Horizon quantum mechanics of rotating black holes
Eur. Phys. J. C
Horizon quantum mechanics of rotating black holes
Roberto Casadio 1 2
Andrea Giugno 0
Andrea Giusti 0 1 2
Octavian Micu 3
0 Arnold Sommerfeld Center, LudwigMaximiliansUniversität , Theresienstraße 37, 80333 Munich , Germany
1 I.N.F.N. , Sezione di Bologna, I.S. FLAG, via B. Pichat 6/2, 40127 Bologna , Italy
2 Dipartimento di Fisica e Astronomia, Università di Bologna , via Irnerio 46, 40126 Bologna , Italy
3 Institute of Space Science , Bucharest, P.O. Box MG23, 077125 BucharestMagurele , Romania
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 spacetimes 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.

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) [1–8], 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
timeindependent configuration, which is the case we shall always
consider here. It is therefore rather straightforward to uplift
this description of the causal structure of spacetime 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 nonspherical spacetime, such as the one
generated by an axially symmetric rotating source, although there
are candidates for the quasilocal 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
spacetimes.
Beside the formal developments, we shall also apply the
extended HQM to specific states with nonvanishing 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 [12–19], 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
1 See also Ref. [11] for an improved version.
2 Horizon quantum mechanics
We start from reviewing the basics of the (global) HQM for
static spherically symmetric sources [1–8], 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 spacetime. 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
The general spherically symmetric metric gμν can be written
as2
ds2 = gi j dx i dx j + r 2(x i )(dθ 2 + sin2 θ dφ2),
where r is the areal coordinate and x i = (x 1, x 2) are
coordinates on surfaces of constant angles θ and φ. The location
of a trapping surface is then determined by the equation
gi j ∇i r ∇ j r = 0,
where ∇i r is perpendicular to surfaces of constant area A =
4 π r 2. If we set x 1 = t and x 2 = r , and if we denote the static
matter density by ρ = ρ (r ), the Einstein field equations tell
us that
grr = 1 − 2 p ( mr/m p) ,
where the Misner–Sharp mass is given by
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.
as if the space inside the sphere were flat. A trapping
surface then exists if there are values of r such that the
gravitational radius rH = 2 p m/m p ≥ r . If this relation
holds in the vacuum outside the region where the source
is located, rH becomes the usual Schwarzschild radius
associated with the total Arnowitt–Deser–Misner (ADM) [20]
mass M = m(∞),
2 We shall use units with c = 1, and the Newton constant G = p/m p,
where p and m p are the Planck length and mass, respectively, and
h¯ = p m p.
and the above argument gives a mathematical foundation to
Thorne’s hoop conjecture [21].
This description clearly becomes questionable for sources
of the Planck size or lighter, for which quantum effects
may not be neglected. The Heisenberg principle introduces
an uncertainty in the spatial localization of the order of
the Compton–de Broglie length, λM p m p/M , and we
could argue that RH only makes sense if RH λM , that
is, M m p. The HQM was precisely proposed in order to
describe cases in which one expects quantum uncertainties
are not negligible. For this purpose, we assume the existence
of two observables, the quantum Hamiltonian corresponding
to the total energy M of the system,3
Hˆ =
where the sum is over the Hamiltonian eigenmodes, and the
gravitational radius with eigenstates
General states for our system can correspondingly be
described by linear combinations of the form
but only those for which the relation (2.5) between the
Hamiltonian and gravitational radius holds are viewed as physical.
In particular, we impose (2.5) after quantization, as the weak
Gupta–Bleuler constraint
0 =
The solution is clearly given by
C (Eα , RHβ ) = C (Eα , 2 p Eα /m p) δαβ ,
which means that Hamiltonian eigenmodes and gravitational
radius eigenmodes can only appear suitably paired in a
physical state. The interpretation of this result is simply
that the gravitational radius is not an independent degree
of freedom in our treatment, precisely because of the
constraint (2.5).4
3 See also Ref. [22] for further clarifications why H is to be taken as
the (super)Hamiltonian of the ADM formalism. We will return to this
important point in Sect. 3.
4 For a comparison with different approaches to horizon quantization,
see Sect. 2.4 in Ref. [8].
By tracing out the gravitational radius part, we recover the
spectral decomposition of the source wave function,
in which we used the (generalized) orthonormality of the
gravitational radius eigenmodes [3]. Note that Eq. (2.10) now
ensures that the result of this operation of integrating out the
gravitational radius is still a pure quantum state.
Conversely, by integrating out the energy eigenstates, we
will obtain the horizon wave function (HWF) [1–3]
RHα  ψH = CS(m p RHα/2 p),
where m p RHα/2 p = E (RHα) is fixed by the
constraint (2.5). If the index α is continuous (again, see
Ref. [3] for some important remarks), the probability
density that we detect a gravitational radius of size RH
associated with the quantum state  ψS is given by PH(RH) =
4 π RH2 ψH(RH)2, and we can define the conditional
probability density that the source lies inside its own gravitational
radius RH as
P<(r < RH) = PS(r < RH) PH(RH) ,
where PS(r < RH) = 4 π 0RH ψS(r )2 r 2 dr .5 Finally, the
probability that the system in the state  ψS is a black hole
will be obtained by integrating (2.14) over all possible values
of RH, namely
PBH =
P<(r < RH) d RH.
Note that now the gravitational radius is necessarily “fuzzy”
and characterized by an uncertainty RH = RH2 − RH 2.
