Historical research guide · 0656 · 2021

Fluid dynamics in the warp drive spacetime geometry

Osvaldo L. Santos-Pereira · Everton M. C. Abreu · Marcelo B. Ribeiro

In one minute

Alcubierre wrote his 1994 warp metric backwards: he drew the geometry he wanted, then read off the matter it would need — and the answer came out negative. Osvaldo Santos-Pereira, Everton Abreu and Marcelo Ribeiro, working in Rio de Janeiro, run the calculation the other way round. They put a fluid into Einstein’s equations as the gravity source and ask which warp geometries that fluid allows. With a perfect fluid, the textbook matter with a single pressure, they get an equation of state in which the pressure is three times the matter density, and four families of solution: two collapse back to the vacuum they had already found with dust, and two permit warp speeds with positive matter density, at the price of a bubble-shaping function that turns complex. They then loosen the fluid so its pressures may differ direction by direction — the parametrized perfect fluid — and find solutions with positive density that satisfy the weak, dominant, strong and null energy conditions outright. Negative matter, they conclude, may not be a strict precondition for a warp bubble.

Connection

Chapter 4 has spent thirty years being told that a warp bubble needs exotic matter, and this paper shows where that conclusion actually came from: solving the problem with the simplest possible source. Give the source more structure — pressures that differ by direction, heat flux, the anisotropic imperfect fluids of chapter 5 — and the requirement for negative density loosens rather than tightens.

From the source

The Alcubierre metric was never advanced as a solution of the Einstein equations. It was proposed as an ad hoc geometry designed to move a massive particle at global superluminal speed while it stays locally subluminal, and the negative energy density came out afterwards, as a consequence. The physical question is therefore the reverse one: which matter-energy sources are capable of creating a warp bubble.Sect. 1, Introduction, paragraph beginning ‘In a previous paper’
With incoherent dust as the source, the authors’ earlier paper found that the Einstein equations for the warp drive metric return the vacuum — dust cannot create a warp bubble — but the solution carries the inviscid Burgers equation with it, so the wall of the warp bubble takes the form of a shock wave moving through a fluid, behaving as a plane wave.Sect. 1, Introduction; Sect. 7.1; Sect. 8, Conclusions
For a perfect fluid source the Einstein equations give the equation of state in which the pressure equals three times the matter density. Two of the four solution subcases reduce to the earlier dust result, a vacuum with a non-homogeneous Burgers equation. The other two allow warp speeds with positive matter density, but then the shift function, and with it the regulating function that shapes the bubble, becomes complex-valued — a caveat the authors name as a possible major stumbling block, whose real part would be what corresponds to a physically viable bubble.Sect. 3.1; Sect. 8, Conclusions, paragraph on Cases 1a and 2a
The parametrized perfect fluid relaxes the perfect fluid by letting the pressure components A, B, C and the momentum density parameter D differ from one another and depend on all four coordinates. Its physically plausible cases give an equation of state linking the matter density to the shift function, the pressures and the momentum density, and the coordinate dependence of the shift becomes a function of time and one transverse coordinate.Sect. 4, Eq. (4.1); Sect. 4.1 and 4.1.1
The weak, dominant, strong and null energy conditions were computed for both fluids. All of them are satisfied by the perfect fluid solutions. For the parametrized perfect fluid they are satisfied under explicit constraints — chiefly that the pressure A be greater than or equal to zero — which do not force the matter density to be negative. The null energy condition also constrains the bubble speed: the shift can take values above the speed of light but cannot take the value of the speed of light exactly, nor its negative.Sect. 6, Sects. 6.4.1 and 6.4.2, Eqs. (6.39) to (6.43); Table 3
The parametrized perfect fluid can be read as an anisotropic fluid with heat flux — a perfect fluid in a warp drive background plus a dissipative fluid — which opens the study of a warp bubble driven by fluid anisotropy with dissipative effects. The open questions the authors name are the amount of mass-energy density, exotic or not, that the drive would actually need, quantum effects, and whether these solutions are stable.Sect. 7.2, Eqs. (7.11) to (7.24); Sect. 7.3

The way in

https://doi.org/10.1140/epjc/s10052-021-08921-3

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

LICENCE. The article carries its own Creative Commons statement — ‘This article is licensed under a Creative Commons Attribution 4.0 International License’ — printed in the Open Access block of the version of record in The European Physical Journal C, volume 81 (2021), article 133, published 6 February 2021. The Unpaywall record for this DOI agrees: gold open access, cc-by. TEXT. This is a 22-page paper of about 15,900 words whose body is largely case-by-case algebra. Reproduced below in full are the abstract, the introduction, the warp drive geometry of Sect. 2.1, the defining energy–momentum tensors of Sects. 3 and 4, the opening of the energy-conditions analysis in Sect. 6, and Sects. 7 and 8 entire. The step-by-step solution of the Einstein equations for each subcase in Sects. 3 to 6, and the three summary tables, are omitted for length; the complete text is at the source. Running heads, page numbers and reference-number markers are dropped as page furniture, and the 42-item bibliography is omitted. The article reached the library as a two-column scan whose columns interleave, so displayed equations arrived with fractions and index structure broken. Equations are given in plain notation with their original numbers only where the scan preserved them intact; the Alcubierre line element (2.6), the ten Einstein tensor components (2.11) to (2.20), the case equations of Sects. 3 to 5 and the energy-condition inequalities of Sect. 6 are given as named results rather than guessed. Inequalities are written in words.

How to cite it

Osvaldo L. Santos-Pereira, Everton M. C. Abreu, Marcelo B. Ribeiro (2021) Fluid dynamics in the warp drive spacetime geometry doi:10.1140/epjc/s10052-021-08921-3

Where it sits in the curriculum

The metric, warp drives and wormholes