Am I understanding the concept of proper frame of reference?

[PAllen]

I'm not sure I see this implication, since the section that discusses this for curved spacetime (13.6) refers to Exercise 6.8 which defines an accelerated rotating frame for flat spacetime, and says that this section will do the same for curved spacetime. There is no reference anywhere that I can see for a use of the term "proper reference frame" to describe a standard inertial frame in flat spacetime.

That said, I agree that zero acceleration and zero rotation can be plugged into the equations as a special case, and yield a standard inertial frame (in curved spacetime, a local inertial frame), and that MTW does include that special case.

I found an explicit statement in this section of MTW (13.6 bottom of p. 331 in my edition) that makes clear that they have the meaning I claim:

" In the case of zero acceleration and zero rotation, (..), the observer's proper reference frame reduces to a local Lorentz frame (...) all along his geodesic world line!"

 

[pervect]

I concur with Pallen's observations. MTW goes into the explicit geometric details of how to construct a proper frame of reference. And I agree one can get an inertial proper frame by setting the acaceleration and rotation to zero - Pallen has quoted the relevant section of MTW on this so perhaps this observation is a bit redundant.

It appears to me that the key concept that makes a frame a "proper frame" of reference are that we start with the worldline of some "observer". Then we label the time coordinate of any event on this worldline of the "observer" with the proper time of said event along this worldline. Additionally, we transport this time coordinates away from the worldine, to points that are not on the worldine of the observer, via geodesics curves that are orthogonal (in the space-time sense) to the worldline of the observer. This is a quick summary of the MTW's longer explanation.

Fermi normal coordinates are a specific and more restrictive example of how one can construct a "proper" reference frame".

Once upon a time I was motivated by this section of MTW to construct such "proper" coordinates for an observer on a "sliding block" which slid on the floor of Einstein's elevator. I chose not to fermi-walker transport the basis vectors, so the resulting frame was both accelerated and rotating, and thus the coordinates I came up with were not Fermi normal coordinates.

In the sliding block example, other people choose and preferred different coordinates that used a different way to define time of events not on the worldline of the observer.

I don't think we should read too much into the use of the word "proper" than as a reference to the use of "proper time" in the coordinate construction. There are a lot of different ways one might assign coordinates in General Relativity which is sort of the whole point of the theory, the amount of freedom it gives one to choose smooth coordinates.

 

[anuttarasammyak]

The Twin Paradox is one example. The traveling twin can be in inertial reference frames for much of his departure and return trips. I

In SR, proper time of the object in motion with velocity v(t) is given by
$$\tau = \int_0^T \sqrt{1-\frac{v(t)^2}{c^2}}dt$$
where once $$\tau = T = 0$$ for a common event.

In the similar way, proper frame of reference, PFR, of the object couuld be made of succesive (instantaneous and local) IFRs as said for astronaut twin above but in approximation, e.g., forgetting about his g experience during U-turn.

One of my concerns is about the rotation of xyz axis during the motion because some combinations of Lorentz transfomation generate rotation.

 

[PeterDonis]

proper frame of reference, PFR, of the object couuld be made of succesive (instantaneous and local) IFRs as said for astronaut twin above but in approximation, e.g., forgetting about his g experience during U-turn.

That's not how the proper frame of reference of an accelerated observer is defined in any of the references that have been given. The part I've bolded above is the part that is not in all those references; an accelerated proper frame of reference as defined there can handle "g experience" of arbitrary acceleration, no "approximation" to avoid it is needed. The main limitation of the proper frame of reference of an accelerated observer is that it can only cover a limited region around the worldline of the observer; the greater the proper acceleration of the observer, the more limited the region. So at the "U-turn" in the standard twin paradox, the proper frame of reference of the traveling twin would only be able to cover a very narrow region around that twin's worldline.