than light any more than someone outside can see you as moving faster than light, as that would violate the symmetry mandated by the absolute relativity of motion, creating a distinct state of absolute motion and absolute rest that could be distinguished by determining who sees things moving faster than light and who doesn't.
