# Geodesics as a local frame of reference

Consider an arbitrary 2D pseudo-Riemannian space (spacetime) with a metric signature of $(1,1)$. I want to construct an orthogonal frame of reference of a free moving body. Obviously my time axis would be a time-like geodesic, along which the body is moving. Would my space axis be along one of space-like geodesics?

More specifically, my question is local. I am only interested in the infinitesimal movement of the body. So my momentary time axis is a tangent vector to the time-like geodesic of the body movement. Is my orthogonal space axis a tangent vector to one of the space-like geodesics intersecting at the point of origin?

Intuitively this seems obvious, but I just wanted to confirm if this is correct.

Note that every vector defined at a point $p$ is tangent to some geodesic through the point $p$. So if you only care about the observer's basis at a single point, there is nothing to answer.
If, however, you care about the observer's coordinate system in some neighbourhood of the point $p$, then the answer to your question depends on what kind of observer we're considering. The coordinates of an observer in a local inertial frame are normal coordinates, which are defined in such a way that the coordinate axes are indeed geodesics.