On the Arbitrary Reduction of Energy in Alcubierre Warp Drive Solutions with Positive Energy in the Warped Region for Superluminal Motion

Building on the article [16] where a superluminal Alcubierre propulsion system with positive
energy density and positive energy is possible as an improvement over [16], we make a small
modification to have a positive energy and positive energy density for a curvature bubble of generic
radius R and introduce a way to arbitrarily reduce such energy sources in the warped region.