Find local orthogonal coordinates based on the global metric and geodesics

Question

Consider a Euclidean space $X={\Bbb R}^2$ equipped with a known global $(1,1)$ pseudo-Riemannian metric in non-Cartesian coordinates. I need to define a local orthogonal Cartesian coordinate system of an observer located at a point $P$. When the observer is in a free flight moving along a timelike geodesic, his momentary time axis is tangent to this geodesic. How can I find his space axis? I am interested in the local limit where the geodesics are asymptotically straight. I would appreciate any insight.

Answer 1

When coordinates are involved, you can think of the metric $g$ as a matrix: for two vectors $v, w$, the inner-product is $$g^{ij}v_iw_j = v_ig^{ij}w_j = \langle v, gw\rangle$$ where $\langle\,\,,\,\rangle$ is the ordinary Euclidean metric on $\Bbb R^n$. This is true whether $g$ is Riemannian or pseudo-Rimannian.

So in your case, just multiply the timelike vector by $g$, then find the Euclidean orthogonal to the result to get the spacelike vector.