View source for Frame of reference (physics)

==Different aspects==

The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in ''Cartesian frame of reference''. Sometimes the state of motion is emphasized, as in ''[[Rotating reference frame|rotating frame of reference]]''. Sometimes the way a frame is related to a family of frames is emphasized, as in ''Galilean frame of reference''. Sometimes frames are distinguished by the scale of their observations, as in ''macroscopic'' and ''microscopic frames of reference''.<ref name=macroscopic>The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where [[Constitutive equation|constitutive relations]] of various time and length scales are used to determine the current and charge densities entering [[Maxwell's equations]]. See, for example, {{cite book |title=Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media |author=Kurt Edmund Oughstun |pages=p. 165 |url=http://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism&lr=&as_brr=0&sig=ACfU3U3J6a2ZwvTOx7T3S6Zunptf9E9nxQ
|isbn=038734599X |year=2006 |publisher=Springer }}. These distinctions also appear in thermodynamics. See {{cite book |title=Classical Theory |author=Paul McEvoy |pages=p. 205 |url=http://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame&lr=&as_brr=0&sig=ACfU3U2JFNgXVpz6Ew6hmp2rmdL6p9O7Ng#PPA205,M1
|isbn=1930832028 |year=2002 |publisher=MicroAnalytix  }}.</ref>

In this article the term ''observational frame of reference'' is used when emphasis is upon the ''state of motion'' rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. In contrast, a ''coordinate system'' may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs ''[[generalized coordinates]]'', ''[[normal modes]]'' or ''[[eigenvectors]]'', which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below: 

*An observational frame (such as an [[Inertial frame of reference|inertial frame]] or [[non-inertial frame of reference]]) is a physical concept related to state of motion.

*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 |pages= p. 429 |year=  |url=http://books.google.com/books?id=JU0DT_wXu2oC&pg=PA429&dq=algebra+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U07j7JfzfwMigYTa2iDVygAb0WKCA
|isbn=2881241336 |publisher=Gordon and Breach|edition=3rd ed }}. 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>,… '''e'''<sub>n</sub>}; see {{cite book |title=Linear Algebra: A Geometric Approach |author= Edoardo Sernesi, J. Montaldi |pages=p. 95 |url=http://books.google.com/books?id=1dZOuFo1QYMC&pg=PA95&dq=algebra+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U1TD6WmY73w4hEYlVFcXK5NxtKSDQ
|isbn=0412406802 |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. Some coordinate systems may be a better choice for some observations than are others. 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.<ref name=Johansson>

This viewpoint can be found elsewhere as well. See, for example,{{cite book |title=Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces |author=J X Zheng-Johansson and Per-Ivar Johansson 
|pages=p. 13 
|url=http://books.google.com/books?id=I1FU37uru6QC&pg=PA13&dq=frame+coordinate+johansson&lr=&as_brr=0&sig=ACfU3U1VAkGbfRjt_GTknoX6WRLWP-AVZw
|isbn=1594542600 
|publisher=Nova Publishers 
|year=2006  }}

</ref> 

*Choice of what to measure and with what observational apparatus is a matter logically separate from the observer's state of motion and choice of coordinate system. The observational apparatus is not necessarily localized to a single observer, but may involve an ''observer team'' relaying observations to a central data collector.<ref name=Johns1>

{{cite book |title=Analytical Mechanics for Relativity and Quantum Mechanics |pages=pp. 318 'ff'' |chapter=§14.4: What is a coordinate system? |author=Oliver Davis Johns |url=http://www.amazon.com/Analytical-Mechanics-Relativity-Quantum-Graduate/dp/019856726X/ref=sr_1_1?s=books&ie=UTF8&qid=1300713166&sr=1-1#reader_019856726X
|isbn=019856726X |year=2005 |publisher=Oxford University Press |edition=2nd ed }}

</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 |pages= p. 9 |author=Jean Salençon, Stephen Lyle |url=http://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22&lr=&as_brr=0&sig=ACfU3U1tEWQICZdsXeuLyfmH2PoLgZnMGA
|isbn=3540414436 |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''}}
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 |editor=Akhlesh Lakhtakia., ed |pages=p. 149 |url=http://books.google.com/books?id=qsOBhKVM1qYC&pg=PA149&dq=coordinate+system+%22reference+frame%22&lr=&as_brr=0&sig=ACfU3U0xhpZ2lI99UPiYQCOL6oJ0ALO5uA
|isbn=9810208545 |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'' (quoted by Patrick Cornille) }}
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=Graham Nerlich |pages=p. 64 |url=http://books.google.com/books?id=fKK7rKOpc7AC&pg=PA64&dq=%22idea+of+a+reference+frame%22&lr=&as_brr=0&sig=ACfU3U2wsO42pqLOJ453eeIzk7ztXTa6uQ
|isbn=0521452619 |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''}}

and from J. D. Norton:<ref name=Norton>

{{cite journal |journal=Rep. Prog. Phys.|author=John D. Norton |year=1993 |volume=vol. 56 |pages=pp. 835-837 |title=General covariance and the foundations of general relativity: eight decades of dispute |url=http://www.pitt.edu/~jdnorton/papers/decades.pdf}}

