  Within the isospin- and momentum-dependent transport model IBUU11, we examine
the relativistic retardation effects of electrical fields on the
$\pi^{-}/\pi^{+}$ ratio and neutron-proton differential transverse flow in
heavy-ion collisions at intermediate energies. Compared to the static Coulomb
fields, the retarded electric fields of fast-moving charges are known to be
anisotropic and the associated relativistic corrections can be significant.
They are found to increase the number of energetic protons in the participant
region at the maximum compression by as much as 25\% but that of energetic
neutrons by less than 10\% in $^{197}$Au+$^{197}$Au reactions at a beam energy
of 400 MeV/nucleon. Consequently, more $\pi^{+}$ and relatively less $\pi^{-}$
mesons are produced, leading to an appreciable reduction of the
$\pi^{-}/\pi^{+}$ ratio compared to calculations with the static Coulomb
fields. Also, the neutron-proton differential transverse flow, as another
sensitive probe of high-density symmetry energy, is also decreased appreciably
due to the stronger retarded electrical fields in directions perpendicular to
the velocities of fast-moving charges compared to calculations using the
isotropic static electrical fields. Moreover, the retardation effects on these
observables are found to be approximately independent of the reaction impact
parameter.
