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Is de Sitter the only vacuum with positive CC

  1. May 20, 2015 #1

    wabbit

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    Assuming a spacetime with zero Weyl curvature and an Einstein tensor proportional to the metric, is it true that in a finite neighborhood of any point, that spacetime must be isometric to a de Sitter vacuum, or are there other possible solutions, and if so how are they classified?

    Thanks
     
  2. jcsd
  3. May 20, 2015 #2

    fzero

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    Einstein tensor proportional to the metric implies constant scalar curvature. Then vanishing of the Weyl curvature implies that the space is maximally symmetric ##R_{mnpq} = k ( g_{mp}g_{nq} - g_{mq}g_{np})## and therefore of constant sectional curvature. Therefore such a space is, by a conformal rescaling of the metric, equivalent to one of the model constant curvature spaces. For signature ##(1,d-1)## and positive cosmological constant, this is indeed de Sitter.
     
  4. May 20, 2015 #3

    wabbit

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    Thanks - you say "by a conformal rescaling", so it isn't isometric, only conformally equivalent ? I must say conformal transformations isn't something I am really familiar with.

    Would you have a source to suggest where I could read more about this ?
     
  5. May 20, 2015 #4

    fzero

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    The issue is that the denominator of the formula for sectional curvature involves precisely the same contractions that correspond to the Riemann tensor of a maximally symmetric manifold. So we can rescale ##g' = e^{2\sigma(x)} g## without changing the sectional curvature. With this relation we say that ##g'## is pointwise conformal to ##g##. If there is a diffeomorphism that pulls ##g'## back to ##g##, then we say that the metrics are conformally equivalent and there is a genuine isometry. I think this is a stronger condition than the assumptions warrant.

    It is probably overkill and yet might not even answer all questions that you might have, but the most specific reference I know of is Besse, Einstein Manifolds. Some results are discussed in the first few pages of this lecture.
     
  6. May 21, 2015 #5

    wabbit

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    Ah, the situation seems more complex than I thought - will check these, thanks for the references.
     
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