The "problem" with General Relativity is that it is so general it can describe arbitrary universes exactly, but doesn't in itself provide a way to tell if an exact description is a good match to our universe. So we have to use some heuristics to reject candidates.
For example, general curved spacetimes have any number of space and time dimensions. General Relativity will let you exactly solve the geometry of a universe with one dimension of space and one of time, or ten dimensions of space and two of time, and so on. However, we have excellent (and so far never contradicted) evidence that our entire observed universe must be in one of a very particular family of general curved spacetimes, those that are Lorentzian. Lorentzian spacetimes have three dimensions of space and one of time. When we examine a solution to the Einstein Field Equations and see only two dimensions of space, or twenty-five dimensions of space, we can reject it as unphysical.
Next, there are an infinite number of possible Lorentzian solutions to the Einstein Field Equations, and we can make some very silly ones. We can reject any that are spatially smaller than a breadbox, for example, or which have a total existence of about ten seconds, or in which literally nothing ever happens for eternity. Indeed, we can reject a Lorentzian spacetime as unphysical if its properties could never produce galaxies (since we see those, and live in one) and the conditions for creating galaxies includes chemistry, nuclear physics, and so on. There are lots of possible Lorentzian universes that can produce "us", but lots more that cannot, and we can reject those as unphysical (because we exist; this is a flavour of the Anthropic Principle).
Relativists, however, like to work with simplified models to make calculations tractable, even if those simplified Lorentzian universes are obviously unphysical. Most of them are vacuum solutions (i.e., there is zero matter content at all, so therefore no galaxies), or use some very simple splash of matter (the Standard Cosmology models the matter content of our Lorentzian universe as a handful of smooth fluids, ignoring the relatively small-scale lumpiness of stars and planets). These models are manifestly unphysical, but [a] still usefully approximate observed physical systems and [b] typically can be made to more and more closely approximate our physical universe (or at least a region of it) perturbatively, meaning that we can add things to the simple model making it much less simple and a little more in accord with observation and experiment.
But how do we connect a manifestly unphysical model with an apparently physical one? Heuristically, again. Among the things we expect to decide whether a Lorentzian spacetime is physical or not are various energy conditions [1]. When a region of some arbitrary Lorentzian spacetime is determined to violate some energy condition, its physicality is in doubt. It may be possible to rescue such a region of spacetime in some cases, but most of the time one would not bother. "Seemed promising until I found it violated some energy condition at galactic scales (kiloparsecs)" can still be good for developing intuitions or numerical methods or whatnot, and might even work as an effective theory -- reasonably modelling our physical universe -- at a much smaller scale (like in a star system, at the scale of tens of astronomical units or microparsecs) or a much larger scale (like at the scale of tens of billions of light-years or gigaparsecs).
"Warp drive" solutions similar to Alcubierre's are vacuum solutions which are almost always a bump function on an otherwise perfectly flat Lorentzian spacetime. For short trips from near some outer planet of our solar system to near some outer planet of a nearby star system (scale of parsecs), the presence of gravitating matter nearby the path of travel can be treated as small perturbations on this model. So heuristically, we look OK. However, the "warp bubble" in these solutions are eternal features of the spacetime, and that's both a giant warning sign of unphysicality (is any complex structure eternally old in our universe, given that everything we know of is consistent with a hot big bang in the finite past?) and also completely useless for the purposes of on-demand travel. So, we ask the question: how does one go from a vacuum flat spacetime to a spacetime with a suitable bump function?
Here one takes a "sources-first" approach, where sources are matter and anything else that generates nonzeroes in the stress-energy tensor that sits on the right hand side of the Einstein Field Equations. As a slogan one would put this as "matter tells spacetime how to curve". So what type of matter can create a bubble?
As one investigates this question, one generically runs into a violation of the positive energy condition. I would describe the postive energy condition as "the joint ground state of all the spacetime-filling fields serves as a floor, and no local perturbation of this state leads to a globally-lower energy", or "empty space is really really cold, and adding things to it can only make it hotter rather than colder". A solution which upsets this -- negative energy -- is something never seen in any experiment since the discovery of the second law of thermodynamics, and negative energy is so torturously hard to sweep under carpets like non-isolation or vacuum energy that after a while it must begin to feel self-deceitful to keep pursuing a region of spacetime that contains it as anything other than an unphysical model.
