It is shown that the geodesics with negative energy for rotating black holes cannot originate or terminate inside the ergosphere. Their length is always finite and this leads to conclusion that they must originate and terminate inside the gravitational radius of the ergosphere.Comment: 9 pages, no figures, published versio

The gravitational collapse in generalized Vaidya space-time is considered. It is known that the end state of gravitational collapse, as to whether a black hole or a naked singularity is formed, depends on the mass function M (v, r). Here we give conditions for the mass function which correspond to the equation of the state P = αρ. where α ∈ (0, 1/3], and according to these conditions we obtain either a black hole or a naked singularity as the end state of gravitational collapse. We also give the conditions for the mass function under which the singularity is gravitationally strong. We present simple examples showing when the result of gravitational collapse is a naked singularity and when this singularity is strong.

In this paper, we consider the gravitational collapse of generalized Vaidya space–time when the matter satisfies the equation of the state either [Formula: see text] or [Formula: see text], where [Formula: see text]. We show that in the case when type I of matter field is dust, then the apparent horizon will never appear but there is no a family of null radial future-directed geodesics which terminate at the central singularity in the past. Also, we show that in the case of negative pressure, the result of the gravitational collapse might be the naked singularity and the apparent horizon appears and in very short time disappears again. In the case of the negative pressure, we show that the result of the gravitational collapse might be the eternal naked singularity.

According to Penrose's effect, particles with negative energy can exist in the ergospheres of rotating black holes. We analyze geodesics for such particles and show that there are no circular and elliptic orbits in the ergosphere of a rotating black hole. We also show that there are geodesics which begin and terminate at the singularity and present the conditions under which such geodesics do not begin and terminate at the singularity.

The gravitational collapse of generalized Vaidya spacetime is considered. It is known that the endstate of gravitational collapse, as to whether a black hole or a naked singularity is formed, depends on the mass function M (v, r). Here we give conditions for the mass function which corresponds to the equation of the state P = αρ where α ∈ (0, ] and according to these conditions we obtain either a black hole or a naked singularity at the endstate of gravitational collapse. Also we give conditions for the mass function when the singularity is gravitationally strong.