Mass inflation in the loop black hole

In classical general relativity the Cauchy horizon within a two-horizon black hole is unstable via a phenomenon known as mass inflation, in which the mass parameter (and the spacetime curvature) of the black hole diverges at the Cauchy horizon. Here we study this effect for loop black holes—quantum gravitationally corrected black holes from loop quantum gravity—whose construction alleviates the $r=0$ singularity present in their classical counterparts. We use a simplified model of mass inflation, which makes use of the generalized Dray-’t Hooft relation, to conclude that the Cauchy horizon of loop black holes indeed results in a curvature singularity similar to that found in classical black holes. The Dray-’t Hooft relation is of particular utility in the loop black hole because it does not directly rely upon Einstein’s field equations. We elucidate some of the interesting and counterintuitive properties of the loop black hole, and corroborate our results using an alternate model of mass inflation due to Ori.

Figure 1

A Penrose diagram displaying the spacetime outside of the black hole as well as between the horizons. We illustrate the ingoing null radiation and label the direction of increasing advanced time $v$.

Figure 2

Two colliding null shells separate spacetime into four distinct regions: $A$, $B$, $C$ and $D$. In general the shells before the collision, 3 and 4, will not have the same properties as the shells after collision, 1 and 2. The two-sphere on which they collide is labeled $S$.

Figure 3

The colliding null shells split the loop spacetime into four regions, the metrics in each of which can be related via the generalized DTR relation. We consider the scenario in which the ingoing shell is arbitrarily close to the Cauchy horizon.

Figure 4

$F_A$ plotted as a function of $m_A$ with fixed values $r=1$, $P=0.6$ and $a_0=0.5$. The top figure gives a large view of the function; note that $F_A$ diverges at $m_A=-r/2P=-5/6$. Although it is not obvious from the figure, $F_A$ asymptotes to a value of $r^4/(r^4+a_0^2)=0.8$ when $m_A\to \pm \infty$. The bottom figure displays the range of $m_A$ from 0 to 3 and we include a horizontal line to emphasize that, given a value of $F_A$, there are two possible values of $m_A$. As $F_A$ increases we see that $m_A$ will either decrease or increase depending on whether the initial value is taken to be on the left or right side of the minimum in $F_A$, respectively.

Figure 5

Solution $r(v)$ of an outgoing null geodesic, with parameters $M=1$, $P^2=0.1$ and $a_0^2=0.15$, giving $r_{-}=0.2$. The solution quickly passes by the radius 0.2 before coming to a stop (this is where it intersects the inner apparent horizon) and starts to increase, eventually settling onto the Cauchy horizon at $v=\infty$. The vector field indicates the slope of $r(v)$ at each point $(v,r)$. Specifically, this is given by $dr/dv=f/2$.

Figure 6

Schematic illustrating the situation considered in the Ori model. We see both the continuous ingoing radiation as well as the outgoing shell $\Sigma$. This shell acts to split the spacetime into two regions.

Figure 7

The results of numerically integrating Eq. (a44) with parameters $M=1$, $P^2=0.25$ and $a_0^2=0.1$; this gives $-r_{-}/2P=-MP=-0.5$. For the top graph the initial conditions were $m_{+}=0.2$, which results in $m_{+}(v)$, asymptotically approaching $-r_{-}/2P=-0.5$. For the bottom graph the initial conditions were $m_{+}=0.3$, which results in $m_{+}(v)$ diverging to infinity at finite $v$. These are the two solutions obtained from the DTR analysis.