  Spin inertia measurements are a novel experimental tool to study long-time
spin relaxation processes in semiconductor nanostructures. We develop a theory
of the spin inertia effect for resident electrons and holes localized in
quantum dots. We consider the spin orientation by short optical pulses with
arbitrary pulse area and detuning from the trion resonance. The interaction
with an external longitudinal magnetic field and the hyperfine interaction with
the nuclear spin bath is considered in both the ground and excited (trion)
states of the quantum dots. We analyze how the spin inertia signal depends on
the magnetic field (polarization recovery) and on the modulation frequency of
the helicity of the pump pulses as well as on their power and detuning. In
particular, we elaborate how approaching the saturation limit of the spin
polarization influences the measurements. The quantitative description of spin
inertia measurements will enable the determination of the parameters of spin
dynamics such as the spin relaxation times in the ground and excited states and
the parameters of the hyperfine interaction. Finally, we predict the emergence
of resonant spin amplification due to the transverse components of the nuclear
spin fluctuations, which manifests itself as oscillations of the spin
polarization as a function of the longitudinal magnetic field.
