Non-inertial reference frame: Difference between revisions

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{{Short description|Reference frame that undergoes acceleration with respect to an inertial frame}}

{{Expand Spanish|Sistema de referencia no inercial|date=September 2022}}

{{Classical mechanics}}

A '''non-inertial reference frame''' (also known as an '''accelerated reference frame'''<ref>{{Cite web | url = https://www.ux1.eiu.edu/~cfadd/1350/06CirMtn/AccFrames.html | title=Accelerated Reference Frames | access-date=2023-09-06}}</ref>) is a [[frame of reference]] that is undergoingundergoes [[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. While Inthe alaws curvedof [[spacetime]] all framesmotion are non-inertial{{Clarifythe |same date=Novemberin 2012}}.all inertial The laws of motionframes, 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>

In [[classical mechanics]] it is often possible to explain the motion of bodies in non-inertial reference frames by introducing additional [[fictitious forces]] (also called inertial forces, [[Pseudo-force|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's principle|d'Alembert forces]]) to [[Newton's laws of motion|Newton's second law]]. Common examples of this include the [[Coriolis force]] and the [[centrifugal force (fictitious)|centrifugal force]]. In general, the expression for any fictitious force can be derived from the acceleration of the non-inertial frame.<ref name=Shadowitz>{{cite book |author=Albert Shadowitz |url=https://archive.org/details/specialrelativit0000shad |url-access=registration |title=Special relativity |isbn=0-486-65743-4 |page=[https://archive.org/details/specialrelativit0000shad/page/4 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>

In the theory of [[general relativity]], the curvature of [[spacetime]] causes frames to be [[Local reference frame|locally]] inertial, but globally non-inertial. Due to the [[Introduction to the mathematics of general relativity|non-Euclidean geometry of curved space-time]], there are no global inertial reference frames in general relativity. More specifically, the fictitious force which appears in general relativity is the force of [[gravity]].

==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 |isbn=0-201-56518-8 }}{{Dead link|date=November 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> This approach avoids the 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_Ω'' 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>

|publisher=, Addison-Wesley |year=1992 |url=https://books.google.com/?id=c5UAAAAACAAJ&dq=isbn=0201565188 |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_Ω'' 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==

[[File:Noninertial reference frames.gif|thumb|300x300px|An example of a non-inertial reference frame- a rotating reference frame. The point feels a centrifugal force which needs to be compensated to keep rotating. Otherwise, it does not feel [[Euler force]] as the rotation rate is constant and it does not feel Coriolis force as it is not moving relative to the rotating frame.]]

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=[https://archive.org/details/discoveringnatur0000roth/page/23 23] |url=https://booksarchive.google.comorg/?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22details/discoveringnatur0000roth

|url-access=registration |quote=reference laws of physics. |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&pg=PA4 |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#PPA115,M1&pg=PA46 |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, the prediction of the measured tension in the string based uponon 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 thatalthough 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)}}

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==See also==

*[[Rotating reference frame]]

*[[Fictitious force]]

*[[Centrifugal force]]