Slides of 5 lectures at XV Brazilian School on Cosmology and ...

The 3 + 1 decompositionFor a globally hyperbolic manifold, we can make a 3 + 1-splitwhereds 2 = −ω 0 ⊗ ω 0 + g ij (t,x) ω i ⊗ ω j ,ω 0ω i= N(t,x) dt= dx i + N i (t,x) dt.N(t,x k ) is the lapse function: measures differencebetween coordin<strong>at</strong>e time, t, and proper time, τ, oncurves normal to hypersurfaces Σ t , n α = (−N, 0, 0, 0)N i (t,x k ) is the shift vector: measures the differencebetween a sp<strong>at</strong>ial point, p, and the point reached byfollowing the normal n.15th <strong>Brazilian</strong> <strong>School</strong> on Cosmology and Gravit<strong>at</strong>ion, August 2012 – p.32/143