Transformation of connection coefficients

• Apr 20, 2022

• #1

accdd

96

20

I don't understand why the highlighted term is there.
This image was taken from Sean Carroll's notes available here: preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf

 

Physics news on Phys.org

• Research suggests how turbulence can be used to generate patterns
• High-quality microwave signals generated from tiny photonic chip
• Quantum talk with magnetic disks

• Apr 20, 2022

• #2

Orodruin

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Gold Member

21,086

12,145

This follows directly (together with the first term) from writing out the transformation relation ##V^{\nu’} = \frac{\partial x^{\nu’}}{\partial x^\nu} V^\nu## and applying the Leibniz rule.

 

• Apr 20, 2022

• #3

accdd

96

20

Thanks for the answer, could you show me the steps to get to the result?

 

• Apr 20, 2022

• #4

Orodruin

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Gold Member

21,086

12,145

I think it would be more instructive if you do what I proposed starting from the first line in (3.3) and show us the steps until you get stuck.

 

• Apr 20, 2022

• #5

accdd

96

20

$$V^{\nu'}=\frac{\partial x^{\nu'}}{\partial x^{\nu}}V^{\nu}$$
$$
\nabla_{\mu'}V^{\nu'}=\partial_{\mu'}(\frac{\partial x^{\nu'}}{\partial x^{\nu}}V^{\nu})+\Gamma^{\nu'}_{\mu'\lambda'}(\frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda}) =\partial_{\mu'}(\frac{\partial x^{\nu'}}{\partial x^{\nu}})V^{\nu}+\frac{\partial x^{\nu'}}{\partial x^{\nu}} \partial_{\mu'}V^{\nu}+\Gamma^{\nu'}_{\mu'\lambda'}\frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda}=
$$

 

• Apr 20, 2022

• #6

Orodruin

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Gold Member

21,086

12,145

Ok. So the piece that you're missing is expressing ##\partial_{\mu'}## in terms of ##\partial_\mu##. Are you familiar with any way to relate those two partial derivatives?

 

• Apr 20, 2022

• #7

accdd

96

20

Is this the correct transformation?
$$
\partial_{\mu'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\partial_\mu
$$
$$
\partial_{\mu'}(\frac{\partial x^{\nu'}}{\partial x^{\nu}})V^{\nu}+\frac{\partial x^{\nu'}}{\partial x^{\nu}} \partial_{\mu'}V^{\nu}+\Gamma^{\nu'}_{\mu'\lambda'}\frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda}= \frac{\partial x^\mu}{\partial x^{\mu'}}\partial_\mu(\frac{\partial x^{\nu'}}{\partial x^{\nu}})V^{\nu}+\frac{\partial x^{\nu'}}{\partial x^{\nu}} \frac{\partial x^\mu}{\partial x^{\mu'}}\partial_\mu V^{\nu}+\Gamma^{\nu'}_{\mu'\lambda'}\frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda}
$$

so the first line of 3.3 should be more clearly:
$$
\nabla_{\mu'}V^{\nu'}=(\frac{\partial x^\mu}{\partial x^{\mu'}}\partial_\mu)(\frac{\partial x^{\nu'}}{\partial x^{\nu}}V^{\nu})+\Gamma^{\nu'}_{\mu'\lambda'}(\frac{\partial x^{\lambda'}}{\partial x^{\lambda}}V^{\lambda})
$$

 

Last edited: Apr 20, 2022

• Apr 20, 2022

• #8

Orodruin

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Gold Member

21,086

12,145

accdd said:

Is this the correct transformation?
$$
\partial_{\mu'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\partial_\mu
$$

Yes. That is the chain rule for partial derivatives.

 

1. What is the transformation of connection coefficients?

The transformation of connection coefficients is a mathematical process used to convert the connection coefficients, which are components of the metric tensor, from one coordinate system to another. This is necessary in order to describe the same physical phenomenon in different reference frames or coordinate systems.

2. Why is the transformation of connection coefficients important?

The transformation of connection coefficients is important because it allows us to understand and analyze physical phenomena from different perspectives. It also helps us to make predictions and calculations in different coordinate systems, which is essential in many fields of science, such as physics, engineering, and mathematics.

3. How is the transformation of connection coefficients performed?

The transformation of connection coefficients is performed using a mathematical formula known as the Christoffel symbol transformation law. This formula involves the use of partial derivatives and the metric tensor to convert the connection coefficients from one coordinate system to another.

4. What are the applications of the transformation of connection coefficients?

The transformation of connection coefficients has various applications in different fields of science, such as general relativity, differential geometry, and fluid mechanics. It is used to study and understand the behavior of physical systems in different reference frames and to make accurate predictions and calculations.

5. Are there any limitations to the transformation of connection coefficients?

Yes, there are limitations to the transformation of connection coefficients. It can only be applied to smooth and continuous coordinate systems, and it may not be valid for systems with singularities or discontinuities. Additionally, the transformation may become more complex for higher-dimensional spaces and may require advanced mathematical techniques.