Editing Sign convention

{{Short description|Agreed-upon meaning of a physical quantity being positive or negative}}

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 the [[imaginary unit]] {{mvar|i}} and {{math|2{{pi}}}}, rather than just choices of sign.

== Relativity ==

===Metric signature===
In [[General relativity|relativity]], the [[metric signature]] can be either {{math|(+,−,−,−)}} or {{math|(−,+,+,+)}}. (Throughout this article, the signs of the eigenvalues of the metric are displayed in the order that presents the timelike component first, followed by the spacelike components). A similar convention is used in higher-dimensional relativistic theories; that is, {{math|(+,−,−,−,...)}} or {{math|(−,+,+,+,...)}}. A choice of signature is associated with a variety of names, physics discipline, and notable graduate-level textbooks:

{| class="wikitable plainrowheaders"
|+ Comparison of metric signatures in [[general relativity]]
|-
! scope="row" | Metric signature
! scope="col" | {{abbr|1={{math|(+,−,−,−)}}|2=plus minus minus minus signature}}
! scope="col" | {{abbr|1={{math|(−,+,+,+)}}|2=minus plus plus plus signature}}
|-
! scope="row" | [[Spacetime interval]] convention
| [[timelike]], <math>\tau^2 = x^\mu x_\mu</math>
| [[spacelike]], <math>\tau^2 = -x^\mu x_\mu</math>
|-
! scope="row" | Subject area primarily using convention
| [[Particle physics]] and [[General relativity|Relativity]]
| [[General relativity|Relativity]]
|- style="text-align: center" 
! scope="row" | Corresponding [[metric tensor]]
| <math display="inline">\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}</math>
| <math display="inline">\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}</math>
|- style="text-align: center" 
! scope="row" | Mass–[[four momentum]] relationship
| <math display="inline">m^2 = p^{\mu}p_{\mu}</math>
| <math display="inline">m^2 = -p^{\mu}p_{\mu}</math>
|-
! scope="row" | Common names of convention
|
* [[West Coast of the United States|West coast]] convention
* "Mostly minuses"
* [[Lev Landau|Landau]]–[[Evgeny Lifshitz|Lifshitz]] sign convention
|
* [[East Coast of the United States|East coast]] convention
* "Mostly pluses"
* Pauli convention
|-
! scope="row" | Graduate textbooks using convention
|
* [[Course_of_Theoretical_Physics#English_editions|Landau & Lifshitz]]
* ''The Mathematical Theory of Black Holes'' ([[Subrahmanyan Chandrasekhar]])
* ''Gravitation: an introduction to current research'' ([[Louis Witten|L. Witten]])
* ''Introducing Einstein's relativity'' (Ray D'Inverno)
* ''General relativity'' (Michael P. Hobson, [[George Efstathiou]] & Anthony N. Lasenby)
|
* ''[[Gravitation (book)|Gravitation]]'' (Misner, Thorne, and Wheeler)
* ''Spacetime and Geometry: An Introduction to General Relativity'' ([[Sean M. Carroll]])
* ''[[General Relativity (book)|General Relativity]]'' (Wald) (Wald changes signature to the timelike convention for Chapter 13 only)
|}

===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]].
* The sign of the weight of a [[tensor density]], such as the weight of the determinant of the covariant metric tensor.
* The active and [[passive sign convention]] of [[Electric current|current]], [[voltage]] and [[Electric power|power]] in [[electrical engineering]].
* A sign convention used for [[curved mirror]]s assigns a positive focal length to concave mirrors and a negative focal length to convex mirrors.

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)]]
* [[Symmetry (physics)]]
* [[Gauge theory]]
* [[Logic level#Active state|Negative logic]]

==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]]