Historical research guide · 0445 · 2020
Dust content solutions for the Alcubierre warp drive spacetime
Osvaldo L. Santos-Pereira · Everton M. C. Abreu · Marcelo B. Ribeiro
In one minute
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.
Connection
Chapter 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.
From the source
The 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
Taking 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
Because 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)
The 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
Read 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
A 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)
The way in
https://doi.org/10.1140/epjc/s10052-020-8355-2How 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