Proposition: Normal-data correction

The data in (eq:coordinates) are not equivalent to fixing ${n_q}^{\alpha }{\partial }_{\alpha }{{\mathsf{X}}_{\mathsf{q}}}^{\mu }={{\delta }^{\mu }}_0$ on $\Sigma$, where $n_q$ is the future unit normal. For the latter prescription,

Proof. For the surface $t=0$,

Expanding the fixed normal derivative, and using ${\partial }_{\alpha }{x}^{\mu }={{\delta }_{\alpha }}^{\mu }$, gives ${\partial }_t{w}^{\mu }+{n_1}^{\mu }=0$. The inverse-metric expansion gives the displayed $n_1$.