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Gen. Rel. - Simple tensor stuff

  1. Apr 20, 2007 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    This is a problem that is solved in my course notes, but I don't follow how it's done.

    The problem is to find the general solution to Einstein's equations for a Robertson-Walker cosmology with a stress-energy tensor corresponding to the fluctuations of the void in QFT. This only means that [tex]T_{\mu\nu}=\Lambda g_{\mu\nu}[/tex] where lambda is a scalar.

    So it start with

    [tex]G_{\mu\nu}=8\pi GT_{\mu\nu}[/tex]

    [tex]R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G \Lambda g_{\mu\nu}[/tex]

    And then the next line is

    [tex]R_{\mu\nu}=8\pi G(\Lambda - \frac{1}{2}g_{\mu\nu}T^{\lambda}_{\lambda})[/tex]

    Since when is

    [tex]R=-8\pi GT^{\lambda}_{\lambda}[/tex]

    ???
     
  2. jcsd
  3. Apr 20, 2007 #2

    Dick

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    Since:

    [tex]R^\lambda_\lambda-\frac{1}{2}g^\lambda_\lambda R=8\pi G T^\lambda_\lambda[/tex]

    and [tex]g^\lambda_\lambda=\delta^\lambda_\lambda=4[/tex] and [tex]R^\lambda_\lambda=R[/tex].

    That's since when.
     
    Last edited: Apr 20, 2007
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