All Questions
5
questions
8
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How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 103
3
votes
2
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166
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What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 374
3
votes
2
answers
171
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?
Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
- 1,815
9
votes
3
answers
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Intuition behind differential operators as the basis vectors of a manifold (space-time)
I understand that in order to provide a basis for every point in space-time, the differential operators, $\partial_\mu$ (or partial derivative operator with respect to each one of the curvilinear ...
- 902
5
votes
3
answers
875
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Motivation for covariant derivative axioms in the context of General Relativity
In General Relativity the idea of a covariant derivative on a manifold is quite important and it is usually defined by a set of axioms:
Let $M$ be a smooth manifold. A covariant derivative $\nabla$ ...
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