Constant Curvature and about its meaning

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    Constant Curvature
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Discussion Overview

The discussion revolves around the concept of constant curvature in the context of cosmology, particularly focusing on the implications of curvature values (κ) for the universe, the cosmological principle, and the nature of homogeneous and isotropic spaces. Participants explore theoretical aspects, mathematical relationships, and the assumptions underlying these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that constant curvature implies homogeneity and isotropy, leading to only three possible values for κ: -1, 0, and +1.
  • Others challenge the assumption that the universe must adhere to the cosmological principle, questioning the necessity of dark energy for certain κ values.
  • A participant suggests that κ may vary over time, prompting further inquiry into the implications of such variability.
  • There is a discussion about the nature of κ, with some arguing that only its sign matters, while others inquire about the curvature of non-homogeneous shapes like a torus.
  • Participants discuss the relationship between κ and a parameter representing the radius of curvature, with questions about their mathematical connection.
  • Concerns are raised about the limitations of the cosmological principle, noting that the universe is not homogeneous on small scales and that this principle is based on the assumption of no special places or directions.
  • Some participants clarify that while constant curvature is associated with homogeneous and isotropic spaces, curvature can exist in other forms as well.
  • There is a debate about the implications of the term "constant" in relation to homogeneity and isotropy, with differing views on the relationship between these concepts.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the assumptions of the cosmological principle, the nature of curvature, and the implications of constant curvature. The discussion remains unresolved, with no consensus reached on several key points.

Contextual Notes

Limitations include the dependence on definitions of homogeneity and isotropy, as well as the unresolved nature of how curvature might behave in non-standard cosmological models.

Arman777
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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
 
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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.
 
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Orodruin said:
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 ?
 
Arman777 said:
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.
 
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Hmm I understand it I think. Only homogeneous and isotropic spaces can take kappa values which there are 3 of them.
Orodruin said:
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 ?
 
Arman777 said:
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.
 
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Orodruin said:
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
 
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.
 
Orodruin said:
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 ?
 
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Arman777 said:
The "constant" implies homogeneity and isotropy ?
The other way around. Homogeneity implies that any scalar quantity must be a constant.
 
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