How to Calculate the Metric and Christoffel Symbols for GR Flat and Curved Space

[RestlessRiver]

1.a) Concider the 2-space consisting of a spherical shell at constan radius, r. In polar coordinates the line element on the surface can be written (a, b,∈ 1,2)

Homework Equations

ds²=g_abdx^adx^b=r²dθ²+sin²θdφ²
calculate g^ab, Γ_ab^c, R¹₂₁₂, R²₁₂₁, R₁₁, R₂₂, R

The Attempt at a Solution

I don't have an attempt on a solution cause I honestly have no idea how. Our teacher has said that it's possible to calculate the Christoffel sybols and the metric but he never showd how =/

really apprisiate any help

thanks a lot

 

[xboy]

Well, do you not have any textbook or something where the formula for Christoffel symbol is given in terms of the metric tensor. There is such a formula, and you will need to know it.

 

[RestlessRiver]

We got the defenition of the christoffelsymbol

Γ_rs^a=½g^al(g_lr,s+g_ls,r-g_rs,l)

and in flat space g_ab=(1 0 0 0 || 0 -1 0 0 || 0 0 -1 0 || 0 0 0 -1)

and R is Riemann's tensor

I'm supposed to calcuate the same thing for 2-space of the cylinder and then determin which is flat and which is curved, and I know it's the cylinder that is flat and the sphere that's curved.

And we only have recomended books, we don't need to buy them, and for a poor student like me, well yea, I don't have the money to buy the books.

 

[xboy]

Yes, so you have the formula.You can now use it to find all the desired quantities. Similarly for the cylinder you can write down the metric tensor for a cylinder and work it out. I suppose you know the condition for a manifold being curved or not?

Well, if you can't buy books, there are some online resources. Sean Carroll's lecture notes on GTR are I think freely available. There may be other stuff at a more elementary level, you can look around.