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Nature of the cosmological constant

  1. May 7, 2004 #1


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    If one adds a scalar to the Hilbert action without considering any matter fields,

    [tex] S = \int {d^nx {\sqrt -g} (R - 2 \Lambda) [/tex]

    one gets the Einstein equations as:

    [tex] R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = 0 [/tex]

    Now, one can take

    [tex] T_{\mu \nu} = - \frac{\Lambda}{8 \pi G} g_{\mu \nu} [/tex]

    as an energy of space-time, and get

    [tex] G_{\mu \nu} = 8 \pi G T_{\mu \nu} [/tex]

    I have read several times that this energy is considered to be the energy density of empty space. Calculations are then made considering contributions of the ground state of quantum fields (bosons and fermions) leading to different values depending on different assumptions for this calculation.

    What I do not understand is why [tex]\inline \Lambda g_{\mu \nu} [/tex] is considered to be related to matter fields, since the defined action above did not include them (did it?). Shouldn’t this term be an energy of, lets say, ‘pure’ space-time?
    Last edited: May 7, 2004
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  3. May 7, 2004 #2


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    Sorry, but my intention was to post this in the Special & General Relativity forum. Is it possible to shift the thread? Thanks.
  4. May 7, 2004 #3


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    no problem...
  5. May 7, 2004 #4


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    I have always seen [tex]\Lambda[/tex] interpretted as a geometrical (as opposed to material) parameter. My only vague idea would be that it might have something to do with the antimatter "sea" that I have heard about regarding Dirac's formulation of QM.
  6. May 8, 2004 #5


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    You're right that it doesn't necessarily have anything to do with matter. Its just interpretation right now. You can take the point of view that the constant is fundamental to gravity, or that it is induced in cosmological situations by a field with the stress energy you wrote down.

    In other words, your action might represent just gravity or gravity+other field. Is it ever possible to have an experimental situation with lambda=0? There is no evidence one way or the other.

    Given that, many people still think that just accepting the constant as a law of nature is inelegant, so they try to explain it (and have failed spectacularly).

    My personal opinion is that it is an overrated problem. There might be something interesting there, but I'm not convinced that there has to be.
  7. May 8, 2004 #6


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    I see, this seams reasonable, but in light of this interpretation, shall I assume that adding up the contribution of bosonic and fermionic fields should lead to the net effect of a scalar field? How can this be proved?

  8. May 8, 2004 #7


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    I don't know the details on the QFT side, but given a Lagrangian, you can compute a stress-energy tensor. If that's proportional to the metric, then it acts like an effective cosmological constant.
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