Is the Alcubierre drive consistent with conservation of energy? (Gravitational potential in particular)

Question

Consider a ship in a circular orbit around a planet. It activates an Alcubierre drive. Do the physics of this drive allow the ship to reach a greater distance from the planet, and if so, where does the energy difference between new and old gravitational potential come from?

(My maths skill level is UK A-Level, roughly the equivalent of first-year university in the USA. I can follow some calculus, but I don't know half the symbols used in GR. My intuition, which I know is a terrible way of approaching physics, says the ship in my question would remain in a circular orbit around the planet, but complete those orbits at whatever speed the drive was set to run at).

Answer 1

There are two things you need to bear in mind when considering the Alcubierre metric. First, it is a time independent metric i.e. it describes an arrangement of the (exotic) matter that has existed for an infinite time and will continue to exist for an infinite time. Secondly it assumes the drive is the only object in the universe and it does not include any interaction with curvature due to the presence of other matter like the planet you are orbiting.

This means the Alcubierre metric is insufficient to do the calculation you describe. We need a way to describe the evolution of the geometry as the ring of exotic matter is assembled to start the drive then dismantled to stop the drive. We also need to describe what happens if the drive is assembled and dismantled in an already curved spacetime rather than in flat space.

As far as I know there are no exact solutions to describe either of these, so the calculation has to be done numerically. I also don't know of any attempts at a numerical calculation like this. So we can at most guess at the answer to your question.

However it seems very likely to me that while the energy necessary to assemble the drive is probably equal to the energy necessary to dismantle it in flat spacetime, the two energies are probably different if the assembling and dismantling steps are done in spacetime backgrounds with different curvatures. And it seems likely that the difference in the energies would be equal to the change in the gravitational potential energy and no violation of energy conservation occurs.

But we should note that in GR conservation of energy only applies when there is a timelike Killing vector. For example energy is not conserved in an expanding universe. So it is possible for energy conservation to be violated when you create then destroy an Alcubierre drive. Whether this would be the case in your example I don't know.

Answer 2

Alcubierre drives require an enormous amount of negative energy to operate. If an Alcubierre drive existed, this is where the energy would come from if it boosted to a higher orbit. Unfortunately, no one even knows if negative energy exists, and even if it did, the amazing properties of an Alcubierre drive have been exaggerated. No matter what anyone tells you, you cannot get from one planet to another faster than the speed of light under the known laws of physics, and it would be absolutely revolutionary to the whole field of physics if this was ever discovered to not be the case.

You say your intuition is "the ship in my question would remain in a circular orbit around the planet, but complete those orbits at whatever speed the drive was set to run at". There is some confusion here. Any object will remain in a circular orbit around a planet if it starts in one and no energy is added to it. If you add energy to it, it goes to a higher orbit, and if you remove energy, it goes to a lower orbit. Activating the Alcubierre drive while in orbit would send you into a higher orbit or else break you out of orbit entirely if it provided enough acceleration.