How do I properly use Ricci calculus in this example?

In summary, the conversation is about substituting A_\mu + \partial_\mu \lambda for A_\mu and also using a contravariant form of the substitution for A^\mu. The speaker is new to Ricci calculus and is seeking an explanation for raised and lowered indices. They have found online explanations to be unnecessarily complicated and have attempted a proof for Proca Lagrangian local gauge invariance. They provide a link for anyone with the same issue in the future.
  • #1
jdbbou
4
0
upload_2016-7-19_15-25-43.png


Do I substitute [itex]A_\mu + \partial_\mu \lambda[/itex] everywhere [itex]A_\mu[/itex] appears, then expand out? Do I substitute a contravariant form of the substitution for [itex]A^\mu[/itex] as well? (If so, do I use a metric to convert it first?)

I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered indices here would be greatly appreciated. Everything I've found online so far has been (unnecessarily, I feel) complicated.
 
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  • #2
This is what I've tried so far. Any suggestions? Was my justification for step (1) correct?

upload_2016-7-20_8-24-17.png
 

Related to How do I properly use Ricci calculus in this example?

1. What is Ricci calculus and how is it used in this example?

Ricci calculus is a mathematical framework used to describe the curvature of space in Einstein's theory of general relativity. In this example, we are using Ricci calculus to calculate the curvature of spacetime in a specific situation.

2. What are the key concepts and equations involved in using Ricci calculus?

The key concepts in Ricci calculus include the metric tensor, the Christoffel symbols, and the Ricci curvature tensor. The main equation used in Ricci calculus is the Einstein field equations, which relate the curvature of spacetime to the energy and matter distribution within it.

3. How do I determine the metric tensor in a given spacetime?

The metric tensor is determined by the geometry and gravitational fields present in the given spacetime. It can be calculated using the Einstein field equations or by solving the geodesic equation for a particle moving in the spacetime.

4. Can Ricci calculus be applied to any type of spacetime?

Yes, Ricci calculus can be applied to any type of spacetime, including flat, curved, and even expanding spacetimes. It is a fundamental tool in understanding the geometry and dynamics of the universe.

5. Are there any limitations to using Ricci calculus in this example?

While Ricci calculus is a powerful tool for describing the curvature of spacetime, it is based on the assumptions and equations of general relativity. Thus, it may not accurately describe extreme conditions, such as those near a black hole or in the early universe, where other theories may be needed.

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