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General relativity: constant curvature, characterizing equation

  1. May 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Show, that a three-dimensional space with constant curvature K is charaterized by the following equation for the Riemann curvature tensor:

    [tex]R_{abcd} = K \cdot \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)[/tex]

    2. Relevant equations



    3. The attempt at a solution

    Hi folks,

    I would like to give an own attempt, but I have no Idea how to start.

    We haven't defined the curvature K in lecture. How is it defined?
    Has anybody an idea, how to start?

    --
    derivator
     
  2. jcsd
  3. May 19, 2010 #2
    anybody an idea?
     
  4. May 19, 2010 #3

    gabbagabbahey

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    Homework Helper
    Gold Member

    Wald defines the Ricci tensor as

    [tex]R_{ac}=R_{abc}{}^{b}[/tex]

    And then the scalar curvature is the trace of the Ricci tensor

    [tex]R=R_{a}{}^{a}[/tex]

    Your text should have similar definitions.

    So, I think want you want to do is to show that if [itex]R_{abcd} = K\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right)[/itex], where [itex]K[/itex] is a constant, then the scalar curvature is [itex]R=K[/itex].
     
  5. May 20, 2010 #4
    ah, thx for your input.

    i haven't seen this exercise from this point of view, but it makes sense.

    I have found: R=6K, thus R is constant, thus we have a constant curvature.
     
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