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The Cosmological (constant)

  1. Jun 22, 2011 #1
    Hi,

    I am wondering if the cosmological constant is a constant in the sense that it can only have one value, ie some constant element of the reals, or if it can be a scalar function too dependent on the coordinate variables, eg [itex]\Lambda(r,t)[/itex].

    Thanks in advance,
     
  2. jcsd
  3. Jun 22, 2011 #2

    atyy

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    Usually it's constant to maintain that the divergence of the stress-energy tensor is zero.

    I think there are ways of adding it not to the field equations, but to the action, and varying with respect to it, to also maintain energy conservation. http://arxiv.org/abs/gr-qc/0505128
     
    Last edited: Jun 22, 2011
  4. Jun 22, 2011 #3
    That makes sense. If I may, would the satisfaction of
    [tex]\nabla_{\beta}\left( T^{\alpha\beta}-g^{\alpha\beta}\Lambda\right)=0[/tex]
    justify the inclusion of a cosmological constant that was a scalar function?
     
    Last edited: Jun 22, 2011
  5. Jun 22, 2011 #4

    atyy

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    Do you mean something like the potential of a scalar field forming part of the stress-energy tensor of matter (http://ned.ipac.caltech.edu/level5/Carroll2/Carroll1_3.html" [Broken])?
     
    Last edited by a moderator: May 5, 2017
  6. Jun 22, 2011 #5
    Thanks for your help,

    I think this answers my question.
    This sounds like it is ok to include a cosmological constant of the form
    [tex]\Lambda(x_{\alpha})=\Lambda_0+V\,[\phi(x_{\alpha})][/tex]
    that consists of an initial cosmological constant, [itex]\Lambda_0[/itex], summed with a scalar function [itex]V[/itex]. Have I interpreted this correctly?
     
    Last edited by a moderator: May 5, 2017
  7. Jun 23, 2011 #6

    haushofer

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    The LHS of the Einstein equation should be divergenceless, because the right hand side is (energy momentum conservation). This brings one to the addition of a term

    [tex]
    \Lambda g^{\mu\nu} \ \ \rightarrow \nabla_{\mu}(\Lambda g^{\mu\nu}) = \nabla^{\nu}\Lambda = 0
    [/tex]

    So,

    [tex]
    \partial_{\mu} \Lambda = 0
    [/tex]

    Hence, lambda must be a constant.
     
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