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{{short description|Abstract coordinate system
{{distinguish|Inertial frame of reference}}▼
{{Other uses|Frame of reference (disambiguation)}}
{{Classical mechanics |Fundamentals}}
In [[physics]] and [[astronomy]], a '''frame of reference''' (or '''reference frame''')
An important special case is that of an ''[[inertial reference frame]]'', a stationary or uniformly moving frame.
For ''n'' dimensions, {{nowrap|''n'' + 1}} reference points are sufficient to fully define a reference frame. Using [[Cartesian coordinate system|rectangular
In [[Theory of relativity|Einsteinian relativity]], reference frames are used to specify the relationship between a moving [[Observer (special relativity)|observer]] and the phenomenon
== Definition ==
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* A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.<ref name =Pontriagin>
In very general terms, a coordinate system is a set of arcs ''x''<sup>i</sup> = ''x''<sup>i</sup> (''t'') in a complex [[Lie group]]; see {{cite book |author=Lev Semenovich Pontri͡agin |title=L.S. Pontryagin: Selected Works Vol. 2: Topological Groups |page= 429 |year= 1986|url=https://books.google.com/books?id=JU0DT_wXu2oC&q=algebra+%22coordinate+system%22&pg=PA429|isbn=2-88124-133-6 |publisher=Gordon and Breach|edition=3rd }}. Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>,
{{cite book |title=Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces |author1=J X Zheng-Johansson |author2=Per-Ivar Johansson |page=13
|url=https://books.google.com/books?id=I1FU37uru6QC&q=frame+coordinate+johansson&pg=PA13|isbn=1-59454-260-0
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* Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.
{{efn|Here is a quotation applicable to moving observational frames <math>\mathfrak{R}</math> and various associated Euclidean three-space coordinate systems [''R'', ''R′'', ''etc.'']:<ref name=Lyle >{{cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= 9 |author1=Jean Salençon |author2=Stephen Lyle |url=https://books.google.com/books?id=H3xIED8ctfUC&q=physical+%22frame+of+reference%22&pg=PA9|isbn=3-540-41443-6 |year=2001 |publisher=Springer}}</ref>▼
▲Here is a quotation applicable to moving observational frames <math>\mathfrak{R}</math> and various associated Euclidean three-space coordinate systems [''R'', ''R′'', ''etc.'']:<ref name=Lyle >{{cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= 9 |author1=Jean Salençon |author2=Stephen Lyle |url=https://books.google.com/books?id=H3xIED8ctfUC&q=physical+%22frame+of+reference%22&pg=PA9|isbn=3-540-41443-6 |year=2001 |publisher=Springer}}</ref>
{{Cquote|We first introduce the notion of ''reference frame'', itself related to the idea of ''observer'': the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted <math>\mathfrak{R}</math>, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame <math>\mathfrak{R}</math> by establishing a ''coordinate system'' ''R'' with origin ''O''. The corresponding set of axes, sharing the rigid body motion of the frame <math>\mathfrak{R}</math>, can be considered to give a physical realization of <math>\mathfrak{R}</math>. In a frame <math>\mathfrak{R}</math>, coordinates are changed from ''R'' to ''R′'' by carrying out, at each instant of time, the same coordinate transformation on the components of ''intrinsic'' objects (vectors and tensors) introduced to represent physical quantities ''in this frame''.| Jean Salençon, Stephen Lyle ''Handbook of Continuum Mechanics: General Concepts, Thermoelasticity'' p. 9}}
and this on the utility of separating the notions of <math>\mathfrak{R}</math> and [''R'', ''R′'', ''etc.'']:<ref name= Lakhtakia>{{cite book |title=Essays on the Formal Aspects of Electromagnetic Theory |author=Patrick Cornille (Akhlesh Lakhtakia, editor) |page=149 |url=https://books.google.com/books?id=qsOBhKVM1qYC&q=coordinate+system+%22reference+frame%22&pg=PA149|isbn=981-02-0854-5 |year=1993 |publisher=World Scientific}}</ref>
{{Cquote|As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified.|L. Brillouin in ''Relativity Reexamined'' (as quoted by Patrick Cornille in '' Essays on the Formal Aspects of Electromagnetic Theory'' p. 149) }}
and this, also on the distinction between <math>\mathfrak{R}</math> and [''R'', ''R′'', ''etc.'']:<ref name= Nerlich>{{cite book |title=What Spacetime Explains: Metaphysical essays on space and time |author-last=Nerlich |author-first=Graham |author-link=Graham Nerlich |page=64 |url=https://books.google.com/books?id=fKK7rKOpc7AC&q=%22idea+of+a+reference+frame%22&pg=PA64|isbn=0-521-45261-9 |year=1994 |publisher=Cambridge University Press}}</ref>
{{Cquote|The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different ''spaces'' (sets of ''rest'' points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to ''classes'' of coordinate systems.|Graham Nerlich: ''What Spacetime Explains'', p. 64}}
