Effects of retarded electrical fields on observables sensitive to the high-density behavior of nuclear symmetry energy in heavy-ion collisions at intermediate energies

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.