# Horizon quantum mechanics of rotating black holes

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## Abstract

The horizon quantum mechanics is an approach that was previously introduced in order to analyze the gravitational radius of spherically symmetric systems and compute the probability that a given quantum state is a black hole. In this work, we first extend the formalism to general space-times with asymptotic (ADM) mass and angular momentum. We then apply the extended horizon quantum mechanics to a harmonic model of rotating corpuscular black holes. We find that simple configurations of this model naturally suppress the appearance of the inner horizon and seem to disfavor extremal (macroscopic) geometries.

### Keywords

Black Hole Angular Momentum Total Angular Momentum Naked Singularity Horizon Radius## 1 Introduction

Astrophysical compact objects are known to be usually rotating, and one correspondingly expects most black holes formed by the gravitational collapse of such sources be of the Kerr type. The formalism dubbed horizon quantum mechanics (HQM) [1, 2, 3, 4, 5, 6, 7, 8], was initially proposed with the purpose of describing the gravitational radius of spherically symmetric compact sources and determining the existence of a horizon in a quantum mechanical fashion. It therefore appears as a natural continuation in this research direction to extend the HQM to rotating sources. Unfortunately, this is not at all a conceptually trivial task.

In a classical spherically symmetric system, the gravitational radius is uniquely defined in terms of the (quasi-)local Misner–Sharp mass and it uniquely determines the location of the trapping surfaces where the null geodesic expansion vanishes. The latter surfaces are proper horizons in a time-independent configuration, which is the case we shall always consider here. It is therefore rather straightforward to uplift this description of the causal structure of space-time to the quantum level by simply imposing the relation between the gravitational radius and the Misner–Sharp mass as an operatorial constraint to be satisfied by the physical states of the system [3].

In a non-spherical space-time, such as the one generated by an axially symmetric rotating source, although there are candidates for the quasi-local mass function that should replace the Misner–Sharp mass [9], the locations of trapping surfaces, and horizons, remain to be determined separately. We shall therefore consider a different path and simply uplift to a quantum condition the classical relation of the two horizon radii with the mass and angular momentum of the source obtained from the Kerr metric. This extended HQM is clearly more heuristic than the one employed for the spherically symmetric systems, but we note that it is indeed fully consistent with the expected asymptotic structure of axially symmetric space-times.

Beside the formal developments, we shall also apply the extended HQM to specific states with non-vanishing angular momentum of the harmonic black hole model introduced in Ref. [10].^{1} This model can be considered as a working realization of the corpuscular black holes proposed by Dvali and Gomez [12, 13, 14, 15, 16, 17, 18, 19], and it turns out to be simple enough, so as to allow one to determine explicitly the probability that the chosen states are indeed black holes. Furthermore, we will investigate the existence of the inner horizon and likelihood of extremal configurations for these states.

The paper is organized as follows: at the beginning of Sect. 2, we briefly summarize the HQM and recall some of the main results obtained for static spherically symmetric sources; the extension of the existing formalism to the case of stationary axisymmetric sources, which are both localized in space and subject to a motion of pure rotation, is presented in Sect. 2.2; a short survey of the harmonic model for corpuscular black holes is given in Sect. 3, where we then discuss some elementary applications of the HQM to rotating black holes whose quantum state contains a large number of (toy) gravitons; finally, in Sect. 4, we conclude with remarks and hints for future research.

## 2 Horizon quantum mechanics

We start from reviewing the basics of the (global) HQM for static spherically symmetric sources [1, 2, 3, 4, 5, 6, 7, 8], and then extend this formalism to rotating systems by means of the Kerr relation for the horizon radii in terms of the asymptotic mass and angular momentum of the space-time. In particular, we shall rely on the results for the “global” case of Ref. [3] and follow closely the notation therein.

