# The "Warp Drive" Equations

The Alcubierre drive is a speculative idea based on a solution of Einstein's field equations in General Relativity. It describes a method of bending space-time in a way that allows for faster-than-light travel. This involves creating a "warp bubble" where space-time is contracted in front of the bubble and expanded behind it. Inside this bubble, a spacecraft could travel faster than light relative to an observer outside the bubble. This device can't actually be built, but it's a fun thought experiment.

So let's derive the equations for the Alcubierre drive.

import sympy as sp

t, x, y, z = sp.symbols('t x y z')
R, vs = sp.symbols('R vs', real=True)  # R is the radius of the bubble, vs is the velocity of the bubble

def f(X):
return sp.Piecewise((sp.exp(1 - 1 / (1 - X**2)), sp.Abs(X) < 1), (0, True))


Using spherical coordinates (r, theta, phi) for simplicity and setting theta = 0 for a linear path

r = sp.sqrt(x**2 + y**2 + z**2)
s = vs * t - r  # Reference parameter


Then we define the metric function for the Alcubierre drive. We define the 4-dimensional metric tensor $g_uv$ (here only components needed)

fs = f(s / R)

g00 = fs - vs**2 * fs
g01 = -vs * fs
g02 = g03 = g10 = g20 = g30 = 0
g11 = g22 = g33 = 1


Now, we attempt to calculate the Einstein tensor $G_uv$, which requires the Christoffel symbols and Ricci tensor

g = sp.Matrix([
[g00, g01, g02, g03],
[g10, g11, g02, g03],
[g20, g10, g22, g03],
[g30, g10, g02, g33]
])

g_inv = g.inv() # The inverse metric tensor


We can calculate the Christoffel symbols using the following function:

def christoffel_symbols(g, g_inv):
n = g.shape[0]
symbols = sp.zeros(n, n, n)
for k in range(n):
for i in range(n):
for j in range(n):
symbols[k, i, j] = 0.5 * sum([
g_inv[k, m] * (sp.diff(g[m, j], symbols[i]) +
sp.diff(g[m, i], symbols[j]) -
sp.diff(g[i, j], symbols[m]))
for m in range(n)
])
return symbols

gamma = christoffel_symbols(g, g_inv)


And then we can calculate the Ricci tensor $R_uv$

n = g.shape[0]
R = sp.zeros(n, n)
for alpha in range(n):
for beta in range(n):
R[alpha, beta] = sum([sp.diff(gamma[alpha, mu, beta], t) -
sp.diff(gamma[alpha, t, mu], beta) +
sum([gamma[alpha, t, sigma] * gamma[sigma, mu, beta] -
gamma[alpha, mu, sigma] * gamma[sigma, t, beta]
for sigma in range(n)])
for mu in range(n)]
)

R_scalar = sum([R[m, n] * g_inv[m, n] for m in range(n) for n in range(n)])

G = R - 0.5 * g * R_scalar
G_simplified = sp.simplify(G)


Which gives us the Einstein tensor $G_uv$

print("Einstein Tensor Components:")
for i in range(n):
for j in range(n):
print(f"G[{i},{j}] = {G_simplified[i,j]}")

fs = f(s / R)

# Show the function
print("Warp factor f(s/R) =", fs)


Which gives us the warp factor $f(s/R)$. Of course now we just need some exotic matter to make this happen.