Metric tensor in spherical coordinates

[GRstudent]

stevendaryl,

One of the components of the Riemann tensor in Schwarzschild solution is

R^{r}_{\theta \theta r}{} = \dfrac{M}{R}

What does this component mean in real sense? Susskind told that, here, two \theta \theta indexes (which are downstairs) have something to do with two vectors which define a plane at some point in space. What other two r-r represent? Basically, Riemann tensor takes as input 4 vectors and outputs the curvature value, right?

Thank you.

Is this a true picture of basis vectors? http://www.springerimages.com/Images/RSS/1-10.1007_978-1-4614-0706-5_6-0

 

[Dale]

R^{r}_{\theta \theta r}{} = \dfrac{M}{R}

What does this component mean in real sense?

I am not completely certain that I have this right, but I believe that it means that if you move a dr vector around a closed differential d\theta, d\theta loop then it will change by a differential amount M/R in the dr direction.

 

[GRstudent]

^
This makes some sense.

 

[stevendaryl]

I am not completely certain that I have this right, but I believe that it means that if you move a dr vector around a closed differential d\theta, d\theta loop then it will change by a differential amount M/R in the dr direction.

I'm not sure about the convention for the order of the indices, but if you have a loop (parallelogram) in which the two sides are parallel (both d\theta), then the loop has area 0, and so the parallel transport gives 0.

I think that the correct interpretation is this: Make a parallelogram with one side equal to dr\ e_r and another side equal to d\theta \ e_\theta. March the vector V = e_\theta around the parallelogram to get a new vector V' that will be slightly different from e_\theta. Then letting \delta V = V' - V,

\delta V^r = R^r_{\theta \theta r} dr d\theta

 

[GRstudent]

This is an excellent representation of basis vectors in Spherical coordinates:

http://en.citizendium.org/wiki/File:Spherical_unit_vectors.png

I tried to reproduce the Riemann tensor's plane according to stevendaryl's suggestions here

http://img826.imageshack.us/img826/5614/605pxsphericalunitvecto.png