We investigate the wave properties for isothermal plasma state around to the de Sitter black hole’s horizon using 3+1 split of spacetime. The corresponding Fourier analyzed perturbed perfect GRMHD equations are used to obtain the complex dispersion relations. We obtain the real values of the wave number k, from these relations, which are used to evaluate the quantities like phase and group velocities etc. These have been analyzed graphically in the neighborhood of the horizon.

It is suggested that magnetosonic waves (also known as equatorial noise) can scatter radiation belt electrons in the Earth’s magnetosphere. Therefore, it is important to understand the global distribution of these waves between the proton cyclotron frequency and the lower hybrid resonance frequency. In this study, we developed an empirical model for estimating the global distribution of magnetosonic wave amplitudes and wave normal angles. The model is based on the entire mission period (approximately 2012–2019) of observations of Van Allen Probes A and B as a function of the distance from the Earth (denoted by L*), magnetic local time (MLT), magnetic latitude (λ), and geomagnetic activity (denoted by the Kp index). In previous studies the wave distribution inside and outside the plasmasphere were separately investigated and modeled. Our model, on the other hand, identifies the wave distribution along with the ambient plasma environment—defined by the ratio of the plasma frequency (fpe) to the electron cyclotron frequency (fce)—without separately determining the wave distribution according to the plasmapause location. The model results show that, as Kp increases, the dayside wave amplitude in the equatorial region intensifies. It thereby propagates the intense region towards the wider MLT and inward to L* < 4. In contrast, the fpe/fce ratio decreases with increasing Kp for all regions. Nevertheless, the decreasing aspect differs between regions above and below L* = 4. This finding implies that the particle energy and pitch angle that magnetosonic waves can effectively scatter vary depending on the locations and geomagnetic activity. Our model agrees with the statistically observed wave distribution and ambient plasma environment with a coefficient of determination of > 0.9. The model is valid in all MLTs, 2 ≤ L* < 6, |λ| < 20°, and Kp ≤ 6.