Suppose $(M,g)$ is a Lorentzian manifold with signature $(-,+,\ldots,+)$ and a positive curvature. Let $p \in M$. Let $U$ be a sufficiently small neighborhood of $p$ in the exterior of the double null cone emanating from the point $p$. Given each $q \in U$ we define $r(q)$ to be the geodesic distance of the point $q$ to $p$ in $U$.

Is it true that given any $q \in U$ and $X \in T_qM$ with $\langle X,X\rangle_g=0$ and $\langle X,\nabla r\rangle_g=0$, there holds $\mathrm{Hess}\, r(X,X)\geq 0$?

I am trying to use a Hessian comparison theorem here, but have not been successful since $X$ is a null-vector.


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