Progress in evaluating a possible electromagnetic interaction energy in a gravitational field

Abstract: The Lorentz-Poincar\'e interpretation of special relativity (SR) keeps the classical concepts of separated space and time, at the price of postulating an indetectable preferred inertial frame or ether". But SR does not contain gravity. The presence of gravity could make the ether detectable. This is one idea behind thescalar ether theory of gravitation" (SET), which coincides with SR if the gravity field vanishes, and passes a number of tests. However, the coupling of SET with the Maxwell electromagnetic (EM) field needs to use the theory's dynamical equation for the energy tensor in a non-trivial way. It cannot be assumed that the energy tensors of the charged matter and the EM field add to give the total energy tensor, source of the gravitational field. Thus, an additional, ``interaction" energy tensor ${\bf T}\mathrm{inter}$ has to be postulated. Asking that ${\bf T}\mathrm{inter}$ is Lorentz-invariant in the situation of SR, fixes its form. It depends only on a scalar field $p$. ${\bf T}_\mathrm{inter}$ is an exotic kind of matter and is distributed in the whole space, hence it could contribute to dark matter. For a weak gravitational field, $p$ obeys a first-order partial differential equation (PDE) involving the EM field and the Newtonian potential. However, the EM field varies on the scale of the wavelength, which is extremely small. To get the field $p$ in a galaxy, some averaging has to be done. After several attempts based on the homogenization theory, a simpler way has been found recently: If the macro-averages of $p$ and the EM field vary smoothly, it can be shown that the PDE for $p$ remains valid in the same form with spacetime-averaged fields. The current stage of calculations will also been shown.