Is an accelerating frame the same as an inertial frame at a point?

[Sciencemaster]

TL;DR Summary

Is an accelerating observer's reference frame the same as an inertial frame with the same instantaneous velocity?

If we have an observer that is accelerating in one direction (perhaps a rocket ship accelerating towards the sun), would its reference frame be identical to an observer at the same point that is not accelerating, but has the same instantaneous velocity? In other words, is an accelerating observer's reference frame identical to an inertial frame at an infinitesimal point (where the only difference between the two observers is that the former one will 'switch' between inertial frames as time passes)? If so, is this still the case if the acceleration is not constant?
I know that in General Relativity, spacetime is locally flat. Given the equivalence principle, I'm wondering if this has anything to do with my question (i.e. an accelerating observer's reference frame is locally identical to an inertial observer's that is moving at the same instantaneous velocity).

 

[Dale]

TL;DR Summary: Is an accelerating observer's reference frame the same as an inertial frame with the same instantaneous velocity?

would its reference frame be identical to an observer at the same point that is not accelerating, but has the same instantaneous velocity?

No. The transformation between the two is the Rindler transform, not the Lorentz transform nor the identity transformation

 

[Ibix]

You can always choose an orthonormal frame at an event (not a point). So yes, at one event, and up to arbitrary rotations.

If your worldline is not an inertial one, trying to patch together local orthogonal frames into a global one will typically fail somewhere.

 

[PeterDonis]

@Sciencemaster, your question assumes that there is such a thing as "the" reference frame for a non-inertial observer (or, in your follow-up about GR, for any observer in a curved spacetime). That is not the case.

In flat spacetime, where there are global inertial frames, any inertial observer will be at rest in some specific global inertial frame, and that frame can be considered to be "the" reference frame (more precisely the rest frame) for that observer. But for non-inertial observers, there is no unique way to define a "rest frame".

In curved spacetime, there are no global inertial frames, so even for inertial observers there is no unique way to define a rest frame. The best you can do is to define a local inertial frame centered on the observer for a small enough patch of spacetime that the effects of spacetime curvature are not significant. (And for a non-inertial observer, you have to pick some particular event on their worldline and construct a local inertial frame centered on just that event.)

 

[Sciencemaster]

Alright, I think I understand. While you can have an inertial frame at a given point with the same position and instantaneous velocity as an accelerating observer, that's not the same as the accelerating observer having that frame of reference at a given point because accelerating observers don't have a frame where they're stationary and do not have global frames. Also, representing this observer switching between inertial frames (as a patchwork of inertial frames) isn't as accurate as I thought it was. Am I correct in my thought process?

 

[PeterDonis]

While you can have an inertial frame at a given point with the same position and instantaneous velocity as an accelerating observer

Yes.

that's not the same as the accelerating observer having that frame of reference at a given point

No. If you're talking about "at a given point", a local inertial frame centered on that point can be "the" frame for any observer whose 4-velocity at that point is at rest in the frame.

accelerating observers don't have a frame where they're stationary

Sure they can. It just won't be an inertial frame if you want them to be stationary in it at more than just one point.

and do not have global frames.

Sure they can. It just won't be a global inertial frame.

 

[Sciencemaster]

I see. So if we don't care about following the Worldline of the accelerating observer before or after a given event, we could say that the inertial frame centered that that event that has the same instantaneous velocity and position as the observer could be considered "the" reference frame of the observer (at that specific point)? (While if we want to follow that observer's Worldline, we have to use a non-inertial frame)

 

[PeterDonis]

if we don't care about following the Worldline of the accelerating observer before or after a given event, we could say that the inertial frame centered that that event that has the same instantaneous velocity and position as the observer could be considered "the" reference frame of the observer (at that specific point)?

Yes.

if we want to follow that observer's Worldline, we have to use a non-inertial frame

Yes.

 

[Sciencemaster]

Alright, got it! Thank you all for the help, I really appreciate it!

 

[PeterDonis]

Alright, got it! Thank you all for the help, I really appreciate it!

You're welcome!

 

[Dale]

I see. So if we don't care about following the Worldline of the accelerating observer before or after a given event, we could say that the inertial frame centered that that event that has the same instantaneous velocity and position as the observer could be considered "the" reference frame of the observer (at that specific point)? (While if we want to follow that observer's Worldline, we have to use a non-inertial frame)

The standard term for this frame is “the momentarily co-moving inertial frame”

 

[vanhees71]

You can always choose an orthonormal frame at an event (not a point). So yes, at one event, and up to arbitrary rotations.

If your worldline is not an inertial one, trying to patch together local orthogonal frames into a global one will typically fail somewhere.

One should note that local inertial reference frames of a non-inertial observer can be defined by constructing corresponding tetrades along the (necessarily timelike) worldline of this observer. A "natural choice" is to use the four-velocity of the observer, normalized to 1, as the time-like tetrad vector. Then you need to choose the 3 space-like tetrad vectors. This can be done by some convenient construction. One is to start with an arbitrary tetrad at one point of the world-line with the normalized four-velocity at this point as the time-like tetrad vector and then Fermi-Walker transport this tetrad along the worldline of the observer. If this worldline is not a straight line, you'll get a set of local inertial frames where the light-like tetrad vectors rotate although Fermi-Walker transport is defined such that for an "infinitesimal move along the worldline" these vectors don't rotate. That is due to the fact that the Fermi-Walker transport in this "infinitesimal step" involves rotation-free Lorentz boosts, but these boosts change direction if you proceed further, if the world line is not a straight line. This is the geometric description of the so-called "Thomas precession", which was historically very important to understand the correce finestructure splitting of atomic spectral lines.

For more on Fermi-Walker transport, see

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf