The sign of the cosmological term

[sweet springs]

I want to know the sign of the cosmological term. It seems that in the left hand side of the Einsetein equation the cosmological term is $-\Lambda g_{ik}$ for (+---) convetion or $+\Lambda g_{ik}$ for (-+++) convetion. Right?
Best

 

[marcus]

That makes intuitive sense to me. I hope someone shows up who can confirm this more authoritatively. The reason it makes sense to me as a non-expert is that Λ shows up with a minus sign in the Friedmann equation in cosmology. The Friedmann is supposed to be derived from the "0 0" component of the GR equation , that is from the "t t" components term by term. And the Friedmann equation looks like this:
H² - Λc²/3 = [constant] ρ

where H is a fractional growth rate with dimension Time^-1 so H² has dimension Time^-2, and rho is the density.
Λ has dimension Length^-2 so Λc² also has dimension Time^-2, so that is consistent.
And Λc²/3 = H_∞² the square of the longterm Hubble expansion rate to which H(t) is converging.

So I think of Λ as appearing with a + sign on the left side of the GR equation, in the (-+++) convention and therefore appearing with a - sign on the left side of the Friedmann.

 

[Ibix]

According to https://en.m.wikipedia.org/wiki/Einstein_field_equations#Sign_convention, yes. Please be aware that sign conventions in general relativity are complex. Misner, Thorne and Wheeler classify eight different conventions...

 

[ShayanJ]

Misner, Thorne and Wheeler classify eight different conventions

Could you tell which section?

 

[Ibix]

Could you tell which section?

The very first page (it's red in my copy) is a table of sign convention by author. There are eight possibilities in their classification scheme; I haven't actually checked that they have an example of every one in use.

Edit: The above isn't directly related to the OP's question - it was more in the vein of a general caution.

 

[sweet springs]

Thanks for your enlightenmental replies. I would likt to know more about dark energy, i.e. cosmological term transposed to right hand side, referring

It seems that in the left hand side of the Einsetein equation the cosmological term is $-\Lambda g_{ik}$ for (+---) convetion or $+\Lambda g_{ik}$ for (-+++) convetion.

i.e. $\Lambda g_{ik}$ for (+---) convetion or $-\Lambda g_{ik}$ for (-+++) convetion. The positive energy and the negative momentum are not usual.
I would like get some interpretation or explanation of this combination. If you know something good please let me share it.
Best.

 

[bcrowell]

The positive energy and the negative momentum are not usual.

It's negative pressure, not negative momentum. The momentum three-vector is a vector, so it doesn't make sense to talk about whether it's positive or negative.

 

[sweet springs]

Is pressure the momentum flux? Force/area = Momentun density * velocity of fluid = momentum flux

 

[bcrowell]

Is pressure the momentum flux? Force/area = Momentun density * velocity of fluid = momentum flux

Pressure is the momentum flux along a spacelike axis.

WP has a nice summary: https://en.wikipedia.org/wiki/Stress–energy_tensor#/media/File:StressEnergyTensor_contravariant.svg