![]() In that co-ordinate system the position of the spaceship at time t is given by x_s(t), and the velocity is v_s := dx_s/dt. The basic Alcubierre spacetimeĪlcubierre imagines a spaceship moving along the x axis of some Cartesian co-ordinate system. But with that said: I learned a lot by studying the Alcubierre solution as an example of spacetime design. However, it's become something of a crank magnet: any time someone says something about faster than light travel being impossible, someone will say: “But what about the Alcubierre drive?” And, so far as we can currently tell the correct response is: “Yes, so what about the Alcubierre drive? It requires exotic matter with properties we believe are likely to be unphysical”. It should be noted up-front: Alcubierre's solution is very likely not realizable in our universe. The result, Alcubierre showed, was that the bubble (and its contents) can follow essentially any trajectory you want, including trajectories faster-than-light. To restate the above in my own terms: this bubble kind of “pulls” on the surrounding region of space, contracting the region in front of its motion, and expanding the region behind. In particular, Alcubierre constructs a spacetime which has a little moving “bubble” of space inside it. One can then invert the process to come back to Earth, taking an arbitrarily small time to complete the round trip. ![]() In this way, the spaceship will be pushed away from the Earth and pulled towards a distant star by spacetime itself. This is the basis of the model for hyperfast space travel that I wish to present here: create a local distortion of spacetime that will produce an expansion behind the spaceship, and an opposite contraction ahead of it. In the same way, one can use a contraction of spacetime to approach an object at any speed. The previous example shows how one can use an expansion of spacetime to move away from some object at an arbitrarily large speed. The enormous speed of separation comes from the expansion of spacetime itself. This doesn't mean that our observers will be travelling faster than light: they always move inside their local light-cones. It is easy to convince oneself that, if we define this relative speed as the rate of change of proper spatial distance over proper time, we will obtain a value that is much larger than the speed of light. The basic idea can be more easily understood if we think for a moment in the inflationary phase of the early Universe, and consider the relative speed of separation of two comoving observers. As Alcubierre notes in motivating background discussion, this effect can be used to achieve what seems (from a global point of view) like faster-than-light travel: However, the mutable nature of spacetime in general relativity means that spacetime itself may expand and contract. It is true that locally it's not possible to travel faster than light, in general relativity. A better description of the universe is provided by general relativity, and there the situation is more complicated. Of course, special relativity is only an approximate description of reality. For this reason, faster-than-light travel is usually regarded as forbidden. It is well known that in special relativity the ability to travel faster than light leads to paradoxes – the ability to send information backward in time, to violate causality, and so on. However, in these brief working notes I review a stimulating 1994 proposal for a striking spacetime, from the physicist Miguel Alcubierre 2. I do not know the answer to these questions. In this viewpoint, fundamental questions are: what spacetimes are possible? What spacetimes can we, in principle, engineer? Or, from a different point of view: what spacetimes can we design and then realize 1? It's as though the action of the players changes the arena in which they play. But in general relativity this is not the case: spacetime changes in response to mass and energy and motion it is a mutable, dynamic quantity. Intuitively, we think of space and time as fixed – an arena for action, but not the action itself. There's also a lot I don't understand about the Alcubierre solution, so these notes contain many questions, as a placeholder for future work./ My own understanding of general relativity is very patchy, so correction of misapprehensions is appreciated. Requires a basic background in general relativity. Working notes by Michael Nielsen, released September 30, 2021. Rough working notes on the Alcubierre 'warp drive' spacetime
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