Historical research guide · 0319 · 1997
The unphysical nature of "Warp Drive"
Michael J. Pfenning · L. H. Ford
In one minute
Michael Pfenning and Larry Ford, at Tufts, do the accounting Miguel Alcubierre’s 1994 warp metric left out: how much energy below the ambient vacuum level the quantum field will actually lend you, and for how long. The tool is the quantum inequality — Ford and Roman’s published bound on how deep and how long such a dip can run — applied inside a region small enough to count as flat. Sampled that way, the wall of the warp bubble turns out to be forced extraordinarily thin, of order a hundred Planck lengths times the bubble’s speed. Integrate the energy density across a wall that thin, for a bubble big enough to hold a ship, and the bill comes to roughly ten to the twentieth galaxy masses. Going faster thickens the wall but raises the bill in exact proportion. The paper does not close the subject. It prices it — and every warp design since has been built against this number.
Connection
This is the specification chapter 4 works against: not a door closed but a bill itemised, denominated in the one quantity chapter 2 says the vacuum is known to supply — regions below its own ambient level. Every serious warp design since is an answer to this number, from Chris van den Broeck’s topological bottle to Alexey Bobrick and Gianni Martire’s positive-energy subluminal shells.
From the source
The energy density of the Alcubierre warp metric measured by any geodesic observer is always negative, and it is not spread through the bubble but concentrated in a toroidal band perpendicular to the direction of travel — the same way negative energy concentrates in a thin band around a wormhole throat.Section 2, Equation 8 and Figure 3
Applying the flat-space quantum inequality for a massless scalar field to the warp metric, with the sampling time held below the smallest local radius of curvature, bounds the wall thickness of the bubble: taking the sampling time at one tenth of that radius, the wall can be no thicker than about a hundred Planck lengths times the bubble velocity.Section 3, Equations 9 to 23
Integrating the energy density over a wall of that permitted thickness, for a bubble of one hundred metre radius, gives a total of order minus six times ten to the sixty-fifth grams per unit bubble velocity — roughly ten to the twentieth galaxy masses, which the authors describe as physically unattainable for human transport.Section 4, Equations 25 to 31
Speed buys nothing: raising the bubble velocity does thicken the permitted wall, but the total negative energy required rises by the same factor. Shrinking the bubble instead helps, though not enough — a bubble the radius of one electron Compton wavelength still costs of order four hundred solar masses, and relaxing the wall to a full metre still leaves about a quarter of a solar mass.Sections 4 and 5, Equation 28 and Summary
The bound is not an artefact of the field chosen to supply the negative energy: for a massive scalar field the quantum inequality is more restrictive still, forcing the walls thinner, and for the quantized electromagnetic field the permitted thickness improves only by a factor of the square root of two.Section 5, Summary
The result is obtained by importing a flat-spacetime inequality into a curved geometry with the sampling time restricted; the authors say an exact treatment would need the mode functions of the scalar field on the warp background, call that exceptionally difficult, and do not attempt it — so the exact curved-space inequality for this metric is still open, and so is whether some other member of the warp family carries a bill the vacuum can pay.Section 1, Introduction; Section 3, opening; Section 5, Summary
The way in
https://arxiv.org/abs/gr-qc/9702026How to cite it
Michael J. Pfenning, L. H. Ford (1997) The unphysical nature of "Warp Drive" doi:10.1088/0264-9381/14/7/011