Black hole singularities: a numerical approach

TL;DR

The singularity structure of charged spherical collapse is studied by considering the evolution of the gravity-scalar field system and suggests the validity of the mass-inflation scenario.

Abstract

The singularity structure of charged spherical collapse is studied by considering the evolution of the gravity-scalar field system. A detailed examination of the geometry at late times strongly suggests the validity of the mass-inflation scenario~\cite{PI:90}. Although the area of the two-spheres remains finite at the Cauchy horizon, its generators are eventually focused to zero radius. Thus the null, mass-inflation singularity {\em generally}\/ precedes a crushing $r=0$ singularity deep inside the black hole core. This central singularity is spacelike.

Figures (4)

• Figure 1: A spacetime diagram showing the setting of the numerical integration. The spacetime is Reissner-Nordström for $v<v_0$. Scalar field falls into the black hole across the event horizon, $\Gamma$. $PEH$ is the past event horizon, located at $v=-\infty$. $AH$ is the outer apparent horizon, and $IAH$ is the inner apparent horizon of the charged black hole. A couple of lines of constant $r$ are shown in light gray. The Cauchy horizon ($CH$) is a singular hypersurface which contracts to meet the central singularity at $r=0$. All singularities are indicated by thick lines in the diagram.
• Figure 2: The null rays in the $rv$-plane. The inner-most rays terminate at $r=0$ in a finite advanced time, however there exist many geodesics which approach a finite radius at large $v$. Subsequent figures show $\overline{g}$ and $g$ along the five outermost rays indicated by different line types. The initial data is exponential with $\beta = 0.26$, $p = 1$ and $e^2 = 0.4$.
• Figure 3: These plots show $\ln|\overline{g}|$ along a selection of outgoing null rays which intersect the CH. The top graph shows $\ln|\overline{g}|$ against $\ln|v|$ for the inverse power data in Table I. The late time fall-off is clearly a power law too. The graph is $\ln|\overline{g}|$ against $v$ for the exponential data. The linear relation in this graph indicates an exponential fall-off at late times. The cusp in these figures corresponds to a change in the sign of $\overline{g}$ where the outgoing null ray intersects the outer apparent horizon of the black hole.
• Figure 4: $\ln |g|$ against advanced time along the outgoing null rays. The asymptotic form $g \sim e^{-\gamma v}$ is in remarkable agreement with predictions based on simplified models.