From here on, it is useful to define m = G M / c 2 . For McVittie metrics with the general expanding FLRW solutions properties H ( t ) = a ˙ ˙ ( t ) / a ( t ) > 0 and lim t → → ∞ ∞ H ( t ) = H 0 = 0 , the spacetime has the property of containing at least two singularities. One is a cosmological, null-like naked singularity at the smallest positive root r − − of the equation 1 − − 2 m / r − − H 0 2 r 2 = 0 . This is interpreted as the black hole event-horizon in the case where H 0 > 0 . For the H 0 > 0 case, there is an event horizon at r = r − − , but no singularity, which is extinguished by the existence of an asymptotic Schwarzschild-De Sitter phase of the spacetime.