Wave functions for the Schwarzschild black hole interior

Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for preclassical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $\gamma$ to the quantum ambiguity $\delta$ appearing in the model. This bound is violated when we use a natural value for $\delta$ based on loop quantum gravity together with a recent proposal for $\gamma$. We also present numerical solutions of the constraint.

Figure 1

A numerical simulation of an $m$-symmetric Gaussian wave function as it approaches the singularity of the black hole. The black line in the middle represents the horizon and the classical singularity is to the upper right.

Figure 2

A comparison between the classical black hole and the quantum wave function. The green (smooth) line indicates the classical trajectory given by the relations (13), while the red (wiggly) line is the averaged middle point of a Gaussian wave packet numerically evolved using the quantum constraint (2).

Figure 3

The sequence ${\alpha }_m$ for the case $\kappa =3/2$ and $\lambda =5$.

Figure 4

The sequence ${\alpha }_m$ for the case $\kappa =3/2$ and $\lambda =-1$. Notice the period of the oscillations is the same as Fig. 3, because the value of $\kappa$ is the same.