</ref>

{{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}}

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 |author=Katherine Brading & Elena Castellani |pages=p. 417 |url=http://books.google.com/books?id=SnmBN64cAdYC&pg=PA417&dq=%22idea+of+a+reference+frame%22&lr=&as_brr=0&sig=ACfU3U1PdXJdmFyMRiDb7xPDAI_dy9MgJg |isbn=0521821371 |year=2003 |publisher=Cambridge University Press}}

</ref> Extension to coordinate systems using generalized coordinates underlies the [[Hamilton's principle|Hamiltonian]] and [[Lagrangian]] formulations of [[quantum field theory]], [[classical mechanics|classical relativistic mechanics]], and [[quantum gravity]].<ref name=generalized>

The application of coordinates in these areas can be found, for example, in: {{cite book |title=Classical dynamics |author=Donald T Greenwood |pages=p. 313 |year=1997 |edition=Reprint of 1977 Prentice-Hall ed |publisher=Courier Dover Publications |url=http://books.google.com/books?id=x7rj83I98yMC&pg=PA313 |isbn=0486696901}}, 
{{cite book |title=Classical Relativistic Many-Body Dynamics |author=Matthew A. Trump & W. C. Schieve |pages= pp. 99 ''ff'' |chapter=§4.2.1 The generalized coordinates and velocities |url=http://books.google.com/books?id=g2yfLOp0IzwC&pg=PA99&dq=relativity+%22generalized+coordinates%22&lr=&as_brr=0&sig=ACfU3U230ux_i1Ov3QHf_xy1dAA_oLSKpw#PPA99,M1 
|year=1999 |publisher=Springer |isbn= 079235737X }}, {{cite book |author=A S Kompaneyets |title=Theoretical Physics |url=http://books.google.com/books?id=CQ2gBrL5T4YC&pg=PA118&dq=relativity+%22generalized+coordinates%22&lr=&as_brr=0&sig=ACfU3U3OOAmAAh2wV46vG0gUN0wSIPiIww
|pages=p. 118 |isbn=0486495329 |year=2003 |publisher=Courier Dover Publications |edition=Reprint of 2nd 1962 ed }}, {{cite book |title=How is quantum field theory possible? |chapter= Chapter 9: Section ''Lagrangian Field Theory'' |author=Sunny Y. Auyang |publisher=Oxford University Press |year=1995 |isbn=0195093453 |url=http://books.google.com/books?id=rcrf0P8DSe0C&pg=PA49 |pages=pp. 49 ''ff'' }},
{{cite book |title=Quantum Gravity |author=Carlo Rovelli |pages= pp. 98 ''ff'' |url=http://books.google.com/books?id=HrAzTmXdssQC&pg=PA179&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian&lr=&as_brr=0&sig=ACfU3U3TLyr3CXsHYKFUGDe1dpq5ZWm_kg#PPA98,M1
|isbn=0521837332 |year=2004 |publisher=Cambridge University Press}}</ref>

===Observational frames of reference===

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 |author=Arvind Kumar & Shrish Barve |pages=p. 115 |title=How and Why in Basic Mechanics |url=http://books.google.com/books?id=czlUPz38MOQC&pg=PA115&dq=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar&lr=&as_brr=0&sig=ACfU3U36HY3RerJYLWRlfJaGxRw7EqzIeA
|isbn=8173714207 |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 and Lasenby.<ref name=Doran>{{cite book |url=http://books.google.com/books?id=VW4yt0WHdjoC&pg=PA133 |title=Geometric Algebra for Physicists |author= Chris Doran & Anthony Lasenby |pages= p. 133 ''ff'' |chapter=§5.2.2: Spacetime frames |isbn=0521480221 |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''<sup>i</sup> in four-space…." {{cite book |author=C. Møller |title=The Theory of Relativity |pages=p. 222 and p. 233 |year=1952 |publisher=Oxford University Press}}</ref><ref name=Lightman>{{cite book |title=Problem Book in Relativity and Gravitation  |author=Alan P. Lightman, R. H. Price & William H. Press |pages=p. 15  |url=http://books.google.com/books?id=YtxGYnnP1PEC&pg=PA15&dq=relativistic++%22general+coordinates%22&lr=&as_brr=0&sig=ACfU3U28SuvzlWcCJ7PXCY71TVn33GeBvA
|isbn=069108162X |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=http://books.google.com/books?id=2XxDatZt-RQC&pg=PA211
|pages=p. 211 ''ff'' |chapter=Chapter III, §8: The Schwarzchild solution|isbn=082471749X |year=1983 |publisher=CRC Press  }}</ref>).
 
