Black-Hole Singularities: A New Critical Phenomenon

The singularity inside a spherical charged black hole, coupled to a spherical, massless scalar field is studied numerically. The profile of the characteristic scalar field was taken to be a power of advanced time with an exponent $\alpha >0$. A critical exponent ${\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ exists. For exponents below the critical one ($\alpha <{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$) the singularity is a union of spacelike and null sectors, as is also the case for data with compact support. For exponents greater than the critical one ($\alpha >{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$) an all-encompassing, spacelike singularity evolves, which completely blocks the “tunnel” inside the black hole, preventing the use of the black hole as a portal for hyperspace travel.

Figure 1

The Penrose diagrams for the two types of singularities. Upper diagram: an all-encompassing spacelike singularity. Lower diagram: The singularity includes both a spacelike sector and a null sector at the CH.

Figure 2

Spacetime diagrams with variable $A$. Lower diagram: Spacetime diagram for $\alpha >{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$. Upper diagram: Spacetime diagram for $\alpha <{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$. In both cases spacelike singularities are depicted by a bold solid line. The outgoing null ray (dashed line) at $u_{*}$ intersects with the spacelike singularity at a corresponding affine parameter (along $u_0$) ${\gamma }^{*}$ (dotted lines). With lower value of $A$, the event horizon (EH) moves inward, and the spacelike singularity occurs later.

Figure 3

Affine parameter at $u_0$, ${\gamma }_{*}$, at which a spacelike singularity is intersected along an outgoing null ray $u_{*}$ as a function of $A$. Here, $\alpha =0.3$. Numerical data are depicted by circles, and the solid line is proportional to $A^{-5.926}$.

Figure 4

Affine parameter at $u_0$, ${\gamma }_{*}$, at which a spacelike singularity is intersected along an outgoing null ray $u_{*}$ as a function of $A$. Here, $\alpha =0.1$. Numerical data are depicted by circles.

Figure 5

The exponent $\beta$ as a function of $\alpha$. The circles are the data points, and the solid curve is proportional to ${\alpha }^{-1.37}$.