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In some PSE questions or answers such as here (and comments below) there appears the notion of "accelerating frame" or (more or less equivalently) "noninertial frame".

What's the definition of this notion?,
i.e.
How are given participants (or, if you prefer, "point particles"$\,\!^{(*)}$) who "keep sight of each other" supposed to determine whether they (pairwise) belonged to the same "noninertial frame", or not?

$(*$: Cmp. the notion of "inertial frame", in distinction to "inertial coordinate system", of http://www.scholarpedia.org/article/Special_relativity:_kinematics

We should, strictly speaking, differentiate between an inertial frame and an inertial coordinate system, although in sloppy practice one usually calls both IFs. An inertial frame is simply an infinite set of point particles sitting still in space relative to each other.
$)$.

Follow-up:
The new (follow-up) question to be asked to fully address this question has been submitted as How should observers determine whether they can be described as being "defined on a Lorentzian manifold"?

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user12262
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    This is a question that could be answered by a quick search engine search. The wikipedia entry for non-inertial reference frame comes up as top-3 hit on at least Google, Bing, Yahoo and DuckDuckGo. I recommend the question is closed. – Thriveth Jun 26 '13 at 21:28
  • @Thriveth Would you please point out one of the numerous references of "noninertial frame" (or, indeed, "non-inertial frame") as definitive; so I could refer to its terminology in case of further questions. BTW, presently there doesn 't seem to be a section Definition at http://en.wikipedia.org/wiki/Non-inertial_reference_frame – user12262 Jun 26 '13 at 21:48
  • @user12262, you might consider telling us what you think it might be, having done some research and all, and then ask about any specific concept you don't "get". – Alfred Centauri Jun 26 '13 at 21:48
  • @user12262 It doesn't say the word "definition:" in front of it, but for example the very first sentence in the Wikipedia article is pretty much spot on. – Thriveth Jun 26 '13 at 21:51
  • Please note the corresponding answer submitted by Alfred Centauri, and the comments following it. – user12262 Jun 26 '13 at 22:13
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    possible duplicate of Inertial frames of reference –  Jun 27 '13 at 23:41
  • Doubtful. I'll consider commenting there (perhaps next week) -- asking for instance about the distinction (quoted in the footnote of my question) between "inertial frame" and "coordinate system" ... – user12262 Jun 28 '13 at 13:31

2 Answers2

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What's the definition of this notion?

From Wiki:

A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will in general detect a non-zero acceleration.

Now, that's pretty straightforward and easy to find so, assuming the above doesn't satisfactorily address your inquiry, what is it that you're asking?