### 2.1 Spherically symmetric systems

^{2}

*r*is the areal coordinate and \(x^i=(x^1,x^2)\) are coordinates on surfaces of constant angles \(\theta \) and \(\phi \). The location of a trapping surface is then determined by the equation

*r*such that the gravitational radius \(r_\mathrm{H}= 2\,\ell _{{p}}\,{m}/{m_{{p}}}\ge r\). If this relation holds in the vacuum outside the region where the source is located, \(r_\mathrm{H}\) becomes the usual Schwarzschild radius associated with the total Arnowitt–Deser–Misner (ADM) [20] mass \(M=m(\infty )\),

*hoop conjecture*[21].

*M*of the system,

^{3}

^{4}

^{5}Finally, the probability that the system in the state \(\mid \! {\psi }_{\mathrm{S}}\rangle \) is a black hole will be obtained by integrating (2.14) over all possible values of \(R_\mathrm{H}\), namely

This quantum description for the total ADM mass *M* and global gravitational radius \(R_\mathrm{H}\) will be next extended to rotating sources by appealing to the asymptotic charges of axially symmetric space-times. We would like to recall that in Ref. [3] a local construction was also introduced based on the quasi-local mass (2.4), which allows one to describe quantum mechanically any trapping surfaces. However, that local analysis cannot be extended to rotating sources without a better understanding of the relation between quasi-local charges and the corresponding casual structure [9].

### 2.2 Rotating sources: Kerr horizons

Our aim is now to extend the HQM to rotating sources, for which there is no general consensus about the proper quasi-local mass function to employ, and how to determine the causal structure from it. For this reason, we shall explicitly consider relations that hold in space-times of the Kerr family, generated by stationary axisymmetric sources which are both localized in space and subject to a motion of pure rotation in the chosen reference frame.

^{6}

*discrete*set of labels that can be either finite of infinite.

*M*and, following Ref. [3] as outlined in the previous subsection, we can replace this classical quantity with the expectation value of our Hamiltonian,

^{7}

*J*of the Kerr space-time. However, in our description of the quantum source, we have two distinct notions of angular momentum, i.e. the total angular momentum

*z*axis, it is reasonable to consider \(\hat{J}^2\) as the quantum extension of the classical angular momentum for a Kerr black hole,

*M*and

*J*. First of all, the term \(J^{2} / M^{2}\) tells us that we should assume \(\hat{H}\) to be an

*invertible self-adjoint*operator, so that

### Corollary 2.1

## 3 Corpuscular harmonic black holes

*N*in the ground state, effectively forming Bose–Einstein condensates. As also derived in Ref. [22] from a post-Newtonian analysis of the coherent state of gravitons generated by a matter source, the virtual gravitons forming the black hole of radius \(R_\mathrm{H}\) are “marginally bound” by their Newtonian potential energy

*U*, that is,

*d*and \(\omega \) will have to be so chosen as to ensure the highest energy mode available to gravitons is just marginally bound [see Eq. (3.1)]. If we neglect the finite size of the well, the Schrödinger equation in spherical coordinates,

*n*is the radial quantum number. It is important to remark that the quantum numbers

*l*and

*m*here must not be confused with the total angular momentum numbers

*j*and

*m*of Sect. 2.2, as the latter are the sum of the former. At the same time, the “energy” eigenvalues \({\mathcal {E}}_{nl}\) must not be confused with the ADM energy \(E_{aj}\) of that section, here equal to \(N\,\mu \) by construction.

*n*and

*j*is indeed in agreement with the post-Newtonian analysis of the “maximal packing condition” for the virtual gravitons in the black hole [22].

^{8}

In the following, we shall consider a few specific states in order to show the kind of results one can obtain from the general HQM formalism of Sect. 2.2 applied to harmonic models of spinning black holes.

### 3.1 Rotating black holes

We shall now consider some specific configurations of harmonic black holes with angular momentum and apply the extended HQM described in the previous section. We first remark that the quantum state of *N* identical gravitons will be a *N*-particle state, i.e. a vector of the *N*-particle Fock space \(\mathcal {F} = \mathcal {H} ^{\otimes N}\), where \(\mathcal {H}\) is a suitable 1-particle Hilbert space. However, both the Hamiltonian of the system \(\hat{H}\) and the gravitational radius \(\hat{R}_\mathrm{H}\) are global observables and act as *N*-body operators on \(\mathcal {F}\).

