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X. Curvature
From the
Riemann (10.2), which contains all information about curvature, one defines
other tensors, with less detailed curvature information, by a contraction of
indices. The most important in GRT are the Ricci tensor (Ricci for
short), either in the coordinate or tetrad basis:
\[{R_{\mu \nu }} : = {g^{\kappa \lambda
}}{R_{\mu \kappa \lambda \nu }}{\qquad} {R_{ab}}: = {\eta ^{cd}}{R}_{acdb}\]
and the scalar curvature, the simplest measure of curvature, obtained
by a further contraction:
\[R: = {g^{\mu \nu }}{R_{\mu \nu }} =
{\eta ^{ab}}{R_{ab}}\]
Note that the Ricci is symmetric on account of the interchange and
skew symmetries (10.4a,b,c).
In the
GA-formalism, the Ricci is defined from equation (10.2) by the contraction
\[{\text{ }}{{\mathbf{R}}_\mu }: =
{{\mathbf{g}}^\nu } \cdot {{\mathbf{R}}_{\mu \nu }} = {{\mathbf{g}}^\nu }
\cdot \operatorname{Riem} ({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu }): =
\operatorname{Ric} ({{\mathbf{g}}_\mu })\]
Thus, in GA, the Ricci is a linear operator on the tangent space that
maps vectors to vectors, consistent with the tensor definitions (10.17):
\[\operatorname{Ric} ({{\mathbf{g}}_\mu
}) = {R_{\mu \nu }}{{\mathbf{g}}^\nu }{\qquad}\operatorname{Ric}
({{\mathbf{e}}_a}) = {R_{ab}}{{\mathbf{e}}^b}\]
Similarly, consistent with (10.18), the scalar curvature is obtained by
contracting the Ricci, or directly the Riemann
\[R = {{\mathbf{g}}^\mu } \cdot
{{\mathbf{R}}_\mu } = ({{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }) \cdot
{{\mathbf{R}}_{\mu \nu }} = {g^{\mu \nu }}{R_{\mu \nu }}\]
Both the
Ricci and the scalar curvature have the dimension of an inverse squared
length. This means that the square root of the inverse of the curvature
gives a typical length scale that one may use to estimate gravitational
effects.