Newtonian limits of warp drive spacetimes
José Natário
Abstract and summary · read the original at the source
In one page
José Natário asks a deceptively simple question about the Alcubierre warp drive: what does it look like as ordinary gravity? A Newtonian limit is the picture you get when the speed of light is allowed to run to infinity — the familiar world of one gravitational potential, with forces and masses. Natário shows that a whole family of spacetimes, warp drives among them, can be written so that free-falling observers are simply carried along over a flat Euclidean background by a velocity field, and that whenever that velocity field is the gradient of something, those observers move exactly as Newton’s would in a potential he writes down. Applied to a warp bubble the potential splits in two: a steady term present whenever the drive is running, and an acceleration term that appears only while the bubble is speeding up. Outside the bubble wall there is vacuum, and for a cruising bubble no gravitational field at all. The same method returns the point-mass potential from the Schwarzschild solution.
Why it matters hereChapter 4 needs the warp bubble to be readable, not just soluble, and this paper supplies the translation: a metric engineer can now point at the potential term that is doing the carrying, see that the exotic requirement lives on the bubble wall rather than everywhere, and note that a cruising bubble leaves flat spacetime and no field behind it. That the same construction hands back Newton’s point mass from Schwarzschild, with the horizon appearing exactly where the carried observer reaches the speed of light, is the check that the tool is sound.
What it claims
01A metric written as minus dt squared plus the flat spatial metric acting on the coordinate displacement minus a velocity field times dt describes free-falling Eulerian observers drifting over a flat Euclidean background, and when that velocity field is a gradient the observers follow Newtonian free-fall, to first order in their velocities, in the potential given by minus the time derivative of the velocity potential minus half the velocity squared; Natário names such a solution a Newtonian spacetime.Section 2, equation 3, Definition 2.1 and Proposition 2.4
Published and peer-reviewed02In a Newtonian spacetime the potential satisfies a Poisson equation sourced by the energy-momentum tensor and the cosmological constant, and when the source is dust comoving with the Eulerian observers this reduces exactly to the Newtonian Poisson equation with mass density and cosmological constant, with the ordinary continuity equation holding as well.Proposition 2.3; Section 3, equation 12 and Proposition 3.1
Published and peer-reviewed03A warp drive spacetime whose bubble moves at speed u of t, with a smooth shape function that is zero inside the bubble and one outside, is a Newtonian spacetime whose potential contains two terms: an acceleration term present only when the speed is changing, and a steady term present whenever the drive is on.Section 5.1, Newtonian warp drive
Published and peer-reviewed04Outside the bubble wall region there is vacuum, and if the bubble is not accelerating the gravitational field vanishes there; while it accelerates, a uniform gravitational field of magnitude equal to the bubble’s acceleration does the accelerating, and spacetime curvature outside the bubble stays zero.Section 5.1, paragraph following the potential
Published and peer-reviewed05Ordinary potential theory forbids the Laplacian of the potential from vanishing identically and makes its weighted integral over space zero, so there must be points on the bubble wall where it is negative; there the strong energy condition is not satisfied, which in the Newtonian limit reads as a negative mass density on the wall itself rather than anywhere else in the solution.Section 5.1, final paragraph
Published and peer-reviewed06Applied to the Schwarzschild metric in Painlevé-Gullstrand form the same construction gives the Newtonian potential of a point mass, minus M over r, and the horizon appears where the carried observer’s speed reaches one — which is why the Newtonian formula for the Schwarzschild radius agrees with the general-relativistic one; spatially flat Friedmann-Robertson-Walker dust models come out of the identical calculation.Sections 5.2 and 5.3; Section 4, equation 14 and Proposition 4.1
Settled physics
The way in
https://doi.org/10.1007/s10714-006-0234-0Published as General Relativity and Gravitation 38, 475-484 (2006) under the publisher’s copyright; the preprint at arxiv.org/abs/gr-qc/0408085 carries the arXiv non-exclusive distribution licence, which is not a Creative Commons licence. This sheet therefore carries the author’s own abstract and sends the reader to the source. Claim locators cite the section and proposition numbers of the preprint, which the published paper keeps. The work was partially supported by FCT, POCTI and FEDER.
How to cite it
José Natário (2006) Newtonian limits of warp drive spacetimes. doi:10.1007/s10714-006-0234-0
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