Derivation for Alcubierre Drive

Is there any online derivation for the Alcubierre Drive? I have looked at several papers and none of them actually derive the equations. They simply point them out. Particularly, I would like a derivation of how he arrives at his metric,

$$\mathrm{d}s^2 ~ = ~ -\left(\alpha^2 - \beta_i\beta^i\right)\mathrm{d}t^2 + 2\beta_i \, \mathrm{d}x^i \, \mathrm{d}t + \gamma_{ij} \, \mathrm{d}x^i \, \mathrm{d}x^j \,.$$

I am unsure how he arrived at this equation. I have studied some tensor calculus and analysis and know enough about the metric tensor and other variables to get semi close to deriving Einstein's field equations. However, I don't understand this. Did he use the 3+1 formalism to arrive at this equation? If this is the case, can anyone point me to where it is derived or could someone show me the derivation so I can practice.

• This is just the standard ADM form of the metric... – Akoben Mar 3 '17 at 19:24
• @Cosmoman See lanl.arxiv.org/pdf/gr-qc/0009013v1. – udrv Mar 3 '17 at 19:30
• Without specifying $\alpha$, $\beta^i$ and $\gamma_{ij}$ this equation is meaningless. Any metric can be cast into this form! – Prof. Legolasov Mar 6 '17 at 14:59

The metric written,

$$ds^2 = -(\alpha^2 - \beta_i\beta^i)dt^2 + 2\beta_i dx^i dt + \gamma_{ij}dx^i dx^j$$

is the highly general expression for the metric of a manifold which can be foliated by space-like hypersurfaces. That is to say, one can think of the spacetime as a timeline of space-like manifolds, with $\alpha$ specifying the proper time between each.

The Alcubierre metric is the case for which $\alpha= 1$, $\beta^x = -v_s(t)f(r(t))$ and $\gamma_{ij} = \delta_{ij}$ is the Euclidean metric, with,

$$r(t)^2 = (x-x_s(t))^2 + y^2+z^2, \quad f(r) = \frac{\tanh \sigma(r+R)-\tanh\sigma(r-R)}{2\tanh\sigma R}$$

which yields the metric,

$$ds^2=(v_s^2f(r)^2-1)dt^2-2v_sf(r)dxdt + dx^2 + dy^2 + dz^2.$$

Alcubierre himself in the original paper does not show how this was derived; it seems to have been more of an educated guess. However the motivation is clear; in the $3+1$ formalism (chosen partially to guarantee that closed timelike curves were not possible) he sought a way to modify the metric in a way to allow a body to be 'pushed' along the trajectory $x_s(t)$.

Notice the metric can be written as,

$$ds^2 = -dt^2 + (dx-v_s f(r)dt)^2 + dy^2 + dz^2.$$

In the limit the parameter $\sigma \to \infty$,

$$f(r) = \Theta(r+R)-\Theta(r-R)$$

where $\Theta$ is the Heaviside step function. Thus, we see that for $r\in[-R,R]$, there is some motion with velocity $v_s$ and outside of this region $f=0$, and we enter purely flat space.

Essentially, Alcubierre defined the metric through the ADM formalism of general relativity, which states that spacetime is foliated into space-like hypersurfaces of constant coordinate time t. Through this, he created his metric such that space was warped in the desired way. I don't have enough reputation to comment and wasn't sure how to format the equations on here properly, but http://zelmanov.ptep-online.com/papers/zj-2009-12.pdf explains how it was derived.