Cartan curvature of a planar warped spacetime ($a=A^{\prime }/A,q=A^{\prime \prime }/A$)

For the orthonormal coframe of a planar warped spacetime, Cartan's first structure equation gives ${\omega }^i{}_3=a{e}^i$ and ${\omega }^3{}_i=-s_{i}a{e}^i$, where $i=0,1,2$ and $(s_0,s_1,s_2)=(-1,1,1)$. All other connection 1-forms vanish. Cartan's second structure equation gives ${\Theta }^i{}_3=-q{e}^i\wedge e^3$ and ${\Theta }^i{}_j=-s_j{a}^2{e}^i\wedge e^j$ for $i\ne j$. Lowering the first index makes both the connection 1-forms and the curvature 2-forms antisymmetric. Contraction yields $R_{00}=q+2a^2$, $R_{11}=R_{22}=-q-2a^2$, and $R_{33}=-3q$.