arXiv2025
We carefully investigate, extend, and shed new light on the McVittie exact solution of Einstein's gravity (EG) with the focus on the implications in the Universe we live in. It turns out that the only known exact solution of EG, which interpolates between an asymptotic homogeneous and isotropic Universe and a Schwarzschild black hole, is actually singular in 2M, namely, the curvature invariants diverge and the spacetime is geodetically incomplete in 2M. Very important: all energy conditions are satisfied except the dominant one (DEC), which is violated inside the radius 8M/3. Notice that 2M is not the event horizon, but a curvature singularity covered by an apparent horizon that, at the actual stage of the Universe, nearly coincides with 2M. Moreover, the curvature singularity is not analytic with respect to the dynamics of the Universe encoded in the Hubble function $H(t)$: for arbitrarily small but not zero $H^\prime(t)$, the curvature invariants are singular, while for $H^\prime(t)$ identically zero, they are regular. Therefore, we can not analytically decouple the black hole from the entire Cosmos, namely, we can not assume the Schwarzschild solution locally and the FRW metric at large scale without violating the analyticity of the metric. Since the spacetime does not exist for $r \leqslant$2M, and since the DEC is violated for r<8M/3, we are allowed to doubt the existence of black holes in our Universe as understood up to now. In particular, the violation of DEC seems catastrophic for the spacetime stability below 8M/3. We build and study a toy model for the gravitational collapse, generalizing the Vaidya to the McVittie-Vaidya metric. Although dynamical, the singularities remain in the same locations. Finally, in order to achieve the curvature smoothness and geodesic completion, we propose two solutions: one in Einstein's conformal gravity, and the other replacing M with M(r).