Loop quantum dynamics of the Schwarzschild interior

We examine the Schwarzschild interior of a black hole, incorporating quantum gravitational modifications due to loop quantum gravity. We consider an improved loop quantization using techniques that have proven successful in loop quantum cosmology. The central Schwarzschild singularity is resolved and the implications for the fate of an in-falling test particle in the interior region is discussed. The singularity is replaced by a Nariai type universe. We discuss the resulting conformal diagram, providing a clear geometrical interpretation of the quantum effects.

Figure 1

These figures show the effective plots of $p_b=p_b(p_c)$ for the case with $\delta =\mathrm{const}$. The dashed lines represent the classical solution while the solid lines correspond to the solution of the effective quantum theory with $\delta =0.1$. Clearly, the initial singularity is removed and the spacetime contains a second black hole. The three plots on the left show the solution for varying mass parameter, $m=1$, 1.5, 2, respectively. On the right $p_b^{(0)}$ is varied in the three plots, $p_b^{(0)}=1$, $2/3$, $1/3$, respectively. One clearly notes (left) that the mass parameter $m$ controls the mass of the first (classical) black only, while (right) the parameter $p_b^{(0)}$ controls the mass of the second black hole.

Figure 2

These figures show the effective plots of $p_b=p_b(p_c)$ for the case where ${\delta }_b$, ${\delta }_c$ are not constants. The dashed lines represent the classical solution while the solid lines correspond to the solution of the effective quantum theory. The initial singularity is removed, however, not with a second black hole. The three plots on the left show the solution for varying mass parameter, $m=1$, 1.5, 2, respectively. On the right $p_b^{(0)}$ is varied in the three plots, $p_b^{(0)}=1$, $2/3$, $1/3$, respectively. One verifies that this quantization scheme does not depend on the choice of $p_b^{(0)}$ and thus the effective theory is independent of this gauge freedom.

Figure 3

In the left figure $b(T)$ (solid) and $p_c(T)$ (dashed) are plotted. Both tends towards constants as $T\to -\infty$. The right figure shows the ratio $c(T)/p_b(T)$ also tends to a constant asymptotically. This asymptotic behavior is indicative of a Nariai type metric of the form given in Eq. (36).

Figure 4

Left: $p_c$ as a function of proper time of radially in-falling geodesic. The black dashed line represents the classical behavior indicating that the particle reaches the singularity in finite proper time. The solid line is the geodesic inside the quantum black hole. The particle approaches the singularity but undergoes damped oscillations at a fixed two-sphere radius eventually settling in at a fixed value ${\overline{p}}_c$. For the natural values of the parameters $\gamma$, $\Delta$, the two-sphere radius is on the order of the Planck length. Right: The logarithm of the Kretschmann invariant as a function of the time coordinate $T$ is plotted. The dashed line is the classical value which behaves as $\exp (-6T)$ and diverges at the singularity $T=-\infty$. The solid line represents the quantum value which asymptotes to a fixed value which again indicates a resolution of the singularity. The numerical asymptotic values obtained for $p_c$ and the Kretschmann invariant $K$ are $p_c\approx 0.182l_{\mathrm{Pl}}^2$ and $K\approx 124.35m_{\mathrm{Pl}}^4$ which are both independent of the black-hole mass.

Figure 5

The left figure shows the Carter-Penrose diagram for the Schwarzschild spacetime. ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$ denote future and past null infinity, respectively. The usual Schwarzschild metric in curvature coordinates only covers region $I$. The Kantowski-Sachs form of the line element covers region $II$. The dotted lines represent the $r=\mathrm{\text{constant}}$ lines in the respective regions. The right figure shows the Carter-Penrose diagram of the nonsingular charged Nariai universe. The $\chi =0$ and $\chi =2\pi$ lines are identified, and the dotted lines stand for the constant two-sphere radii. The solid and dashed diagonal lines represent the disconnected future and past event horizon, respectively. Every point of this diagram represents a sphere of the same area. For later purpose we have indicated a shaded region.

Figure 6

In region $II$ we have indicated a specific $r=\mathrm{\text{constant}}$ line. This line separates the region with decreasing group orbits ($I{I}_S$) from the region where the 2-spheres have (effectively) constant volume ($I{I}_N$), which defines the Nariai spacetime.

Figure 7

These figures show the effective plots of $p_b=p_b(p_c)$. The dashed lines represent the classical solution while the solid lines correspond to the solution of the effective quantum theory. The initial singularity is removed and the spacetime is connected to a second black hole. The three plots on the left show the solution for varying mass parameter, $m=1$, 1.5, 2, respectively. On the right $p_b^{(0)}$ is varied in the three plots, $p_b^{(0)}=1$, $2/3$, $1/3$, respectively.