When I do calculation under Gaussian normal coordinate:
$$
d s^2=-2 d u d v-u^2\alpha(x) d v^2-2u w_i(x) dv dx^i+\gamma_{ij}(x) d x^i d x^j\,\quad x=(uv\,,x^i)
$$
How to get inverse metric and others directly? I don't know how to deal with $w_i$ and $\gamma_{ij}$.