With Zero Expansion

It is commonly believed that Alcubierre’s warp drive works by contracting space in front of the warp bubble and expanding space behind it. We show that this contraction/expansion is but a marginal consequence of the choice made by Alcubierre, and explicitly construct a similar spacetime where no contraction/expansion occurs. Global and optical properties of warp drive spacetimes are also discussed. Introduction Alcubierre’s warp drive spacetime ([Alc94]) allows effective superluminal travel within General Relativity. As remarked by Alcubierre himself, it requires negative energy densities, and there are reasons to believe that all spacetimes allowing superluminal travel must violate energy conditions ([VBL00], [GW00]). Another feature is the existence of horizons ([CHL99]), which has been argued on general grounds to be unavoidable ([Low99]). The traditional heuristic explanation of how the warp drive spacetime works is that space in front of a given region (the “warp bubble”) is contracted, whereas space behind the same region is expanded. In response, the warp bubble moves forwards with a speed determined by the contraction/expansion rate. We will show in this paper that this contraction/expansion is not necessary at all, and that in particular it is possible to construct a similar spacetime where no contraction/expansion occurs. Heuristically, one could best describe the warp drive spacetime as “sliding” the warp bubble region through space; space in front of the bubble may get contracted (and space behind it expanded), or not, depending on the details of the construction. The paper is divided in three sections. In the first section we define the warp drive spacetime generated by any (timedependent) vector field X in Euclidean 3-space. This is a globally hyperbolic spacetime foliated by Euclidean 3-spaces such that the integral lines of X represent timelike geodesics (corresponding to the so-called Eulerian observers). We show that such spacetimes always violate some energy condition, and compute the expansion of the Eulerian observers’ volume element to be ∇ ·X. Finally, we show how Alcubierre’s warp drive can be obtained from a particular choice of X. In the second section, we explicitly construct a volume-preserving warp drive by selecting an appropriate divergenceless vector field X. In the third section, we take advantage of the formalism developed thus far to analyse some simple examples, and find them to possess horizons and infinite blueshifts. These results are similar to those obtained (some numerically) This work was partially supported by FCT (Portugal) through programs PRAXIS XXI and PROCTI. 1