Quantum black hole in Loop Quantum Gravity

Today micro black holes are known (seeing the brief review). However, we do not know what is a quantum black hole?

A quantum black hole is not necessarily a very small black hole. In fact, large black holes may or maynot demonstrate some quantum effects when they touch their outer pace.

In Loop Quantum Gravity, there are two crucial pictures for a quantum black hole.

1- Quantum Isolated Horizon,(aka ABCK model, proposed about 1996-7 by Ashtekar, Baez, Corichi, and Krasnov - major reference).

In this picture, first a spacetime with internal boundary (at the horizon of a black hole) is considered. Then, some conditions are imposed on this boundary in order to make it behave like a black hole horizon from the thermodynamic point of view.

After that, the spacetime is quantized. This is done by promoting the gravitational degrees of freedom (the Ashtekar variables -on holonomies- and their conjugate momenta) to operators. The Hilbert space of such a spacetime is separated into the one of the bulk and the one of the internal boundary.

The Hilbert space associated to the bulk fields contains the spin network states, defined on one dimensional floating graphs embedded into the bulk. The Hilber space of the boundary is simplified by gauge-fixing. The contribution of the boundary in the spacetime action is a Chern-Simons term.

Each bulk edge interesect at the boundary at a puncture. The wave function associated to a puncture is not unique. This is pretty similar to regular quantum mechanics in which when the angular momentum j is projected on the z-axis it takes 2j+1 possible components m, where m=-j, -j+1, ..., j-1, j. In fact, each puncture is a degenerate state. A puncture corresponding the edge of spin j takes 2j+1 degenerate states on the horizon Hilbert space.

On the other hand, each interesecting edge carries the area proportional to the edge spin. An edge of spin j generate the area proportional to [j*(j+1)]^0.5. Thus, associated to the degenerate wave function of a puncture, there is one area. This degeneracy is the root of black hole entropy, this picture says.

These punctures are responcible for the curvature of the horizon and everywhere else on the horizon is flat.

Following picture shows the portrait of a quantum isolated horizon:

A quantum isolated horizon

This model is conceptual restricted and a more fundamental microscopic description should be possible. I can give three reasons for this:

• (i) in general relativity metric extends through a black hole via the junction conditions. However in quantum limits, the spin network states must extend through the horizon. A quantum isolated horizon forbid the extention by treating the black hole as the internal boundary of space.
• (ii) the horizon in this picture is defined by the classical notion of localization. However, it is completely known that the notion of quantum localization is different than its classical version. Thus a quantum horizon must be localized as a ‘quantum boundary’ of its interior states.
• (iii) the bulk edges may only end at the boundary at punctures and no tangential edge is allowed. Why? This is a too strong assumption!

2- Black Hole Spin Network (Proposed recently. Reference)

As a second picture, spacetime is first geometrically quantized, without reference to the horizon. Instead of partial gauge-fixing, a condition that a surface be a horizon of a black hole is imposed on a surface in the full quantum state. The result is that no dynamical constraint is imposed at the horizon. Instead, a black hole horizon is defined by a partition of a spin network.

In this picture all of the above ambiguities are resolved, although the dynamics of this model has not been completed yet. See the papers of Viqar Husain and Oliver Winkler (see 1 and 2), Martin Bojowald (see here), and I (see 1, 2).

The portrait of a Black Hole Spin Network is as the following

A black hole spin network

Two results quickly followed from this picture of black holes.

• 1- The first is a new computation of black hole entropy on the basis of a degeneracy in the spectrum of areas.
  
  These came from states that were excluded from the isolated horizon boundaryconditions. In fact, their exclusion is a result of imposing a classical notion of black hole horizon as a boundary condition, and then quantizing, rather than quantizing and then identifying a surface as an horizon.
  The fact is that the background independent area operator is generically a degenerate operator. The area eigenstates associated to quantum surfaces are not unique! Some of the are eigenstates in the complete spectrum may be associated to one area eigenvalue.
  
  The following graph shows how the degeneracy is correlated
  

Genericness of area degeneracy in loop quantum gravity (from here)

• 2- The second payoff of the new definition of a black hole horizon is the prediction of small, but potentially observable corrections to the Hawking’s thermal black hole radiation formula. This happens by considering the fluctuation of horizon area in the kinematical step of the black hole definition. The full spectra of the area operator have an unexpected symmetry that was previously unnoticed. This leads to a physical effect, which is amplification of some modes of black hole radiation. This conclusion differs from Hawking’s original prediction even for massive black holes.

Let me explain this a bit more.

The complete spectrum of area in LQG is such that the gap between different levels of area eigenvalues become smaller in higher levels. But it is proved mathematically this spectrum can be split exactly into evenly spaced subsets. The gap between levels in each one of the subsets is unique and propostional to square roots of all square-free numbers

(Squarefree numbers are those numbers whose prime number ingridients are not repeated. For example 18=2*3*3 is not a square-free number, but 15=3*5 is. These numbers have been studied heavily in Number theory. You can see some of their amazing propeties here.)

Each one of the evenly spaced subsets, to which the complete spectrum of area eigenvalues are splitted, is called a "generation".

Black hole is a sector of spacetime on which horizon area A is proportional to mass squared M^2. Thus, the quantum of energy and the quantum of area are relevant, (1/M) dA = dM. Consequently, the fluctuations of the horizon are seen as the transition of black hole mass, the emission of energy (energy radiation).

Having splitted the complete spectrum of area into different evenly spaced sets of numbers (with different gaps), the area transitions fall into two categories:

1. Generational Transitions, those transitions occuring between two area levels of the same generations.
2. Inter-generational Transitions, those transitions occuring between two area levels of different generation.

The frequencies emitted via generational transitions are all proportional to each other by integers and thus are called "harmonics". This is not the case for the intergenerational transitions. The following picture discribes this graphically.

The area transitions as how loop quantum gravity discribes. (from here

In one generation, a harmonic frequency can be generated by transitions from many pairs of levels. For example: the frequency of the emission from te level 3 to 1 can be reproduced by the transitions from level 4 to 2, the levels 5 to 3, etc. This is not the case with the intergenerational frequencies.

Therefore the average number of photons at the harmonic frequancies exceeds the other frequencies. Among the harmonic frequencies themelves, the number of photons corresponding to the transitions from nearby levels are more than others. The reason is simple. The more the gap between levels decreases, the more photons are created at each generational transition.

Therefore, a few of the lines in the radiation spectrum are expected to be the brightest lines. These lines are all unblended. Exact calculation predicts the following spectrum for a black hole radiation:

The spectrum of a black hole radiation as how loop quantum gravity predicts (from here)

Here wo is of the order of 10^16/M(kg) electronVolts, which can be of the order of 10keV for a primordial black hole of mass M=10^12kg.

This makes it possible to test loop quantum gravity with black holes well above Planck scale. These predictions will become amenable to experimental check if primordial black holes are ever found. In fact, this prediction has opened a window upon quantum gravity that does not require reaching down to Planck scale physics.