Constant Curvature and about its meaning

[Arman777]

In a book that I am reading it stated that, the constant curvature implies curvature is homogeneous and isotropic, hence only three $κ$ values are possible for our universe
$$κ = -1, 0, +1$$ as we all know these values represent negative, flat and positive curvature respectively.

Now if $κ$ is different from these values than it means that the universe is not obeying cosmological principle (CP) right? My problem is that why we are assuming that the universe is obeying CP? Maybe for some $κ$ value, we will not need "dark energy"? Also is it possible that $κ$ may vary in time ?

Thanks

 

[Orodruin]

You have misunderstood why $\kappa$ only takes three values. The existence of $\kappa$ at all implies an isotropic homogeneous universe. It is just that regardless of its value you can always rescale your coordinates in a way such that $\kappa$ becomes one of those values depending on its sign.

 

[Arman777]

You have misunderstood why $\kappa$ only takes three values. The existence of $\kappa$ a all implies an isotropic homogeneous universe. It is just that regardless of its value you can always rescale your coordinates in a way such that $\kappa$ becomes one of those values depending on its sign.

So only the sign of the kappa matters not the value of it ? If that's the case than for instance what's the sign of the kappa for a torus ?

 

[Orodruin]

So only the sign of the kappa matters not the value of it ? If that's the case than for instance what's the sign of the kappa for a torus ?

A torus is not homogenous and isotropic. It has no ”value of kappa”.

If you restrict the value of kappa to -1, 0, and +1, then there is another parameter (often called R), which is taken to be positive and in essence represents radius of curvature.

 

[Arman777]

Hmm I understand it I think. Only homogeneous and isotropic spaces can take kappa values which there are 3 of them.

which is taken to be positive and in essence represents radius of curvature.

Is there a mathematical relationship between kappa and R or they are unrelated ?

So I want to ask something again. Couldnt we live in a universe which "does not have kappa" ? Why we are so obsessed by using CP principle as a starting point for our equations ? Is it just because its simple ?

 

[Orodruin]

So I want to ask something again. Couldnt we live in a universe which "does not have kappa" ? Why we are so obsessed by using CP principle as a starting point for our equations ? Is it just because its simple ?

Of course it is a possibility and it is being investigated. It is clear that the universe is not homogeneous on small scales. The cosmological principle rests on the (rather convenient) assumption that no place nor direction in the universe is special. The predictions have worked out pretty well so far, but should of course be open to scrutiny from data.

 

[Arman777]

Of course it is a possibility and it is being investigated. It is clear that the universe is not homogeneous on small scales. The cosmological principle rests on the (rather convenient) assumption that no place nor direction in the universe is special. The predictions have worked out pretty well so far, but should of course be open to scrutiny from data.

Thanks

 

[Khashishi]

It's because the form of the equation includes a scale factor parameter "a". A sphere twice as big has half the curvature, but you can just normalize the curvature to 1 and just put all the length dependence into the parameter a. The value of a will change over time.

 

[Arman777]

A torus is not homogenous and isotropic. It has no ”value of kappa”.

Is this curvature different than the Gaussian Curvature ? Like it seems that we are using ##\kappa# for only homogeneous and isotropic spaces.

I guess the crucial point is being "constant " right. The "constant" implies homogeneity and isotropy ?

We can have curvature for any space but only the constant curvature ones will be homogeneous and isotropic ?

 

[Orodruin]

The "constant" implies homogeneity and isotropy ?

The other way around. Homogeneity implies that any scalar quantity must be a constant.