To understand warp drives and why they might really exist, it helps to understand one of the most general phenomena in physics, which is how circumstances beyond our control are often where insights into the universe begin.
Our various physical laws tell us that if we create these circumstances then this motion will result. But the laws don't tell us how to create those circumstances, or even if we can create them.
The simplest case of this is Newton's second law, F=m*a: the acceleration multiplied by the object's mass is equal to the force acting on it. So for any given force we can figure out the acceleration, and once we know the acceleration we can integrate it to find the velocity and position. The force itself is often a function of velocity and position, which makes the math tricky--we end up having to solve things called second order differential equations, often with brute computational force--but the principle remains simple.
In a lot of cases, the force is directly proportional to an object's displacement from some equilibrium position, which results in something called simple harmonic motion. But in other cases it's vastly more complicated: we can easily find materials in nature that have force laws we don't know how to reproduce.
For example, duplicating the complex springiness of cartilage in human joints using synthetic materials is beyond us. Our equations tell us "if you can make a material with this sort of springiness, it will result in a joint motion like that", but we're on our own when it comes to figuring out how to create the material. Newton is no help at all.
We ran into the same problem with Maxwell's electromagnetic equations back when Queen Victoria was in the middle of her long reign. This set of four equations says if we wiggle a charged particle back and forth in just the right way it should generate waves in the electromagnetic field. It took the German physicist Heinrich Hertz twenty years to figure out how to do the wiggling appropriately--oscillating currents in specially-shaped wires called "antenna"--and produce what we now call "radio waves".
Newton and Maxwell are the "simple" cases. Einstein's equations--which describe gravity--consist of sixteen inter-related second-order differential equations, and it's fair to say that after a century of progress we are still figuring out how to wrap our heads around them.
In each of these cases there is a "source term"--forces for Newton, electric charge for Maxwell, and mass or energy for Einstein--and the equations tell us "arrange a source term like this, and motion like that will follow."
Recent work has found a couple of source terms that appear to allow "warp drives": solutions to Einstein's equations that would let us to travel faster than light by stretching space around an object to create a wave it can ride like a surfer.
The first of these ideas was entirely theoretical. In the early '90s Mexican physicist Miguel Alcubierre found an arrangement of "negative energy" that produced a solution to Einstein's equations containing a "bubble" of space-time that could move at any velocity. Because the inside was effectively hidden from the rest of the universe, an object in the bubble could move with it, even faster than light.
This was exciting, but had two problems. The first is that no one knows how to create "negative energy", or even what it means... if anything. Our mathematical descriptions of reality often describe more reality than we actually have.
For example, we routinely throw away "negative time" solutions to wave equations because we never see waves going backward in time. This is the difference between physics and mathematics.
Sticking a negative sign in front of the source term in Einstein's equations tells us what would happen if we lived in a universe where negative energy was a meaningful thing, but it doesn't tell us if we're actually living in that universe. And if we are, it doesn't tell us how to create or control negative energy, which sounds hard, especially as it turns out Alcubierre's configuration requires converting just about the whole universe into "negative energy" to work. Although a few people have found tricks that might bring that down by a lot, it's still on a scale where we'd have to be able to gather up whole galaxies of negative energy, which sounds like the kind of thing you'd need to have a warp drive to do.
So although Alcubierre demonstrated an extremely intriguing loophole in the "nothing goes faster than light" restriction we think we're living with, his work didn't spark any sudden rush in technological development.
But the thing about intriguing loopholes is they lead other people to wonder if there might be more of them.
New work by Edmund Lentz in Germany has removed that negative energy constraint, and shown how to construct a configuration of positive energy that will allow macroscopic objects move at arbitrary velocities by warping space-time around them.
The amount of energy involved is still huge--something like 1/10th the mass of the sun converted into energy in the form of multiple plasma lenses around a volume that's 100 m across and 1 m thick--like the "saucer section" of Star Trek's Enterprise, but squashed really flat--but there may be tricks to bring it down to something closer to what is achievable, and there may even be natural circumstances where similar conditions prevail, in the hot, diffuse plasmas that surround highly magnetized neutron stars. These wouldn't have the right configuration to travel at warp speed, but they could be (distant) laboratories to allow us to study such effects even if we can't yet create them in the laboratory.
Lentz's work has only recently been published, and there may be holes in his argument or insurmountable practical difficulties, but it opens up a promising new frontier for investigation, and suggests that practical interstellar travel might be far closer than anyone dared dream.