Hello friends, I am now interested in Einstein scalar field equations with a very little knowledge about Physics. I would like to ask why vacuum Einstein equations are hyperbolic equations? As far as I know that for [tex]3+1[/tex]-dimensional manifold [tex]V[/tex], we can convert the vacuum Einstein equations as a system of time derivative w.r.t. the induced metric and the second fundamental form of a [tex]3[/tex]-dimensional manifold, called hypersurface [tex]\Sigma[/tex]. What happen to non-vacuum case since we have an extra term, called stress-energy momentum tensor. Can you explain in details? Thank you.