Memorandum for the Record

"Since they were introduced by Alcubierre [1], warp-drive spacetimes have been certainly one of the most studied solutions of the Einstein equations among those requiring exotic matter [2]. They are not only an exciting theoretical test for our comprehension of general relativity and quantum field theory in curved spacetimes, but they might also be, at least theoretically, a way to travel at superluminal speed. The warp drive consists of a bubble containing an almost flat region, moving at arbitrary speed within an asymptotically flat spacetime.

After the proposal of this solution, its most investigated aspect has been the amount of exotic matter (i.e. energy-conditions violating matter) required to support such a spacetime [3, 4, 5, 6]. It has been found, using the so called quantum inequalities (QI), that such a matter must be confined in Planck-size regions at the edges of the bubble. This bound on the wall thickness turns into lower limits on the amount of exotic matter required to support the bubble (at least of the order of 1 solar mass)."

Of course this is much too big in terms of http://www.darpa.mil/news/2010/starshipnewsrelease.pdf

The only hope is amplification of the effective gravity coupling G/c^4 of curvature Guv to applied stress-energy current densities Tuv by at least 40 powers of ten over a small spatial "Yukawa" length, or in some frequency-wavevector region of the EM spectrum with nonlinear transduction to DC EM fields with negative energy density (superconducting meta-material)."

In such a case the required mass-energy drops from 10^33 gm to 10^-7 gm because, roughly

Guv(curvature) ~ 8piGNewton(index of refraction)^4/c^4Tuv(applied electromagnetic field).

"Less effort has been devoted to other important issue regarding the feasibility of these spacetimes: the study of the warp-drive semiclassical stability. In particular, it was studied in [7] the case of an eternal superluminal warp drive by discussing its stability against quantum effects. It was there noticed that, to an observer within the warp-drive bubble, the backward and forward walls (along the direction of motion) look respectively as the future and past event horizon of a black hole. By imposing over the spacetime a quantum state which is vacuum at the null infinities (i.e. what one may call the analogue of the Boulware state for an eternal black hole) it was found that the renormalized stress-energy tensor (RSET) had to diverge on the horizons .

In this contribution we consider the more realistic case of a warp drive which is created with zero velocity at early times and then accelerated up to some superluminal speed in a finite time (a more detailed treatment can be found in [8]). We found, as expected, that in the centre of the bubble there is a thermal flux at the Hawking temperature corresponding to the surface gravity of the black horizon. However, this surface gravity is inversely proportional to the wall thickness, leading to a temperature of the order of the Planck temperature, for Planck-size walls. Even worse, we do show that the RSET does increase exponentially with time on the white horizon (while it is regular on the black one). This clearly implies that a warp drive becomes rapidly unstable once superluminal speed are reached."

The effective Planck length is increased by a factor of ~ 10^20.

Indeed we practical micron-tech fabrication we want not 10^-13 cm, but 10^-3 cm i.e. another factor of 10^20 in G. Ideally we want 10^60G i.e. index of refraction ~ 10^15 slowing down the speed of light to ~ 10^-5 cm/sec. This should improve the situation.

The Hawking temperature is reduced by a whopping factor of 10^30 in this Panglossian best of all possible worlds scenarios - and I suspect that the time it takes for the instability to develop will also be very long so as not to be a problem.

All is not yet lost in my opinion.

On Jan 17, 2011, at 5:33 PM, JACK SARFATTI wrote:

http://iopscience.iop.org/1742-6596/229/1/012018/pdf/1742-6596_229_1_012018.pdf

not the final word, but important to know (semi-classical model)