Is faster than light travel possible?
The ship is not moving through space-time (but moving space-time itself) It is analogous to a surfer riding a wave; the surfer isn't moving rather the wave is moving the surfer.
The Alcubierre drive is a speculative idea by theoretical physicist Miguel Alcubierre which is based on a theoretical solution of Einstein’s field equations in general relativity. The idea is to create a spacecraft that could achieve faster-than-light travel. The ship isn't actually travelling at any speed, but rather, it's moving the space-time around it. And since space-time can move faster than the speed of light, way faster, it creates the effect of the ship travelling faster than light. So, essentially the warp drive is actually curving space-time so much that it takes space-time with it, rather than travelling through space-time. As you can see, the warped space-time that the drive creates is moving the space rather than the ship moving through space.
It is very similar to how gravitational waves work by squeezing and stretching spacetime. The gravitational waves were discovered by Ligo, so observations show that is it possible to warp space and time in nature, therefore the Alcubierre drive, in theory, could work. So, the warp drive, for example, can travel to Proxima Centauri, not in four and a half years, but rather in a few weeks. But it does get a wee bit more complex because we have to take into consideration the fact that the ship does not move at all, so the speed is quite irrelevant as essentially what is occurring is that the warp drive in away is stealing a piece of space-time and then placing it somewhere else.
You may be thinking that this violates the speed of light, but the ship isn't moving through space-time at all. This very odd propulsion system is manipulating space-time by generating a so-called warp bubble. It would expand space on side of the spacecraft and contract it on another. In this way, the spaceship will be pushed away from the Earth and pulled towards a distant star.
But
The explanation about expanding and contracting space is unfortunately an incorrect explanation of how warp drives move. Rather, it is an artefact from the times when the Alcubierre metric was proposed.
When Miguel Alcubierre's paper came out in 1994, it appeared that Alcubierre's solution needed something special to move, and coincidentally Alcubierre metric has expanded and contracted regions behind and in front of the bubble (Alcubierre paper: https://arxiv.org/abs/gr-qc/0009013, https://iopscience.iop.org/article/10.1088/0264-9381/11/5/001).
So, the explanation quickly made it to the popular domain due to, well, lazy journalism. In 2002, Jose Natario had constructed a class of warp drives which could travel like the Alcubierre drive but did not have any contracted or expanded regions around the bubble
(Natario paper: https://arxiv.org/abs/gr-qc/0110086, https://iopscience.iop.org/article/10.1088/0264-9381/19/6/308).
In other words, Natario showed that expansion and contraction have nothing to do with the motion of the warp bubble. Unfortunately, the correct explanation did not make it into the popular domain.
Erik Lentz's solution (Lentz paper: https://arxiv.org/abs/2006.07125, https://iopscience.iop.org/article/10.1088/1361-6382/abe692), just like the warp drives in the Natario class, does not need expansion and contraction to move.
In fact, the recent Bobrick paper (Bobrick paper: https://arxiv.org/abs/2102.06824, https://iopscience.iop.org/article/10.1088/1361-6382/abdf6e) shows that all physical drives move inertially, and can only accelerate by propulsion.
In summary, warp drives move because they already move, or else they need to propel, and their movement is not related to spacetime expansion or contraction.
Eric Lentz came up with a way to generate both the push and the pull force without needing to create negative energy, which as far as we know is impossible to do. Since energy and mass are interchangeable due to the well-known equation E=mc^2, and mass can warp space-time (aka it creates gravity) that in turn means that energy can also warp space-time. What warp drive designs do, contracts the space-time in front of them, aka they create a gravitational attraction in front of them, and expand the space-time behind them, aka ANTI gravity, which since there is no negative mass and there is no way to create any significant amount of negative energy, is impossible.
Lentz warp drive design is arranged in such a way that the ship is propelled forwards without needing the negative energy or reverse gravity behind it. Not all scientists are convinced of his designs validity, but he maintains his belief in it’s authenticity. So, for now, the warp drive remains theoretical.