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Need to find the Ricci scalar curvature of this metric

  1. May 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Need to find the Ricci scalar curvature of this metric:
    ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2

    2. Relevant equations



    3. The attempt at a solution

    I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:

    <The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z.
    [itex]\Gamma\stackrel{x}{xz}[/itex]=[itex]\Gamma\stackrel{x}{zx}[/itex]=a'(z)
    [itex]\Gamma\stackrel{y}{yz}[/itex]=[itex]\Gamma\stackrel{y}{zy}[/itex]=a'(z)
    [itex]\Gamma\stackrel{z}{tt}[/itex]=b'(z)e2b(z)
    [itex]\Gamma\stackrel{z}{xx}[/itex]=[itex]\Gamma\stackrel{z}{yy}[/itex]=-a'(z)e2a(z)
    [itex]\Gamma\stackrel{t}{tz}[/itex]=[itex]\Gamma\stackrel{t}{zt}[/itex]=b'(z)
    [itex]\Gamma\stackrel{}{either}[/itex]=0

    <The Riemann curvature tensor>
    [itex]\R\stackrel{x}{zxz}[/itex]=[itex]\R\stackrel{y}{zyz}[/itex]=-a''(z)-[a'(z)]2
    [itex]\R\stackrel{z}{tzt}[/itex]=b''(z)+[b'(z)]2

    I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?

    Thanks for answering this question~!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 28, 2012 #2
    Why wouldn't it depend on z when your metric does. You can compare this to the FRW-metric. In that case, the metric and the scalar curvature depend on time.

    The results look reasonable but I'm a bit too lazy to check it explicitly.
     
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