NASA is studying the Alcubierre drive as a possible mechanism of faster-than-light travel. Lubos Motl remarked that the drive would break the special theory of relativity:
http://motls.blogspot.fi/2013/07/relativity-bans-faster-than-light-warp.html,
though the drive is consistent with the equations of the general theory of relativity. Let us discuss if faster-than-light travel makes sense, and whether it might be possible.
Faster-than-light travel enables a time machine
The above Wikipedia article states that a faster-than-light rocket would make closed timelike curves possible, that is, the rocket could travel backwards in time. Suppose that we have a rocket that can travel at a "moderate" speed of 2 times the speed of light, relative to the frame of the Earth.Using the formulas of time dilation and length contraction:
http://en.wikipedia.org/wiki/Time_dilation,
http://en.wikipedia.org/wiki/Length_contraction,
we can calculate how we can travel back in time with our rocket.
Suppose that stars A and B are traveling at the speed 0.99c, where c is the speed of light, relative to the Earth. They move to the direction of the vector (A, B):
A ------------ 1 light year ---------- B ------> speed 0.99c
O <--- the Earth
The distance of stars A and B is 1 light year, measured from the reference frame of the Earth.
The time dilation coefficient for A and B is:
1
______________ = 1 / sqr(1 - 0.99^2) = 7.0888.
sqr(1 - v^2 / c^2)
That is, a time interval t measured at star A appears for an observer in the frame of the Earth to last 7.0888 t.
The length contraction coefficient for A and B is the same as the time dilation coefficient. That is, the distance of stars A and B in the frame of A and B is 7.0888 light years (recall it is only 1 light year in the frame of the Earth).
Suppose that we have a rocket that can move at speed 2c relative to the Earth. Let us make the rocket to fly from star A to star B. As the rocket starts from A, we also send a ray of light from A towards B.
The rocket arrives at B after 1 year of the Earth's time. But the ray of light only moves at a relative speed of only 0.01c relative to B (the speed of light observed from the Earth is always c, regardless where the light originated from). Thus, the ray of light arrives at B after 100 years of the Earth's time.
The time interval (rocket arrives at B, light arrives at B) is 99 years, relative to the frame of the Earth.
Since the time dilation coefficient is 7.0888, the time interval of (rocket arrives at B, light arrives at B) is 99 years / 7.0888 = 13.966 years in the reference frame of A and B.
But remember that the distance of A and B in their own reference frame was just 7.0888 light years. Thus, in the frame of A and B, the light arrives at B 7.0888 years later than it left A. Since the rocket arrives at B 13.966 years earlier (in the frame of A and B), we see that the rocket flew 6.877 years backwards in time!
In the calculation above, we assumed that our rocket can fly at speed 2c relative to the frame of the Earth. Since in special theory of relativity, all inertial frames are equivalent, we can let the rocket fly at speed 2c also relative to the frame of A and B. Let us make our rocket fly back from B to A at speed 2c, relative to the frame of A and B. The flight back takes 3.544 years. The rocket moved back in time 6.877 years when it flew from A to B. And it moves forward in time 3.544 years when it flies back to A. That is, the rocket arrives at A 3.433 years BEFORE it left A! Our rocket has acted as a time machine and sent us backwards in time 3.433 years on star A.
Time travel makes faster-than-light travel impossible?
If we allow faster-than-light travel only relative to the frame of the Earth or the Milky Way, then there is no time travel with respect to our frame. Ww just travel immensely fast within our galaxy.But if we allow time travel relative to any frame, like in our calculation above, then time travel is possible, and we have to face the numerous paradoxes involved with time travel. Suppose that I use my fast rocket to travel one week back in time, and then blow up the rocket before it started its journey. How can I (or my copy) start my journey one week later? The rocket was blown up, and does not exist any more!
We instinctively believe in the principle that the past cannot be changed. If the past can be changed, then there essentially does not exist the past, because the past could be changed arbitrarily in the future. Any kind of flexible time travel, where we can transport intelligent robots to carry out missions in the past, breaks the principle that the past cannot be changed.
To avoid the paradoxed of time travel, we could use the many worlds interpretation from the philosophy of quantum mechanics. In that interpretation, the universe is constanly branching according to the results of "experiments" we perform on microscopic systems. For example, in the double-slit experiment, the branching happens according to the spot where a photon hits the screen behind the double slit.
If our time travel machine takes us to another branch of the branching universe, then no paradoxes arise. In that brach I can blow up the copy of my time travel rocket, and that does not spoil my departure with the rocket which I did from another branch of the universe. But we do have the restriction that our time travel rocket cannot take us to the past of our current branch.
