Semiclassical analytic estimation of charged boson nebulae mass and the gravitationally radiated flux

In the present paper, we use the first-order approximate solutions to the non-linear system of Klein-Gordon-Maxwell-Einstein equations describing the minimally coupled charged spinless field to a spherically symmetric spacetime to analyze a becoming boson star. In the far future and long-range approximation, the analytical time-dependent mass formula has allowed us to point out several significant moments in the evolution of the boson nebula and to predict the necessary period to turn the collapsing star into a Schwarzschild black hole. Finally, we have semiclassically derived the transition amplitude for the graviton emission as well as the gravitational analog of the Umov-Poynting vector.