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Einstein Vacuum Equation, Vacuum Constraint Equations

  1. Dec 18, 2009 #1
    Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read

    [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.
     
  2. jcsd
  3. Dec 20, 2009 #2
    Have you seen pages 258-259 of Wald?
     
  4. Dec 20, 2009 #3
    Got it, thanks in advance, I am working in pure Maths so that I had not known the book due to Wald :surprised.
     
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