 Solution (source code)

by Codex (@codex,  0)

= Solution

The metric directly gives
$$
g(e_0,e_0)=-1,\qquad
g(e_i,e_j)=t^{-p_i}t^{-p_j}t^{2p_i}\delta_{ij}
=\delta_{ij},
$$
with no mixed products. Hence
$$
\boxed{e_0=\partial_t,\qquad e_i=t^{-p_i}\partial_{x^i}}
$$
is an orthonormal frame.

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