Involution (mathematics): Difference between revisions
Jump to navigation
Jump to search
en>ZéroBot m r2.7.1) (Robot: Adding ca:Involució |
en>DavidCary add reference, as requested on talk page. |
||
| Line 1: | Line 1: | ||
In [[physics]], a '''sign convention''' is a choice of the physical significance of [[sign (mathematics)|sign]]s (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors in scientific work. In general, a sign convention is a special case of a choice of [[coordinate system]] for the case of one dimension. | |||
Sometimes, the term "sign convention" is used more broadly to include factors of ''[[Imaginary unit|i]]'' and 2[[pi|π]], rather than just choices of sign. | |||
== | == Relativity == | ||
===Metric signature=== | |||
In [[General relativity|relativity]], the [[metric signature]] could either be + − − − or − + + +. A similar dual convention is used in higher-dimensional relativistic theories. The choice of signature is given a variety of names: | |||
+ − − −: | |||
*''[[Timelike]] convention'' | |||
*''[[Particle physics]] convention'' | |||
*''[[West Coast of the United States|West coast]] convention'' | |||
*''Mostly minuses'' | |||
*''[[Lev Landau|Landau]]-[[Evgeny Mikhailovich Lifshitz|Lifshitz]] sign convention''. | |||
− + + +: | |||
*''[[Spacelike]] convention'' | |||
*''[[General relativity|Relativity]] convention'' | |||
*''[[East Coast of the United States|East coast]] convention'' | |||
*''Mostly pluses'' | |||
*''Pauli convention'' | |||
== | Regarding the choice of − + + + versus + − − −, a survey of some [[General relativity resources#Graduate level|classic textbooks]] reveals that [[Gravitation (book)|Misner, Thorne and Wheeler]] (MTW) chose − + + + while Weinberg chose + − − −{{Citation needed|reason=Weinberg uses +++- (with the last dimension being time) both in Gravitation and Cosmology (1972; p. 26) and in The Quantum Theory of Fields, Vol. 1 (1995; p. xxv).|date=November 2012}} (with the understanding that the first sign corresponds to "time"). Subsequent authors writing in particle physics have generally followed Weinberg, while authors of papers in classical gravitation and [[string theory]] have generally followed MTW (as do most Wikipedia articles related to [[relativistic physics]]). Nevertheless, the Weinberg form is consistent with [[Hyperbolic quaternion]]s, a forerunner of [[Minkowski space]]. | ||
The signature + − − − would correspond to the following [[metric tensor]]: | |||
:<math>\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} </math> | |||
whereas the signature − + + + would correspond to this one: | |||
:<math>\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} </math> | |||
===Curvature=== | |||
The [[Ricci tensor]] is defined as the contraction of the [[Riemann tensor]]. Some authors use the contraction <math>R_{ab} \, = R^c{}_{acb}</math>, whereas others use the alternative <math>R_{ab} \, = R^c{}_{abc}</math>. Due to the [[Riemann tensor#Symmetries and identities|symmetries of the Riemann tensor]], these two definitions differ by a minus sign. | |||
= | In fact, the second definition of the Ricci tensor is <math>R_{ab} \, = {R_{acb}}^c</math>. The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor. The second definition just compensates the sign, and it works together with the second definition of the Riemann tensor (see e.g. Barrett O'Neill's Semi-riemannian geometry). | ||
== Other sign conventions == | |||
* The sign choice for [[arrow of time|time]] in frames of reference and proper time: '''+''' for future and '''−''' for past is universally accepted. | |||
* The choice of <math>\pm</math> in the [[Dirac equation]]. | |||
* The sign of the [[electric charge]], [[electromagnetic tensor|field strength tensor]] <math>\, F_{ab}</math> in [[Gauge theory|gauge theories]] and [[Maxwell's equations|classical electrodynamics]]. | |||
* Time dependence of a positive-frequency wave (see, e.g., the [[electromagnetic wave equation]]): | |||
** <math>\, e^{-i\omega t}</math> (mainly used by physicists) | |||
** <math>\, e^{+j\omega t}</math> (mainly used by engineers) | |||
* The sign for the imaginary part of [[Permittivity#Complex permittivity|permittivity]] (in fact dictated by the choice of sign for time-dependence) | |||
* The signs of distances and [[radius of curvature (optics)|radii of curvature]] of optical surfaces in [[optics]] | |||
* The sign of work in the [[first law of thermodynamics]]. | |||
It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article. | |||
== See also == | |||
* [[Orientation (vector space)]], also known as "handedness". | |||
==References== | |||
{{reflist}} | |||
* {{cite book | author=[[Charles Misner]]; [[Kip S Thorne]] & [[John Archibald Wheeler]] | title=[[Gravitation (book)|Gravitation]] | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0|page=cover}} | |||
[[Category:Mathematical physics]] | |||
Revision as of 19:23, 26 November 2013
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors in scientific work. In general, a sign convention is a special case of a choice of coordinate system for the case of one dimension.
Sometimes, the term "sign convention" is used more broadly to include factors of i and 2π, rather than just choices of sign.
Relativity
Metric signature
In relativity, the metric signature could either be + − − − or − + + +. A similar dual convention is used in higher-dimensional relativistic theories. The choice of signature is given a variety of names:
+ − − −:
- Timelike convention
- Particle physics convention
- West coast convention
- Mostly minuses
- Landau-Lifshitz sign convention.
− + + +:
- Spacelike convention
- Relativity convention
- East coast convention
- Mostly pluses
- Pauli convention
Regarding the choice of − + + + versus + − − −, a survey of some classic textbooks reveals that Misner, Thorne and Wheeler (MTW) chose − + + + while Weinberg chose + − − −Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. (with the understanding that the first sign corresponds to "time"). Subsequent authors writing in particle physics have generally followed Weinberg, while authors of papers in classical gravitation and string theory have generally followed MTW (as do most Wikipedia articles related to relativistic physics). Nevertheless, the Weinberg form is consistent with Hyperbolic quaternions, a forerunner of Minkowski space.
The signature + − − − would correspond to the following metric tensor:
whereas the signature − + + + would correspond to this one:
Curvature
The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction , whereas others use the alternative . Due to the symmetries of the Riemann tensor, these two definitions differ by a minus sign.
In fact, the second definition of the Ricci tensor is . The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor. The second definition just compensates the sign, and it works together with the second definition of the Riemann tensor (see e.g. Barrett O'Neill's Semi-riemannian geometry).
Other sign conventions
- The sign choice for time in frames of reference and proper time: + for future and − for past is universally accepted.
- The choice of in the Dirac equation.
- The sign of the electric charge, field strength tensor in gauge theories and classical electrodynamics.
- Time dependence of a positive-frequency wave (see, e.g., the electromagnetic wave equation):
- The sign for the imaginary part of permittivity (in fact dictated by the choice of sign for time-dependence)
- The signs of distances and radii of curvature of optical surfaces in optics
- The sign of work in the first law of thermodynamics.
It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article.
See also
- Orientation (vector space), also known as "handedness".
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534