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Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form $$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$ without knowing EM?

I wonder if it's a degree of freedom thing. Please enlighten! Thanks.

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What do you mean by "Without knowing EM"? In nonlinear theories of electromagnetism your expression is manifestly false. See for example equation (7) on page 7 of arxiv.org/abs/1012.1400. If you assume you are working with Maxwell's linear theory, you have a Lagrangian formulation to which you add the Hilbert-action and assume minimal coupling to get the form above. – Willie Wong Mar 8 '13 at 12:55
    
Also, this question belongs better on physics, so I am migrating it over. Please register on that site so you can edit the question into better form. – Willie Wong Mar 8 '13 at 12:56

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