Effective polymer dynamics of $D$-dimensional black hole interiors

We consider two different effective polymerization schemes applied to $D$-dimensional, spherically symmetric black hole interiors. It is shown that polymerization of the generalized area variable alone leads to a complete, regular, single-horizon spacetime in which the classical singularity is replaced by a bounce. The bounce radius is independent of rescalings of the homogeneous internal coordinate, but does depend on the arbitrary fiducial cell size. The model is therefore necessarily incomplete. It nonetheless has many interesting features: After the bounce, the interior region asymptotes to an infinitely expanding Kantowski-Sachs spacetime. If the solution is analytically continued across the horizon, the black hole exterior exhibits asymptotically vanishing quantum corrections due to the polymerization. In all spacetime dimensions except four, the falloff is too slow to guarantee invariance under Poincaré transformations in the exterior asymptotic region. Hence, the four-dimensional solution stands out as the only example which satisfies the criteria for asymptotic flatness. In this case it is possible to calculate the quantum-corrected temperature and entropy. We also show that polymerization of both phase space variables, the area and the conformal mode of the metric, generically leads to a multiple horizon solution which is reminiscent of polymerized minisuperspace models of spherically symmetric black holes in loop quantum gravity.

Figure 1

Physical conformal mode plotted as a function of ${\Pi }_{\varphi }$ in the models where $n=2$ (solid [red] curve), $n=3$ (dashed [green] curve), $n=4$ (dashed and dotted [yellow] curve) and $n=10$ (dotted [blue] curve). In every model the solution begins from zero (the horizon), increases monotonically to its maximum value, and then decreases until it reaches the end point at $\mu {\Pi }_{\varphi }=\pi$ (where $\varphi \to \infty$). It can be verified that for every $n$ the solution extends toward the same end point, $e^{2\rho }/j(\varphi )=4/{\alpha }^2$, which is strictly positive. Hence, there is no horizon after the bounce. For numerical convenience, we have taken $l$ equal to the Planck length so that $G^{(n+2)}=l^n$, and we have used the numerical values $\alpha =l=L_0=1$, $\mu =0.1$, and $\beta =2$.

Figure 2

(a) Radial coordinate $r$ plotted as a function of the proper time $\tau$ of an observer in a radial free fall (4D). The observer falls from the horizon to the minimum radius $k$ within finite proper time. After the bounce the areal radius $r$ expands without limit, reaching infinity in infinite proper time. Despite its appearance, there is no cusp at the bounce. (b) Close-up near the bounce radius illustrates that the curve is smooth at the bounce. In both figures $M=1$ and $k=0.1$.

Figure 3

Conformal diagram of the partially polymerized Schwarzschild spacetime. The complete spacetime includes two exterior regions (I an I’), the black hole and the white hole interior regions (II and II’), and two “quantum-corrected” interior regions (III and III’). The classical singularity is replaced by a bounce at $r=r_{\min }$ and subsequent expansion to $r=\infty$.