The Painleve-Gullstrand coordinates are extended to describe the black hole in cosmological environment: Schwarzschild-de-Sitter hole, which has two horizons. extension is made using Arnowitt-Deser-Misner formalism. In this extension, describes metric whole range of radial $0<r < \infty$, there point $r=r_0$ at shift function (velocity) changes sign. At observer rest, while observers $r<r_0$ fr...

using the mean-value theorem for integrals we tried to solved the nonlinear integral equations of hammerstein type . the mean approach is to obtain an initial guess with unknown coefficients for unknown function y(x). the procedure of this method is so fast and don&apos;t need high cpu and complicated programming. the advantages of this method are that we can applied for those integral equation...

We discuss the tension between possible existence of Painleve-Gullstrand coordinate systems versus explicit geometrical features Kerr spacetime; a subject interest to Professor Thanu Padmanabhan in weeks immediately preceding his unexpected death. shall carefully distinguish strong and weak systems, conformal variants thereof, cataloguing what we know can cannot be done -- sometimes make global...

We discuss the macroscopic quantum tunneling from black hole to white of same mass. Previous calculations in Ref.[1] demonstrated that probability is $p \propto \exp(-2S_\text{BH})$, where $S_\text{BH}$ entropy Schwarzschild hole. This particular suggests with minus sign hole, $S_\text{WH}(M)=- S_\text{BH}(M)= - A/(4G)$. Here we use a different way calculations. consider three types objects: an...