Spatially homogeneous and isotropic universe. The Robertson-Walker metric
$$\begin{equation*}
g=-d t \otimes d t+a^{2}(t)\left(\frac{d r \otimes d r}{1-k r^{2}}+r^{2}\left(d \theta \otimes d \theta+\sin ^{2} \theta d \phi \otimes d \phi\right)\right) \tag{5.399}
\end{equation*}$$
where $k$ can be $-1,0$, or 1 , describes a spatially homogenous and isotropic spacetime. Calculate the Riemann tensor, the Ricci tensor, and the scalar curvature associated with the metric (5.399).

Data from 5.399

Transcribed Image Text:

dr dr g=-dtdt+a(t) 1- kr +r(d0d0+ sin e do do)