The dynamic Kerr response of noninteracting rigid polar linear molecules in superimposed strong dc and weak ac electric fields including inertial effects is investigated. The calculations are carried out for rotation in space in the context of linear response theory for the extended rotational (J-diffusion) model and extend the results of previous work. The birefringence spectrum is evaluated for typical values of the model parameters. Simple analytical equations for the spectrum are obtained for the limiting cases of rare and frequent collisions. The limit of frequent collisions allows us to compare the J-diffusion model with the Fokker-Planck model in the overdamped limit, where the inertial effects are neglected. It is found just as in dielectric relaxation that the birefringence relaxation times are very similar for the two models, but not identical. Also a crossover region between the low and high damping limits is exhibited by the Kerr effect relaxation time just as in Kramers' escape rate theory. In addition, this crossover region is shifted to higher friction parameter values when the applied dc field is increased. A comparison with previous theories is given.