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{{Cquote|In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers […] To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. […] Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime. […] Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.|John D. Norton: ''General Covariance and the Foundations of General Relativity: eight decades of dispute'', ''Rep. Prog. Phys.'', '''56''', pp. 835-7.}}
}}
The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani.<ref name=Brading>{{cite book |title=Symmetries in Physics: Philosophical Reflections |author1=Katherine Brading |author2=Elena Castellani |page=417 |url=https://books.google.com/books?id=SnmBN64cAdYC&q=%22idea+of+a+reference+frame%22&pg=PA417|isbn=0-521-82137-1 |year=2003 |publisher=Cambridge University Press}}</ref> Extension to coordinate systems using generalized coordinates underlies the [[Hamilton's principle|Hamiltonian]] and [[Lagrangian mechanics|Lagrangian]] formulations<ref name=Johns>{{cite book |title=Analytical Mechanics for Relativity and Quantum Mechanics |page=Chapter 16 |author=Oliver Davis Johns |url=https://books.google.com/books?id=PNuM9YDN8CIC&q=coordinate+observer&pg=PA318▼
|isbn=0-19-856726-X |year=2005 |publisher=Oxford University Press |no-pp=true }}</ref> of [[quantum field theory]], [[classical mechanics|classical relativistic mechanics]], and [[quantum gravity]].<ref name=Greenwood>{{cite book |title=Classical dynamics |author=Donald T Greenwood |page=313 |year=1997 |edition=Reprint of 1977 edition by Prentice-Hall |publisher=Courier Dover Publications |url=https://books.google.com/books?id=x7rj83I98yMC&q=%22relativistic+%22+Lagrangian+OR+Hamiltonian&pg=RA2-PA314▼
|isbn=0-486-69690-1 }}</ref><ref name=Trump>{{cite book |title=Classical Relativistic Many-Body Dynamics |author1=Matthew A. Trump |author2=W. C. Schieve |page= 99 |url=https://books.google.com/books?id=g2yfLOp0IzwC&q=relativity+%22generalized+coordinates%22&pg=PA99 ▼
|year=1999 |publisher=Springer |isbn= 0-7923-5737-X }}</ref><ref name=Kompaneyets>{{cite book |author=A S Kompaneyets |title=Theoretical Physics |url=https://books.google.com/books?id=CQ2gBrL5T4YC&q=relativity+%22generalized+coordinates%22&pg=PA118|page=118 |isbn=0-486-49532-9 |year=2003 |publisher=Courier Dover Publications |edition=Reprint of the 1962 2nd }}</ref><ref name=Srednicki>{{cite book |title=Quantum Field Theory |page= Chapter 4|author=M Srednicki |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-86449-7 |url=https://books.google.com/books?id=5OepxIG42B4C&pg=PA266 |no-pp=true }}</ref><ref name=Rovelli>{{cite book |title=Quantum Gravity |author=Carlo Rovelli |page= 98 ff |url=https://books.google.com/books?id=HrAzTmXdssQC&q=%22relativistic+%22+Lagrangian+OR+Hamiltonian&pg=PA179▼
|isbn=0-521-83733-2 |year=2004 |publisher=Cambridge University Press}}</ref>▼
== Coordinate systems ==
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}} A mathematical definition is: ''A connected [[Hausdorff space]] ''M'' is called an ''n''-dimensional manifold if each point of ''M'' is contained in an open set that is homeomorphic to an open set in Euclidean ''n''-dimensional space.''</ref><ref name=Morita>{{cite book |title=Geometry of Differential Forms |author1=Shigeyuki Morita |author2=Teruko Nagase |author3=Katsumi Nomizu |page=[https://archive.org/details/geometryofdiffer00mori/page/12 12] |url=https://archive.org/details/geometryofdiffer00mori
|url-access=registration |quote=geometry axiom coordinate system. |isbn=0-8218-1045-6 |year=2001 |publisher=American Mathematical Society Bookstore}}</ref> The [[Cartesian coordinate system|coordinates]] of a point '''r''' in an ''n''-dimensional space are simply an ordered set of ''n'' numbers:<ref name=Korn>{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author1=Granino Arthur Korn |author2=Theresa M. Korn |author2-link= Theresa M. Korn |page=169 |url=https://books.google.com/books?id=xHNd5zCXt-EC&q=curvilinear+%22coordinate+system%22&pg=PA169
|isbn=0-486-41147-8 |year=2000 |publisher=Courier Dover Publications}}</ref><ref name=encarta>See [http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html Encarta definition]. [https://
: <math>\mathbf{r} = [x^1,\ x^2,\ \dots,\ x^n].</math>
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: <math> x^j (x, y, z, \dots) = \mathrm{constant},\quad j = 1,\ \dots,\ n.</math>
The intersection of these surfaces define '''coordinate lines'''. At any selected point, tangents to the intersecting coordinate lines at that point define a set of '''basis vectors''' {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>,
|isbn=90-277-0540-2 |year=1974 |publisher=Springer}}</ref>
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As is apparent from these remarks, a coordinate system is a [[Model theory|mathematical construct]], part of an [[axiomatic system]]. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[Coordinate system#Transformations|coordinate transformation]]s.
