  In this paper it is presented a manifestly covariant formulation of the
Aharonov-Bohm (AB) phase difference for the magnetic AB effect . This covariant
AB phase is written in terms of the Faraday 2-form F and using the
decomposition of F in terms of the electric and magnetic fields as
four-dimensional (4D) geometric quantities. It is shown that there is a static
electric field outside a stationary solenoid with resistive conductor carrying
steady current, which causes that the AB phase difference in the magnetic AB
effect may be determined by the electric part of the covariant expression, i.e.
by the local influence of the 4D electric field and not, as generally
accepted,in terms of nonzero vector potential.
