Frame of reference (physics): Difference between revisions

Revision as of 08:57, 23 March 2011

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 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.^[1]

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 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.^[2] 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.^[3]

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.^[4]

Here is a quotation applicable to moving observational frames ${\mathfrak {R}}$ and various associated Euclidean three-space coordinate systems [R, R' , etc.]: ^[5]

“	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 ${\mathfrak {R}}$ , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame ${\mathfrak {R}}$ by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame ${\mathfrak {R}}$ , can be considered to give a physical realization of ${\mathfrak {R}}$ . In a frame ${\mathfrak {R}}$ , 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 ${\mathfrak {R}}$ and [R, R' , etc.]:^[6]

“	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 ${\mathfrak {R}}$ and [R, R' , etc.]:^[7]

“	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:^[8]

“	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.^[9] Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity.^[10]

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 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.^[11] 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.^[12] This restricted view is not used here, and is not universally adopted even in discussions of relativity.^[13]^[14] In general relativity the use of general coordinate systems is common (see, for example, the Schwarzschild solution for the gravitational field outside an isolated sphere^[15]).

There are two types of observational reference frame: inertial and 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 transformations. In Newtonian mechanics, a more restricted definition requires only that 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 transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.

In contrast to the inertial frame, a non-inertial frame of reference is one in which inertial forces 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 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. Coordinates, coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.^[16] If the basis vectors are mutually perpendicular at every point, the coordinate system is an orthogonal coordinate system. These components are discussed further below.

A coordinate system in mathematics is a facet of geometry or of algebra, in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).^[17]^[18] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:^[19]

\mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .

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 robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.^[20] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

x^{j}=x^{j}(x,\ y,\ z,\ \dots )\ ,

j=1,\ \dots \ ,\ n\

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

x^{j}(x,y,z,\dots )=\mathrm {constant} \ ,

j=1,\ \dots \ ,\ n\ .

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 {e₁, e₂, …, e_n} at that point. That is:

\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 }}\ ,

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

An important aspect of a coordinate system is its metric tensor g_ik, which determines the arc length ds in the coordinate system in terms of its coordinates:^[21]

(ds)^{2}=g_{ik}\ dx^{i}\ dx^{k}\ ,

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 transformations and Galilean transformations may be viewed as coordinate transformations.^[22]

Measurement apparatus

A further aspect of a frame of reference is the role of the 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 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, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation.^[23] (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.^[24]

Notes

↑ The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer, p. 165. ISBN 038734599X. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix, p. 205. ISBN 1930832028. .
↑ In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin. L.S. Pontryagin: Selected Works Vol. 2: Topological Groups, 3rd ed. Gordon and Breach, p. 429. ISBN 2881241336. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi, J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press, p. 95. ISBN 0412406802. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
↑ This viewpoint can be found elsewhere as well. See, for example,J X Zheng-Johansson and Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers, p. 13. ISBN 1594542600.
↑ Oliver Davis Johns (2005). “§14.4: What is a coordinate system?”, Analytical Mechanics for Relativity and Quantum Mechanics, 2nd ed. Oxford University Press, pp. 318 'ff. ISBN 019856726X.
↑ Jean Salençon, Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer, p. 9. ISBN 3540414436.
↑ Patrick Cornille (1993). Akhlesh Lakhtakia., ed: Essays on the Formal Aspects of Electromagnetic Theory. World Scientific, p. 149. ISBN 9810208545.
↑ Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press, p. 64. ISBN 0521452619.
↑ John D. Norton (1993). "General covariance and the foundations of general relativity: eight decades of dispute". Rep. Prog. Phys. vol. 56: pp. 835-837.
↑ Katherine Brading & Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press, p. 417. ISBN 0521821371.
↑ The application of coordinates in these areas can be found, for example, in: Donald T Greenwood (1997). Classical dynamics, Reprint of 1977 Prentice-Hall ed. Courier Dover Publications, p. 313. ISBN 0486696901. , Matthew A. Trump & W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer, p. 99. ISBN 079235737X. , A S Kompaneyets (2003). Theoretical Physics, Reprint of 2nd 1962 ed. Courier Dover Publications, p. 118. ISBN 0486495329. , M Srednicki (2007). Quantum Field Theory. Cambridge University Press, Chapter 4. ISBN 978-0-521-86449-7. , Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press, pp. 98 ff. ISBN 0521837332.
↑ See Arvind Kumar & Shrish Barve (2003). How and Why in Basic Mechanics. Orient Longman. ISBN 8173714207.
↑ Chris Doran & Anthony Lasenby (2003). “§5.2.2: Spacetime frames”, Geometric Algebra for Physicists. Cambridge University Press, p. 133 ff. ISBN 0521480221. .
↑ 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 xⁱ in four-space…." C. Møller (1952). The Theory of Relativity. Oxford University Press.
↑ Alan P. Lightman, R. H. Price & William H. Press (1975). Problem Book in Relativity and Gravitation. Princeton University Press. ISBN 069108162X.
↑ Richard L Faber (1983). “Chapter III, §8: The Schwarzchild solution”, Differential Geometry and Relativity Theory: an introduction. CRC Press. ISBN 082471749X.
↑ Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X.
↑ 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." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064. 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.
↑ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.
↑ Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478.
↑ Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176.
↑ A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332.
↑ An essay about using time as a coordinate is found in the preface to Oliver Davis Johns (2005). “Preface”, Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press, pp. vii ff. ISBN 019856726X.
↑ Richard Wolfson (2003). Simply Einstein. W W Norton & Co.. ISBN 0393051544.
↑ See Guido Rizzi, Matteo Luca Ruggiero (2003). Relativity in rotating frames. Springer. ISBN 1402018053. .

