Historical research guide · 0463 · 2021
Curvature Invariants for the Alcubierre and Natário Warp Drives
Brandon Mattingly · Abinash Kar · Matthew Gorban · William Julius · Cooper K. Watson · MD Ali · Andrew Baas · Caleb Elmore · Jeffrey S. Lee · Bahram Shakerin · Eric W. Davis · Gerald B. Cleaver
In one minute
A twelve-author Baylor University team, with Eric Davis of the Institute for Advanced Studies at Austin, does something nobody had done for warp drives: they draw them. The mathematics of Alcubierre’s 1994 bubble and Natário’s 2002 refinement has been settled for decades, but the pictures have not, and the usual map — the York time, which tracks how volume expands — only tells the whole story when the geometry of each time-slice is flat. So the team computes curvature invariants instead. These are numbers built from the curvature tensors that do not care which coordinates you chose, so a plot of them shows the real shape of the spacetime rather than an artefact of the map. Four invariants suffice for warp-drive geometries. Varying velocity, bubble radius and skin depth one at a time, the plots show a flat safe harbour for the ship, a thin shell of curvature at the bubble wall, and no singularity anywhere.
Connection
Chapter 4 needs pictures as much as it needs equations, and this is the paper that supplies them: a coordinate-free map of the curvature a warp bubble carries, with the ship’s safe harbour and the bubble wall visible as features rather than as claims. It also hands chapter 4 two design dials — the bubble radius sets the size of the harbour, the skin depth sets how sharply the wall folds — and a testable signature, since the wake that appears only under acceleration may radiate high-frequency gravitational waves.
From the source
Curvature invariants are scalar products of the Riemann, Ricci and Weyl tensors and their covariant derivatives, and they are independent of the coordinate basis. Plotting them is therefore free of coordinate mapping distortions, divergences, discontinuities and other artefacts of the chosen coordinates — which the existing warp-drive map, the York time, is not, because it only carries the full information when the three-geometry of the hypersurfaces is flat.Section 1, Introduction, final three paragraphs
Four Carminati–McLenaghan invariants — the Ricci scalar, the first two Ricci invariants, and the real component of the Weyl invariant J — form a complete set for class B1 spacetimes, which include all globally hyperbolic spacetimes and therefore the general warp-drive line element. Seventeen invariants are needed in general; the syzygies reduce the warp-drive case to four.Section 1, paragraph 9; Section 2, equations (1) to (4)
Neither the Alcubierre warp drive nor the Natário warp drive at constant velocity contains a curvature singularity anywhere in its spacetime manifold. Because these spacetimes are fully characterised by their invariants, a horizon would have been detected as a curvature singularity; since none exists, the inside and the outside of the warp drive must be connected, and a coordinate transformation removing the known causal discontinuity may exist.Section 4, paragraph 2; Section 5, paragraph 2; Section 6, paragraph 1
Both drives have a safe harbour — a flat region inside the bubble where the invariants are zero and a ship would be unaffected by the bubble’s own curvature. Its spatial size is set directly by the bubble radius parameter: inspect the axis of each plot and the harbour is the same size as that radius, for both the Alcubierre and the Natário geometries.Section 4.3; Section 5.3; Section 6, paragraph 2
The curvature is concentrated in a thin shell at the bubble wall, where the radial coordinate equals the bubble radius, because the shape function converges to Alcubierre’s top-hat function. Increasing the skin depth drops the magnitude of the curvature exponentially — for the Alcubierre drive the Ricci scalar falls from the order of ten to the minus three down to the order of ten to the minus twenty-eight across the range examined — which implies that a thinner skin depth, carrying greater curvature, would propel the bubble at greater velocities.Section 4.2; Section 5.2; Section 6, paragraph 3
The wake and the constant non-zero curvature outside the bubble that appear for the accelerating Natário warp drive are absent at constant velocity, so both are features of acceleration alone. Potentially the wake produced by the constant acceleration of spacetime produces high-frequency gravitational waves — which would be detectable only during the accelerating phase of a flight, not during the constant-velocity phase.Abstract; Section 5, paragraph 4; Section 6, paragraph 4
The way in
https://doi.org/10.3390/universe7020021How to cite it
Brandon Mattingly, Abinash Kar, Matthew Gorban, William Julius, Cooper K. Watson, MD Ali, Andrew Baas, Caleb Elmore, Jeffrey S. Lee, Bahram Shakerin, Eric W. Davis, Gerald B. Cleaver (2021) Curvature Invariants for the Alcubierre and Natário Warp Drives doi:10.3390/universe7020021