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I understand the differences between the various equivalence principles at the level of words, but I'm not sure how they follow from various equations. If I'm given a theory of gravity with metric $g_{\mu\nu}$ and matter fields $\psi$ with couplings specified, is there some easy way to see which equivalence principles the theory will satisfy?

As a more precise question, how does one see that General Relativity satisfies the Strong equivalence principle? Next, I believe that $f(R)$ theories only satisfy the Einstein equivalence principle. What is the important technical difference between $f(R)$ theories and General Relativity that causes the two to obey different equivalence principles?

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