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DavidWhitbeck
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And the last one is optional.
Is the Covariant Derivative of the Metric Tensor an Axiom or a Derivation?
- Context: Graduate
- Thread starter snoopies622
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- Metric Metric tensor Tensor
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SUMMARY
The covariant derivative of the metric tensor is established as zero, a condition known as "metric compatibility." This property is derived from the definition of the covariant derivative, which accounts for the non-flatness of the space. The discussion emphasizes that this relationship is not merely an axiom but can be derived through the parallel transport of 4-vectors, as outlined in standard texts like Wald and Ohanian and Ruffini. The consensus is that while the assumption of no torsion is necessary, the vanishing of the covariant derivative is a derived property rather than an axiom.
PREREQUISITES- Understanding of covariant derivatives in differential geometry
- Familiarity with metric tensors and their properties
- Knowledge of parallel transport and its implications in manifold geometry
- Basic concepts of affine connections and torsion-free manifolds
- Study the derivation of the covariant derivative in the context of general relativity
- Explore the implications of metric compatibility in Riemannian geometry
- Learn about the role of Christoffel symbols in defining connections
- Investigate the relationship between torsion and curvature in differential geometry
Mathematicians, physicists, and students of differential geometry who are interested in the foundational aspects of metric tensors and covariant derivatives in the context of general relativity.
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