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A question of Einstein field equation

  1. May 24, 2012 #1
    I got some trouble from this question:
    For a given metric: ds2 =t-2(dx2-dt2), derive the energy-momentum tensor which satisfies the Einstein equation: Rαβ- 1/2Rgαβ=8[itex]\pi[/itex]GTαβ.

    I got the Ricci scalar R=2, but Tαβ=0 for all α,β. Does this means a curved spacetime without any source(energy-momentum tensor)? Is this possible? Or this result implies that I have made some mistakes in my calculation?

    Thanks for answering this question!
     
    Last edited: May 24, 2012
  2. jcsd
  3. May 24, 2012 #2

    Nabeshin

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    A schwarzschild black hole is a curved spacetime with no stress-energy tensor, so yes.
     
  4. May 24, 2012 #3

    Mentz114

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    Gold Member

    A correction to the EFE as you've written them

    Rαβ- (R/2)gαβ=8πGTαβ.

    I hope it is a typo. (added later ) I see you fixed it after I posted.

    Your result is possible as Nabeshin has said.

    [edit]
    I did the calculation and your results are correct.
     
    Last edited: May 25, 2012
  5. Jun 11, 2012 #4
    I've got a question about this problem. I'm not completely new to GR, but I'm new to actual calculations because I've focused mostly on concepts and I haven't taken a GR class.

    How can one find the value of the Ricci Scalar from a given metric? And what about the Stress Energy tensor?
     
  6. Jun 11, 2012 #5

    Mentz114

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    From the Christoffel symbols,
    [tex]
    {\Gamma ^{m}}_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k})
    [/tex]
    the Riemann tensor follows,
    [tex]
    {R^{r}}_{mqs} = \Gamma ^{r}_{mq,s}-\Gamma ^{r}_{ms,q}+\Gamma ^{r}_{ns}\Gamma ^{n}_{mq}-\Gamma ^{r}_{nq}\Gamma ^{n}_{ms}
    [/tex]
    from which
    [tex]
    R_{ms}={R^{r}}_{mrs}
    [/tex]
    and so
    [tex]
    R=g^{ms}R_{ms}
    [/tex]

    Use Maxima or some other CAS to calculate this stuff - it takes days by hand.
     
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