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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 $R_{33}=-3q$.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

The metric tensor of the planar warped spacetime depends only on $z$. The three translations therefore preserve it, givingThey are linearly independent Killing vector fields everywhere. The associated geodesic conserved quantities from Killing vectors are $-A^2\dot{t}$, $A^2\dot{x}$ and $A^2\dot{y}$, for an affine parameter; the conventional positive energy is $E=A^2\dot{t}$. No restriction on the positive function $A$ is needed for these three symmetries.