Historical research guide · 0701 · 2000
Warp drive space-time
Pedro F. González-Díaz
In one minute
Three years after Miguel Alcubierre wrote down his warp metric, William Hiscock did the quantum accounting on it and found trouble: once the bubble’s apparent speed passes the speed of light an event horizon forms, and the energy of a quantum field piles up without limit right there. Pedro González-Díaz, at the CSIC in Madrid, takes that result seriously and looks for the assumption behind it. His answer is topology. Hiscock treated the inside of the bubble as an ordinary simply connected patch of space. González-Díaz instead extends the warp geometry beyond the horizon and embeds it in a flat three-dimensional space carrying the identifications of Misner space, which makes the ship’s interior multiply connected. Redo the calculation there and the infinity is gone: the renormalised stress-energy stays finite even on what is now a chronology horizon, and the bubble meets Ford’s limit on borrowed negative energy. The price is a bubble near the Planck scale.
Connection
Chapter 4 is written on the arithmetic of warp geometries, and this paper is the moment the argument moved from how much negative energy to what shape of space you are doing the sum in. Hiscock’s divergence settles that a simply connected two-dimensional Alcubierre bubble is not quantum-mechanically stable once it goes superluminal; what it leaves open is everything about the topology of the interior, and González-Díaz shows that changing that single assumption removes the infinity and brings the design inside Ford’s negative-energy limit. It also leaves the next problem stated plainly: at that scale the bubble is microscopic. Read it beside Alcubierre’s original at /library/stm-fc5383ec73 and the Pfenning and Ford energy bill at /library/stm-710bf626ca.
From the source
The warp drive is consistent with relativity because nothing moves faster than light through its own neighbourhood. Two observers at the same point can never have a relative velocity faster than light, but observers at distant locations may have an arbitrarily large relative velocity while each moves inside its own light cone — so the spaceship’s apparent speed can greatly exceed the speed of light while the ship itself never locally does.Sec. I, Introduction, opening paragraph
Hiscock computed the stress-energy tensor of a conformally invariant field in the two-dimensional Alcubierre warp drive and found that it diverges on the event horizon that appears once the apparent velocity exceeds the speed of light, which makes that version of the warp drive unstable. The metric function that carries this behaviour is named in the literature after him.Sec. I, Introduction, the paragraph citing Hiscock; Sec. II, Eqs. 2.4 and 2.5
Extending the geodesically incomplete warp geometry beyond its horizon by the Kruskal procedure and then embedding it in a three-dimensional Minkowski space with the topological identifications of the universal covering of three-dimensional Misner space converts the interior of the spaceship bubble into a multiply connected, nonchronal region.Sec. III and Sec. IV; Sec. V, Summary and conclusions
In that multiply connected case the quantum calculation comes out finite. The most natural vacuum allows fluctuations that induce no divergent behaviour in the renormalised stress-energy tensor, even on the event horizon, which now becomes a regularised Cauchy chronology horizon — and the result holds both for the self-consistent Li and Gott vacuum and for a modified three-dimensional Misner embedding.Abstract; Sec. IV, the two ansätze and their Hadamard functions; Sec. V
The price of the repair is size. The consistent quantum treatment puts the bubble and its closed timelike curves at scales close to the Planck length, and at that scale the warp drive satisfies Ford’s negative energy-time inequality — the same constraint Ford and Pfenning had shown a macroscopic Alcubierre bubble cannot meet, even with a wall only a few hundred Planck lengths thick, and which Van Den Broeck’s modified geometry had reduced but not removed.Abstract; Sec. I, the paragraph citing Ford and Pfenning and Van Den Broeck; Sec. IV, closing paragraphs
A geometric connection worth pursuing: the superluminal two-dimensional warp drive space is related to two-dimensional gravitational kinks, the topological defects of the Finkelstein kinked metric. Generalising that metric to describe the complete one-kink generalises the Alcubierre metric to a standard nonstatic metric that can no longer be written in a single coordinate patch.Abstract; Sec. III, Warp drive spacetime and topological gravitational kinks
The way in
https://doi.org/10.1103/PhysRevD.62.044005How to cite it
Pedro F. González-Díaz (2000) Warp drive space-time doi:10.1103/PhysRevD.62.044005