Narrow Results

In this paper, the structure of the generalized Vaidya space–time when the type-II of the matter field satisfies the equation of the state $P=\rho$ is investigated. Satisfying all energy conditions, we show that this space–time contains the “eternal” naked singularity. It means that once the singularity is formed it will never be covered with the apparent horizon. However. in the case of the apparent horizon formation the resulting object is a white hole. We also prove that this space–time contains only null naked singularity.

We discuss the macroscopic quantum tunneling from the black hole to the white hole of the same mass. Previous calculations in [G. E. Volovik, Universe 6, 133 (2020)] demonstrated that the probability of the tunneling is $p\propto exp(-2S_{\mathrm{\text{BH}}})$, where $S_{\mathrm{\text{BH}}}$ is the entropy of the Schwarzschild black hole. This in particular suggests that the entropy of the white hole is with minus sign the entropy of the black hole, $S_{\mathrm{\text{WH}}}(M)=-S_{\mathrm{\text{BH}}}(M)=-A/(4G)$. Here, we use a different way of calculations. We consider three different types of the hole objects: black hole, white hole and the fully static intermediate state. The probability of tunneling transitions between these three states is found using singularities in the coordinate transformations between these objects. The black and white holes are described by the Painleve–Gullstrand coordinates with opposite shift vectors, while the intermediate state is described by the static Schwarzschild coordinates. The singularities in the coordinate transformations lead to the imaginary part in the action, which determines the tunneling exponent. For the white hole the same negative entropy is obtained, while the intermediate state — the fully static hole — has zero entropy. This procedure is extended to the Reissner–Nordström black hole and to its white and static partners, and also to the entropy and temperature of the de Sitter Universe.