The real question is therefore not, whether a space-time solves einstein's equations but whether the distribution of mass and energy required to make the solution to the equations is physically reasonable.
And for the alcubierre drive the answer is multiple NOs.
First, as I already said it requires negative energy, second, it requires a huge amount of that, third, the energy is not conserved.
Instead, what you actually do when you write down the alcubierre space time is that you just assume you have something that accelerates it beyond the speed of light barrier.
That it's beyond the barrier is why you need negative energies and that it accelerates is why you need to feed energy into the system.
Please check the info below the video for technical comment about just what I mean by energy conservation here.