Generalized permittivity tensor for the description of waves in general relativistic plasma around a Schwarzschild black hole

The effects of gravitation on the permittivity tensor in the relativistic electron-positron or ions plasma in a frame of reformulated relativistic two-fluid equations by gravitational effects due to the event horizon using the 3 + 1 formalism of general relativity are investigated. The plasma is assumed to be freefalling in the radial direction toward the event horizon due to the strong gravitational field of a Schwarzschild black hole. The elements of the generalized permittivity tensor in this configuration are obtained. It is shown that the permittivity tensor could be written as a summation of two parts: Hermitian and non-Hermitian parts. Furthermore, the generalized dispersion relations are investigated for transverse and longitudinal modes. In the absence of gravitation effects, correctness of the obtained results is confirmed.