Frame of Reference

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The content is sourced from: https://handwiki.org/wiki/Astronomy:Frame_of_reference

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame. For n dimensions, n + 1 reference points are sufficient to fully define a reference frame. Using 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 axes. In Einsteinian relativity, reference frames are used to specify the relationship between a moving observer and the phenomenon or phenomena under observation. In this context, the phrase 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. A relativistic reference frame includes (or implies) the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent.

galilean frame of reference reference frame

References

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 0-387-34599-X. https://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix. p. 205. ISBN 1-930832-02-8. https://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame#PPA205,M1. .
In very general terms, a coordinate system is a set of arcs xi = xi (t) in a complex Lie group; see Lev Semenovich Pontri͡agin (1986). L.S. Pontryagin: Selected Works Vol. 2: Topological Groups (3rd ed.). Gordon and Breach. p. 429. ISBN 2-88124-133-6. https://books.google.com/books?id=JU0DT_wXu2oC&pg=PA429&dq=algebra+%22coordinate+system%22. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e1, e2,… en}; see Edoardo Sernesi; J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press. p. 95. ISBN 0-412-40680-2. https://books.google.com/books?id=1dZOuFo1QYMC&pg=PA95&dq=algebra+%22coordinate+system%22. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
J X Zheng-Johansson; Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers. p. 13. ISBN 1-59454-260-0. https://books.google.com/books?id=I1FU37uru6QC&pg=PA13&dq=frame+coordinate+johansson.
Jean Salençon; Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer. p. 9. ISBN 3-540-41443-6. https://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22.
Patrick Cornille (Akhlesh Lakhtakia, editor) (1993). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific. p. 149. ISBN 981-02-0854-5. https://books.google.com/books?id=qsOBhKVM1qYC&pg=PA149&dq=coordinate+system+%22reference+frame%22.
Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press. p. 64. ISBN 0-521-45261-9. https://books.google.com/books?id=fKK7rKOpc7AC&pg=PA64&dq=%22idea+of+a+reference+frame%22.
John D. Norton (1993). General covariance and the foundations of general relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7. http://www.pitt.edu/~jdnorton/papers/decades.pdf
Katherine Brading; Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press. p. 417. ISBN 0-521-82137-1. https://books.google.com/books?id=SnmBN64cAdYC&pg=PA417&dq=%22idea+of+a+reference+frame%22.
Oliver Davis Johns (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. Chapter 16. ISBN 0-19-856726-X. https://books.google.com/books?id=PNuM9YDN8CIC&pg=PA318&dq=coordinate+observer#PPA276,M1.
Donald T Greenwood (1997). Classical dynamics (Reprint of 1977 edition by Prentice-Hall ed.). Courier Dover Publications. p. 313. ISBN 0-486-69690-1. https://books.google.com/books?id=x7rj83I98yMC&pg=RA2-PA314&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian#PRA2-PA313,M1.
Matthew A. Trump; W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer. p. 99. ISBN 0-7923-5737-X. https://books.google.com/books?id=g2yfLOp0IzwC&pg=PA99&dq=relativity+%22generalized+coordinates%22#PPA99,M1.
A S Kompaneyets (2003). Theoretical Physics (Reprint of the 1962 2nd ed.). Courier Dover Publications. p. 118. ISBN 0-486-49532-9. https://books.google.com/books?id=CQ2gBrL5T4YC&pg=PA118&dq=relativity+%22generalized+coordinates%22.
M Srednicki (2007). Quantum Field Theory. Cambridge University Press. Chapter 4. ISBN 978-0-521-86449-7. https://books.google.com/books?id=5OepxIG42B4C&pg=PA266&dq=isbn=9780521864497#PPA31,M1.
Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press. p. 98 ff. ISBN 0-521-83733-2. https://books.google.com/books?id=HrAzTmXdssQC&pg=PA179&dq=%22relativistic+%22+Lagrangian+OR+Hamiltonian#PPA98,M1.
William Barker; Roger Howe (2008). Continuous symmetry: from Euclid to Klein. American Mathematical Society. p. 18 ff. ISBN 978-0-8218-3900-3. https://books.google.com/books?id=NIxExnr2EjYC&pg=PA17&dq=geometry++axiom+%22coordinate+system%22#PPA18,M1.
Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer. p. 11. ISBN 0-387-94339-0. https://archive.org/details/introductiontohy0000rams. "geometry axiom coordinate system."
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 0-521-09906-4. https://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22#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.
Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0-8218-1045-6. https://archive.org/details/geometryofdiffer00mori. "geometry axiom coordinate system."
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 0-486-41147-8. https://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22#PPA169,M1.
See Encarta definition. Archived 2009-10-31. https://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html
Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer. pp. 12–13. ISBN 3-540-20317-6. https://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22#PPA13,M1.
Achilleus Papapetrou (1974). Lectures on General Relativity. Springer. p. 5. ISBN 90-277-0540-2. https://books.google.com/books?id=SWeOggyp1ZsC&pg=PA3&dq=relativistic++%22general+coordinates%22#PPA5,M1.
Wilford Zdunkowski; Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. p. 84. ISBN 0-521-00666-X. https://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22.
A. I. Borisenko; I. E. Tarapov; Richard A. Silverman (1979). Vector and Tensor Analysis with Applications. Courier Dover Publications. p. 86. ISBN 0-486-63833-2. https://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric.
See Arvind Kumar; Shrish Barve (2003). How and Why in Basic Mechanics. Orient Longman. p. 115. ISBN 81-7371-420-7. https://books.google.com/books?id=czlUPz38MOQC&pg=PA115&dq=%22characterized+only+by+its+state+of+motion%22+inauthor:Kumar.
Chris Doran; Anthony Lasenby (2003). Geometric Algebra for Physicists. Cambridge University Press. p. §5.2.2, p. 133. ISBN 978-0-521-71595-9. http://www.worldcat.org/search?q=9780521715959&qt=owc_search. .
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. p. 222 and p. 233.
A. P. Lightman; W. H. Press; R. H. Price; S. A. Teukolsky (1975). Problem Book in Relativity and Gravitation. Princeton University Press. p. 15. ISBN 0-691-08162-X. https://archive.org/details/problembookinrel00ligh. "relativistic general coordinates."
Richard L Faber (1983). Differential Geometry and Relativity Theory: an introduction. CRC Press. p. 211. ISBN 0-8247-1749-X. https://books.google.com/books?id=ctM3_afLuVEC&pg=PA149&dq=relativistic++%22general+coordinates%22#PPA211,M1.
Richard Wolfson (2003). Simply Einstein. W W Norton & Co.. p. 216. ISBN 0-393-05154-4. https://books.google.com/books?id=OUJWKdlFKeQC&pg=PA216&dq=%22gravitational+time+dilation+%22.
See Guido Rizzi; Matteo Luca Ruggiero (2003). Relativity in rotating frames. Springer. p. 33. ISBN 1-4020-1805-3. https://books.google.com/books?id=_PGrlCLkkIgC&pg=PA226&dq=centrifugal+%22+%22+relativity+OR+relativistic#PPA33,M1. .
That is, both descriptions are equivalent and can be used as needed. This equivalence does not hold outside of general relativity, e.g., in entropic gravity.

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