- #1
bookworm_vn
- 9
- 0
Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read
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
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.
\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.
