The Threat to the Equivalence Principle According to Fulton and Rohrlich

To review what has been said, if we assume that the Lorentz-Dirac equation is applicable, this equation of motion for a charged particle is satisfied by hyperbolic motion and differs in no way from the equation of motion of a neutral particle, assuming of course that the operative external forces compelling them to that motion are non-electromagnetic, since otherwise those forces could not affect a neutral particle. Of course, we have to be careful not to state a truism: the particle forced to follow hyperbolic motion will follow hyperbolic motion! The point is that the same external force can get both the neutral and the charged particle undergoing hyperbolic motion. This happens because the extra term in the Lorentz-Dirac equation for the charged particle happens to be zero whenever we arrange for the motion to be a uniform acceleration.

As a consequence, Fulton and Rohrlich conclude that, in a static homogeneous gravitational field (SHGF), a neutral and a charged particle will follow the same trajectory with the same time dependence. If Galileo had dropped a neutron and a proton fromthe leaning tower of Pisa, they would have fallen equally fast! Acharged and a neutral particle in an SHGF behave exactly alike, except for the emission of radiation fromthe charged particle. At least, this is the conclusion from the Lorentz- Dirac equation, with the proviso that we remain within the jurisdiction of special relativity, i.e., they are referring to a special relativistic theory of gravity, as proposed in Chap. 3. The latter is an important proviso here.We apply the dynamical equation of special relativity when we analyse the 4-force due to an SHGF, and we conclude that this SHGF will accelerate the particles uniformly. Then the charged and neutral particle behave exactly alike in this case, except for the emission of radiation from the charged particle.