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The relativistic Cowling approximation, where the metric perturbations areneglected during the fluid oscillations, is often adopted for considering thegravitational waves from the protoneutron stars (PNSs) provided viacore-collapse supernova explosions. In this study, we evaluate how the Cowlingapproximation works well by comparing the frequencies with the Cowlingapproximation to those without the approximation. Then, we find that thebehavior of the frequencies with the approximation is qualitatively the sameway as that without the approximation, where the frequencies with theapproximation can totally be determined within $\sim 20\%$ accuracy. Inparticular, the fundamental mode with the Cowling approximation isoverestimated. In addition, we also discuss the damping time of variouseigenmodes in gravitational waves from the PNSs, where the damping time for thePNSs before the avoided crossing between the $f$- and $g_1$-modes, is quitedifferent from that for cold neutron stars, but it is more or less similar tothat for cold neutron stars in the later phase. The damping time is long enoughcompared to the typical time interval of short-Fourier transformation thatoften used in the analysis, and that ideally guarantees the validity of thetransformation.

Accuracy of relativistic Cowling approximation in protoneutron star asteroseismology