# Locally inertial coordinates on geodesics

1. Aug 9, 2012

### VantagePoint72

It's a standard fact of GR that at a given point in space-time, we can construct a coordinate system such that the metric tensor takes the form of Minkowski spacetime and its first derivatives vanish. Equivalently, we can make the Christoffel symbols vanish at point. Moreover, the fact that, in general, there's no coordinate system that lets us do this simultaneously for the entire manifold is the essence of curvature.

But, is it possible to construct a coordinate system such that the metric is Minkowskian along an entire geodesic? Since geodesics are the generalization in GR of inertial motion, it seems intuitively that it should be possible to do so. According to the equivalence principle, an arbitrarily small 'laboratory' falling along a geodesic should be unable to determine if its in an inertial (in the SR sense) frame in flat spacetime or falling along a geodesic in curved spacetime. It seems to me that finding a coordinate system in which the Christoffel symbols along the geodesic vanish would be mathematical realization of this.

2. Aug 9, 2012

### atyy

http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html [Broken]

Try the section Fermi normal coordinates, and see the remarks after Eq 9.16.

Last edited by a moderator: May 6, 2017
3. Aug 9, 2012

### bcrowell

Staff Emeritus
It sounds like you're describing Fermi coordinates. Fermi originally did this for a Riemannian manifold, but it was later generalized to pseudo-Riemannian manifolds.

4. Aug 10, 2012

### VantagePoint72

Perfect, thank you!