It's Hard to Learn How Gravity and Electromagnetism Couple
  We construct the most general effective Lagrangian coupling gravity and
electromagnetism up to mass dimension 6 by enumerating all possible non-minimal
coupling terms respecting both diffeomorphism and gauge invariance. In all,
there are only two unique terms after field re-definitions; one is known to
arise from loop effects in QED while the other is a parity violating term which
may be generated by weak interactions within the standard model of particle
physics. We show that neither the cosmological propagation of light nor,
contrary to earlier claims, solar system tests of General Relativity are useful
probes of these terms. These non-minimal couplings of gravity and
electromagnetism may remain a mystery for the foreseeable future.
