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Homework Help: General Relativity, Flat isotropic universe

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data
    We are looking at an isotropic flat universe, with the metric
    [itex] ds^2 = dt^2 - b(t)^2(dx^2 + dy^2 + dz^2) [/itex]

    I need to write down the energy conservation equation
    [itex] \frac{dV}{V} = -\frac{d\epsilon}{\epsilon + p} [/itex]

    We have been given the solution to be
    [itex] 3\ln(b) = -\int \frac{d\epsilon}{\epsilon + p}[/itex]

    3. The attempt at a solution
    I have found that
    [itex] 3\frac{b'^2}{b^2} = \kappa\epsilon[/itex]
    by solving the [itex]{}^t_t[/itex] component of the Einstein Equation
    [itex]R^a_b -\frac{1}{2}R\delta^a_b = \kappa T^a_b[/itex].
    [itex] b' = \partial_t b[/itex], and [itex]\epsilon[/itex] is the energy density, after having found the Christoffel symbols, Riemann tensor, Ricci tensor and Ricci Scalar.

    I can't seem to find an equation of V or dV.
  2. jcsd
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