# Defintion of Curvature

1. May 14, 2008

### 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.

Last edited: May 14, 2008
2. May 14, 2008

### 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.

3. May 15, 2008

### 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?

4. May 15, 2008

### pmb_phy

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

Pete

5. May 15, 2008

### Mentz114

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

6. May 16, 2008

### maverick280857

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.

7. May 16, 2008

### maverick280857

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

8. May 16, 2008