Non-inertial reference frame: Difference between revisions

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

{{Classical mechanics}}

A '''non-inertial reference frame''' (also known as an '''accelerated reference. In this the Newton's Second Law of motion is not followed i.e. it is deniedframe'''<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 undergoes [[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-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]].

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==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://archive.org/details/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&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&dq=laws+physics+%22+form%22&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]].

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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, although 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==

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===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 al galong 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.