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bookworm_vn

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[tex]\overline R_{\alpha \beta} - \frac{1}{2}\overline g_{\alpha\beta}\overline R=0[/tex]

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

[tex]R - |k|^2 + ({\rm trace} \; k)^2=0[/tex]

and

[tex]\nabla^\beta k_{\alpha\beta} - \nabla_\alpha {\rm trace} \; k=0[/tex]

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.