A Python implmentation of Alcubierre's metric using Numpy and Matplotlib.
Alcubierre defines the following,
in which
with parameters R>0 and σ>0. Alcubierre's metric is then written as,
Alcubierre later derives an expression using the extrinsic curvature tensor Kij (9) showing the expansion (12) as
θ = vs (xs / rs)(d𝑓 / drs)
In Python I defined the derivatives of r(s) & 𝑓(rs) as
def d_rs(x, rho, xs=15):
return ((x - xs)**2 + rho**2)**(1/2)
def d_frs(rs, sigma=8, R=1):
a = sigma * (np.tanh((R + rs)*sigma)**2 - 1)
b = sigma * ((np.tanh(-(R - rs)*sigma)**2 - 1) / np.tanh(R * sigma))
return (-1/2) * (a - b)where θ
def theta(x, p, xs=15, s=8, R=1):
vs = R
drs = d_rs(x, p, xs)
dfrs = d_frs(drs, s, R)
return vs * ((x - xs) / drs) * dfrs- Alcubierre, Miguel (1994). "The warp drive: hyper-fast travel within general relativity". Classical and Quantum Gravity. 11 (5): L73–L77. arXiv:gr-qc/0009013 Freely accessible. Bibcode:1994CQGra..11L..73A. doi:10.1088/0264-9381/11/5/001.
- https://en.wikipedia.org/wiki/Alcubierre_drive
- Added code to animate the graph.
- Added contours for each axis.