= Cartan curvature of a planar warped spacetime
{c}
{title2=$a=A'/A,\quad q=A''/A$}
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 $\Theta^i{}_j=-s_j a^2e^i\wedge e^j$ 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 $R_{33}=-3q$.
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