Abstract
The Christoffel connection is associated to a covariant derivative acting on tensors. In familiar gauge theories, the partial derivative is replaced by a covariant derivative as \(\partial _{\mu } \rightarrow D_{\mu } = \partial _{\mu }+ ie A_{\mu }(x)\). In general relativity, the covariant derivative acts as \(\partial _{\mu } \rightarrow \nabla _{\mu } = \left( \partial _{\mu }+ \Gamma ^{\nu }_{\mu \lambda }\right) \circ \), where \(\circ \) indicates that multiplication is tensorially non-trivial, see (4.2).
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Lüst, D., Vleeshouwers, W. (2019). Einstein’s Equations. In: Black Hole Information and Thermodynamics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-10919-6_5
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DOI: https://doi.org/10.1007/978-3-030-10919-6_5
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