Derivation of the wave equation on a curved space-time

In summary, the conversation discusses a question about minimal coupling and the EM tensor, specifically whether the answer should be ##\nabla^a\nabla_a F_{bc}=0##. The goal is unclear, but in Minkowski coordinates on flat spacetime, both expressions reduce to ##\partial^\lambda\partial_\lambda F_{\mu\nu}=0##. The conversation also mentions the issue of covariant derivatives not commuting and asks about the level of physics involved.
  • #1
Woolyabyss
143
1
Homework Statement
Problem attached as image
Relevant Equations
## \nabla^a F_{ab} = 0 ##
## \nabla_a F_{bc} + \nabla_b F_{ca} + \nabla_c F_{ab} = 0 ##
I'm confused by this question, from minimal coupling shouldn't the answer simply be ## \nabla^a \nabla_a F_{bc} = 0 ##? Any help would be appreciated.

EDIT: I should also point out ##F_{ab}## is the EM tensor.
 

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  • #2
You have not given the starting point and so the goal is unclear. Both expressions reduce to ##\partial^\lambda\partial_\lambda F_{\mu\nu}=0## in Minkowski coordinates on flat spacetime.
 
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Likes Woolyabyss and dextercioby
  • #3
Orodruin said:
You have not given the starting point and so the goal is unclear. Both expressions reduce to ##\partial^\lambda\partial_\lambda F_{\mu\nu}=0## in Minkowski coordinates on flat spacetime.
Thanks for the reply, I managed to work out the answer my issue turned out to be I wasn't taking into account that covariant derivatives don't commute.
 
  • #4
Out of curiosity - what level of Physics is this?
 
  • #5
Physics?
I thought it was Greek. ;)
 

1. What is the wave equation on a curved space-time?

The wave equation on a curved space-time is a mathematical equation that describes the propagation of waves in a space-time that is not flat, meaning it has curvature. This equation is derived from the general theory of relativity and takes into account the curvature of space-time caused by the presence of massive objects.

2. How is the wave equation on a curved space-time derived?

The wave equation on a curved space-time is derived using the principles of general relativity and differential geometry. It involves manipulating the Einstein field equations, which describe the curvature of space-time, and applying the d'Alembert operator to obtain the wave equation.

3. What are the implications of the wave equation on a curved space-time?

The wave equation on a curved space-time has significant implications in understanding the behavior of waves in the presence of massive objects, such as black holes. It also helps to explain phenomena such as gravitational lensing and gravitational waves.

4. Can the wave equation on a curved space-time be applied to other types of waves?

Yes, the wave equation on a curved space-time can be applied to other types of waves, such as electromagnetic waves and quantum mechanical waves. It is a fundamental equation in physics and has wide-ranging applications in various fields.

5. Are there any limitations to the wave equation on a curved space-time?

While the wave equation on a curved space-time is a powerful tool for understanding the behavior of waves in curved space-time, it does have some limitations. It does not take into account quantum effects and is only applicable in the classical regime. Additionally, it is a linear equation, so it cannot fully describe the behavior of non-linear waves.

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