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Need to calculate Christoffel connection from a given metrics

  1. Jun 13, 2012 #1
    Hi all,
    I am trying to find the Christoffel connections of this metric:

    ds2= -(1+2∅)dt2 +(1-2∅)[dx2+dy2+dz2]
    where ∅ is a general function of x,y,z,t.

    I tried to solve this through the least action principle, but some of my results(t-related terms) were different from the answer with a minus sign. So, I guess it's a problem about the part of t of the action.

    I regarded this part as -1/2(1+2∅)[itex]\dot{t}[/itex]2, should I remove the minus sign to get the correct answer?

    [itex]\dot{t}[/itex]: the derivative of t regard to the affine parameter λ

    Thanks for your help!
     
    Last edited: Jun 13, 2012
  2. jcsd
  3. Jun 13, 2012 #2

    HallsofIvy

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    That's pretty straight forward isn't it? The Christoffel symbols (of the first kind) are given by
    [tex][ij, k]= \frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^k}+ \frac{\partial g_{ik}}{\partial x^j}- \frac{\partial g_{jk}}{\partial x^i}\right)[/tex]

    Here, if we take [itex]x^1= x[/itex], [itex]x^2= y[/itex], [itex]x^3= z[/itex], and [itex]x^4= t[/itex], then [itex]g_{11}= g_{22}= g_{33}= 1- 2\phi[/itex], [itex]g_44= -(1+ 2\phi)[/itex]. Of course, the result will depend upon the partial derivatives of [itex]\phi[/itex]. If [itex]\phi[/itex] can be any function of the variables, then the Christoffel symbols can be just about anything!
     
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