Difference between revisions of "Alcubierre Warp Drive"

Revision as of 15:03, 4 December 2012

Stuff from Sonny White's Paper

<math>\LARGE ds^2 = -c^2dt^2+\left(dx - v_s\left(t\right)f\left(r_s\right)dt\right)^2+dy^2+dz^2 </math>

This uses the familiar coordinates <math>\left(t, x, y, z\right)</math> and curve <math>x = x_s\left(t\right), y = 0, z=0</math> where x is analogous to what is commonly referred to as a spacecraft's trajectory. <math>r_s</math> is the "radial position in the commoving spherical space around the spacecraft's origin."

Sigma and R are arbitrary values that control the bubble thickness and size of the bubble, respectively.
<math>\LARGE f\left(r_s\right) = \frac{tanh\left(\sigma\left(r_s+R\right)\right)-tanh\left(\sigma\left(r_s-R\right)\right)}{2tanh\left(\sigma R\right)} </math>

Courtesy Mouse:
<math>\LARGE \frac{df}{dr_s} = \frac{2tanh\left(\sigma R\right)\left(\sigma sech^2\left(\sigma r_s+\sigma R\right) - \sigma sech^2\left(\sigma r_s - \sigma R\right)\right)}{4tanh^2\left(\sigma R\right)} </math>

Actual energy density:
<math>\LARGE T^{00} = -1\frac{1}{8\pi}\frac{v_s^2\rho^2}{4r_s^2}\left(\frac{df}{dr_s}\right)^2 </math>

• <math>\rho</math> is the mass-energy density (kg/m^3)

Other Stuff

<math>\LARGE ds^2 = \left(v_s^2f\left(r_s\right)^2-1\right)\left(dt-\frac{v_sf\left(r_s\right)}{v_s^2f\left(r_s\right)^2-1}\right)^2-dx^2+dy^2+dz^2 </math>

The Real Deal

Energy density at radial position r_s:

<math>\LARGE T^{00} = \frac{-v_s^2\rho^2}{32\pi r_s^2}\left(\frac{df}{dr_s}\right)^2 </math>

Where ρ is the vacuum energy density (~6E-10 J/m³) and v_s is the velocity of the warp bubble. df/dr_s is the derivative of f(r_s), above.

Total energy density is calculated by integrating T⁰⁰ for values of r_s from 0 to R+σ