Frame of reference: Difference between revisions

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{{short description|Abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurement (s)}}

{{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''') consists ofis an abstract ''[[coordinate system]]'', whose [[origin (mathematics)|origin]], [[orientation (geometry)|orientation]], and [[scale (geometry)|scale]] arehave been specified byin [[physical space]]. It is based on a set of '''reference points''', defined ―as [[point (geometry)|geometric points]] whose [[position (geometry)|position]] is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).<ref name="Kovalevsky Mueller 1989 pp. 1–12">{{cite book | last1=Kovalevsky | first1=J. | last2=Mueller |author-link=Jean Kovalevsky |first2=Ivan I. |author-link2=Ivan I. Mueller |title=Reference Frames | chapter=Introduction | series=Astrophysics and Space Science Library | publisher=Springer Netherlands | publication-place=Dordrecht | year=1989 | volume=154 | issn=0067-0057 | doi=10.1007/978-94-009-0933-5_1 | pages=1–12| isbn=978-94-010-6909-0 }}</ref>

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 (Cartesian) coordinates]], a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the ''n'' coordinate [[Cartesian coordinate system|axes]].{{Citation needed|date=December 2020}}

In [[Theory of relativity|Einsteinian relativity]], reference frames are used to specify the relationship between a moving [[Observer (special relativity)|observer]] and the phenomenon or phenomena under observation. In this context, the phraseterm often becomes "'''observational frame of reference'''" (or "'''observational reference frame'''"), which implies that the observer is at rest in the frame, although not necessarily located at its [[origin (mathematics)|origin]]. A relativistic reference frame includes (or implies) the [[coordinate time]], which does not equate across different reference frames [[relative motion|moving relatively]] to each other. The situation thus differs from [[Galilean invariance|Galilean relativity]], wherein which all possible coordinate times are essentially equivalent.{{Citation needed|date=December 2020}}

== 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''ⁱ = ''x''ⁱ (''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'''₁, '''e'''₂,…... '''e'''_n}; see {{cite book |title=Linear Algebra: A Geometric Approach |author1=Edoardo Sernesi |author2=J. Montaldi |page=95 |url=https://books.google.com/books?id=1dZOuFo1QYMC&q=algebra+%22coordinate+system%22&pg=PA95|isbn=0-412-40680-2 |year=1993 |publisher=CRC Press}} As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.</ref> Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …...) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's ''observational'' frame of reference. This viewpoint can be found elsewhere as well.<ref name=Johansson>

{{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}}

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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://wwwweb.webcitationarchive.org/5kwcKb20f?url=web/20091030054251/http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html Archived] 2009-10-31.</ref>

: <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'''₁, '''e'''₂, …..., '''e'''_n} at that point. That is:<ref name=Papapetrou>{{cite book |title=Lectures on General Relativity |author=Achilleus Papapetrou |page=5 |url=https://books.google.com/books?id=SWeOggyp1ZsC&q=relativistic++%22general+coordinates%22&pg=PA3

|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.

{{main|Inertial== Observational frame of reference}} ==

General and specific topics of coordinate systems can be pursued following the [[#See also|See also]] links below.

{{distinguishfurther|Inertial frame of reference}}

== Physics ==

{{main|Inertial 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 [[Observer (special relativity)|observer]] and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized ''only by its state of motion''.<ref name=Kubar>See {{cite book |author1=Arvind Kumar |author2=Shrish Barve |page=115 |title=How and Why in Basic Mechanics |url=https://books.google.com/books?id=czlUPz38MOQC&q=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar&pg=PA115|isbn=81-7371-420-7 |year= 2003 |publisher =Orient Longman}}</ref> However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an ''observer'' and a ''frame''. According to this view, a ''frame'' is an ''observer'' plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.<ref name=Doran>{{cite book |url=http://www.worldcat.org/search?q=9780521715959&qt=owc_search |title=Geometric Algebra for Physicists |author1=Chris Doran |author2=Anthony Lasenby |page= §5.2.2, p. 133 |isbn=978-0-521-71595-9 |year=2003 |publisher=Cambridge University Press}}.</ref> This restricted view is not used here, and is not universally adopted even in discussions of relativity.<ref name=Moller>For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates ''x''ⁱ in four-space…." {{cite book |author=C. Møller |title=The Theory of Relativity |page=222 and p. 233 |year=1952 |publisher=Oxford University Press}}</ref><ref name=Lightman>{{cite book |title=Problem Book in Relativity and Gravitation |author1=A. P. Lightman |author2=W. H. Press |author3=R. H. Price |author4=S. A. Teukolsky |page=[https://archive.org/details/problembookinrel00ligh/page/15 15] |url=https://archive.org/details/problembookinrel00ligh|url-access=registration |quote=relativistic general coordinates. |isbn=0-691-08162-X |publisher=Princeton University Press |year=1975}}</ref> In [[general relativity]] the use of general coordinate systems is common (see, for example, the [[Karl Schwarzschild|Schwarzschild]] solution for the gravitational field outside an isolated sphere<ref name= Faber>{{cite book |title=Differential Geometry and Relativity Theory: an introduction |author=Richard L Faber |url=https://books.google.com/books?id=ctM3_afLuVEC&q=relativistic++%22general+coordinates%22&pg=PA149

|page=211 |isbn=0-8247-1749-X |year=1983 |publisher=CRC Press}}</ref>).

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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 [[standardStandard modelModel]] and that must be corrected for [[gravitational time dilation]].<ref name= Wolfson>{{cite book |author= Richard Wolfson |title=Simply Einstein |url=https://books.google.com/books?id=OUJWKdlFKeQC&q=%22gravitational+time+dilation+%22&pg=PA216|page=216 |isbn=0-393-05154-4 |publisher=W W Norton & Co. |year=2003}}</ref> (See [[second]], [[meter]] and [[kilogram]]).

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=AAlexander SSolomonovich 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>

== Instances ==

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* [[Moving frame|Moving frame in mathematics]]

== See also ==

<div style="column-count: 2; column-width: 20em;">

{{div col}}

* [[Analytical mechanics]]

* [[Applied mechanics]]

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* [[Principle of relativity]]

* [[Quantum reference frame]]

{{</div end}}>

<div style="clear: both;"></div>

== Notes ==

{{notelist}}

== References ==

{{reflist}}

{{Relativity}}

{{Authority control}}