= Null energy condition for a planar warped spacetime
{title2=$(\log A)''\le0$}
In a <planar warped spacetime>, contraction of the <Ricci tensor> with a <null vector> in an <orthonormal frame> gives $R_{ab}k^ak^b=-2(\log A)''(k^3)^2$. The <Einstein field equations> identify this with $8\pi T_{ab}k^ak^b$; 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.
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