Dust content solutions for the Alcubierre warp drive spacetime

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 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 out 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.

Warp drive dynamic solutions considering different fluid sources

“…In previous studies 26,27 it was found an intrinsic relationship between the WD concept and shock waves via the Burgers equation arising from vacuum solutions of Einstein equations for the Alcubierre WD metric. Such a result may mean that the warp bubble could be interpreted as shock waves in vacuum.…”

“…Starting with incoherent fluid, or dust, vacuum solutions of the Einstein equations for the WD metric (6) were recovered, which led to the connection of the WD with shock waves via Burgers equation. 26 Vacuum solutions mean that all the energy conditions are trivially satisfied, but the possibility of vacuum shock waves that could physically represent warp bubbles indicated that other known matter-energy distributions could result in more complex solutions of Einstein equations. Under this perception we proposed two other energy-momentum tensors, the perfect and "parametrized" fluids, where the latter is, in fact, an anisotropic fluid with heat flux.…”

“…(34) to be as identically zero we arrive at vacuum solutions for the WD spacetime. 26,27 However, if we include the cosmological constant in the Einstein equations the expression (34) is changed, yielding,…”

“…New results such as the vacuum solutions connecting the WD to shock waves via the Burgers equation are presented. 26 We used the perfect fluid energy-momentum tensor 27 (EMT) which disclosed new exact solutions of the Einstein equations. We also demonstrated that starting with simple forms of mass and energy sources, vacuum solutions for the WD spacetime arise when we impose that the WEC is trivially satisfied by a choice of gauge on the shift vector and the spacetime coordinates.…”

“…Section 3 depicts the perfect and anisotropic fluid cases 27 and Section 4 discusses the relationship between the WD spacetime, vacuum solutions, and shock waves described by the Burgers equations. 26 Section 5 presents a discussion on the role of the cosmological constant in the Einstein equation solutions for the WD spacetime. 29 Section 6 presents the conclusions.…”