Alfred Centauri
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  • Thanks; well ... Is any set of "accelerometers [while they] detect a non-zero acceleration" therefore a "non-inertial reference frame"?, for instance. (Please note the word "same" in my question, which has been present, and emphasized, there from the outset.) – user12262 Jun 26 '13 at 22:09
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    @user12262, I'm still not clear on what your essential question is. Are you satisfied with the definition of non-inertial reference frame? If so, is your question how two observers might determine if they share the same non-inertial reference frame? – Alfred Centauri Jun 26 '13 at 22:18
  • "Are you satisfied with the definition of [NIRF]?" I believe I can comprehend what you stated. Trying to do so, however, I can imagine sets of "accelerometers [each detecting] non-zero a" which are so obviously not "sitting still relative to each other" (cmp. footnote) that I wonder whether they are generally referred to as "(non-inertial) frame". (That's, of course, presuming some particular operational notion of pairwise "sitting still relative to each other"; leading to the part on determination.) – user12262 Jun 26 '13 at 22:40
  • p.s. @Alfred Centauri Keeping in the comment character budget I was perhaps not explicit enough. Therefore: I'm thinking of certain (infinite) sets of point particles/accelerometers/observers who each find non-zero acceleration; but not necessarily equal acceleration(s), nor necessarily constant acceleration(s) -- since your answer doesn't require that (yet). Again: Would you call any such set a "frame", or would you supplement your definition? – user12262 Jun 26 '13 at 22:56
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    @user12262, no, I wouldn't think of such a general set as defining a "frame" at all. It might be that you're not clear on the notion of "frame of reference" or "observer". – Alfred Centauri Jun 26 '13 at 23:20
  • @user12262, have you perused the Wiki article on Rindler Coordinates? http://en.wikipedia.org/wiki/Rindler_coordinates See, in particular, the section on "Rindler Observers" – Alfred Centauri Jun 27 '13 at 00:25
  • "have you perused the Wiki article on Rindler Coordinates?" -- Not really, having been encouraged for instance by the reference from the footnote of my question to first persue geometric relations between participants (point particles, observers). (OMG!, that reference was authored by Rindler! ;) My persuit had led to an answer here, and this question, mostly ... – user12262 Jun 27 '13 at 04:57
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    What's missing in this answer is the distinction between an inertial frame as defined in Newtonian mechanics and an inertial frame as defined in relativity. In the former, the earth's surface is nearly inertial. In the latter, it's noninertial. –  Jun 27 '13 at 18:01
  • @BenCrowell, I find it highly unlikely that that distinction is even remotely relevant in this context. As you might be able to glean from the comments, it still isn't clear what the actual question is. Would you prefer that I remove this answer? Or, shall we wait for the OP to further clarify his question? – Alfred Centauri Jun 27 '13 at 19:09
  • What's missing from @Alfred Centauri's answer, as it stands, should be obvious from his comment above (#5): "no, I wouldn't think of such a general set as defining a "frame" at all." So Alfred C. might want to strengthen his answer/definition so that any sets he considers "[too] general" become excluded. Of course, instead I might try to research/ask "What's a frame? - How to comprehend and generalize Rindler's notion of sitting still relative to each other?". (However, getting mixed in with all sorts of questions about "inertial frames" didn't seem a good idea, at the time.) – user12262 Jun 28 '13 at 13:17
  • @user12262, any editing of my answer will await a further clarification of your question. The purpose of my "answer" was to discover what precisely you didn't find satisfactory with the given definition. From what you've just written, it seems that you aren't clear on the notions of "frame of reference", "observers", and "coordinate system". So, I would suggest editing to your question to reflect that. – Alfred Centauri Jun 28 '13 at 13:44
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In classical mechanics, an inertial frame of reference is one in which a free point mass floating in space does not experience any outside force. If it is moving relative to the frame, it traces a straight line in that frame in accordance with the laws of motion.

A frame that is not inertial is either accelerating or rotating, or some combination of the two, or else it is bathed in a gravitational field (and is not free falling in that field).

For instance, a frame that is fixed with regard to the surface of the Earth appears to be one which accelerates at 9.81 $m/s^2$ away from the center of the planet.

The frame does not consist of its contents; the frame is a the coordinate system selected for tracking the positions of whatever is of interest that frame. It is itself not an object, but a frame can move relative to other frames, accelerate and so on.

Kaz
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  • A free falling frame "bathed" in a gravitational field is (locally) an inertial frame of reference, i.e., the accelerometers read zero. A frame fixed to the surface of the Earth is not free falling and that's why acceleration is felt. – Alfred Centauri Jun 26 '13 at 23:26
  • @AlfredCentauri Ah, yes, that needs to be rolled in. – Kaz Jun 27 '13 at 00:13
  • @Kaz Same question as regarding Alfred Centauri's answer: Is any set of participants (or if you prefer, "point masses") who are "not free falling" therefore a "non-inertial frame"? (Perhaps, any two "free" protagonists don't necessarily belong to one and the same "inertial frame", either?) Also: is it helpful or even necessary to invoke dynamics (such as mentioning of "force", and of "free ness") at all? (In contrast, Alfred Centauri's answer appeared to be restricted to geometry/kinematics.) – user12262 Jun 27 '13 at 05:15
  • The masses are not the frame. The frame is the coordinate system with respect to which you measure the position of the masses. – Kaz Jun 27 '13 at 05:29
  • @user12262 : A frame is a coordinate system with an origin and an orientation. For each observer (or point mass, or participant), you have to define a different frame, where the position of the observer (or point mass, or participant) is just zero. – Trimok Jun 27 '13 at 11:37
  • Trimok, @Kaz: "[...] frame is a coordinate system [...]" -- Would you both relate your statement(s) to the statement quoted in the footnote of my question ("We should, strictly speaking, differentiate between an inertial frame and an inertial coordinate system [...]"), please? Do you deny that such a distinction can be made and expressed? (Meta: Or should I attempt to ask this as a separate question? ...) – user12262 Jun 27 '13 at 16:36