“Warp drives” originated in the science fiction shows “Star Trek” in the 1960s, but in 1994, the Mexican Miguel Alcubierre published a paper arguing that under certain conditions exceeding light speed was not forbidden by Einstein’s General Relativity. Alcubierre reached his solution by assuming it was possible, then working backwards to see what was required while rejecting those awkward points that arose. The concept is that the ship sits in a bubble, and spacetime in front of the ship is contracted, while that behind the ship is expanded. In terms of geometry, that means the distance to your destination has got smaller, while the distance from where you started gets longer, i.e. you moved relative to the starting point and the destination. One of the oddities of being in such a bubble is you would not sense you are moving. There would be no accelerating forces because technically you are not moving; it is the space around you that is moving. Captain Kirk on the enterprise is not squashed to a film by the acceleration! Since then there have been a number of proposals. General relativity is a gold mine for academics wanting to publish papers because it is so difficult mathematically.
There is one small drawback to these proposals: you need negative energy. Now we run into definitions, and before you point out the gravitational field has negative energy it is generated by positive mass, and it contracts the distance between you and target, i.e. you fall towards it. If you like, that can be at the front of your drive. The real problem is at the other end – you need the repulsive field that sends you further from where you started, and if you think gravitationally, the opposite field, presumably generated from negative mass.
One objection often heard to negative energy is if quantum field theory were correct, the vacuum would collapse to negative energy, which would lead to the Universe collapsing on itself. My view is, not necessarily. The negative potential energy of the gravitational field causes mass to collapse onto itself, and while we do get black holes in accord with this, the Universe is actually expanding. Since quantum field theory assumes a vacuum energy density, calculations of the relativistic gravitational field arising from this are in error by ten multiplied by itself 120 times, so just maybe it is not a good guideline here. It predicts the Universe has long since collapsed, but here we are.
The only repulsive stuff we think might be there is dark energy, but we have no idea how to lay hands on it, let alone package it, or even if it exists. However, all may not be lost. I recently saw an article in Physics World that stated that a physicist, Erik Lentz, had claimed there was no need for negative energy. The concept is that energy could be capable of arranging the structure of space-time as a soliton. (A soliton is a wave packet that travels more like a bubble, it does not disperse or spread out, but otherwise behaves like a wave.) There is a minor problem. You may have heard that the biggest problem with rockets is the mass of fuel they have to carry before you get started. Well, don’t book a space flight yet. As Lentz has calculated it, a 100 m radius spacecraft would require the energy equivalent to hundreds of times the mass of Jupiter.
There will be other problems. It is one thing to have opposite energy densities on different sides of your bubble. You still have to convert those to motion and go exactly in the direction you wish. If you cannot steer as you go, or worse, you don’t even know for sure exactly where you are and the target is, is there a point? Finally, in my science fiction novels I have steered away from warp drives. The only times my characters went interstellar distances I limited myself to a little under light speed. Some say that lacks imagination, but stop and think. You set out to do something, but suppose where you are going will have aged 300 years before you get there. Come back, and your then associates have been dead for 600 years. That raises some very awkward problems that make a story different from the usual “space westerns”.