Suppose we have a semi-Riemannian manifold $(M^n,g)$ with metric signature $(n-k,k)$. By definition, each $p \in M$ the map $g_p : T_pM \times T_pM \to \Bbb{R}$ is a non-degenerate, symmetric, bilinear form.
How to find a local orthonormal frame on a neighbourhood $U \subseteq M$ of a point $p \in M$ ? That is $n$-tuple of local vector fields $(X_1,\dots,X_n)$ defined on a neighbourhood of a point such that for any $p \in M$, $(X_1|_p,\dots,X_n|_p)$ is linearly independent, $\text{span }(X_1|_p,\dots,X_n|_p) = T_pM$ and $g_p(X_i|_p,X_j|_p) = \pm \delta_{ij}$. I've seen such construction for Riemannian manifold in Lee's smooth manifolds which using Gram-Schmidt procedure. Using similar idea as in Lee's, i came up with this following construction. The problem with using the same strategy (Gram-Schimdt) is that since $g_p$ is indefinite, the normalizing function $1/|\cdot| = 1/\sqrt{|g(\cdot,\cdot)|}$ is not automatically smooth function since $g(\cdot,\cdot)$ can vanish somewhere. However, i may get a way with this issue but not really sure. I need some opinion and justification for this.
$\textbf{Choosing a local frame}$
Let $p \in M$ be arbitrary point and $\{v_1,\dots,v_n\}$ be an orthonormal basis for $T_pM$. Having this, we can find a smooth local frame $(X_1,\dots,X_n)$ on some neighbourhood $U\subseteq M$ of $p$ such that $X_i|_p=v_i$ for all $i =1,\dots,n$ (See John Lee's Introduction to Smooth Manifolds 2nd Ed, Proposition 8.11).
$\textbf{Normalizing}$
We normalize them as follows: Since $g(X_1,X_1): U \to \Bbb{R}$ is a smooth function with $g(X_1,X_1)(p) = g_p(X_1|_p,X_1|_p) =g_p(v_1,v_1)= \pm 1$, then by continuity we have a smaller neighbourhood $U_1 \subseteq U$ of $p$ such that $g(X_1,X_1)$ is strictly positive or strictly negative in this region, depends on the sign $g_p(v_1,v_1)=\pm 1$. Because of this $|X_1| = \sqrt{|g(X_1,X_1)|}$ is a non-vanishing smooth vector field on $U_1$, then we can normalize $X_1$, by setting
$$
E_1 = \frac{X_1}{|X_1|}.
$$
Similarly, the vector $V_2 = X_2 – \varepsilon_1 g(X_2,E_1)E_1$ is a smooth vector field with $g(V_2,V_2)(p) = g_p(V_2|_p,V_2|_p) = g_p(v_2,v_2) = \pm 1$. So $g(V_2,V_2) $ has a neighbourhood $U_2 \subseteq U$ of $p$ such that it has only positive or negative values. So we have normalized smooth vector field
$$
E_2 = \frac{X_2 – \varepsilon_1 g(X_2,E_1)E_1}{|X_2 – \varepsilon_1 g(X_2,E_1)E_1|}, \quad \varepsilon_1=g(E_1,E_1)=\pm1
$$
defined on $U_1 \cap U_2$ and with $g(E_2,E_1) = 0$.
Working inductively, we have $n$-tuple of smooth vector fields $(E_1,\dots,E_n)$ defined on $W = U_1 \cap \cdots \cap U_n$ given by
$$
E_j = \frac{X_j – \sum_{i=1}^{j-1} \varepsilon_i \, g(X_j,E_i)E_i}{|X_j – \sum_{i=1}^{j-1} \varepsilon_i \, g(X_j,E_i)E_i|}
$$
with $\varepsilon_i=g(E_i,E_i)=\pm1$. By construction $g(E_i,E_j) = 0$ for $i \neq j$ and $g(E_i,E_i) = \pm 1$, and $\text{span}(X_1,\dots,X_n) = \text{span}(E_,\dots,E_n)$.
Is this correct ? Is there any simpler approach ? Any help will be appreciated. Thank you.
$\textbf{EDIT}$
I just want to tell that i've found a similar construction using GS method in Darling's Differential Form and Connection section 7.7
Best Answer
Yes, this is basically fine. One small correction: You wrote
But $|X_1|$ is not a vector field. It should say