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Register a new userEffective dynamics of the Schwarzschild black hole interior with inverse triad corrections
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We reconsider the study of the interior of the Schwarzschild black hole now including inverse triad quantum corrections within loop quantization. We derive these corrections and show that they are are related to two parameters $delta_b, delta_c$ associated to the minimum length in the radial and angular directions, that enter Thiemanns trick for quantum inverse triads. Introduction of such corrections may lead to non-invariance of physical results under rescaling of the fiducial volume needed to compute the dynamics, due to noncompact topology of the model. So, we put forward two prescriptions to resolve this issue. These prescriptions amount to interchange $delta_b, delta_c$ in classical computations in Thiemanns trick. By implementing the inverse triad corrections we found, previous results such as singularity resolution and black-to-white hole bounce hold with different values for the minimum radius-at-bounce, and the mass of the white hole.
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We present the detailed analyses of a model of loop quantum Schwarzschild interior coupled to a massless scalar field and extend the results in our previous rapid communication arXiv:2006.08313 to more general schemes. It is shown that the spectrum of the black hole mass is discrete and does not contain zero. This indicates the existence of a black hole remnant after Hawking evaporation due to loop quantum gravity effects. Besides to show the existence of a stable black hole remnant in the vacuum case, the quantum dynamics for the non-vacuum case is also solved and compared with the effective one.
We study the interior of a Reissner-Nordstrom Black-Hole (RNBH) using Relativistic Quantum Geometry, which was introduced in some previous works. We found discrete energy levels for a scalar field from a polynomial condition for the Heun Confluent functions expanded around the effective causal radius $r_*$. From the solutions it is obtained that the uncertainty principle is valid for each energy level of space-time, in the form: $E_n, r_{*,n}=hbar/2$, and the charged mass is discretized and distributed in a finite number of states. The classical RNBH entropy is recovered as the limit case where the number of states is very large, and the RNBH quantum temperature depends on the number of states in the interior of the RNBH. This temperature, depending of the number of states of the RNBH, is related with the Bekeinstein-Hawking (BH) temperature: $T_{BH} leq T_{N} < 2,T_{BH}$.
The investigation about the volume of a black hole is closely related to the quantum nature of the black hole. The entropy is a significant concept for this. A recent work by Majhi and Samanta [Phys. Lett. B 770 (2017) 314] after us presented a similar conclusion that the entropy associated with the volume is proportional to the surface area of the black hole, but the proportionality coefficient is different from our earlier result. In this paper, we clarify the difference and show that their calculation is unrelated to the interior of the black hole.
Hawking-Bekenstein entropy formula seems to tell us that no quantum degrees of freedom can reside in the interior of a black hole. We suggest that this is a consequence of the fact that the volume of any interior sphere of finite surface area simply vanishes. Obviously, this is not the case in general relativity. However, we show that such a phenomenon does occur in various gravitational theories which admit a spontaneously induced general relativity. In such theories, due to a phase transition (one parameter family degenerates) which takes place precisely at the would have been horizon, the recovered exterior Schwarzschild solution connects, by means of a self-similar transition profile, with a novel hollow interior exhibiting a vanishing spatial volume and a locally varying Newton constant. This constitutes the so-called hollowgraphy driven holography.
The spacetime in the interior of a black hole can be described by an homogeneous line element, for which the Einstein--Hilbert action reduces to a one-dimensional mechanical model. We have shown in [SciPost Phys. 10, 022 (2021), [2010.07059]] that this model exhibits a symmetry under the $(2+1)$-dimensional Poincare group. Here we explain how this can be understood as a broken infinite-dimensional BMS$_3$ symmetry. This is done by reinterpreting the action for the model as a geometric action for BMS$_3$, where the configuration space variables are elements of the algebra $mathfrak{bms}_3$ and the equations of motion transform as coadjoint vectors. The Poincare subgroup then arises as the stabilizer of the vacuum orbit. This symmetry breaking is analogous to what happens with the Schwarzian action in AdS$_2$ JT gravity, although in the present case there is no direct interpretation in terms of boundary symmetries. This observation, together with the fact that other lower-dimensional gravitational models (such as the BTZ black hole) possess the same broken BMS$_3$ symmetries, provides yet another illustration of the ubiquitous role played by this group.
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