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.

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:

*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.*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.

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.

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