Historical research guide · 0543 · 2023
ADM mass in warp drive spacetimes
Sebastian Schuster · Jessica Santiago · Matt Visser
In one minute
A warp drive is a patch of spacetime engineered so a bubble of ordinary space carries its contents along while space flows around it. Sebastian Schuster, Jessica Santiago and Matt Visser ask a question nobody had formalised: what happens when the bubble has mass? They separate three cases that usually get run together — the payload inside, the bubble itself, and a background the bubble flies past — and show the first is the simplest. Because the metric is built by design, a payload’s mass is screened by the bubble, and the mass an outside astronomer measures depends only on how the flow falls off far away. For the zero-vorticity family they make precise what it means for the bubble to move while the fixed stars stand still, compute the stress-energy exactly, and find the average pressure changes sign between the front and the back. One whole hemisphere therefore breaks the null energy condition — which the authors read as an invitation to work out the underlying physics, not as a prohibition.
Connection
Chapter 4 is where the site’s propulsion argument lives, and this is the paper that says precisely what a warp bubble’s mass is, where the energy-condition violation sits, and how big it has to be. The authors’ own framing is the one a builder wants: these violations are not an absolute prohibition but a signpost to think carefully about the source — which is where chapter 2’s question about what the vacuum can actually supply takes over.
From the source
The generic Natário warp drive line element has flat spatial slices and unit lapse, so all of its physics sits in the flow vector, which is the negative of the ADM shift; Schwarzschild in Painlevé–Gullstrand form and every spatially flat FLRW spacetime also fit this form, so asymptotic flatness plus non-staticity is needed before a metric in these coordinates counts as a warp drive at all.Sect. 2.1, Eqs. (2.1)–(2.5)
A massive payload inside a warp bubble is the least troublesome of the three ways mass can enter: because the metric is reverse-engineered, a smooth bump function splices the payload’s metric into the bubble’s, and the mass of the whole spacetime then depends on the bubble alone, which screens whatever mass is inside it.Sect. 2.3, case A, Eq. (2.10)
The ADM mass of a warp drive is read off the large-distance falloff of the flow vector, the term going as the square root of 2M over r; the Alcubierre and zero-expansion warp drives have identically zero ADM mass, while a zero-vorticity warp drive can carry a finite non-zero one, and the mass wants to be both non-negative and finite.Sect. 2.3, case B, Eq. (2.11)
Switching the warp field on changes only the spatial part of the stress-energy, and changes it linearly in the warp velocity: the energy density and the momentum flux are exactly those of the base spacetime, while the average pressure picks up a term proportional to the warp velocity contracted with the gradient of the Laplacian of the flow potential.Sect. 3.2, Eqs. (3.19)–(3.24)
For a Schwarzschild-based warp drive the base energy density is zero, so the average pressure is proportional to the warp velocity dotted into the radial unit vector; that product changes sign between the front and the back of the bubble, giving an entire hemisphere where the sum of density and average pressure is negative, so the null energy condition — and with it the weak, strong and dominant conditions — is explicitly violated.Sect. 4.3, Eqs. (4.8)–(4.12); Sect. 5.2, Eq. (5.4)
The open problem the authors name is a self-consistent warp drive: reverse-engineering the Einstein equations says nothing about the matter that would source the geometry, and the steps they identify are sourcing it with Einstein–Klein–Gordon or Einstein–Maxwell fields, and then the full semiclassical Einstein equations with a quantised field on a backreacting background.Sect. 6, closing paragraph
The way in
https://doi.org/10.1007/s10714-022-03061-9How to cite it
Sebastian Schuster, Jessica Santiago, Matt Visser (2023) ADM mass in warp drive spacetimes doi:10.1007/s10714-022-03061-9