Christoffel Symbols: Intuitive Proof for Covariant Derivative of Metric Tensor

In summary, there are several sources that discuss the representation of Christoffel symbols as a linear combination of products of the metric tensor and its derivative. However, the derivation of this representation is often done in a confusing manner. The covariant derivative of the metric tensor is equal to zero, but this does not make it a tensor. To make it a tensor, the Christoffel symbols must be subtracted from the covariant derivative. There are also sources that provide a more physically intuitive proof, such as chapter 3 of Wald's GR book and "Riemannian manifolds: an introduction to curvature" by John Lee. Additionally, the proof can be found in MTW exercise 8.15.
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I am learning about christoffel symbols and there is a pretty standard representation of christoffel symbols as a linear combination of products of the metric tensor and the metric tensors derivative. However when this is derived it is always done in a hoakey manner. Something along the lines of ... do these permutations add this subtract that and walllaaa. I am trying to make a more physically intuitive proof based off the covariant derivative of the metric tensor being equal to zero. Has anyone seen this proof somewhere i haven't got it to work out and i am looking go help.
 
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  • #2
Check out chapter 3 of Wald's GR book.
 
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I think the best place to read about connections is "Riemannian manifolds: an introduction to curvature", by John Lee. But I don't remember how he did this particular thing.
 
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The ordinary derivative of a tensor is NOT a tensor. In order to make it one, the "covariant derivative", you have to subtract off the Christoffel symbols- or, to put it another way, the Chrisoffel symbols are the covariant derivative minus the ordinary derivative.
 
  • #5
zwoodrow said:
I am trying to make a more physically intuitive proof based off the covariant derivative of the metric tensor being equal to zero. Has anyone seen this proof somewhere i haven't got it to work out and i am looking go help.

Yes, you can find it in MTW exercise 8.15. It has an outline solution too.
 

1. What are Christoffel symbols?

Christoffel symbols are mathematical quantities used in differential geometry to describe the curvature of a space. They are used in the covariant derivative of a metric tensor, which is a way to measure how a vector changes as it moves along a curve.

2. Why are Christoffel symbols important?

Christoffel symbols are important because they allow us to describe the curvature of a space and calculate how vectors change as they move along a curve. They are also used in the equations of general relativity, which describe the behavior of gravity.

3. How do you intuitively prove the covariant derivative of a metric tensor using Christoffel symbols?

The intuitive proof for the covariant derivative of a metric tensor using Christoffel symbols involves showing that the change in a vector along a curve can be calculated by subtracting the effect of parallel transport along the curve from the total change in the vector. This is done by using the Christoffel symbols to calculate the parallel transport term.

4. Can you explain the geometric interpretation of Christoffel symbols?

Geometrically, Christoffel symbols represent the connection between different tangent spaces in a curved space. They describe how vectors change as they move from one point to another on a curved surface.

5. How can I use Christoffel symbols in my research?

Christoffel symbols are used in various fields of science, such as physics and engineering, to study the behavior of objects in curved spaces. They can be used to calculate the curvature of a space, solve differential equations, and make predictions about the behavior of physical systems. They are also used in the development of new technologies, such as space travel and GPS systems.

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