There are two types of observational reference frame: [[Inertial frame of reference|inertial]] and [[non-inertial reference frame|non-inertial]].
 
An ''inertial frame of reference'' is defined as one in which all laws of physics take on their simplest form. In [[special relativity]] these frames are related by [[Lorentz transformation]]s. In Newtonian mechanics, a more restricted definition requires only that [[Classical_mechanics#Newton.27s_laws_of_motion|Newton's first law]] holds true; that is, a Newtonian inertial frame is one in which a [[free particle]] travels in a [[straight line]] at constant [[speed]], or is at rest. These frames are related by [[Galilean transformation]]s. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of [[Representation theory|representations]] of the [[Representation theory of the Poincaré group|Poincaré group]] and of the [[Representation theory of the Galilean group|Galilean group]].

In contrast to the inertial frame, a ''non-inertial frame of reference'' is one in which [[inertial 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 Earth's axis, which introduces an inertial force known as the [[Coriolis force]] (among others).

===Coordinate systems===

Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well. A coordinate ''system'' in mathematics is a facet of [[geometry]] or of [[algebra]], in particular, a property of [[Manifold (geometry)|manifold]]s (for example, in physics, [[configuration space]]s or [[phase space]]s).<ref name=Hawking>

According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." {{cite book |title=The Large Scale Structure of Space-Time |author=Stephen W. Hawking & George Francis Rayner Ellis |isbn=0521099064 |year=1973 |publisher=Cambridge University Press |pages=p. 11 |url=http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&lr=&as_brr=0&sig=ACfU3U1q-iaRTBDo6J8HMEsyPeFi8cJNWg#PPA11,M1
}} 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 |author=Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu |pages=p. 12 |url=http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1
|isbn=0821810456 |year=2001 |publisher=American Mathematical Society Bookstore  }}

</ref> ''Coordinates'', coordinate ''surfaces'', coordinate ''lines'', and [[Basis (linear algebra)|basis vectors]] are components of a ''coordinate system''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |pages=p. 84  |isbn=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig  }}</ref> If the basis vectors are mutually perpendicular at every point, the coordinate system is an [[Orthogonal coordinates|orthogonal coordinate system]]. These components are discussed further below.

The ''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 |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1
|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}}

</ref>
:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ ,  x^n] \ .</math>

In a general [[Banach space]], these numbers could be (for example) coefficients in a functional expansion like a [[Fourier series]]. In a physical problem, they could be [[spacetime]] coordinates or [[normal mode]] amplitudes. In a [[Robotics|robot design]], they could be angles of relative rotations, linear displacements, or deformations of [[linkage (mechanical)|joints]].<ref name=Yamane>

{{cite book |author=Katsu Yamane |title=Simulating and Generating Motions of Human Figures |isbn=3540203176 |year=2004 |publisher=Springer  |pages=pp. 12–13 |url=http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22&lr=&as_brr=0&sig=ACfU3U3LRGJJTAHs21CHdOvuu08vw0cAuw#PPA13,M1   }}

</ref> 

To introduce ''coordinate surfaces'', for simplicity, let us restrict consideration to [[Manifold_(geometry)#Differentiable_manifold|differentiable manifolds]] based upon real numbers.<ref name=Morrow>

A technical discussion of the case using complex numbers can found, for example, in {{cite book |title=Encyclopedia of Distances |chapter=§7.3 Hermitian metrics and generalizations|author=Michel M. Deza, Elena Deza |url=http://books.google.com/books?id=LXEezzccwcoC&pg=PA148 |pages=p. 148 ''ff'' |isbn=3642002331 |year=2009 |publisher=Springer}}

</ref> Suppose the coordinates can be related to a [[Cartesian coordinate]] system by a set of functions:
:<math>x^j = x^j (x,\  y,\  z,\  \dots)\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  </math>

where ''x'', ''y'', ''z'', ''etc.'' are the ''n'' Cartesian coordinates of the point. Given these functions,  coordinate surfaces are defined by the relations:

:<math> x^j (x, y, z, \dots) = \mathrm{constant}\ , </math>&ensp; &ensp; <math> j = 1, \ \dots \ , \ n\  .</math>

The ''coordinate lines'' are the intersections of the coordinate surfaces. 

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>, …, '''e'''<sub>n</sub>} at that point. That is:

:<math>\mathbf{e}_i(\mathbf{r}) =\lim_{\epsilon \rightarrow 0} \frac{\mathbf{r}\left(x^1,\  \dots,\  x^i+\epsilon,\  \dots ,\  x^n \right) - \mathbf{r}\left(x^1,\  \dots,\  x^i,\  \dots ,\  x^n \right)}{\epsilon }\ ,</math>

which can be normalized to be of unit length. For more background see [[tangent space]].

