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Tetrads in General Relativity

X. Curvature

Ricci Tensor

   From the Riemann (10.2), which contains all information about curvature, one defines other tensors important in GRT, with less detailed curvature information. First of all the Ricci tensor (Ricci for short), either in the coordinate or tetrad basis, is obtained by the contraction of the first and third indices of the Riemann:

10.17

\[{R_{\mu \nu }} : = {g^{\kappa \lambda }}{R_{\kappa \mu \lambda \nu }}{\qquad} {R_{ab}}: = {\eta ^{cd}}{R}_{cadb}\]

The Ricci is symmetric on account of the interchange and skew symmetries (10.4a,b,c). Note that the alternative contraction over indices 1 and 4 differs by a minus sign. It is a matter of choice what convention to use, because there is no generally accepted convention for the sign of either the Riemann tensor or the Ricci tensor.

   By a further contraction of the Ricci with the metric one obtains the Ricci scalar (also called scalar curvature), the simplest measure of curvature:

10.18

\[R: = {g^{\mu \nu }}{R_{\mu \nu }} = {\eta ^{ab}}{R_{ab}}\]

Both the Ricci and the Ricci scalar 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.

   In the GA-formalism, the Ricci is defined from equation (10.2) by the contraction

10.19

\[{\text{ }}{{\mathbf{R}}_\mu }: = {{\mathbf{g}}^\nu } \cdot {{\mathbf{R}}_{\nu \mu }} = {{\mathbf{g}}^\nu } \cdot \operatorname{Riem} ({{\mathbf{g}}_\nu } \wedge {{\mathbf{g}}_\mu }): = \operatorname{Ric} ({{\mathbf{g}}_\mu })\]

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):

10.20

\[\operatorname{Ric} ({{\mathbf{g}}_\mu }) = {R_{\mu \nu }}{{\mathbf{g}}^\nu }{\qquad}\operatorname{Ric} ({{\mathbf{e}}_a}) = {R_{ab}}{{\mathbf{e}}^b}\]

   Similarly, consistent with (10.18), the Ricci scalar

10.21

\[R = {{\mathbf{g}}^\mu } \cdot {{\mathbf{R}}_\mu } = ({{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }) \cdot {{\mathbf{R}}_{\mu \nu }} = {g^{\mu \nu }}{R_{\mu \nu }}\]

is obtained by contracting the Ricci, or directly the Riemann.