  In this article we investigate the beam energy dependence of relativistic
retardation effects of electrical fields on the single and double
$\pi^{-}/\pi^{+}$ ratios in three heavy-ion reactions with an isospin- and
momentum-dependent transport model IBUU11. With the beam energy increasing from
200 to 400 MeV/nucleon, effects of the relativistically retarded electrical
fields on the $\pi^{-}/\pi^{+}$ ratio are found to increase gradually from
negligibly to considerably significant as expectedly; it is however, the
interesting observation is the relativistic retardation effects of electrical
fields on the $\pi^{-}/\pi^{+}$ ratio are becoming gradually insignificant as
the beam energy further increasing from 400 to 800 MeV/nucleon. Moreover, we
also investigate the isospin dependence of relativistic retardation effects of
electrical fields on the $\pi^{-}/\pi^{+}$ ratio in two isobar reaction systems
of $^{96}$Ru+$^{96}$Ru and $^{96}$Zr+$^{96}$Zr at the beam energies from 200 to
800 MeV/nucleon. It is shown that the relativistic retardation effects of
electrical fields on the $\pi^{-}/\pi^{+}$ ratio are independent of the isospin
of reaction. Furthermore, we also examine the double $\pi^{-}/\pi^{+}$ ratio in
reactions of $^{96}$Zr+$^{96}$Zr over $^{96}$Ru+$^{96}$Ru at the beam energies
from 200 to 800 MeV/nucleon with the static field and retarded field,
respectively. It is shown the double $\pi^{-}/\pi^{+}$ ratio from two reactions
is still an effective observable of symmetry energy without the interference of
electrical field due to using the relativistic calculation compared to the
nonrelativistic calculation.
