## FANDOM

153 Pages

The FRW metric is the generic form of a metric for a matter distribution that is homogeneous and isotropic. We begin the derivation with an equation for a four dimensional sphere

$x^2 + y^2 + z^2 + w^2 = r^2 + w^2 = R^2$

where $R$ is a constant. Taking the differential of this equation yields

$r dr + w dw = 0$

The spatial part of the metric becomes

$d l^2 = dw^2 + dr^2 + r^2 d \Omega^2 = \frac{d r^2}{1-\left(r/R\right)^2} + r^2 d \Omega^2$

where $d\Omega^2 = d \theta^2 + \sin^2 \theta d \phi^2$. The complete metric is often written in the following form

$d s^2 = c^2 dt^2 - R^2 \left( \frac{dr^2}{1 - k r^2} - r^2 d \Omega \right)$