Matching the Alcubierre and Minkowski spacetimes

TL;DR

This paper investigates the conditions required to match an Alcubierre warp drive spacetime with Minkowski space, revealing that the warp drive is not globally flat and depends on solutions to a Burgers equation at the junction.

Contribution

It provides a detailed analysis of the junction conditions between warp drive and flat spacetime, linking the shift vector to shock wave solutions and showing the warp drive's non-global flatness.

Findings

The shift vector must satisfy an inviscid Burgers equation at the junction.

Not all curvature components vanish at the joining hypersurface.

Warp drive geometry is not globally flat, depending on the shift vector solution.

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.