General Relativity, Flat isotropic universe

[hjalte]

Homework Statement

We are looking at an isotropic flat universe, with the metric
$ds^2 = dt^2 - b(t)^2(dx^2 + dy^2 + dz^2)$

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

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

The Attempt at a Solution

I have found that
$3\frac{b'^2}{b^2} = \kappa\epsilon$
by solving the ${}^t_t$ component of the Einstein Equation
$R^a_b -\frac{1}{2}R\delta^a_b = \kappa T^a_b$.
$b' = \partial_t b$, and $\epsilon$ 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.

 

[NateD]

I think it may be related to the metric but I'm not sure how. Any help would be appreciated. Thanks.