== Observational frame of reference ==
[[File:Minkowski diagram - 3 systems.svg|thumb|right|256px|Three frames of reference in special relativity. The black frame is at rest. The primed frame moves at 40% of light speed, and the double primed frame at 80%. Note the scissors-like change as speed increases.]]
An '''observational frame of reference''', often referred to as a ''physical frame of reference'', a ''frame of reference'', or simply a ''frame'', is a physical concept related to an
|page=211 |isbn=0-8247-1749-X |year=1983 |publisher=CRC Press}}</ref>).
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In contrast to the inertial frame, a non-inertial frame of reference is one in which [[fictitious force]]s must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the [[Coriolis force]], [[centrifugal force]], and [[gravitational force]]. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)
== Measurement apparatus ==
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In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the [[laboratory frame]] or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the [[center of momentum frame]] "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles.
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect [[metrology]] that is connected to the nature of the [[vacuum]], and uses [[atomic clocks]] that operate according to the [[
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.<ref name=Rizzi>See {{cite book |title=Relativity in rotating frames |page=33 |url=https://books.google.com/books?id=_PGrlCLkkIgC&q=centrifugal+%22+%22+relativity+OR+relativistic&pg=PA226
|isbn=1-4020-1805-3 |year=2003 |publisher=Springer |author1=Guido Rizzi |author2=Matteo Luca Ruggiero }}.</ref>
==Generalization==
▲The discussion is taken beyond simple space-time coordinate systems by [[Katherine Brading|Brading]] and Castellani.<ref name=Brading>{{cite book |title=Symmetries in Physics: Philosophical Reflections |author1=Katherine Brading|author1-link=Katherine Brading |author2=Elena Castellani |page=417 |url=https://books.google.com/books?id=SnmBN64cAdYC&q=%22idea+of+a+reference+frame%22&pg=PA417|isbn=0-521-82137-1 |year=2003 |publisher=Cambridge University Press}}</ref> Extension to coordinate systems using generalized coordinates underlies the [[Hamilton's principle|Hamiltonian]] and [[Lagrangian mechanics|Lagrangian]] formulations<ref name=Johns>{{cite book |title=Analytical Mechanics for Relativity and Quantum Mechanics |page=Chapter 16 |author=Oliver Davis Johns |url=https://books.google.com/books?id=PNuM9YDN8CIC&q=coordinate+observer&pg=PA318
▲|isbn=0-19-856726-X |year=2005 |publisher=Oxford University Press |no-pp=true }}</ref> of [[quantum field theory]], [[classical mechanics|classical relativistic mechanics]], and [[quantum gravity]].<ref name=Greenwood>{{cite book |title=Classical dynamics |author=Donald T Greenwood |page=313 |year=1997 |edition=Reprint of 1977 edition by Prentice-Hall |publisher=Courier Dover Publications |url=https://books.google.com/books?id=x7rj83I98yMC&q=%22relativistic+%22+Lagrangian+OR+Hamiltonian&pg=RA2-PA314
▲|isbn=0-486-69690-1 }}</ref><ref name=Trump>{{cite book |title=Classical Relativistic Many-Body Dynamics |author1=Matthew A. Trump |author2=W. C. Schieve |page= 99 |url=https://books.google.com/books?id=g2yfLOp0IzwC&q=relativity+%22generalized+coordinates%22&pg=PA99
▲|year=1999 |publisher=Springer |isbn= 0-7923-5737-X }}</ref><ref name=Kompaneyets>{{cite book |author=
▲|isbn=0-521-83733-2 |year=2004 |publisher=Cambridge University Press}}</ref>
== Instances ==
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* [[Moving frame|Moving frame in mathematics]]
==
<div style="column-count: 2; column-width: 20em;">
* [[Analytical mechanics]]
* [[Applied mechanics]]
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* [[Curvilinear coordinates]]
* [[Datum reference]]
* [[Dynamics (physics)]]
* [[Frenet–Serret formulas]]
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* [[Principle of relativity]]
* [[Quantum reference frame]]
<div style="clear: both;"></div>
== Notes ==
{{notelist}}
== References ==
{{reflist}}
{{Relativity}}
{{Authority control}}
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