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Homework Help: GR flat and curved space

  1. Feb 13, 2009 #1
    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=gabdxadxb=r22+sin2θdφ2
    calculate gab, Γabc, 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
     
  2. jcsd
  3. Feb 14, 2009 #2
    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.
     
  4. Feb 14, 2009 #3
    We got the defenition of the christoffelsymbol

    Γrsa=½gal(glr,s+gls,r-grs,l)

    and in flat space gab=(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.
     
  5. Feb 14, 2009 #4
    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.
     
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