In one minute

Santos-Pereira, Abreu and Ribeiro solve the Einstein equations for the Alcubierre warp metric with fluid matter as the source. They use a perfect fluid and a more flexible parametrised perfect fluid, chosen to explore warp solutions with positive matter density. Two of four perfect-fluid subcases reduce to vacuum, as dust did, while others allow a warp drive with positive matter density at the cost of a complex regulating function. The parametrised fluid also gives solutions in which warp speeds arise with positive matter density. All four energy conditions are satisfied for the perfect fluid and constrain the parametrised fluid, and more complex matter distributions may make negative matter density unnecessary.

Fluid dynamics in the warp drive spacetime geometry