Historical research guide · 0632 · 2002
Warp drive with zero expansion
José Natário
In one minute
Everyone repeats the same picture of Alcubierre’s warp drive: space contracts in front of the bubble and expands behind it, and the bubble rides that. José Natário, a mathematician at Instituto Superior Técnico in Lisbon, shows the picture is optional. He writes any warp drive as a single vector field on ordinary flat space — the field tells you how fast each point of space is being carried along — and proves that the expansion of space is just the divergence of that field. Choose a field with zero divergence and you get a bubble that carries a traveller at any speed you like while no volume anywhere is stretched or squeezed. His own image for what a warp drive really does is sliding a region of space through the rest. He also works out what the trip looks like: inside and outside the bubble space is flat, the traveller floats freely, and light arriving from ahead is blueshifted while the drive builds a horizon it cannot see past.
Connection
Chapter 4 is about engineering the metric rather than pushing on propellant, and this paper strips the warp drive down to the one object an engineer would actually specify — a velocity field for space itself — then shows that the most-quoted feature of Alcubierre’s solution, the stretching and squeezing of space, is a choice and not a requirement.
From the source
A warp drive spacetime is completely specified by one time-dependent vector field on flat Euclidean three-space. The line element is minus the square of the time interval plus, for each of the three space directions, the square of the coordinate displacement minus that component of the field times the time interval. The spatial slices are ordinary flat Euclidean space; the observers moving along the field’s integral lines, which Natário calls Eulerian observers, are in free fall. Wherever the field is spatially constant the spacetime is flat, so the inside of the bubble and the far exterior are both ordinary flat space, even though observers inside move at arbitrary speed relative to observers outside.Section 1, Definition 1.1, Proposition 1.3, Corollary 1.6 and Example 1.8
The expansion of space is simply the divergence of that field, and a warp drive does not need any. Natário constructs an explicit divergenceless field, built as the exterior derivative of a potential in spherical coordinates with a shape function equal to one half far away and zero near the centre, which still carries a bubble at whatever velocity is prescribed. In front of the bubble the wall is compressed radially, and that compression is exactly balanced by expansion in the two perpendicular directions, leaving every volume element unchanged. The better heuristic, he writes, is that the warp drive slides the bubble region through space; contraction ahead and expansion behind may happen or not, depending on the construction.Section 1, Corollary 1.5; Section 2, from the divergenceless construction to its closing paragraph
Every non-flat warp drive spacetime violates either the weak or the strong energy condition. The proof is short: the energy density an Eulerian observer measures is the square of the expansion minus the square of the extrinsic curvature, all divided by sixteen pi, so if the expansion vanishes the density cannot be positive, and it is zero only when the extrinsic curvature vanishes everywhere, which means flat spacetime. For the zero-expansion drive he gives the density in closed form: it is negative, it scales with the square of the bubble velocity, and it is concentrated in the wall where the shape function varies.Section 1, Theorem 1.7; Section 2, the energy-density expression
Above light speed the drive builds a horizon, and its shape is the Mach cone. For a bubble travelling faster than light with a steady field, a flash of light outside the bubble spreads at speed one while being carried along at the bubble speed, so events inside cannot causally influence events far ahead. The boundary is the cylindrically symmetric surface whose angle to the field satisfies sine of the angle equals one over the field magnitude — far from the bubble, exactly the familiar Mach cone angle of one over the bubble velocity.Section 3, the horizon construction and Figure 1
The view from inside is calculable. Because interior and exterior are flat, light travels in straight lines and only refracts at the wall, with the refraction angle given by the incidence angle divided by one plus the bubble velocity times the cosine of the angle between them; in the thin-wall limit there is no aberration, and any finite wall thickness restores some. The observer at the centre sees light from directly ahead blueshifted by one plus the bubble velocity, falling to no shift at ninety degrees and to zero at what Natário calls the visibility horizon, beyond which nothing is seen at all. Light he sends backwards is redshifted by the same factor, and light he sends toward the horizon is blueshifted without bound.Section 3, refraction footnote and the redshift computation; Figures 2 and 3
The interior of the bubble is causally disconnected from part of the bubble’s own wall, which Natário notes is unavoidable. That is the control problem in one line: the region that has to be shaped to steer the craft cannot be signalled to from inside once the bubble is superluminal, so the wall would have to be laid out in advance. Whether a shift field exists that keeps the wall reachable, and what matter would source it, is what the numerical warp-metric programmes are computing now.Section 3, the paragraph following the horizon construction
The way in
https://doi.org/10.1088/0264-9381/19/6/308How to cite it
José Natário (2002) Warp drive with zero expansion doi:10.1088/0264-9381/19/6/308