Mass inflation and curvature divergence near the central singularity in spherical collapse

We study spherical scalar collapse toward a black hole formation and examine the asymptotic dynamics near the central singularity of the formed black hole. It is found that, in the vicinity of the singularity, due to the strong backreaction of a scalar field on the geometry, the mass function inflates and the Kretschmann scalar grows faster than in the Schwarzschild geometry. In collapse, the Misner-Sharp mass is a locally conserved quantity, not providing information on the black hole mass that is measured at asymptotically flat regions.

Figure 1

Solutions near to the central singularity in neutral scalar collapse. (a,b) apparent horizon and singularity curve of the formed black hole. (c) Misner-Sharp mass function along the slices $(x=1,t=t)$ and $(x=2.5,t=t)$. (d,e) are results for the slice $(x=1,t=t)$. (d) $\ln r=a\ln \zeta +b$, $a=0.49948\pm 0.00004$, $b=0.2172\pm 0.0005$. $\sigma =a\ln \zeta +b$, $a=-0.1651\pm 0.0001$, $b=-0.083\pm 0.001$. (e) $\psi =a\ln \zeta +b$, $a=-0.18129\pm 0.00003$, $b=0.1322\pm 0.0003$. $\ln m=a\ln r+b$, $a=-1.670\pm 0.002$, $b=-1.258\pm 0.009$.

Figure 2

Spatial vs temporal variations near the central singularity in spherical collapse. $\mathrm{\Delta }\lambda$ and $-|a|\widehat{r}$ are orthogonal to the $r=r_1$ and $r=r_2$ constant curves. The value of $a$ is such that the magnitude of $-|a|\widehat{r}$ has the length as shown in the figure.

Figure 3

Verification of Eq. (21): $K\ge 32m^2/r^6$ for $r<r_{\mathrm{AH}}$. The sample slice is $(x=2.5,t=t)$. (a) results for the time range $t\in [0{;t}_0]$ with grid spacings $\mathrm{\Delta }x=\mathrm{\Delta }t=0.005$, where $t_0$ is the coordinate time for $r=0$. (b) results near $r=0$ obtained via mesh refinement. In the figure, $\text{Ratio}\equiv K/(32m^2/r^6)$.