[macroscopic-1] The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer, p. 165. ISBN 038734599X. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix, p. 205. ISBN 1930832028. .

[Pontriagin-2] In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin. L.S. Pontryagin: Selected Works Vol. 2: Topological Groups, 3rd ed. Gordon and Breach, p. 429. ISBN 2881241336. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi, J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press, p. 95. ISBN 0412406802. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.

[Johansson-3] This viewpoint can be found elsewhere as well. See, for example,J X Zheng-Johansson and Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers, p. 13. ISBN 1594542600.

[Johns1-4] Oliver Davis Johns (2005). “§14.4: What is a coordinate system?”, Analytical Mechanics for Relativity and Quantum Mechanics, 2nd ed. Oxford University Press, pp. 318 'ff. ISBN 019856726X.

[Lyle-5] Jean Salençon, Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer, p. 9. ISBN 3540414436.

[Lakhtakia-6] Patrick Cornille (1993). Akhlesh Lakhtakia., ed: Essays on the Formal Aspects of Electromagnetic Theory. World Scientific, p. 149. ISBN 9810208545.

[Nerlich-7] Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press, p. 64. ISBN 0521452619.

[Norton-8] John D. Norton (1993). "General covariance and the foundations of general relativity: eight decades of dispute". Rep. Prog. Phys. vol. 56: pp. 835-837.

[Brading-9] Katherine Brading & Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press, p. 417. ISBN 0521821371.

[generalized-10] The application of coordinates in these areas can be found, for example, in: Donald T Greenwood (1997). Classical dynamics, Reprint of 1977 Prentice-Hall ed. Courier Dover Publications, p. 313. ISBN 0486696901. , Matthew A. Trump & W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer, p. 99. ISBN 079235737X. , A S Kompaneyets (2003). Theoretical Physics, Reprint of 2nd 1962 ed. Courier Dover Publications, p. 118. ISBN 0486495329. , M Srednicki (2007). Quantum Field Theory. Cambridge University Press, Chapter 4. ISBN 978-0-521-86449-7. , Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press, pp. 98 ff. ISBN 0521837332.

[Kubar-11] See Arvind Kumar & Shrish Barve (2003). How and Why in Basic Mechanics. Orient Longman. ISBN 8173714207.

[Doran-12] Chris Doran & Anthony Lasenby (2003). “§5.2.2: Spacetime frames”, Geometric Algebra for Physicists. Cambridge University Press, p. 133 ff. ISBN 0521480221. .

[Moller-13] 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 xⁱ in four-space…." C. Møller (1952). The Theory of Relativity. Oxford University Press.

[Lightman-14] Alan P. Lightman, R. H. Price & William H. Press (1975). Problem Book in Relativity and Gravitation. Princeton University Press. ISBN 069108162X.

[Faber-15] Richard L Faber (1983). “Chapter III, §8: The Schwarzchild solution”, Differential Geometry and Relativity Theory: an introduction. CRC Press. ISBN 082471749X.

[Zdunkowski-16] Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X.

[Hawking-17] 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." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064. 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.

[Morita-18] Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.

[Korn-19] Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478.

[Yamane-20] Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176.

[Borisenko-21] A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332.

[Johns2-22] An essay about using time as a coordinate is found in the preface to Oliver Davis Johns (2005). “Preface”, Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press, pp. vii ff. ISBN 019856726X.

[Wolfson-23] Richard Wolfson (2003). Simply Einstein. W W Norton & Co.. ISBN 0393051544.

[Rizzi-24] See Guido Rizzi, Matteo Luca Ruggiero (2003). Relativity in rotating frames. Springer. ISBN 1402018053. .

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@@ Line 88: / Line 88: @@
 ===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.
+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. ''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 |page=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.
-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>
+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
@@ Line 115: / Line 115: @@
 :<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:
+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 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>, …, '''e'''<sub>n</sub>} at that point. That is:
+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>, …, '''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>
@@ Line 125: / Line 125: @@
 which can be normalized to be of unit length. For more detail see [[curvilinear 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 |page=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 orthogonal at every point, the coordinate system is an [[Orthogonal coordinates|orthogonal coordinate system]].
+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 |page=86 |url=http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q
-An important aspect of a coordinate system is its [[Metric (mathematics)|metric]] ''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 |page=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>

Frame of reference (physics): Difference between revisions

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