A Reissner-Nordstr\"om$+\Lambda$ black hole in the Friedman-Robertson-Walker universe

A charged, non-rotating, spherically symmetric black hole which has cosmological constant $\Lambda$ (Reissner-Nordstr\"om+$\Lambda$ or RN+$\Lambda$), active gravitational mass $M$ and electric charge $Q$ is studied with exterior Friedman-Robertson-Walker (FRW) universe in (2+1) dimensional spacetime. We find a new classes of exact solutions of the charged black hole where the generalized Birkoff's theorem is assumed to be valid. It is found that the cosmological constant is negative inside the black hole. We confirm it from the geodesic equations too. The cosmological constant is found to be dependent on $R, Q$ and $a(v)$ which correspond to the areal radius, charge, of the black hole and the scale factor of the universe respectively. We note that the expansion of the universe affects the size and the mass of the black hole. An important observation is that, for an observer at infinity, both the mass and charge of black hole increase with the contraction of the universe and decrease with the expansion of the universe.