The nonequilibrium character of the spin glass dynamics has been studied in Monte Carlo simulations of two- and three-dimensional short-range Ising spin glass systems. The spin system was quenched in zero field to a temperature T. After equilibrating for a waiting time tw a weak magnetic field was applied and the time dependence of the magnetization M(t) was studied. Our results show that the relaxation rate, S(t) = ∂M(t)/∂log t, exhibits a maximum at a time t ≈ tw, which is consistent with results from experiments. If the temperature is increased or decreased prior to the field application, the apparent age ta of the system, as defined by the maximum in relaxation rate, is changed. The results are qualitatively in agreement with experiments as well as with theoretical models for the nonequilibrium spin glass dynamics