Do Black Holes Exist?

TL;DR

The paper interrogates whether Einstein gravity permits traditional black holes within our FRW-like Universe by analyzing the McVittie solution, revealing a curvature singularity at $r=2M$ that is not analytic in $H'(t)$ and is only hidden behind an apparent horizon, casting doubt on the standard Schwarzschild interpretation in a cosmological setting. It demonstrates geodesic incompleteness at $r=2M$ and explores dynamical collapse via a McVittie-Vaidya extension, showing the singular surface remains at $r=2M(v)$. To resolve these pathologies, the authors propose two routes: (i) conformal gravity via conformalon-type conformal rescalings that push the singular surface to unattainable regions and remove the Big-Bang singularity, and (ii) a radial-mass profile $M(r)$ that eliminates the $2M$ surface at the cost of violating energy conditions. The work highlights the non-perturbative coupling between local black-hole structure and cosmic expansion and suggests that genuinely non-singular, horizon-structured spacetimes may require non-standard gravity or matter profiles, with potential cosmological implications for black-hole physics and cosmic evolution.

Abstract

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).

Figures (5)

• Figure 1: The plots on the left represents the pressure for three different values of $t$. The presssure $p$ is positive for $r>2M$ and diverges as $r$ tends to $2M$. The plot on the right shows the DEC and its violation for all values of $r$ lower than a certain radius, namely $8M/3$ as explained in the main text. In the the plots we made the choice $M = 10$.
• Figure 2: Plot of the affine parameter $\lambda$ as a function of $R$. The dashed line represents the solution of the geodesic equation without rescaling, whereas the continuous line represents the solution with the lowest (integer) power $n=2$ in $S(r)$ of the form \ref{['fattore_conforme_generico']} ($n = 2$). Since the latter diverges as $R$ tends to 0, it comes that now the spacetime is geodesically complete in $r = 2M$. ${\rm const.} = 10$
• Figure 3: The first two plots at the top show the energy density and the pressure as functions of the radial coordinate $r$, for three different values of $t$, for a $M(r)$ shaped like \ref{['M(r)']} with $M=1$ and $b=2$. The two plots at the bottom show the NEC (or WEC, if we combine the condition $\rho + p\ge0$ with the requirement that $\rho\ge0$) and the DEC are not satisfied.
• Figure 4: The two apparent horizons $r_{-}$ and $r_{+}$, along with the curvature singularity in $r = 2M$ and the radius $r = 8M/3$, are plotted as functions of the Hubble parameter $H(t)$ (here $M=2000$). Hence, the time coordinate $t$ increases from right to left. A thin gray vertical line near zero marks the value of $H(t)$ at the present time. We can infer two features: $2M$ is almost a naked singularity, and the instability region marked by $r = 8M/3$ is outside the horizon.
• Figure 5: The cosmological horizon which results from plotting together the two solutions of \ref{['eq_app_hor_with_M(r)']} that are physically relevant. The two roots are: $r_a$ and $r_b$, for $M(r)$ given by \ref{['M(r)']}. Here $M = 1$ and $b = 2$.