#### 3.1.1 Single eigenstates

*N*constituents of effective mass \(\mu \sim m_{{p}}/\sqrt{N}\) cannot exceed the classical bound for black holes, or that naked singularities cannot be associated with such multi-particle states. However, a naked singularity has no horizon and we lose the condition (3.1) from which the effective mass \(\mu \) is determined. If naked singularities can still be realized in the quantum realm, they must be described in a qualitatively different way from the present one.

^{9}

*N*and \(n_+\).

*N*(albeit very slowly). For instance, if we define \(\gamma _\mathrm{c}\) as the value at which \(P_\mathrm{BH}(n_+,N)\simeq 0.99\), we obtain the values of \(\gamma _\mathrm{c}\) plotted in Fig. 5. It is also interesting to note that, for \(\gamma =1\), which we saw can realize the extremal Kerr geometry, we find

#### 3.1.2 Superpositions

*N*constituents with quantum numbers \(n_1=0\), \(l_1=2\) and \(m=\pm 2\) in the state (3.16), here denoted with \(\mid \! g_1\rangle \); the same number

*N*of gravitons with quantum numbers \(n_2=1\), \(l_2=2\) and \(m=\pm 2\) in the state

The probability (3.49) can be computed explicitly and is shown in Fig. 6 for \(N=100\), with \(a=b=1\). Beside the specific shape of those curves, the overall result appears in line with what we found in the previous subsection for an Hamiltonian eigenstate: the system is most certainly a black hole provided the Compton/de Broglie length is sufficiently shorter than the possible outer horizon radius (that is, for sufficiently large \(\gamma _1\) and \(\gamma _2\)).

## 4 Conclusions

After a brief review of the original HQM for static spherically symmetric sources, we have generalized this formalism in order to provide a proper framework for the study of quantum properties of the causal structure generated by rotating sources. We remark once more that, unlike the spherically symmetric case [1, 2, 3], this extension is not based on (quasi-)local quantities, but rather on the asymptotic mass and angular momentum of the Kerr class of space-times. As long as we have no access to local measurements on black hole space-times, this limitation should not be too constraining.

In order to test the capabilities of the so extended HQM, one needs a specific (workable) quantum model of rotating black holes. For this purpose, we have considered the harmonic model for corpuscular black holes [10], which is simple enough to allow for analytic investigations. Working in this framework, we have been able to design specific configurations of harmonic black holes with angular momentum and confirm that they are indeed black holes according to the HQM. Some other results appeared, somewhat unexpected. For instance, whereas it is reasonable that the probability of realizing the inner horizon be smaller than the analogous probability for the outer horizon, it is intriguing that the former can indeed be negligible for cases when the latter is close to one. It is similarly intriguing that (macroscopic) extremal configurations do not seem very easy to achieve with harmonic states.

The results presented in this work are overall suggestive of interesting future developments and demand considering more realistic models for self-gravitating sources and black holes. For example, it would be quite natural to apply the HQM to regular configurations of the kinds reviewed in Refs. [23, 24, 25].

## Footnotes

- 1.
See also Ref. [11] for an improved version.

- 2.
We shall use units with \(c=1\), and the Newton constant \(G={\ell _{{p}}}/{m_{{p}}}\), where \(\ell _{{p}}\) and \({m_{{p}}}\) are the Planck length and mass, respectively, and \(\,{\hbar }={\ell _{{p}}} \,{m_{{p}}}\).

- 3.
- 4.
For a comparison with different approaches to horizon quantization, see Sect. 2.4 in Ref. [8].

- 5.
One can also view \(\mathcal {P}_<(r<R_\mathrm{H})\) as the probability density that the sphere \(r=R_\mathrm{H}\) is a horizon.

- 6.
For later convenience, we rescale the standard angular momentum operators \({\hat{{j}}}^2\) and \({\hat{{j}}}_z\) by factors of \(G_{N}\) so as to have all operators proportional to \({m_{{p}}}\) to a suitable power.

- 7.
See footnote 3.

- 8.
It becomes positive if we consider the effective mass \(\mu <0\) for virtual gravitons.

