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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. While the laws of motion are the same in all inertial frames, in non-inertial frames, they 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>
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-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) 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://
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]].
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==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=[https://archive.org/details/discoveringnatur0000roth/page/23 23] |url=https://
|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/?id=GfCil84YTm4C&pg=PA4&dq=%22in+accelerated+systems,+we+must%22 |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/?id=cFQ7AAAAIAAJ&pg=PA46&dq=laws+physics+%22+form%22#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>
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