Dust content solutions for the Alcubierre warp drive spacetime
Osvaldo L. Santos-Pereira · Everton M. C. Abreu · Marcelo B. Ribeiro
Open licence · full text · Creative Commons Attribution 4.0 International (CC BY 4.0)
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Alcubierre wrote his warp geometry down in 1994 without saying what matter would produce it — he drew the shape of the ride, not the engine. Osvaldo Santos-Pereira, Everton Abreu and Marcelo Ribeiro, working in Rio de Janeiro, do the missing half: they put the Alcubierre metric into the Einstein field equations and ask what the simplest imaginable source, pressureless dust, would have to do. The answer is sharp and useful. Dust alone does not hold the bubble up — every solution runs back to vacuum, the matter density comes out zero, and precisely because it does, the weak, strong, null and dominant energy conditions are all satisfied. What survives is the surprise. The one equation left standing for the bubble’s shape is the inviscid Burgers equation, the classic equation for shock waves in a frictionless fluid, so the warp bubble travels through spacetime the way a shock front travels through air, and the energy that makes the wave is purely geometrical. The source to try next, they say, carries electromagnetic terms.
Why it matters hereChapter 4 needs the warp drive stated as a solution of the Einstein equations rather than as a drawing, and this paper is the step that makes the demand exact: name your matter content, and the field equations will tell you what it can and cannot hold up. Its second result belongs to chapter 5 — the warp bubble comes out obeying the equation of a shock wave in an inviscid fluid, which is the vacuum-as-fluid picture arriving from the geometry side.
What it claims
01The Alcubierre metric was never advanced as a solution of the Einstein field equations. It was proposed as a geometry whose properties are equivalent to a propulsion system, so the question of what matter or field source could produce it was left open, and answering it means coupling the metric to the field equations and solving what results.Sect. 1, Introduction; Sect. 3, opening paragraph
Settled physics02Taking the dust energy-momentum tensor as the source, the Einstein equations for the Alcubierre metric reduce to the single condition that the matter density vanishes: a warp bubble cannot be created with a pressureless dust distribution as its source, and the solutions run back to vacuum.Sect. 4.1, Eqs. (4.4) and (4.5); Table 1
Published and peer-reviewed03Because the matter density vanishes, the energy density of Eqs. (3.14) and (3.16) is zero, and the weak, strong, null and dominant energy conditions are all trivially satisfied — the contraction of the stress-energy tensor with the Eulerian observers’ four-velocity is allowed to be zero, not only negative.Sect. 3.2, closing paragraph; Sect. 4.1, after Eq. (4.10)
Published and peer-reviewed04The one field equation left standing for the warp bubble’s shape is the conservative form of the inviscid Burgers equation, the equation of shock waves in a frictionless fluid. The warp drive metric can therefore be understood as spacetime motion equivalent to a shock wave moving in a fluid — here a plane wave — and the energy needed to create that shock wave is purely geometrical.Sect. 4.1, Eqs. (4.11) to (4.14); Sect. 4.2
Published and peer-reviewed05Read term by term, the surviving equation is a conservation law along the direction of travel: the time derivative of the shift function acts as a force per unit mass, and one half the x-derivative of its square acts as the divergence of a total energy that is entirely kinetic, which fits a source with no interaction between its particles.Sect. 4.1, final paragraph; Sect. 4.2
Published and peer-reviewed06A matter distribution richer than simple dust is what a warp bubble requires — the authors name electromagnetic components in the energy-momentum tensor, or the cosmological constant in the Einstein equations, as the next sources to try, and record that this is ongoing research. If the free function h(t) takes the form of a diffusion term, the equation becomes the viscous Burgers equation and h(t) may act as the source that generates the shock waves.Sect. 5, Conclusions; Eq. (5.1)
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Abstract
The Alcubierre metric is a spacetime geometry where a massive particle inside a spacetime distortion, called warp bubble, is able to travel at velocities arbitrarily higher than the velocity of light, a feature known as the warp drive. This is a consequence of general relativity, which allows for global superluminal velocities but restricts local speeds to subluminal ones as required by special relativity. In this work we solved the Einstein equations for the Alcubierre warp drive spacetime geometry considering the dust matter distribution as source, since the Alcubierre metric was not originally advanced as a solution of the Einstein equations, but as a spacetime geometry proposed without a source gravity field. We found that all Einstein equations solutions of this geometry containing pressureless dust lead to vacuum solutions. We also concluded that these solutions connect the Alcubierre metric to the Burgers equation, which describes shock waves moving through an inviscid fluid. Our results also indicated that these shock waves behave as plane waves.
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https://doi.org/10.1140/epjc/s10052-020-8355-2Licence confirmed on the source itself: the article carries the Springer Open Access statement, ‘This article is licensed under a Creative Commons Attribution 4.0 International License’, and is funded by SCOAP3. ABSTRACT ONLY on this sheet for now: the licence permits the full text, which is available at the source; a later pass may add the cleaned body.
How to cite it
Osvaldo L. Santos-Pereira, Everton M. C. Abreu, Marcelo B. Ribeiro (2020) Dust content solutions for the Alcubierre warp drive spacetime. doi:10.1140/epjc/s10052-020-8355-2
Where it sits in the curriculum
The metric, warp drives and wormholesThe vacuum as a quantum fluid