Frame of reference: Difference between revisions

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{{short description|Abstract coordinate system}}

{{distinguish|Inertial frame of reference}}

{{Other uses|Frame of reference (disambiguation)}}

{{Classical mechanics |FundamebntalsFundamentals}}

{{Merge from|Observer (special relativity)|discuss=Talk:Frame of reference#Proposed merge of Observer (special relativity) into Frame of reference|date=November 2022}}

{{Classical mechanics |Fundamebntals}}

In [[physics]] and [[astronomy]], a '''frame of reference''' (or '''reference frame''') is 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 [[AxisCartesian (mathematics)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 under observation. In this context, the term 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]], in which all possible coordinate times are essentially equivalent.{{Citation needed|date=December 2020}}

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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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: <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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[[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

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==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

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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]]

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{{Relativity}}

{{Authority control}}

== Notes ==

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== References ==

{{reflist}}

{{Relativity}}

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[[Category:Frames of reference|*]]