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In
physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spacetime, spa ...
, 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
for the case of one dimension. Sometimes, the term "sign convention" is used more broadly to include factors of '' i'' and 2 π, rather than just choices of sign.


Relativity


Metric signature

In relativity, the
metric signatureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
can be either (+,−,−,−) or (−,+,+,+). (Note that throughout this article we are displaying the signs of the eigenvalues of the metric 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, (+,−,−,−,...) or (−,+,+,+,...). A choice of signature is associated with a variety of names: + − − −: *'' Timelike convention'' *''
Particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that ...
convention'' *'' West coast convention'' *''Mostly minuses'' *''
Landau Landau, officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße Südliche Weinstraße ( en, Southwest Wine Route) is a district (''Kreis'') in the south of Rhineland-Palatinate, Germany. Neigh ...
Lifshitz sign convention''. − + + +: *'' Spacelike convention'' *'' Relativity convention'' *'' East coast convention'' *''Mostly pluses'' *''Pauli convention'' We catalog the choices of various authors of some graduate textbooks: (+,−,−,−): *'' Landau & Lifshitz'' *
Gravitation: an introduction to current research
' (Louis Witten, L. Witten) *
Ray D'Inverno, Introducting Einstein's relativity
'' (−,+,+,+): *''Gravitation (book), Misner, Thorne and Wheeler'' *
Spacetime and Geometry: An Introduction to General Relativity
' *''General Relativity (book), General Relativity (Wald)'' (Note that Wald changes signature to the timelike convention for Chapter 13 only.) The signature + − − − corresponds to the metric tensor: :\begin 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end whereas the signature − + + + corresponds to: :\begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end


Curvature

The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction R_ \, = R^c_, whereas others use the alternative R_ \, = R^c_. 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 R_ \, = ^c. 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 \pm in the Dirac equation. * The sign of the electric charge, electromagnetic tensor, field strength tensor \, F_ 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): ** \, e^ (mainly used by physicists) ** \, e^ (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 the determinant of the metric tensor when dealing with tensor density. It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article. The sign of spherical mirrors are also represented by sign convention


See also

* Orientation (vector space), also known as "handedness" * Symmetry (physics) * Gauge theory


References

* {{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 Mathematical physics