Let $(M,g)$ be a globally hyperbolic spacetime, $S \subset M$ Cauchy hypersurface with unit normal vector field $n$, $\Sigma \subset S$ compact 2-dim. submanifold with unit normal vector field $\nu$.
I have a defined function $\exp: (- \epsilon, \epsilon) \times \sigma \rightarrow M$ defined via $\exp(r,p)=c_p(r)$ where $c_p$ is the null geodesic with $c^{'}_p(0)=n_p+\nu_p$. Let $q=\exp(r_0,p)$ where $q$ is not a critical value for $\exp$. One can define local coordinates $(t,u,x^2,x^3)$ on some neighborhood $V$ of $q$ with $\varphi$ a local parametrization around $p$ and using the map
$(u,r,x^2,x^3) \mapsto \exp(r, \psi_u(\varphi(x^2,x^3)))$
with $\psi_u$ as the flow along the timelike geodesics orthogonal to $S$. Now we have $g(\partial_r,\partial_r)=0$ and we can compute using properties of the Levi-Civita connection, that
$\dfrac{\partial g(\partial_r, \partial_{\mu})}{\partial r}=0$ for $\mu=0,1,2,3$. Now my book says "Since $g_{ru}=-1$ and $g_{r2}=g_{r3}=0$ on $\psi_u(\Sigma)$ we get $g_{ru}=-1$ and $g_{r2}=g_{r3}=0$ on $V$" but I just can't see where this comes from.. why is $g_{ru}=-1$ on $\psi_u(\Sigma)$??