Introduction to the 3+1 Einstein equations
The 3-metric gij and extrinsic curvature (second fundamental form) Kij are the fundamental variables describing the geometry in any space+time decomposition of the Einstein equations. (g,K) describe the local geometry of a single space-like hypersurface M , and it is then natural to describe the evolution of the space-time geometry by a 1-parameter family (g(t), K(t)), describing the local geometry of the (space-like) hypersurfaces Mt. In order to piece these hypersurfaces together, however, we must also specify the lapse N and shift vector X , which describe the relation between the time evolution vector ∂t 1 and the space-time vector n normal to the hypersurfaces: