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Tetrads in General Relativity

X. Curvature

Riemann Tensor

   One of the characteristics of a flat manifold is that two coderivatives ${D_\mu }{D_\nu }$ commute; in curved space they do not:

10.1

\[\left[ {{D_\mu },{D_\nu }} \right]{{\mathbf{g}}_\kappa }: = {{\mathbf{R}}_{\mu \nu }} \cdot {{\mathbf{g}}_\kappa }\]

This defines the curvature bivector ${{\mathbf{R}}_{\mu \nu }}$ as a measure for the noncommutativity of the coderivative. Geometrically, the bivector calculates the difference in a vector that is parallel transported around a loop. If and only if the curvature is zero everywhere, space is flat. Physically, the curvature bivector fully encodes the gravitational interaction.

   The curvature bivector may be expanded with respect to a coordinate or tetrad basis

10.2

\[{{\mathbf{R}}_{\mu \nu }} = \frac{1}{2}{R_{\mu \nu \kappa \lambda }}{{\mathbf{g}}^\kappa } \wedge {{\mathbf{g}}^\lambda } = \frac{1}{2}{R_{\mu \nu ab}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b}\]

The mixed components tensor is the one defined in (6.3). The coefficients ${R_{\mu \nu \kappa \lambda }}$ are the components of the Riemann curvature tensor (Riemann for short) as it is usually presented in the tensorial formalism of GRT. By explicitly evaluating the commutator (10.1), one derives an expression for the Riemann in terms of the Levi-Civita connection coefficients (1.15):

10.3

\[{R^\mu }_{\nu \kappa \lambda } = 2{\partial _{[\kappa }}\Gamma _{\lambda ]\nu }^\mu + 2\Gamma _{[\kappa |\rho }^\mu \Gamma _{\lambda ]\nu }^\rho \]

This is a complicated expression of $\Gamma $’s that involves first and second order derivatives of the metric tensor, which makes explicit calculation a daunting task. However, there are a number of symmetries and constraints that greatly reduce the number of independent components.

   The components of the Riemann are anti-symmetric in both their first and second pairs of indices. One can also interchange the two pairs of indices:

10.4

\[\begin{gathered} &{R_{\mu \nu \kappa \lambda }} = {R_{[\mu \nu ]\kappa \lambda }}{\text{ (a) }}\qquad {R_{\mu \nu \kappa \lambda }} = {R_{\mu \nu [\kappa \lambda ]}}{\text{ (b)}} \hfill \\ &{R_{\mu \nu \kappa \lambda }} = {R_{\kappa \lambda \mu \nu }}{\text{ (c) }}\qquad \qquad {R_{\mu [\nu \kappa \lambda ]}} = 0{\text{ (d)}} \hfill \\ \end{gathered} \]

The anti-symmetry propriety (a) is obvious from definition (10.1) and the anti-symmetry (b) can be read off from (10.2) or (10.3). Propriety (c) may be proven by considering a local inertial frame such that the $\Gamma$’s in the last term of equation (10.3) vanish. From this simplified expression, with (1.29) inserted, one can prove symmetry relations (c) and (d). The latter only holds in spaces without torsion. Together, these symmetries reduce the number of independent components of the Riemann to 20.