Curvature Invariants for the Alcubierre and Nat\'ario Warp Drives

TL;DR

This paper employs curvature invariants to visualize and analyze the complex spacetimes of Alcubierre and Natario warp drives, revealing how parameters affect their curvature and identifying unique features of accelerating warp drives.

Contribution

It introduces a novel method of visualizing warp drive spacetimes using curvature invariants, providing new insights into their geometric properties and parameter effects.

Findings

Curvature invariants are coordinate-independent and reveal warp drive structures.

Warp drive parameters like velocity and radius uniquely affect curvature.

Accelerating Natario warp drives lack a wake and show constant curvature features.

Abstract

A process for using curvature invariants is applied to evaluate the metrics for the Alcubierre and the Natario warp drives at a constant velocity.Curvature invariants are independent of coordinate bases, so plotting these invariants will be free of coordinate mapping distortions. As a consequence, they provide a novel perspective into complex spacetimes such as warp drives. Warp drives are the theoretical solutions to Einstein's field equations that allow the possibility for faster-than-light (FTL) travel. While their mathematics is well established, the visualisation of such spacetimes is unexplored. This paper uses the methods of computing and plotting the warp drive curvature invariants to reveal these spacetimes. The warp drive parameters of velocity, skin depth and radius are varied individually and then plotted to see each parameter's unique effect on the surrounding curvature. For each warp drive, this research shows a safe harbor and how the shape function forms the warp bubble. The curvature plots for the constant velocity Natario warp drive do not contain a wake or a constant curvature indicating that these are unique features of the accelerating Natario warp drive.