The Textile editor

So I found a neat-o little tool which turns human-readable code into HTML, and tried out a couple of paragraphs of my final year project on it. The results are given in the full post. Textile's a bit buggy, but it doesn't half save time. I'd suggest using it to get the basic code and then weeding through by hand to clean it up.

3. The Alcubierre metric

3.1 The (3+1) formalism

York¹³ explains how space-time may be described by what is termed a (3+1) formalism. Alcubierre’s paper uses this formalism, so we shall spend some time looking at it. In such a formalism, a space time V is thought of as being divided (or foliated) into disjoint three-surfaces Σ_τ, each arising as the level surface of some scalar function, so f(t,x) = τ. We look for space-like foliations, which is to say that if T_xΣ_τ is the tangent space to Σ_τ at x∈Σ_τ, then g(X,X) > 0 for all X∈T_xΣ_τ and all x∈Σ_τ.

…

Let us define β^ε = −α´(t) Y^ε: then

Y^a = (α´(t))⁻¹(1,−β^ε) and Y_a = −α´(t)(1,0).

¹³ J. W. York, Jr., Kinematics and dynamics of general relativity in Sources of gravitational radiation (Larry Smarr, ed.), Cambridge University Press, 1979, pp. 83—126.