# Constant Curvature and about its meaning

• I
• Arman777
In summary: If not, the point is a special one in the universe, and that contradicts the cosmological principle.In summary, the constant curvature of the universe implies that it is homogeneous and isotropic, with only three possible values for the curvature: -1, 0, and +1. The existence of curvature at all implies a homogeneous and isotropic universe, but it can always be rescaled to fit one of those three values depending on its sign. The sign of kappa is what matters, not its value. For a torus, which is not homogeneous and isotropic, there is no value for kappa. Instead, there is another parameter, often called R, which represents the radius of curvature. The cosmological principle, which assumes that no
Arman777
Gold Member
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

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.

Arman777 and Klystron
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.

Arman777
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 ?

Arman777 said:
The "constant" implies homogeneity and isotropy ?
The other way around. Homogeneity implies that any scalar quantity must be a constant.

Arman777

## 1. What is constant curvature?

Constant curvature refers to a property of a geometric shape or surface where the curvature remains the same at every point. This means that the amount of bending or curving of the shape or surface is uniform throughout.

## 2. What does constant curvature mean in mathematics?

In mathematics, constant curvature is a characteristic of a space or manifold that has the same curvature at every point. This can be described by the constant scalar curvature or the constant sectional curvature, depending on the specific context.

## 3. How is constant curvature related to the curvature of space?

In the theory of general relativity, constant curvature is used to describe the curvature of space. This is because in a space with constant curvature, the curvature is the same at every point and does not change over time. This is known as a homogeneous and isotropic space.

## 4. What are some examples of objects or surfaces with constant curvature?

Some examples of objects or surfaces with constant curvature include spheres, cylinders, and cones. In these shapes, the curvature is the same at every point and does not change. Other examples include the surface of a torus and the hyperbolic plane.

## 5. How is constant curvature used in real-world applications?

Constant curvature is used in various real-world applications, such as in the design of lenses and mirrors for optical devices, in computer graphics and animation to create smooth surfaces, and in the study of curved spaces in physics and cosmology. It also has applications in engineering, such as in the design of curved bridges and roads for efficient transportation.

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