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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.

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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.

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  • $\begingroup$ Thank you, this is helpful. I will review the link you provided. Yes, I am interested in the neighbourhood of the point where the geodesics are asymptotically straight. Also could you please clarify what you mean by "what kind of observer"? The observer is in a free flight along a timeline geodesic. What additional information can I provide to specify its "kind"? Thanks again! $\endgroup$ – safesphere Dec 16 '17 at 6:55
  • $\begingroup$ I have posted a follow-up question. I would appreciate if you take a look. Thanks! math.stackexchange.com/questions/2568882/… $\endgroup$ – safesphere Dec 16 '17 at 7:28
  • $\begingroup$ Dependent on whether your observer follows a geodesic or not, and dependent on whether that observer is rotating or not, the above statement about spatial axes may not be true. I was largely covering my back here, because my suspicion is that all physical observers have spacelike geodesics for coordinate axes, but don't quote me on that. You might like to read my answer to this question: physics.stackexchange.com/questions/307466/…. $\endgroup$ – gj255 Dec 16 '17 at 14:53
  • $\begingroup$ Yes, my observer follows a timeline geodesic and is not rotating in this question. $\endgroup$ – safesphere Dec 16 '17 at 18:04
  • $\begingroup$ Thanks again for all your help. I've poseted a related follow-up question and would appreciate if you have any insight: math.stackexchange.com/questions/2572957/… $\endgroup$ – safesphere Dec 20 '17 at 3:44

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