How to transfer a metric to the orthonormal coordinate?

Question

Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ This is a diagonal metric. How can I transform this metric to the orthonormal frame? For that case, what will be the components of this metric in the orthonormal frame?

Answer 1

Since the metric is diagonal, we can do compute this quickly: (Assuming $1 \pm 2 \psi(t, x)$ are both positive, and suppressing the arguments of $a, \psi$) we can rewrite the metric as $$ds^2 = -(\sqrt{1 + 2 \psi} \,dt)^2 + [a \sqrt{1 - 2 \psi} \,dx]^2, $$ so in terms of the $1$-forms $$\alpha = \sqrt{1 + 2 \psi} \,dt, \qquad \beta = a \sqrt{1 - 2 \psi} \,dx$$ the metric has the form $$ds = - \alpha^2 + \beta^2 .$$ In particular, the basis dual to $(\alpha, \beta)$, namely, $$\left(\frac{1}{\sqrt{1 + 2 \psi}} \partial_t, \frac{1}{a \sqrt{1 - 2 \psi}} \partial_x\right),$$ is pseudo-orthonormal.