Singularity resolution in LQG inspired black holes

It is expected that quantum gravity will somehow resolve the singularities that are generically present in classical gravitational theories. For example, this may be the Big Bang singularity that one encounters when applying Einstein's theory of General Relativity all the way to the beginning of the universe. An example of how this singularity is resolved in the context of loop quantum cosmology was discussed in this post.

Another place where singularities prominently occur is inside black holes. Matter that falls through the horizon of a black hole will eventually hit this singularity and the theory describing its propagation breaks down in this instant. On the other hand, a compete theory of quantum gravity should provide a well-defined description of such a process.

Penrose diagram of a maximally extended Schwarzschild spacetime. Physical motion bounded by the speed of light is possible only within the future light-cones drawn in orange. Due to the structure of the spacetime, any observer crossing the black hole horizon will eventually hit the black hole singularity.  

While the general problem of dynamical collapse, leading classically to the formation of singularities, is a very hard problem to solve in quantum gravity, some insights can be gained by considering the simpler situation of eternal black holes, in particular the Schwarzschild solution. It turns out that the interior of a Schwarzschild black hole can be described by a cosmological spacetime so that the black hole singularity is mapped to a cosmological singularity of the Big Crunch (future Big Bang) type. This in particular allows to apply techniques from quantum cosmology to such spacetimes.

Within loop quantum gravity, this line of thought has a long history with many publications subsequently improving the involved effective models. Most work is at the level of effective classical equations that capture quantum effects as $\hbar$-corrections. The main problem that researchers are currently faced with is to make a good educated guess about the effective theory that captures the main relevant quantum effects.

Key insights for how to construct physically viable theories capturing the main aspects of full loop quantum gravity were previously obtained within loop quantum cosmology, leading to the so-called notion of $\bar{\mu}$-schemes. They should ideally be understood and derived as coarse grained versions of the fundamental quantum dynamics. So far however, existing derivations mainly involve heuristic steps that may or may not reflect the physics of full loop quantum gravity. In any case, the effective models obtained along these lines are interesting in their own right and can simply be considered as modified gravity theories with interesting phenomenology.

In a recent paper together with my PhD students Fabio Mele and Johannes Münch, we proposed a new effective model that is inspired by two main ideas:

1. Physical viability of the dynamics: most previous works were facing severe problems in that quantum effects were large in low curvature regions were this wasn't expected. This was due to an unsuitable choice of quantisation scheme, ultimately due to a choice of variables that weren't tailored to the situation. We proposed a new set of variables that directly link quantum effects to high curvature regions when applied in a loop quantum gravity type quantisation. 
2. Existence of a corresponding quantum theory: one is ultimately interested in a full theory embedding of the quantum theory corresponding to the effective model used. For this, the effective Hamiltonian should exist as a well-defined operator. We showed that this is the case in our model due to being able to work in a simple so-called $\mu_0$ scheme due to our adapted choice of variables, reminiscent of the $b,v$-variables in loop quantum cosmology.

The physics captured in the model is as follows: an observer falling into the black hole crosses the horizon like in classical general relativity as quantum effects are small in this low-curvature region. The singularity that would eventually be hit in the classical theory is replaced by a transition surface to a new spacetime region where the observer will emerge from a white hole. This journey can be repeated an infinite number of times, each time to a new spacetime region.

Quantum corrected maximally extended Schwarzschild spacetime according to our (and other) papers.  The diagram can be extended an infinite number of times into the future and past. In the text, we refer to the lower region as the black hole region and the upper region as the white hole region, as this would be nomenclature used by the indicated observer. The masses of the spacetimes oscillate between black and white hole masses. The dotted line is a regular surface of high but finite curvature denoted as the transition surface. 

This picture is not new to our article and was presented before. The main new input was to achieve both goals 1 and 2 above simultaneously via an adapted choice of variables. Moreover, our paper identifies a new Dirac observable in systems of this type that can be interpreted as the white hole mass (in addition to the black hole mass) and exists non-trivially only in the quantum theory. This allows to (in principle) already measure the white hole mass on the black hole side, although its signature is very faint. This new observable and suitable analogues thereof in other effective theories should be relevant to future investigations on the topic.