Principles of equivalence, Eötvös experiments, and gravitational red-shift experiments: The free fall of electromagnetic systems to post-post-Coulombian order

We analyze free fall in an external static gravitational field of a composite test body consisting of electromagnetically interacting charged particles. The computation employs the formalism developed by Lightman and Lee (LL) to encompass a wide class of metric and nonmetric theories of gravitation. By working to "post-Coulombian" order [$O\left(\frac{g{v}^2}{c^2}\right)\sim O\left(\frac{g{e}^2}{\mathrm{mr}{c}^2}\right)$, where $g$ is the acceleration of gravity and $v$, $e$, $m$, and $r$ are the velocity, charge, mass, and separation of the charged particles], LL found violations of the composition independence of free fall (weak equivalence principle) predicted by nonmetric theories of gravity that depend on the electrostatic structure of the body. We extend their calculation to "post-post-Coulombian" order [$O\left(\frac{g{v}^4}{c^4}\right)\sim O\left(\frac{g{v}^2{e}^2}{\mathrm{mr}{c}^4}\right)\sim O\left(\frac{g{e}^4}{m^2{r}^2{c}^4}\right)$] and find violations that depend on the magnetostatic structure of the body. We show that results from the current generation of Eötvös experiments when combined with our estimates for nuclear magnetostatic energies yield an upper limit on the nonmetric parameter $\left|{\Lambda }_0\right|<\mathrm{}6\mathrm{}\times {10}^{-6\mathrm{}}$ (${\Lambda }_0=0\mathrm{}$ in all metric theories of gravity). The Lightman-Lee analysis did not constrain this parameter strongly. This limit is significantly tighter than that provided by the SAO-NASA hydrogen-maser rocket gravitational red-shift experiment, performed in 1976. We also discuss possible tests of other nonmetric parameters by improved Eötvös experiments.