Sign convention

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:

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).