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.