An important aspect of a coordinate system is its [[metric tensor]] ''g''<sub>ik</sub>, which determines the [[arc length]] ''ds'' in the coordinate system in terms of its coordinates:<ref name=Borisenko>{{cite book |title=Vector and Tensor Analysis with Applications |author= A. I. Borisenko, I. E. Tarapov, Richard A. Silverman |pages=p. 86 |url=http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q
|isbn=0486638332 |publisher=Courier Dover Publications |year=1979 |pages=pp. 86 ''ff'' |chapter=§2.8.4 Arc length. Metric coefficients |edition=Reprint of Prentice-Hall 1968 ed   }}</ref>

:<math>(ds)^2 = g_{ik}\ dx^i\ dx^k \ , </math>

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a 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 be used to describe motion by interpreting one coordinate as time. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[coordinate transformation]]s.<ref name=Johns2>

An essay about using time as a coordinate is found in the preface to {{cite book |title=Analytical Mechanics for Relativity and Quantum Mechanics |url=http://www.amazon.com/Analytical-Mechanics-Relativity-Quantum-Graduate/dp/019856726X/ref=sr_1_1?s=books&ie=UTF8&qid=1300713166&sr=1-1#reader_019856726X |chapter=Preface |pages=pp. ''vii ff'' |isbn=019856726X |publisher=Oxford University Press |year=2005 |editions=2nd ed |author=Oliver Davis Johns }}

</ref>

Lest one underestimate the subtlety of coordinate systems, [[Ehrenfest's paradox]] may be mentioned. The paradox applies to a rigid rotating circular cylinder as examined in special relativity. Each element of the perimeter is aligned with its direction of motion due to the rotation and so suffers [[Lorentz-FitzGerald_contraction/Definition|Lorentz contraction]], but the radius to each element on the perimeter from the center of rotation is perpendicular to its motion and so does not contract. Thus, the circumference as measured around the perimeter differs from the calculated perimeter 2&pi;''R'' using the radius ''R'' measured from the axis of rotation, and Euclidean geometry apparently does not apply. This paradox has been discussed for decades, even quite recently. Proposed resolutions of the paradox involve the synchronizing of clocks among non-inertial frames of reference located at the perimeter.<ref name=Ehrenfest> See:

{{cite journal |title=What is wrong with relativistic kinematics? |author=V Cantoni |journal=Il Nuovo Cimento |volume=vol LVII B |issue= N. 1 |date= 1968-09-11 |pages=pp. 220 ''ff'' |url=http://www.springerlink.com/content/m03412p042301379/ |doi=10.1007/BF02710332}}, {{cite journal |title=Space geometry of rotating platforms: an operational approach |author=Guido Rizzi, Matteo Luca Ruggiero |url=http://arxiv.org/abs/gr-qc/0207104 |journal=Foundations of physics |volume=vol. 32 |issue=No. 10 |pages=1525-1556 |doi=10.1023/A:1020427318877 |year=2002}}, {{cite journal |title=Does anything happen on a rotating disk? |author=A Tartaglia |journal=Fundamental theories of physics |volume=vol. 135 |pages=pp. 261-274 |year=2003 |url=http://areeweb.polito.it/ricerca/relgrav/solciclos/tartaglia_d.pdf}}


</ref>

===Measurement apparatus===
A further aspect of a frame of reference is the role of the [[metrology|measurement apparatus]] (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in [[quantum mechanics]], where the relation between observer and measurement is still under discussion (see [[Measurement in quantum mechanics|measurement problem]]).

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 (quantum electrodynamic)|vacuum]], and uses [[atomic clocks]] that operate according to the [[standard model]] and that must be corrected for [[gravitational time dilation]].<ref name= Wolfson>{{cite book |author=  Richard Wolfson |title=Simply Einstein |url=http://books.google.com/books?id=OUJWKdlFKeQC&pg=PA216&dq=%22gravitational+time+dilation+%22&lr=&as_brr=0&sig=ACfU3U0_wc8IuNJdGCLnsaO-SyqXYaRapw
|pages=pp. 216 ''ff'' |chapter=Warping time |isbn=0393051544 |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=Rizzi0>See {{cite book |title=Relativity in rotating frames |pages=pp. 33 ''ff'' |url=http://books.google.com/books?id=_PGrlCLkkIgC&pg=PA33 |chapter=§4. Space and time without rods and clocks
|isbn=1402018053 |year=2003 |publisher=Springer |author=Guido Rizzi, Matteo Luca Ruggiero   }}.</ref>