Definition of Curvature for Sphere in Relativistic Cosmology

[maverick280857]

Hi.

I am reading "An Introduction to Modern Astrophysics" by Carroll and Ostlie, for a summer project. In section 27.3 (Relativistic Cosmology) the curvature of a sphere is given by

$$6\pi \frac{C_{exp}-C_{meas}}{C_{exp}A_{exp}}$$

The situation is as follows:

Consider a sphere of radius R. An ant is moving on the sphere at a fixed polar angle $\theta$. The ant measures the circumference $C_{meas} = 2\pi R\sin\theta$ whereas the expected value of circumference is $C_{exp} = 2\pi D$ where $D = R\theta$. The expected area of the circle is $A_{exp} = \pi D^2$.

I am not sure how the above expression leads to the curvature of the sphere...

Any thoughts?

Thanks.
Cheers
Vivek.

 

[Mentz114]

My understanding of this is that the ant walks in a circle, not a great circle. In this case the expression you quote gives the curvature of the sphere. Imagine the ant walking is a circle on a flat surface, then let the flat surface curve. The ratio of the diameter to the radius changes.

 

[maverick280857]

Mentz114, thanks for your reply. I'm not clear about the way the curvature has been written in terms of the two circumferences and the area, and also the $6\pi$ factor. How does all that come in?

 

[pmb_phy]

Hi.

I am reading "An Introduction to Modern Astrophysics" by Carroll and Ostlie, for a summer project. In section 27.3 (Relativistic Cosmology) the curvature of a sphere is given by

$$6\pi \frac{C_{exp}-C_{meas}}{C_{exp}A_{exp}}$$

The situation is as follows:

Consider a sphere of radius R. An ant is moving on the sphere at a fixed polar angle $\theta$. The ant measures the circumference $C_{meas} = 2\pi R\sin\theta$ whereas the expected value of circumference is $C_{exp} = 2\pi D$ where $D = R\theta$. The expected area of the circle is $A_{exp} = \pi D^2$.

I am not sure how the above expression leads to the curvature of the sphere...

Any thoughts?

Thanks.
Cheers
Vivek.

That relation is the Gaussian Curvature for a two dimensional surface. Gaussian curvature is not defined for spaces higher than two.

Pete

 

[Mentz114]

maverick280857 , if you want a derivation, check out Wiki on 'Gaussian curvature'. There's even an expression in terms of Christoffel symbols.

 

[maverick280857]

maverick280857 , if you want a derivation, check out Wiki on 'Gaussian curvature'. There's even an expression in terms of Christoffel symbols.

Thanks, I'm still learning these things and I don't know much about them. The page you have referred to does not explain the origin of the particular definition of curvature used here. But I will keep looking.

 

[maverick280857]

Ok, so I guess its a special case of the expression given there, for 2 dimensions.

 

[robphy]

check out:
http://books.google.com/books?id=IP...nce&sig=kV8lYa5yKzTPaxjxzkWrBaWVA_M#PPA163,M1
(corrected to point to the specific page)