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Tetrads in General Relativity

I. Metric and Connection

Metric Compatibility

   To establish a relation between the connection coefficients and the metric one may apply the coderivative to both sides of eq. (1.10). Since the metric is a scalar function

1.27

\[ \begin{gathered} {D_\kappa }{g_{\mu \nu }} = {\partial _\kappa }{g_{\mu \nu }} = ({D_\kappa }{{\mathbf{g}}_\mu }) \cdot {{\mathbf{g}}_\nu } + {{\mathbf{g}}_\mu } \cdot ({D_\kappa }{{\mathbf{g}}_\nu }) \hfill \\ \qquad \qquad {\text{ }} = \Gamma _{\kappa \mu }^\lambda {{\mathbf{g}}_\lambda } \cdot {{\mathbf{g}}_\nu } + \Gamma _{\kappa \nu }^\lambda {{\mathbf{g}}_\lambda } \cdot {{\mathbf{g}}_\mu } =  \Gamma _{\kappa \mu }^\lambda {g_{\lambda \nu }} + \Gamma _{\kappa \nu }^\lambda {g_{\lambda \mu }} \hfill \\ \end{gathered} \]

It is then seen that the coderivative of the metric vanishes:

1.28

\[{g_{\mu \nu ;\kappa }}: = {\partial _\kappa }{g_{\mu \nu }} - \Gamma _{\kappa \mu }^\lambda {g_{\lambda \nu }} - \Gamma _{\kappa \nu }^\lambda {g_{\lambda \mu }} = 0\]

It follows that the operations of covariant differentiation and raising and lowering of indices commute. This result is called the ‘metric compatibility’ of the coderivative, i.e., the connection preserves the scalar inner product induced by the metric ${g_{\mu \nu }}$.

   Add to equation (1.28) the same equation with $\kappa $ and $\mu $ interchanged and subtract the same equation with $\kappa $ and $\nu $ interchanged. By defining ${\Gamma _{\kappa \mu \nu }}: = \Gamma _{\kappa \mu }^\lambda {g_{\lambda \nu }}$ one then obtains the Christoffel formula: 

1.29

\[  \Gamma _{\mu \nu \kappa }^{} = \frac{1}{2}({\partial _\mu }{g_{\nu \kappa }} + {\partial _\nu }{g_{\kappa \mu }} - {\partial _\kappa }{g_{\mu \nu }})\qquad \Gamma _{\mu \nu }^\mu = \frac{1}{2}{g^{\mu \kappa }}{\partial _\nu }{g_{\mu \kappa }} \]

If the metric is postulated or can be determined on physical grounds, then a knowledge of the embedding is not necessary for calculating the connection coefficients.

   Given a metric connection on a Lorentzian (pseudo-Riemannian) manifold with torsion, one can always find a single connection that is torsion free. This is the Levi-Civita connection, uniquely defined as torsion-free and metric-compatible. The Levi-Civita connection has as many independent components as ${g_{\mu \nu ;\kappa }}$ and is therefore fully determined by the metric compatibility equation (1.28).