Connection & Affine Geodesics: Q&A

[GR191511]

1 Does the connection vanish along a affine geodesic?
2 In《Introducing Einstein's Relativity》Ed2 on page 96"It can be shown that the result(the connection vanishes at P) can be extended to obtain a coordinate system in which the connection vanishes along a curve,but not in general to a neighbourhood of P."...
Is the curve a affine geodesic?Thanks.

 

[Orodruin]

1 What do you mean by ”the connection vanishing”? The connection tells you how nearby tangent spacee relate to each other. There is no such thing as a vanishing connection. The connection coefficients may vanish but only in a particular coordinate system.

 

[Dale]

Does the connection vanish along a affine geodesic?

The connection coefficients have nothing to do with the curve, whether geodesic or not. The connection coefficients are related to the coordinate system.

 

[GR191511]

1 What do you mean by ”the connection vanishing”?

The connection coefficients vanish

 

[GR191511]

The connection coefficients have nothing to do with the curve, whether geodesic or not. The connection coefficients are related to the coordinate system.

If it is in the "geodesic coordinate system"(Riemann Normal Coordinates system)?

 

[Dale]

If it is in the "geodesic coordinate system"(Riemann Normal Coordinates system)?

Yes, even so. Suppose you have constructed a Riemann normal coordinate system around some geodesic path. Suppose also that you have a non-geodesic path that intersects with the geodesic path at some event. Then, at that event, the connection coefficients vanish for both the geodesic path and the non-geodesic path.

 

[pervect]

If it is in the "geodesic coordinate system"(Riemann Normal Coordinates system)?

I assume you're talking about a space-time manifold. If you're not, this reply may not be useful.

Christoffel symbols do vanish in a Riemann Normal coordnate system at some event labelled p. In space-time, p is an event, so it is one particular location at one particular instant of time. At events in the neighborhood of p, the Christoffel symbols are "small", but not necessarily zero. Thus, if you consider an arbitrary point q in the space-time manifold, q must be close to p both in space and in time for the Christoffel symbols to vanish.

Perhaps more usefully, in Fermi Normal coordinates, Christoffel symbols can be made to vanish near some geodesic curve. Which is what I think you might have been asking about when you asked

1 Does the connection vanish along a affine geodesic?"

Riemann normal coordinates won't make the Christoffel symbols vanish along a curve, but Fermi-Normal coordinates can make them vanish along the whole worldline, as long as the worldine is a geodesic. If the worldline is not a geodesic, not all of the Christoffel symbols can be made to vanish by this coordinate choice. In some world-tube "near" the geodesic worldline, the Christoffel symbols will be small.

Going back from a discussion of space-time to a discussion of space, it's useful to note that the Christoffel symbols are coordinate dependent. If you have a flat plane with cartesian coordinates (x,y), the line element for the metric is dx^2 + dy^2 and the Christoffel symbols vanish everywhere. But if you use plolar coordinates (r,theta) for the plane, the Christoffel symbols do not vanish. The same geometry, a plane, has different Christoffel symbols depending on your coordinate choice. So asking about the symbols vanishing doesn't make sense unless you include information about the coordinates you are using.

 

[GR191511]

I assume you're talking about a space-time manifold. If you're not, this reply may not be useful.

Christoffel symbols do vanish in a Riemann Normal coordnate system at some event labelled p. In space-time, p is an event, so it is one particular location at one particular instant of time. At events in the neighborhood of p, the Christoffel symbols are "small", but not necessarily zero. Thus, if you consider an arbitrary point q in the space-time manifold, q must be close to p both in space and in time for the Christoffel symbols to vanish.

Perhaps more usefully, in Fermi Normal coordinates, Christoffel symbols can be made to vanish near some geodesic curve. Which is what I think you might have been asking about when you asked

Riemann normal coordinates won't make the Christoffel symbols vanish along a curve, but Fermi-Normal coordinates can make them vanish along the whole worldline, as long as the worldine is a geodesic. If the worldline is not a geodesic, not all of the Christoffel symbols can be made to vanish by this coordinate choice. In some world-tube "near" the geodesic worldline, the Christoffel symbols will be small.

Going back from a discussion of space-time to a discussion of space, it's useful to note that the Christoffel symbols are coordinate dependent. If you have a flat plane with cartesian coordinates (x,y), the line element for the metric is dx^2 + dy^2 and the Christoffel symbols vanish everywhere. But if you use plolar coordinates (r,theta) for the plane, the Christoffel symbols do not vanish. The same geometry, a plane, has different Christoffel symbols depending on your coordinate choice. So asking about the symbols vanishing doesn't make sense unless you include information about the coordinates you are using.

Thank you!