Apparently people who try also run into comparable problems when trying to (for want of a better word) steer a warp bubble that exists (ignoring how to make it exist in the first place, or how to make it cease existing at some later time). I'd wager the model runs into trouble for trips between galaxy clusters, by virtue of interacting with the metric expansion of space: having to make big changes to simple models to accommodate the cosmological constant is a common pattern in the history of General Relativity.
There is something of a literature on problems like this, and while it's interesting in the sense of hunting for heuristics to shoot down other seemingly physical solutions to the Einstein Field Equations, I don't think it's very interesting to keep looking for loopholes to try to save the family of Alcubierre-like spacetimes. Beating a dead horse into some sleek-looking and seemingly aerodynamic shape won't make a stagecoach be more likely to reach Mach 6.
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[1] https://en.wikipedia.org/wiki/Energy_condition#Motivation
The lack of interest hangs on the fact that we have highly-tested descriptions of our universe which have relativity baked right in. Even Alcubierre's "drive" paper [1] is written in the language of relativity. And thus we have to take seriously the problem that we have in relativity no obvious way to induce a warp bubble onto a patch of flat spacetime by scattering any configuration of known matter through that region.
We can create stars, including relativistic stars (neutron stars are an example) and black holes by depositing practically an infinite variety of matter into a region of flat spacetime. (In fact, if we use a solar mass of hydrogen gas then Raychaudhuri's focusing theorem makes it hard to avoid it coming together as a star eventually, and then probably a white dwarf).
Likewise, "large scale structure formation" is nearly inevitable with a handful of parameters which in the standard cosmology we capture and try to measure. There's lots of ways in an expanding universe to wind up with galaxies.
We can even get silly and contrive extremely unlikely initial configurations of known matter and simulate what happens. We can even simplify known matter by removing physical features that complicate the picture. This can even be useful. Historically one started with perfectly spherical arrangements of gas that does not self-interact (doesn't clump together, doesn't repel, doesn't induce twisting or spiralling) to try to understand the formation of spheroidal objects like stars, planets, and black holes. Then as tools to simulate such collapses became better and faster, adding back in the stripped-out physical aspects of real interacting matter (which clumps into molecules, and those in turn break apart as they get hot, etc., and probably never found in perfect spheres rather than discoids or blobs) for a better approximation of such systems.
No known matter has the properties necessary to create negative energy.
If we start with an extremely simple form of a purely classical non-interacting negative-energy-density non-relativistic dust that we can drop into the stress-energy tensor of the Einstein Field Equations, we can do some simulations and see what happens. [3] But we are complicating something wholly fictional. Nothing known has the requisite property (negative energy), and so there is no path to adding more complex behaviours to such a basic test set up to try to approach a physically plausible system.
Such an unknown substance is politely called "exotic matter" for the simple reason that it foreign to our solar system. "Unobtanium" is just as fair, if a little less polite.
And that, for me, leads to it being an uninteresting family of solutions.
In the very unlikely event we find some mechanism to generate any negative energy, then great, general interest will revive (and we'll start asking questions about what suppresses that negative energy such that it is undetected in our solar system and in various-sized systems (galaxy clusters, etc) we see in our sky).
Until then, I will tend to see such solutions to the Einstein Field Equations as starting with highly contrived metrics ab initio. I don't object to anyone investigating such things on strictly that basis: Schwarzschild's solution was highly contrived too, and was a matter-free eternal setup. But there were already pretty spherical gravitating systems around to examine (Schwarzschild lived on one!) and Newtonian methods to study those were useful for validating the Schwarzschild solution for objects wholly outside their own Schwarschild radius. But we should also think about the work of Kerr on axisymmetric rotating systems : it too was a contrived vacuum solution, but almost nobody (and certainly not Kerr himself [1]) thinks that we should think of the inside-the-horizons region as physical (and probably most relativists don't, for various reasons).
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[1] Alcubierre has left a copy at https://arxiv.org/abs/gr-qc/0009013v1 -- note the very first sentence of the abstract ("within the framework of relativity").
[2] "... negative mass instead of positive: that's NOTHING to do with physics." Around 28:05 at https://youtu.be/jji2pgfq7oE?t=1685 -- this lecture also shows that a very simple complication of an exact solution, just adding angular momentum to Schwarzschild -- is both very hard work and will often produce obvious garbage. Even Kerr's solution, as he himself says, is part garbage. Fortunately the garbage in Kerr's solution is hidden behind an event horizon, so we can decide not to care at all about that for systems that aren't dense enough to have event horizons, and can say that the inner parts of black holes are for all practical purposes purely a theoretical problem, since we know no way of investigating what goes on in there. Relatedly, Kerr at around that point in the video points out that we know how to arrange matter to form a rotating black hole with the features of the Kerr solution, and frankly, to me, since that is a true statement, that overrides any theoretical objections about the interior garbage. (We don't know how to arrange matter to form a warp bubble, and warp bubbles have nice quiet interiors and distant exteriors, but have tremendous garbage in the thin shell).
