Matching the Alcubierre and Minkowski spacetimes

TL;DR

This paper analyzes whether an interior Alcubierre warp-drive spacetime can be smoothly joined to an exterior Minkowski geometry using Darmois junction conditions. By defining a joining surface Σ and enforcing continuity of the first and second fundamental forms, it shows that the shift vector on Σ must be independent of the transverse coordinates and satisfy a Burgers-type equation, with two sign choices corresponding to barβ and β. The results reveal that flatness and vacuum at the boundary occur only under these constrained conditions, while in general the Riemann and Ricci tensors remain nonzero, indicating surface gravity at the wall; the exterior region remains locally flat, decoupled from the interior. The work highlights a deep link between warp-drive boundary dynamics and Burgers/shock-wave equations, clarifying the geometric constraints required for a consistent WD–Minkowski match and outlining avenues to engineer interior profiles that avoid curvature imprints on the exterior.

Abstract

This work analyzes the Darmois junction conditions matching an interior Alcubierre warp drive spacetime to an exterior Minkowski geometry. The joining hypersurface requires that the shift vector of the warp drive spacetime must satisfy the solution of a particular inviscid Burgers equation, namely, the gauge where the shift vector is not a function of the $y$ and $z$ spacetime coordinates. Such a gauge connects the warp drive metric to shock waves via a Burgers-type equation, which was previously found to be an Einstein equations vacuum solution for the warp drive geometry. It is also shown that not all Ricci and Riemann tensors components are zero at the joining hypersurface, but for that to happen they depend on the shift vector solution of the inviscid Burgers equation at the joining wall. This means that the warp drive geometry is not globally flat.