The Cosmological (constant)

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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,
 

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  • #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
 
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  • #3
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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
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?
 
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  • #4
atyy
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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?
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])?
 
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  • #5
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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])?
Thanks for your help,

I think this answers my question.
Classically, then, the effective cosmological constant is the sum of a bare term [itex]\Lambda_0[/itex] and the potential energy [itex]V(\phi)[/itex], where the latter may change with time as the universe passes through different phases.
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?
 
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  • #6
haushofer
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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,
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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