Quastions about Zee's book EGR in a nutshell

1. Nov 16, 2013

sergiokapone

Quastions about Zee's book "EGR in a nutshell""

Now I'm reading a book Einstein Gravity in a Nutshell by A. Zee, and I had some misunderstanding.
Zee writes (page 89):
before that, he wrote,
Trying to solve the problem. I expanded in a series metrics as mentioned $g_{\theta\theta}(r) = r^2_*+2r_*(r-r_*)+2(r-r_*)^2$
I see that $B_{\mu\nu, \lambda \sigma}=B_{\theta\theta,r r}=B_{22,11}=1$.

How can I get intrinsic curvature?
Below, he still writes
I turns out that the curvature is not zero, it should be.

2. Nov 16, 2013

Bill_K

Should be $g_{\theta\theta}(r) = r^2_*+2r_*(r-r_*)+(r-r_*)^2$

3. Nov 16, 2013

sergiokapone

Yes, right, $B's=1$ as I sad.

4. Nov 16, 2013

Staff: Mentor

You've only expanded $g_{\theta \theta}$. What about the other components?

5. Nov 16, 2013

sergiokapone

But $g_{rr}=1$.

6. Nov 16, 2013

Staff: Mentor

It is in the global metric, but is it in the locally flat coordinates centered on the point (r*, 0)?

7. Nov 16, 2013

Staff: Mentor

To expand on this some more: "locally flat coordinates" means Cartesian coordinates, i.e., you are supposed to set up a local chart centered on the point (r*, 0) in which the metric at that point is $ds^2 = dx^2 + dy^2$. So the metric coefficients you should be expanding to 2nd order are really not $g_{rr}$ and $g_{\theta \theta}$ anyway; they are $g_{xx}$, $g_{yy}$, and $g_{xy} = g_{yx}$. You know that $g_{xx} (0) = g_{yy} (0) = 1$ and $g_{xy} (0) = g_{yx} (0) = 0$ since the (x, y) chart is locally flat at its origin. So in order to expand out the first and second-order terms for the locally flat metric coefficients, you just need to know the coordinate transformation from the global $(r, \theta)$ coordinates to the local $(x, y)$ coordinates, so you can transform the line element in general--i.e., not just at the single point (r*, 0).

8. Nov 17, 2013

sergiokapone

Ok, assuming that I do not know that the above metric is flat, how do I know this transformation?

9. Nov 17, 2013

Bill_K

Zee apparently tells you how he wants you to do this. Your quote from the book says,

10. Nov 17, 2013

sergiokapone

Yes, but how to do it? That is the question.I have this difficulty. I need to give step by step, then I'll do.

11. Nov 17, 2013

Bill_K

I think you must go on to the next page. Normally the curvature would depend on both the A's and B's. Since the expression you gave involves B's only, Zee must expect you to transform the coordinates further to eliminate the A's.

12. Nov 17, 2013

sergiokapone

Well, put the problem in another way. Let us forget for a while about Zee. How to find locally-flat coordinates (x,y) on the sphere at a north pole, say? What it means to find? It is clear that they are Cartesian, so I need to find transition formulas.

13. Nov 17, 2013

Bill_K

From what you've said, I still don't know just what Zee is planning to do. What he's calling "locally flat coordinates" are more usually called quasi-Cartesian coordinates or normal coordinates at a given point P. To find them, one can

a) Choose a set of orthonormal vectors Xi at P.
b) Solve the geodesic equations to construct a set of geodesics starting at P with initial tangent Xi.
c) Use these geodesics as a set of coordinate axes in the neighborhood of P, defining the normal coordinates as distance measured along each axis.

Just a roundabout way of finding coordinates in which the Christoffel symbols vanish at P, which is what Zee's A coefficients are.