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Abstract

Consider an observer surrounded by a charged, conducting elevator (assume
that the charge is isolated from the observer). In the presence of the external
electric field the elevator will accelerate however, due to the screening
effect, the observer will not be able to detect any electromagnetic field.
According to the equivalence principle, the observer may identify the cause of
the acceleration with the external gravitational field. However the elevator's
motion is given by Lorentz-force equation. Therefore there should exist a
metric, depending on electromagnetic potential, for which the geodesics
coincide with the trajectories of the charged body in the electromagnetic
field.
We give a solution to this problem by finding such metric. In doing so one
must impose a constraint on the electromagnetic field in a certain way. That
constraint turns out to be achievable by marginal gauge transformations whose
phase is closely related to the Hamilton-Jacobi function.
Finally we show that for weak fields the Einstein-Hilbert action for the
proposed metric results in the Stueckelberg massive electrodynamics. For strong
fields (e.g. at small scales) the correspondence is broken by a term that at
the same time makes the theory non-renormalizable. We conjecture the existence
of a quantum theory whose effective action reproduces the non-renormalizable
term and hence the Einstein-Hilbert action.