Sign convention

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 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 convention
West coast convention
Mostly minuses
Landau–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:

(+,−,−,−):

(−,+,+,+):

Misner, Thorne and Wheeler
Spacetime and Geometry: An Introduction to General Relativity
General Relativity (Wald) (Note that Wald changes signature to the timelike convention for Chapter 13 only.)

The signature + − − − corresponds to the metric tensor:

{\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}

whereas the signature − + + + corresponds to:

{\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}

Curvature

The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction $R_{ab}\,=R^{c}{}_{acb}$ , whereas others use the alternative $R_{ab}\,=R^{c}{}_{abc}$ . 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 $R_{ab}\,={R_{acb}}^{c}$ . 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 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, field strength tensor $\,F_{ab}$ in gauge theories and classical electrodynamics.
Time dependence of a positive-frequency wave (see, e.g., the electromagnetic wave equation):
- $\,e^{-i\omega t}$ (mainly used by physicists)
- $\,e^{+j\omega t}$ (mainly used by engineers)
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.
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.

References

Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. p. cover. ISBN 0-7167-0344-0.

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