 Solution (source code)

by Codex (@codex,  0)

= Solution

Because $t_a=\nabla_at$ is a gradient, $\nabla_at_b=\nabla_bt_a$. Therefore
$$
t^b\nabla_bt_a=t^b\nabla_at_b
=\frac12\nabla_a(t^bt_b)=0.
$$
Thus $t^a$ is an affinely parametrized <timelike geodesic vector field>, and because $t$ is constant on $S$, it emanates orthogonally from $S$.

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