[3] Results tend to follow a pattern that we can think about using the idea of a gravitational charge. Unlike the electromagnetic charge, where like charges to repel, the gravitational charge causes like charges to attract. All known matter -- including dark matter -- has the same gravitational charge. So protons, neutrinos, photons, whatever the hell dark matter is, and so on all tend to collapse into structures like galaxies. "Negative" stuff has the opposite gravitational charge. Things with the opposite elecromagnetic charge tend to attract one another, but things with the opposite gravitational charge tend to repel one another. So if we brought a small amount of "negative" matter, with its opposite gravitational charge, into our solar system, it would be thrown out by all the normally-charged stuff (the sun, the earth, etc). Things get really absurd when one couples equally-massed equally-but-oppositely-gravitationally-charged matter together: you can create infinite power in some other charge (like the electromagnetic one) for instance. Again, this oppositely-gravitationally-charged negative stuff is called exotic because if there wasn't much of it in the early universe, it's all been thrown out of it since, and if there was much of it in the early universe, where the hell are all the side-effects ? (if we're talking about a tiny percent of negative stuff, a fraction of baryonic matter, then wormholes and warp bubbles should fill our sky almost as much as stars do, making all sorts of weird observables from gravitational lensing far from galaxies to odd emissions and absorption lines on light from distant quasars).
As the Casimir effect pulls two surfaces toward each other, the net motion is zero. Not very useful.
(One might also try to use a Casimir setup to lift some heavy object off the floor, or do other gravimetry experiments).
The usual explanation for the Casimir results involves modes in the fields of the Standard Model, all of which couple in the same way to the curvature of spacetime.
In the language of particles and charges (which we conventionally get to by considering the spin statistics of gravitational waves in General Relativity), all the Standard Model particles have the same gravitational charge. That includes everything involved in a Casimir experiment, including "vacuum energy" if any.
However, so far most of the thought that has gone into "how do we scatter matter around flat spacetime to induce an Alcubierre metric on it" requires stuff with the opposite gravitational charge.
Back to the language of metric theories of gravitation: if there is only one gravitational charge that everything possesses, then everything couples to a single metric tensor. If we allow for a gravitational charge that has both positive and negative signs, so far as we know (from among other things analysis under the PPNF[1]) we are forced (by the Universality of Free Fall) into having matter couple to a metric tensor according to the sign of its charge. This is what we would call a bimetric theory of gravity, and such theories have been studied for decades motivated by understanding the very early universe https://en.wikipedia.org/wiki/Bimetric_gravity
Generically, one asks "where's all the stuff that falls differently?" and almost inevitably has to say "it fell out of our universe much earlier than the formation of the cosmic microwave background, or decayed into ordinary stuff, or dilute away much faster during the metric expansion than ordinary stuff and became undetectably sparse before the first stars shone", or in other words that either the second metric tensor has decayed to all zeroes everywhere in the observable universe, or that nothing is in the observable universe that couples to it. Otherwise there are lots and lots of bullets to bite about why we don't see any observables that would support the second metric.
Alcubierre's "drive" paper (he has left a copy at https://arxiv.org/abs/gr-qc/0009013v1 ) uses one metric, not two. The warp bubble is also eternal. Importing a whole extra metric tensor into our fundamental theory of gravity is a really high price to pay for having a mechanism which can produce a non-eternal warp bubble (one that forms in the finite past and un-forms in the finite future). And then one has to really spread around a lot of matter that couples with (i.e., it sources as well as follows) the second metric if one hopes to steer the warp bubble. We probably end up in some ratholes of inventing entire chemistries and nuclear physics of "negatively gravitationally charged" matter, and then figure out how it interacts with normal matter when they're confined together in some system.
So although a terse "well maybe it's hiding in the Casimir effect" massively underplays what is needed, or what the consequences would be of reproducible Cavendish (or other gravimeter) experiments on Casimir apparatuses that show that when the plates are being drawn together they are also lighter (in a weight-is-the-quantity-that-bathroom-scales-measure sense). Show it happens at all first, even if it's a very very weak effect, then take seriously thoughts about scaling it up.
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[1] PPNF, or PPN formalism: https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_f...