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 Στ, 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 TxΣτ is the tangent space to Στ at x∈Στ, then g(X,X) > 0 for all X∈TxΣτ and all x∈Στ.
Let us define βε = −α´(t) Yε: then
Ya = (α´(t))−1(1,−βε) and Ya = −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.