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Sign convention

## Summary

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 the imaginary unit i and 2π, rather than just choices of sign.

## Relativity

### Metric signature

In relativity, the metric signature can be either (+,−,−,−) or (−,+,+,+). (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, (+,−,−,−,...) or (−,+,+,+,...). A choice of signature is associated with a variety of names, physics discipline, and notable graduate-level textbooks:

Comparison of metric signatures in general relativity
Metric signature (+,−,−,−) (−,+,+,+)
Spacetime interval convention timelike spacelike
Subject area primarily using convention Particle physics and Relativity Relativity
Corresponding metric tensor ${\textstyle {\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}$  ${\textstyle {\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$
Mass–four momentum relationship ${\textstyle m^{2}=p^{\mu }p_{\mu }}$  ${\textstyle m^{2}=-p^{\mu }p_{\mu }}$
Common names of convention
• East coast convention
• "Mostly pluses"
• Pauli convention

### Curvature

The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction ${\displaystyle R_{ab}\,=R^{c}{}_{acb}}$ , whereas others use the alternative ${\displaystyle R_{ab}\,=R^{c}{}_{abc}}$ . 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 ${\displaystyle R_{ab}\,={R_{acb}}^{c}}$ . 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

It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article.

• Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. p. cover. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link)