Black hole horizons from within loop quantum gravity

In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. Similarly, in quantum gravity, the quantized horizon degrees of freedom should result from restricting, or pulling back, the quantized bulk degrees of freedom. This is not yet fully realized in the—otherwise very successful—quantization of isolated horizons in loop quantum gravity. In this work we outline a setting in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom. There is no need to quantize them separately. We present evidence that for a horizon of sphere topology, the resulting horizon theory is remarkably similar to what has been found before.

Figure 1

A spin network edge punctures $H$ that results in nontrivial holonomy around $\partial S=\alpha$.

Figure 2

Relation between a charge network edge and its transformation under a diffeomorphism.

Figure 3

Relation between a charge network and its tranformation under a diffeomorphism.

Figure 4

A surface the flux of which does not have a well-defined action on surface states.

Figure 5

Graphs of two gauge invariant charge nets, for the case of $H\simeq S^2$ (left) and $H\simeq T^2$ (right).

Figure 6

Two ways to permute two punctures: Left, the original configuration, middle and right, the two final states of the permutation.

Figure 7

Gauge invariance gives a condition on $n_1$, $n_2$, and $n_3$, but does not fix the flux $m=-(n_1+n_2)$ (“outside” of $H$ is to the right).

Figure 8

Change in decomposition of an $H$-charge net.

Figure 9

The graph used in the discussion of diffeomorphism invariance in the SU(2) case.

Figure 10

Two ways to obtain a quantized isolated horizon.