The Cosmological (constant)

1. Jun 22, 2011

jfy4

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 $\Lambda(r,t)$.

2. Jun 22, 2011

atyy

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
3. Jun 22, 2011

jfy4

That makes sense. If I may, would the satisfaction of
$$\nabla_{\beta}\left( T^{\alpha\beta}-g^{\alpha\beta}\Lambda\right)=0$$
justify the inclusion of a cosmological constant that was a scalar function?

Last edited: Jun 22, 2011
4. Jun 22, 2011

atyy

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
5. Jun 22, 2011

jfy4

I think this answers my question.
This sounds like it is ok to include a cosmological constant of the form
$$\Lambda(x_{\alpha})=\Lambda_0+V\,[\phi(x_{\alpha})]$$
that consists of an initial cosmological constant, $\Lambda_0$, summed with a scalar function $V$. Have I interpreted this correctly?

Last edited by a moderator: May 5, 2017
6. Jun 23, 2011

haushofer

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

$$\Lambda g^{\mu\nu} \ \ \rightarrow \nabla_{\mu}(\Lambda g^{\mu\nu}) = \nabla^{\nu}\Lambda = 0$$

So,

$$\partial_{\mu} \Lambda = 0$$

Hence, lambda must be a constant.