So lets say an observer outside of any warp uses coordinates [cT, X, Y, Z], define where will be a function of T for this will be a standard for the distance from somewhere to the center of a warp ship. The spacetime given by will be an Alcubierre warp drive spacetime given boundary conditions A is called the lapse function because its value at the ship's location determines time dilation for the ship during the trip. The ship will have a speed of For the case that A=1 everywhere, time does not lapse for the ship with respect to the frame for which it was initially at rest when So what I'm interested in is the energy density according to coordinates that are actually appropriate for the ship. After all it is the ship that we want to generate the warp. So what we start out doing is Let A=1 everywhere and then do the transformation The expression for the spacetime becomes And so for simplicity of calculation we define a new function g g=1-f So it becomes defining Yields the boundary conditions to be From the perspective of the ship observer, he floats at zero velocity with no pushes or pulls in a locally inertial frame while it is the stars that are at warp in the opposite direction. Now here's the first trick. I am going to reintroduce what was called the lapse function term into this standard of coordinates with the modifying exception that I am going to change its boundary conditions. The modified warp drive spacetime will be with You calculate or compute from the metric , and use to arrive at the ship frame energy density term to be So from the perspective of the ship, which is what we want to produce the stress-energy tensor for the warp, the energy density can be arbitrarily reduced by allowing A to be large in the region of the warp matter where the negative energy density would have otherwise been large. Yet with its boundary conditions there need be no lapse in time between the ship observer, and the inertial frame observers he was initially at rest with respect to before going into warp. More later.