Inertial Reference Frames- circular?

[MeJennifer]

(I'm going to lapse into the differential geometry terminology I know)

Coordinate charts aren't required to represent the correct curvature.

A frame (at a point) is nothing more than a choice of basis for the tangent space at that point. (which, I suppose, can be intuitively thought of as an "infinitessimal" coordinate chart)

Only the orthonormal frames are inertial.

I fail to see how this is in any way a response to:

There is no such thing as a non inertial frame in GR (except for a very small region).

Do you or do you not agree that there is no such thing as a non inertial frame in GR (except for a very small region that may be considered flat)?

Depending on exactly what you mean by the word "relative", either all three of "position, velocity, acceleration" are relative, or none of the three are. Since you have said earlier that position and velocity are relative, then in order to be internally consistent, you must also have acceleration as a relative quantity.

It seems we can agree to disagree.
Whille position and velocity are relative acceleration is absolute in GR.

Feel free to explain how the principle of equivalence would hold if acceleration were not absolute. It would follow that a gravitational field would not be absolute as well.

 

[Hurkyl]

Do you or do you not agree that there is no such thing as a non inertial frame in GR (except for a very small region that may be considered flat)?

No. If by "frame" you mean "coordinate system", then by the definition of manifold, you have to have frames that are defined on "large" regions, and they will be non-inertial.

If by "frame" you mean "a choice of basis vector for the tangent space", then again you can construct a frame over a large region -- you can often do it over the entire manifold. These will almost always be non-inertial (though some inertial ones exist)

If by "frame" you mean something else, then you have to say what it is.

Feel free to explain how the principle of equivalence would hold if acceleration were not absolute.

I can't even figure out why you think there's a problem.

 

[JesseM]

Do you or do you not agree that there is no such thing as a non inertial frame in GR (except for a very small region that may be considered flat)?

Do you think that Schwarzschild coordinates, which are defined on a large region, qualify as "inertial"? By what criteria? Certainly the path of an object which is at rest in these coordinates (in the sense that its position coordinates are not changing with time) is not following a geodesic.

 

[MeJennifer]

Do you think that Schwarzschild coordinates, which are defined on a large region, qualify as "inertial"? By what criteria? Certainly the path of an object which is at rest in these coordinates (in the sense that its position coordinates are not changing with time) is not following a geodesic.

The selection of coordinates have absolutely nothing to do with something being inertial or not.

Something that is resisting the gravitational pull is obviously not traveling on a geodesic. For instance when you stand on the Earth you are not traveling a geodesic, you are actually accelerating away from the enter of gravity.

Do you realize that the Schwarzschild solution is a model from the perspective of a distant observer who is away from the gravitational pull?

 

[JesseM]

The selection of coordinates have absolutely nothing to do with something being inertial or not.

Sure, but I thought we were talking about non-inertial vs. inertial frames, not the question of whether a path is inertial (ie a geodesic) or not. Are you using "frame" to mean something different than "coordinate system"?

Something that is resisting the gravitational pull is obviously not traveling on a geodesic. For instance when you stand on the Earth you are not traveling a geodesic, you are actually accelerating away from the enter of gravity.

Of course I know this, wasn't it obvious that this was exactly my point? Things which are at rest in Schwarzschild coordinates are not moving on a geodesic, therefore Schwarzschild coordinates cannot be considered an "inertial coordinate system". This is exactly the same standard you use to judge whether a coordinate system is inertial or non-inertial in SR--an object moving on a geodesic path will be moving in a straight line in coordinate terms (constant dx/dt, dy/dt and dz/dt) in an inertial coordinate system, and that path will still be a geodesic in a non-inertial coordinate system, but the system is called "non-inertial" precisely because the path would no longer appear straight in terms of the coordinates (dx/dt, dy/dt and dz/dt wouldn't be constant), and because objects which are at rest in this coordinate system (constant x, y, and z coordinate) are moving on non-geodesic paths.

If this is not the standard that you use to judge whether a coordinate system in SR is inertial or non-inertial, than what is?