Einstein Vacuum Equation, Vacuum Constraint Equations

bookworm_vn

Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read


where [tex]\overline R[/tex] the scalar curvature, [tex]\overline g_{\alpha\beta}[/tex] the metric tensor and [tex]\overline R_{\alpha\beta}[/tex] the Ricci tensor.

By using the twice-contracted Gauss equation and the Codazzi equations of the Riemannian submanifold [tex]M[/tex], one finds that the normal-normal and normal-tangential components of the above Einstein vacuum equation are


and

where [tex]R[/tex] is the scalar curvature of [tex]M[/tex], and [tex]k[/tex] its second fundamental form. These equations, called the Vacuum Constraint Equations involve no time derivatives and hence are to be considered as restrictions on the data [tex]g[/tex] and [tex]k[/tex].

The point is how to derive these Vacuum Constraint Equations. Thank you very much.

hamster143

Have you seen pages 258-259 of Wald?

bookworm_vn

Got it, thanks in advance, I am working in pure Maths so that I had not known the book due to Wald .