Black hole formation from the gravitational collapse of a nonspherical network of structures

We examine the gravitational collapse and black hole formation of multiple nonspherical configurations constructed from Szekeres dust models with positive spatial curvature that smoothly match to a Schwarzschild exterior. These configurations are made of an almost spherical central core region surrounded by a network of “pancake-like” overdensities and voids with spatial positions prescribed through standard initial conditions. We show that a full collapse into a focusing singularity, without shell crossings appearing before the formation of an apparent horizon, is not possible unless the full configuration becomes exactly or almost spherical. Seeking for black hole formation, we demand that shell crossings are covered by the apparent horizon. This requires very special fine-tuned initial conditions that impose very strong and unrealistic constraints on the total black hole mass and full collapse time. As a consequence, nonspherical nonrotating dust sources cannot furnish even minimally realistic toy models of black hole formation at astrophysical scales: demanding realistic collapse time scales yields huge unrealistic black hole masses, while simulations of typical astrophysical black hole masses collapse in unrealistically small times. We note, however, that the resulting time-mass constraint is compatible with early Universe models of primordial black hole formation, suitable in early dust-like environments. Finally, we argue that the shell crossings appearing when nonspherical dust structures collapse are an indicator that such structures do not form galactic mass black holes but virialize into stable stationary objects.

Figure 1

Comparison of density profiles. The panels show the profile of the mass density $\rho (r,t,\theta )$ evaluated along the curve of the angular maxima of $\mathbf{W}$ (red and solid) and the q-density average ${\rho }_q(r,t)$ [black and dotted, Eq. (7)]. The left panel shows a typical initial setup of multiple overdensity structures. Evolved with the growing mode, the Szekeres regions II and III will eventually present shell crossings. On the other hand, the right panel shows the evolution with a dominant decaying mode which flattens the profile and allows for a collapse free of shell crossings (at least at late times).

Figure 2

Density mass in the equatorial plane in units of $M_{\odot }/{\mathrm{Mpc}}^3$. Equatorial projection of the density mass distribution at a time close, but before to the time of shell crossings. The “$x$” and “$y$” axes respectively correspond to $R\cos \varphi$ and $R\sin \varphi$, with $R=ar$.

Figure 3

Collapse and shell crossings times. Grey, light red and blue curves respectively represent the times of collapse and shell crossing along the direction of the maxima and minima of the dipole for the case $\mathrm{\Lambda }=0$. Black, dark red and violet lines respectively represent the time of collapse and shell crossings considering $\mathrm{\Lambda }$.

Figure 4

Hiding the shell crossing singularities behind the apparent horizon surface. The apparent horizon (black curve) hides the shell crossings thus they are already inside the black hole by the time they appear. Notice that in the yellow-shaded area the apparent horizon first appears at the point B, where $t_{\mathrm{AH}}$ has a local minimum, $t_{\mathrm{AH}}^{(\mathrm{B})}$. At all times in the interval ($t_{\mathrm{AH}}^{(\mathrm{B})}$, $t_{\mathrm{AH}}^{(\mathrm{A})}$), the mass swallowed up by the singularity is necessarily smaller than the mass into the AH.

Figure 5

Black hole formation time as function of the mass. The left panel shows with a black line the time in which the entire region of Szekeres has been hidden behind the apparent horizon as a function of black hole mass. For the values of time and masses shaded in red, which include the masses of astrophysical black holes, the shell crossings are formed outside the apparent horizon. The curve in the right panel represents the collapse time as a function the of the mass demanding the condition (23) to be satisfied. We have also indicated with dashed grey lines the typical masses of active galactic nuclei (AGN, $\sim {10}^9{M}_{\odot }$), galaxies ($\sim {10}^{11}{M}_{\odot }$) and superclusters ($\sim {10}^{15}{M}_{\odot }$), as a reference.