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)



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


3. 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.