- 9.

## Notes

### Acknowledgements

R. C. and A. G. are partially supported by the INFN grant FLAG. The work of A. G. has also been carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM). O. M. was supported by the Grant LAPLAS 4.

### References

- 1.R. Casadio, Localised particles and fuzzy horizons: a tool for probing quantum black holes (2013). arXiv:1305.3195 [gr-qc]
- 2.R. Casadio, What is the Schwarzschild radius of a quantum mechanical particle? in
*1st Karl Schwarzschild Meeting on Gravitational Physics*, vol. 170, ed. by P. Nicolini, M. Kaminski, J. Mureika, M. Bleicher. Springer Proceedings in Physics (Springer, Switzerland, 2016), p. 225Google Scholar - 3.R. Casadio, A. Giugno, A. Giusti, Gen. Relat. Gravit.
**49**, 32 (2017). arXiv:1605.06617 [gr-qc] - 4.R. Casadio, F. Scardigli, Eur. Phys. J. C
**74**, 2685 (2014). arXiv:1306.5298 [gr-qc]ADSCrossRefGoogle Scholar - 5.R. Casadio, O. Micu, F. Scardigli, Phys. Lett. B
**732**, 105 (2014). arXiv:1311.5698 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 6.R. Casadio, O. Micu, D. Stojkovic, JHEP
**1505**, 096 (2015). arXiv:1503.01888 [gr-qc]ADSCrossRefGoogle Scholar - 7.R. Casadio, O. Micu, D. Stojkovic, Phys. Lett. B
**747**, 68 (2015). arXiv:1503.02858 [gr-qc]ADSCrossRefGoogle Scholar - 8.R. Casadio, A. Giugno, O. Micu, Int. J. Mod. Phys. D
**25**, 1630006 (2016). arXiv:1512.04071 [hep-th]ADSCrossRefGoogle Scholar - 9.L.B. Szabados, Living Rev. Relat.
**12**, 4 (2009)ADSCrossRefGoogle Scholar - 10.
- 11.
- 12.
- 13.G. Dvali, C. Gomez, Black hole’s information group (2013). arXiv:1307.7630
- 14.G. Dvali, C. Gomez, Eur. Phys. J. C
**74**, 2752 (2014). arXiv:1207.4059 [hep-th]ADSCrossRefGoogle Scholar - 15.G. Dvali, C. Gomez, Phys. Lett. B
**719**, 419 (2013). arXiv:1203.6575 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 16.G. Dvali, C. Gomez, Phys. Lett. B
**716**, 240 (2012). arXiv:1203.3372 [hep-th]ADSCrossRefGoogle Scholar - 17.G. Dvali, C. Gomez, Fortsch. Phys.
**61**, 742 (2013). arXiv:1112.3359 [hep-th]ADSCrossRefGoogle Scholar - 18.G. Dvali, C. Gomez, S. Mukhanov, Black hole masses are quantized (2011). arXiv:1106.5894 [hep-ph]
- 19.R. Casadio, A. Giugno, O. Micu, A. Orlandi, Entropy
**17**, 6893 (2015). arXiv:1511.01279 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 20.R.L. Arnowitt, S. Deser, C.W. Misner, Phys. Rev.
**116**, 1322 (1959)ADSMathSciNetCrossRefGoogle Scholar - 21.K.S. Thorne, Nonspherical gravitational collapse: a short review. in
*Magic Without Magic*, ed. by J.R. Klauder (San Francisco, 1972), p. 231Google Scholar - 22.R. Casadio, A. Giugno, A. Giusti, Phys. Lett. B
**763**, 337 (2016). arXiv:1606.04744 [hep-th]ADSCrossRefGoogle Scholar - 23.P. Nicolini, Int. J. Mod. Phys. A
**24**, 1229 (2009). arXiv:0807.1939 [hep-th]ADSCrossRefGoogle Scholar - 24.
- 25.E. Spallucci, A. Smailagic, Regular black holes from semi-classical down to Planckian size (2017). arXiv:1701.04592 [hep-th]

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