Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]​


    [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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook