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{{Classical mechanics}}
A '''non-inertial reference frame''' is a [[frame of reference]] that is undergoing [[acceleration]] with respect to an [[Inertial frame of reference|inertial frame]].<ref name=Tocaci>{{cite book |title=Relativistic Mechanics, Time, and Inertia |author=Emil Tocaci, Clive William Kilmister |page=251 |url=https://books.google.com/books?id=7dVRL51JRI0C&pg=PA251 |isbn=90-277-1769-9 |year=1984 |publisher=Springer }}</ref> An [[accelerometer]] at rest in a non-inertial frame will in general detect a non-zero acceleration. In a curved [[spacetime]] all frames are non-inertial{{Clarify | date=November 2012}}. The laws of motion in non-inertial frames do not take the simple form they do in inertial frames, and the laws vary from frame to frame depending on the acceleration.<ref>{{cite book |title=Essential Relativity |author=Wolfgang Rindler |page=25 |url=https://books.google.com/books?id=0J_dwCmQThgC&pg=PT43 |isbn=3-540-07970-X |year=1977 |publisher=Birkhäuser}}</ref><ref>{{cite book |title=Basics of Space Flight |author= Ludwik Marian Celnikier |page=286 |url=https://books.google.com/books?id=u2kf5uuaC6oC&pg=PA286 |isbn=2-86332-132-3 |year=1993 |publisher=Atlantica Séguier Frontières}}</ref> To explain the motion of bodies entirely within the viewpoint of non-inertial reference frames, [[fictitious forces]] (also called inertial forces, pseudo-forces<ref name=Iro>{{cite book |author=Harald Iro |title=A Modern Approach to Classical Mechanics |page=180 |url=https://books.google.com/books?id=-L5ckgdxA5YC&pg=PA179 |isbn=981-238-213-5 |year=2002 |publisher=World Scientific }}</ref> and d'Alembert forces) must be introduced to account for the observed motion, such as the [[Coriolis force]] or the [[centrifugal force (fictitious)|centrifugal force]], as derived from the acceleration of the non-inertial frame.<ref name=Shadowitz>{{cite book |author=Albert Shadowitz |url=https://books.google.com/books?id=1axfJqUT6R0C&pg=PA4 |title=Special relativity |isbn=0-486-65743-4 |page=4 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988}}</ref>
As stated by Goodman and Warner, "One might say that '''F''' {{=}} ''m'''''a''' holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."<ref name=Goodman>{{cite book |title=Dynamics |author=Lawrence E. Goodman & William H. Warner |url=https://books.google.com/books?id=2z0ue1xk7gUC |isbn=0-486-42006-X |publisher=Courier Dover Publications |year=2001 |edition=Reprint of 1963|page=358}}</ref>

==Avoiding fictitious forces in calculations==
{{see also|Inertial frame of reference|Fictitious force}}
In flat spacetime, the use of non-inertial frames can be avoided if desired. Measurements with respect to non-inertial reference frames can always be transformed to an inertial frame, incorporating directly the acceleration of the non-inertial frame as that acceleration as seen from the inertial frame.<ref name=Alonzo>{{cite book |author= M. Alonso & E.J. Finn |title=Fundamental university physics
|publisher=, Addison-Wesley |year=1992 |url=https://books.google.com/books?id=c5UAAAAACAAJ&dq=isbn=0201565188&lr=&as_brr=0 |isbn= 0-201-56518-8}}</ref> This approach avoids use of fictitious forces (it is based on an inertial frame, where fictitious forces are absent, by definition) but it may be less convenient from an intuitive, observational, and even a calculational viewpoint.<ref name=Price>“The inertial frame equations have to account for ''V<sub>Ω</sub>'' and this very large centripetal force explicitly, and yet our interest is almost always the small relative motion of the atmosphere and ocean, ''V' '', since it is the relative
motion that transports heat and mass over the Earth. … To say it a little differently—it is the relative velocity that we measure when [we] observe from Earth’s surface, and it is the relative velocity that we seek for most any practical purposes.” [http://ocw.mit.edu/ans7870/resources/price/index.htm MIT essays] by James F. Price, Woods Hole Oceanographic Institution (2006). See in particular §4.3, p. 34 in the [http://ocw.mit.edu/ans7870/resources/price/essay2.pdf Coriolis lecture]</ref> As pointed out by Ryder for the case of rotating frames as used in meteorology:<ref name=Ryder>{{cite book |title=Classical Mechanics |author=Peter Ryder |url=https://books.google.com/books?id=j1Y5FfdQHsQC&pg=PA80 |isbn=978-3-8322-6003-3 |publisher=Aachen Shaker |year=2007 |pages=78–79 }}</ref>
{{quote|A simple way of dealing with this problem is, of course, to transform all coordinates to an inertial system. This is, however, sometimes inconvenient. Suppose, for example, we wish to calculate the movement of air masses in the earth's atmosphere due to pressure gradients. We need the results relative to the rotating frame, the earth, so it is better to stay within this coordinate system if possible. This can be achieved by introducing ''fictitious'' (or "non-existent") forces which enable us to apply Newton's Laws of Motion in the same way as in an inertial frame.|Peter Ryder|''Classical Mechanics'', pp. 78-79}}

==Detection of a non-inertial frame: need for fictitious forces==
That a given frame is non-inertial can be detected by its need for fictitious forces to explain observed motions.<ref name=Serway>{{cite book |title=Physics for scientists & engineers |author=Raymond A. Serway |year=1990 |publisher=Saunders College Publishing |edition=3rd |isbn=0-03-031358-9 |page=135 |url=https://books.google.com/books?lr=&as_brr=0&q=%22fictitious+forces+do+not+exist+when+the+motion+is+observed+in+an+inertial+frame.+The+fictitious+forces+are+used+only+in+an+accelerating%22&btnG=Search+Books}}</ref><ref name="ArnoldQuote">{{cite book |title=Mathematical Methods of Classical Mechanics |page=129 |author=V. I. Arnol'd |isbn=978-0-387-96890-2 |year=1989 |url=https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=|publisher=Springer}}</ref><ref name=Rothman>{{cite book |title=Discovering the Natural Laws: The Experimental Basis of Physics |author= Milton A. Rothman |page=23 |url=https://books.google.com/books?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22&lr=&as_brr=0&source=gbs_search_s&cad=5&sig=ACfU3U33YE3keeD7lDVtQvt-ltW87Lsq2Q
|isbn=0-486-26178-6 |publisher=Courier Dover Publications |year=1989 }}</ref><ref name=Borowitz>{{cite book |title=A Contemporary View of Elementary Physics |page=138 |publisher=McGraw-Hill |year=1968 |url=https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=The+effect+of+his+being+in+the+noninertial+frame+is+to+require+the+observer+to&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn= |asin= B000GQB02A |author=Sidney Borowitz & Lawrence A. Bornstein }}</ref><ref name=Meirovitch>{{cite book |author=Leonard Meirovitch |page=4 |isbn=0-486-43239-4 |publisher=Courier Dover Publications |year=2004 |edition=Reprint of 1970 |url=https://books.google.com/books?id=GfCil84YTm4C&pg=PA4&dq=%22in+accelerated+systems,+we+must%22&lr=&as_brr=0&sig=ACfU3U0UrA5jcOx4pB9QIlyA7BQiXwAV5Q |title =Methods of analytical Dynamics}}</ref> For example, the rotation of the [[Earth]] can be observed using a [[Foucault pendulum]].<ref name=diFrancia>{{cite book |title=The Investigation of the Physical World |author=Giuliano Toraldo di Francia |page=115 |url=https://books.google.com/books?id=cFQ7AAAAIAAJ&pg=PA46&dq=laws+physics+%22+form%22&lr=&as_brr=0&sig=ACfU3U0zI1ZXjyB3G6Z3AI3zM_Z2YfYN6g#PPA115,M1 |isbn=0-521-29925-X |publisher=CUP Archive |year=1981 }}</ref> The rotation of the Earth seemingly causes the pendulum to change its plane of oscillation because the surroundings of the pendulum move with the Earth. As seen from an Earth-bound (non-inertial) frame of reference, the explanation of this apparent change in orientation requires the introduction of the fictitious [[Coriolis effect|Coriolis force]].

Another famous example is that of the tension in the string between [[rotating spheres|two spheres rotating about each other]].<ref>
{{cite book |title=Analytical Mechanics |page=324 |url=https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA324 |isbn=0-521-57572-9 |publisher=[[Cambridge University Press]] |year=1998 |author=Louis N. Hand, Janet D. Finch}}</ref><ref>{{cite book |title=The Cambridge companion to Newton |url =https://books.google.com/books?id=3wIzvqzfUXkC&pg=PA43 |author=I. Bernard Cohen, George Edwin Smith |page=43 |isbn=0-521-65696-6 |year=2002 |publisher=Cambridge University Press}}</ref> In that case, prediction of the measured tension in the string based upon the motion of the spheres as observed from a rotating reference frame requires the rotating observers to introduce a fictitious centrifugal force.

In this connection, it may be noted that a change in coordinate system, for example, from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of fictitious forces, despite the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another.

==Fictitious forces in curvilinear coordinates==
{{see also|Mechanics of planar particle motion}}
A different use of the term "fictitious force" often is used in [[curvilinear coordinates]], particularly [[polar coordinates]]. To avoid confusion, this distracting ambiguity in terminologies is pointed out here. These so-called "forces" are non-zero in all frames of reference, inertial or non-inertial, and do ''not'' transform as vectors under rotations and translations of the coordinates (as all Newtonian forces do, fictitious or otherwise).

This incompatible use of the term "fictitious force" is unrelated to non-inertial frames. These so-called "forces" are defined by determining the acceleration of a particle within the curvilinear coordinate system, and then separating the simple double-time derivatives of coordinates from the remaining terms. These remaining terms then are called "fictitious forces". More careful usage calls these terms "[[generalized forces|<u>generalized</u> fictitious forces]]" to indicate their connection to the [[generalized coordinates]] of [[Lagrangian mechanics]]. The application of Lagrangian methods to polar coordinates can be found [[Mechanics of planar particle motion#Lagrangian approach|here]].

==Relativistic point of view==
{{unreferenced section|small=y|date=April 2017}}
===Frames and flat spacetime===
{{See|Proper reference frame (flat spacetime)}}
If a region of spacetime is declared to be [[Euclidean space|Euclidean]], and effectively free from obvious gravitational fields, then if an accelerated coordinate system is overlaid onto the same region, it can be said that a ''uniform fictitious field'' exists in the accelerated frame (we reserve the word gravitational for the case in which a mass is involved). An object accelerated to be stationary in the accelerated frame will "feel" the presence of the field, and they will also be able to see environmental matter with inertial states of motion (stars, galaxies, etc.) to be apparently falling "downwards" in the field along curved [[trajectory|trajectories]] as if the field is real.

In frame-based descriptions, this supposed field can be made to appear or disappear by switching between "accelerated" and "inertial" coordinate systems.

===More advanced descriptions===
As the situation is modeled in finer detail, using the [[general principle of relativity]], the concept of a ''frame-dependent'' gravitational field becomes less realistic. In these [[Mach's principle|Machian]] models, the accelerated body can agree that the apparent gravitational field is associated with the motion of the background matter, but can also claim that the motion of the material as if there is a gravitational field, causes the gravitational field - the accelerating background matter "[[light-dragging effects|drags light]]". Similarly, a background observer can argue that the forced acceleration of the mass causes an apparent gravitational field in the region between it and the environmental material (the accelerated mass also "drags light").
This "mutual" effect, and the ability of an accelerated mass to warp lightbeam geometry and lightbeam-based coordinate systems, is referred to as [[frame-dragging]].

Frame-dragging removes the usual distinction between accelerated frames (which show gravitational effects) and inertial frames (where the geometry is supposedly free from gravitational fields). When a forcibly-accelerated body physically "drags" a coordinate system, the problem becomes an exercise in warped spacetime for all observers.

==See also==
*[[Fictitious force]]
*[[Centrifugal force]]
*[[Coriolis effect]]
*[[Inertial frame of reference]]
*[[Free motion equation]]

==References and notes==
{{reflist|30em}}

{{DEFAULTSORT:Non-Inertial Reference Frame}}
[[Category:Frames of reference]]
[[Category:Classical mechanics]]

Non-inertial reference frame - Revision history

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