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Weak field newtonian limit

  1. Nov 5, 2012 #1
    Hello,

    I have to calculate the gravitational field strength g given by the De Sitter-Schwarzschild spacetime.

    If G=c=1 I get:

    [itex]R_{00}\simeq -kT_{00}+\frac{1}{2}kT\eta_{00}+\Lambda \eta_{00}\simeq [/itex]

    [itex]-kT_{00}+\frac{1}{2}kT_{00}\eta^{00}\eta_{00} +\Lambda \eta_{00}\simeq -\frac{1}{2}kT_{00}+ \Lambda =-\frac{1}{2}k\rho+ \Lambda [/itex]

    On the other side:

    [itex]R_{00}\simeq -\Gamma_{00,j}^j \simeq -\frac{1}{2}\bigtriangledown^2g_{00}\simeq -\bigtriangledown^2 \phi[/itex]

    Equaling:

    [itex]\bigtriangledown^2 \phi=\frac{1}{2}k\rho-\Lambda=4piG\rho-\Lambda [/itex]

    The first problem I have is that I don't know how to get this result in normal units, and the second problem is that, from here, I have to find the gravitational field strength g, and I don't know how to do it.

    thanks for any help!
     
  2. jcsd
  3. Nov 5, 2012 #2

    haushofer

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    Science Advisor

    What precisely are "normal units"? I would say the units as you use here are just fine. Second, the gravitational field strength is given by

    [tex]
    g^i \equiv \partial^i \phi
    [/tex]

    whereas you have found an expression for [itex] \partial_i \partial^i \phi [/itex]. Looking at the general method to solve the Poisson equation for lambda=0 should help you, e.g. here:

    http://cass.ucsd.edu/~ppadoan/new_website/physics105b/Lecture3.pdf [Broken]

    You have an extra constant, which modifies this solution. See also

    http://arxiv.org/abs/gr-qc/0004037,

    eqn.2.6.
     
    Last edited by a moderator: May 6, 2017
  4. Nov 5, 2012 #3
    With 'normal' units I mean G=c=1
     
  5. Nov 5, 2012 #4
    Thank you haushofer, those links have helped me a lot.
     
    Last edited by a moderator: May 6, 2017
  6. Nov 5, 2012 #5

    haushofer

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    Science Advisor

    Well, there is no c in your equations, so that's easy: just put G=1 :P

    Glad I could help :)
     
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