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I got the same results as you for Gamma, but for R I’m different.

For R theta-phi-theta-phi I’m the same as you, but I used symmetry to identify it as equal to R phi-theta-phi-theta.

I have both of those as sin^2 theta.

Same thing with theta-phi-phi-theta – it should be equal to phi-theta-theta-phi and I believe both are -sin^2 theta

That…[Read more]

Good point, Mike; I hadn’t caught that.

For what it’s worth, here’s a summary of the values I got for the Christoffel symbol and the Riemann tensor, for a spherical surface. I found it useful to have these handy while working the problems:

Gamma.png
R.pnf

Hopefully these are correct. Any comments?

Also, maybe I’m reading it wrong, but it seems to me that in part (b) of 6.7 he “shows…[Read more]

We’ll review the solutions to problems 6.6, 6.7, and 6.8

In the event we have time, we can also review 6.9

It will be my wife’s birthday, so it’s unlikely I’ll be able to join.

We’re having an informal meeting tomorrow to discuss anything on our minds.
Meeting link: https://us06web.zoom.us/j/87696074341?pwd=N17bmYVaABjibjrFsyEF3QIZB4Ds4o.1
Usual time (3pm Vancouver, 4pm Boulder/Denver)

@eduard
@mike
@scott
@maryann
@tina
@stefan

I probably created some unnecessary confusion with my comments about the covariant derivative. Essentially, I was just confused by a matrix full of derivatives being called “the derivative.” I guess I would have thought it would be called a “vector gradient” or something like that (as the $Nabla$ notation implies). Anyway, I won’t complain about…[Read more]

In the meeting Stefan brought up the question of under what circumstances we would take the (covariant) derivative of a tensor with rank greater than one. I couldn’t think of anything at the time. But later it occurred to me that the metric tensor itself (potentially) changes as we move from one point to another in spacetime. So we could…[Read more]

I want to put something we discussed in my own words to see if I understand it.

Suppose we have a vector V in Cartesian coordinates, and we want to transform it to Polar coordinates.
We denote its Polar version with
V ¯ α

[𝑣𝛼]

I want to put something we discussed in my own words to see if I understand it.

Suppose we have a vector V in Cartesian coordinates, and we want to transform it to Polar coordinates.
We denote its Polar version with

[𝑣𝛼]