- #1
DuckAmuck
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- TL;DR Summary
- F_mu_nu does not change in the presence of a gravitational field?
It seems a gravitational field does not alter the electromagnetic field strength. Is this correct?
My reasoning:
With no gravity, field strength is:
[tex] F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu [/tex]
Introduce gravity:
[tex]\partial_\mu A_\nu \rightarrow \nabla_\mu A_\nu = \partial_\mu A_\nu + \Gamma_{\mu\nu}^{\alpha} A_\alpha [/tex]
Field strength then becomes:
[tex] F_{\mu\nu} = \nabla_\mu A_\nu - \nabla_\nu A_\mu [/tex]
[tex] = \partial_\mu A_\nu - \partial_\nu A_\mu + \Gamma_{\mu\nu}^{\alpha} A_\alpha - \Gamma_{\nu\mu}^{\alpha} A_\alpha [/tex]
[tex] = \partial_\mu A_\nu - \partial_\nu A_\mu [/tex]
The Christoffel symbol terms cancel. So you end up with an unchanged field strength.
Is this correct? It seems to make sense as field strength is essentially the number of photons, and gravity should not change that.
Some further insight would be much appreciated.
My reasoning:
With no gravity, field strength is:
[tex] F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu [/tex]
Introduce gravity:
[tex]\partial_\mu A_\nu \rightarrow \nabla_\mu A_\nu = \partial_\mu A_\nu + \Gamma_{\mu\nu}^{\alpha} A_\alpha [/tex]
Field strength then becomes:
[tex] F_{\mu\nu} = \nabla_\mu A_\nu - \nabla_\nu A_\mu [/tex]
[tex] = \partial_\mu A_\nu - \partial_\nu A_\mu + \Gamma_{\mu\nu}^{\alpha} A_\alpha - \Gamma_{\nu\mu}^{\alpha} A_\alpha [/tex]
[tex] = \partial_\mu A_\nu - \partial_\nu A_\mu [/tex]
The Christoffel symbol terms cancel. So you end up with an unchanged field strength.
Is this correct? It seems to make sense as field strength is essentially the number of photons, and gravity should not change that.
Some further insight would be much appreciated.