# Derivation of the wave equation on a curved space-time

#### Woolyabyss

Problem 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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#### Orodruin

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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.

#### Woolyabyss

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.

#### Professor Hulk

Out of curiosity - what level of Physics is this?

#### Ralph Rotten

Physics?
I thought it was Greek. ;)

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