Einstein Vacuum Equation, Vacuum Constraint Equations

[bookworm_vn]

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

$$\overline R_{\alpha \beta} - \frac{1}{2}\overline g_{\alpha\beta}\overline R=0$$​

where $$\overline R$$ the scalar curvature, $$\overline g_{\alpha\beta}$$ the metric tensor and $$\overline R_{\alpha\beta}$$ the Ricci tensor.

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

$$R - |k|^2 + ({\rm trace} \; k)^2=0$$​

and

$$\nabla^\beta k_{\alpha\beta} - \nabla_\alpha {\rm trace} \; k=0$$​

where $$R$$ is the scalar curvature of $$M$$, and $$k$$ 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 $$g$$ and $$k$$.

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]

Have you seen pages 258-259 of Wald?

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