Fundamental limitations on ‘warp drive’ spacetimes
Francisco S. N. Lobo · Matt Visser
Abstract and summary · read the original at the source
In one page
Francisco Lobo and Matt Visser take Alcubierre’s warp bubble, and José Natário’s variant of it, and ask what general relativity requires of them before any question of going fast comes up. Their first move is bookkeeping. In the original construction the geometry is exactly flat outside the bubble, so the whole spacetime carries zero total mass — the ship and its generators are precisely cancelled by the stress-energy of the warp field. Their second move is to build the honest model: put a real ship of finite mass inside the bubble and treat the field as weak. Now the leading energy density is the ship’s own, and it is positive. What the authors most want to say, and say nobody had remarked on before, is what such a spacetime would be if you built it: a reaction-less drive. The bubble moves by interacting with the geometry of spacetime rather than by expending reaction mass, and the ship is simply carried along inside it.
Why it matters hereChapter 4 is metric engineering, and this is the paper that names the prize in the authors’ own words — a reaction-less drive, a craft that moves by interacting with the geometry of spacetime instead of throwing propellant overboard. It is also the paper that sets the bill chapter 6 has to pay, and it sets it in the most useful form anyone has: not a number of galaxy masses but a ratio, the warp field’s own energy against the mass of the ship you are carrying. The Alcubierre and Lobo textbook statement of the geometry is at /library/stm-ad0758f024, Alcubierre’s original 1994 letter at /library/stm-fc5383ec73, the Pfenning and Ford quantum-inequality bill at /library/stm-710bf626ca, Chris van den Broeck’s bottle-shaped answer to it at /library/stm-3ce9e507bb, and Brendan McMonigal, Geraint Lewis and Philip O’Byrne on what the bubble does to the matter it sweeps up at /library/stm-ae326cd1a7.
What it claims
01Warp drive spacetimes are shift-only spacetimes. In the ADM description space itself is flat and the lapse function is exactly one, so every bit of the structure sits in the shift vector. The Alcubierre drive takes that shift along the direction of motion; Natário’s takes it divergence-free, so a radial compression at the front is balanced exactly by an expansion perpendicular to it and the bubble pushes space aside and slides through it.Section II, Equations 1 to 3; Section II B, Equation 21
Published and peer-reviewed02In the original warp drive the total ADM mass of the spacetime is exactly zero, because the geometry is asymptotically Minkowski. The mass of the ship and of the warp-field generators is therefore compensated exactly by the stress-energy of the warp field itself — which is why energy below the ambient vacuum level is unavoidable in that version, and why the useful question is not whether it appears but where it is localised and how much of it there is.Section II A, discussion following Equation 8
Published and peer-reviewed03The energy-condition violations are not a side effect of going faster than light. For both the Alcubierre and the Natário bubble they persist at arbitrarily low bubble velocity, and the volume integral quantifier gives the same estimate for both: about minus the square of the velocity times the square of the bubble radius, divided by the wall thickness. Cost scales with the square of speed, with the square of size, and inversely with wall thickness.Section II A, Equations 13 and 14; Section II B, Equations 22 to 24
Published and peer-reviewed04If even a weak-field warp drive can be realised in nature, such a spacetime is an example of a reaction-less drive: the warp bubble moves by interacting with the geometry of spacetime instead of expending reaction mass, and the spaceship — which linearised theory lets you treat as a finite mass object rather than a test particle — is simply carried along with it. The authors present this as the feature of the warp drive that had not previously been remarked upon.Abstract; Section IV, Summary and Discussion
Published and peer-reviewed05Put a finite-mass ship inside the bubble and work to first order in the bubble velocity, and the picture improves markedly: the energy density an observer at rest measures is the ship’s own ordinary density, manifestly positive, and the total mass of the spacetime reduces to the mass of the ship. The energy below ambient vacuum level does not vanish — it moves. It sits in a thin shell in the immediate interior neighbourhood of the bubble wall, where the ship’s own mass is not, and observers boosted along the direction of travel are the ones who measure it.Section III A and III B, Equations 39 to 49; Section III E, Equations 65 to 67 and 77 to 78
Published and peer-reviewed06Carrying the calculation to second order in velocity gives the design constraint the whole paper turns on, and it is a ratio rather than an absolute figure: the square of the velocity times the square of the bubble radius, divided by the wall thickness, must stay below the mass of the ship. Written out, the allowed speed squared goes as the ship’s mass over its radius, times the ship radius times the wall thickness, over the square of the bubble radius. Any strong-field warp drive has to be built up through this weak-field regime, so the open engineering question is which member of the warp family carries a bill the vacuum can pay — thick walls, small bubbles, and geometries that put the ship somewhere other than at the centre of a large one.Section III E, second order approximation, Equations 90 to 102; Section IV
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Abstract
"Warp drive" spacetimes are useful as "gedanken-experiments" that force us to confront the foundations of general relativity, and among other things, to precisely formulate the notion of "superluminal" communication. We verify the non-perturbative violation of the classical energy conditions of the Alcubierre and Natario warp drive spacetimes and apply linearized gravity to the weak-field warp drive, testing the energy conditions to first and second order of the non-relativistic warp-bubble velocity. We are primarily interested in a secondary feature of the warp drive that has not previously been remarked upon, if it could be built, the warp drive would be an example of a "reaction-less drive". For both the Alcubierre and Natario warp drives we find that the occurrence of significant energy condition violations is not just a high-speed effect, but that the violations persist even at arbitrarily low speeds. An interesting feature of this construction is that it is now meaningful to place a finite mass spaceship at the center of the warp bubble, and compare the warp field energy with the mass-energy of the spaceship. There is no hope of doing this in Alcubierre's original version of the warp-field, since by definition the point in the center of the warp bubble moves on a geodesic and is "massless". That is, in Alcubierre's original formalism and in the Natario formalism the spaceship is always treated as a test particle, while in the linearized theory we can treat the spaceship as a finite mass object. For both the Alcubierre and Natario warp drives we find that even at low speeds the net (negative) energy stored in the warp fields must be a significant fraction of the mass of the spaceship.
The way in
https://doi.org/10.1088/0264-9381/21/24/011LICENCE. Published as Classical and Quantum Gravity 21 (2004) 5871 to 5892; the Unpaywall record calls the publisher copy free to read but carries no licence for it. The authors’ manuscript is public on arXiv as gr-qc/0406083, submitted 18 June 2004 and revised 24 October 2004, but that deposit falls under arXiv’s assumed licence for 1991 to 2003 and legacy submissions rather than a Creative Commons statement, so this page holds the summary, the claims and the authors’ own abstract and sends the reader to the source. The paper runs to about twenty pages with three appendices — the full Einstein tensor for the Alcubierre bubble, the same for the Natário bubble, and a summary of linearised gravity — from the Centro de Astronomia e Astrofísica da Universidade de Lisboa and the School of Mathematical and Computing Sciences at Victoria University of Wellington. The abstract below is the authors’ own, as posted with the arXiv version.
How to cite it
Francisco S. N. Lobo, Matt Visser (2004) Fundamental limitations on ‘warp drive’ spacetimes. doi:10.1088/0264-9381/21/24/011
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