Metric signature edit

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

Cataloged below are the choices of various authors of some graduate textbooks:

(+,−,−,−):

• Landau & Lifshitz
• Gravitation: an introduction to current research (L. Witten)
• Ray D'Inverno, Introducing Einstein's relativity.

(−,+,+,+):

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

The signature (+,−,−,−) corresponds to the metric tensor:

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

and gives m² = p^μp_μ as the relationship between mass and four momentum

whereas the signature (−,+,+,+) corresponds to:

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

and gives m² = −p^μp_μ.

Curvature edit

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).

• The sign choice for time in frames of reference and proper time: + for future and − for past is universally accepted.
• The choice of ${\displaystyle \pm }$  in the Dirac equation.
• The sign of the electric charge, field strength tensor ${\displaystyle \,F_{ab}}$  in gauge theories and classical electrodynamics.
• Time dependence of a positive-frequency wave (see, e.g., the electromagnetic wave equation):
  • ${\displaystyle \,e^{-i\omega t}}$  (mainly used by physicists)
  • ${\displaystyle \,e^{+j\omega t}}$  (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 a tensor density, such as the weight of the determinant of the covariant metric tensor.
• The active and passive sign convention of current, voltage and power in electrical engineering.
• A sign convention used for curved mirrors 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.

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

• 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)