This quantum description for the total ADM mass M and
global gravitational radius RH will be next extended to
rotating sources by appealing to the asymptotic charges of
axially symmetric spacetimes. We would like to recall that in
Ref. [3] a local construction was also introduced based on the
quasilocal mass (2.4), which allows one to describe
quantum mechanically any trapping surfaces. However, that local
5 One can also view P<(r < RH) as the probability density that the
sphere r = RH is a horizon.
analysis cannot be extended to rotating sources without a
better understanding of the relation between quasilocal 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
quasilocal mass function to employ, and how to determine the
causal structure from it. For this reason, we shall explicitly
consider relations that hold in spacetimes 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.
We assume the existence of a complete set of commuting
operators {H , J 2, Jz } acting on a Hilbert space H connected
with the quantum nature of the source. We also consider only
the discrete part of the energy spectrum [3], and denote with
α = {a, j, m} the set of quantum numbers parametrizing
the spectral decomposition of the source, that is,
where the sum formally represents the spectral
decomposition in terms of the common eigenmodes of the operators
{Hˆ , Jˆ2, Jˆz }. In particular, we have6
Hˆ =
Jˆ2 ≡
Jˆz ≡
From the previous discussion, one can also easily infer that
j ∈ N0/2, m ∈ Z/2, with m ≤ j , and a ∈ I, where I is a
discrete set of labels that can be either finite of infinite.
Let us first note that Eq. (2.16) stems from the idea that
the spacetime should reflect the symmetries of the source.
Therefore, our first assumption is that the source should
obviously have an angular momentum in order to describe a
rotating black hole. Now, for a stationary asymptotically flat
spacetime, we can still define the ADM mass M and,
following Ref. [3] as outlined in the previous subsection, we
6 For later convenience, we rescale the standard angular momentum
operators jˆ2 and jˆz by factors of G N so as to have all operators
proportional to m p to a suitable power.
a, j,m b,k,n
In general relativity, we can also define a conserved
classical charge arising from the axial symmetry by means of
the Komar integral. This will be the total angular
momentum J of the Kerr spacetime. However, in our description of
the quantum source, we have two distinct notions of angular
momentum, i.e. the total angular momentum
and the component of the angular momentum along the axis
of symmetry
Since, at least classically, we can always rotate our reference
frame so that the axis of symmetry is along the z axis, it is
reasonable to consider Jˆ2 as the quantum extension of the
classical angular momentum for a Kerr black hole,
J 2 → ψS  Jˆ2  ψS =
In the following, we will further assume that Jˆz is maximum
in our quantum states, so that the proper (semi)classical limit
is recovered, that is,
for h¯ j = p m p j held constant.
For the Kerr spacetime we have two horizons given by
p
= m p ⎝
provided J 2 < M 4. Let us then introduce two operators Rˆ (±)
and, for the sake of brevity, write their eigenstates as
Rˆ H(±)  β ± = RH(β±)  β ±.
The generic state for our system can now be described by a
triply entangled state given by
can replace this classical quantity with the expectation value
of our Hamiltonian,7
but Eq. (2.25) tells us that in order to be able to define the
analog of the condition (2.9) for the rotating case, we have
to assume some mathematical restrictions on the operator
counterparts of M and J . First of all, the term J 2/M 2 tells
us that we should assume Hˆ to be an invertible selfadjoint
operator, so that
J 2/M 2 → Jˆ2 (Hˆ −1)2 = (Hˆ −1)2 Jˆ2.
For this purpose, it is useful to recall a corollary of the spectral
theorem:
Corollary 2.1 Let Aˆ be a selfadjoint positive semidefinite
operator. Then Aˆ has a positive semidefinite square root Sˆ,
that is, Sˆ is selfadjoint, positive semidefinite, and
Sˆ2 = Aˆ.
If Aˆ is positive definite, then Sˆ is positive definite.
It follows that the operator Hˆ 2 − Jˆ2 (Hˆ −1)2 should be,
at least, a positive semidefinite operator. On defining the
operators
we see that the physical states of the system are those
simultaneously satisfying
RHβ(−))  a j m  α +  β −,
7 See footnote 3.
(Rˆ H(+) − Oˆ +) 
(Rˆ H(−) − Oˆ −) 
phys = 0
phys = 0.
These two conditions reduce to
C ({a j m}, RH(α+), RH(β−))
C ({a j m}, RH(α+), RH(β−))
from which we obtain
C (Ea j , { j m}, RH(α+), RH(β−))
= C (Ea j , { j m},
= C (Ea j , { j m}, RHa(+j)(Ea j ), RHβ(−)) δα,{a, j},
= C (Ea j , { j m}, RH(α+), RHa(−j)(Ea j )) δβ,{a, j},
RHa(+j)(Ea j ), RHa(−j)(Ea j )) δα,{a, j} δβ,{a, j}.
By tracing out the geometric parts, we should recover the
matter state, that is,
RHa(−j)(Ea j ))  a j m ,
which implies
= C (Ea j , λ j , ξm , RHa(+j)(Ea j ), RHa(−j)(Ea j )).
Now, by integrating out the matter state, together with one of
the two geometric parts, we can compute the wave function
corresponding to each horizon,
ψ±(RH(±)) = C (Ea j (RH(±)), λ j (RH(±)), ξm (RH(±))).
It is also important to stress that the Hamiltonian constraints
imply a strong relation between the two horizons, indeed we
have R±H = R±H (R∓H ).
3 Corpuscular harmonic black holes
In the corpuscular model proposed by Dvali and Gomez [12–
19], black holes are macroscopic quantum objects made
of gravitons with a very large occupation number N in
the ground state, effectively forming Bose–Einstein
condensates. As also derived in Ref. [22] from a postNewtonian
analysis of the coherent state of gravitons generated by a
matter source, the virtual gravitons forming the black hole
of radius RH are “marginally bound” by their Newtonian
potential energy U , that is,
where μ is the graviton effective mass related to their
quantum mechanical size via the Compton/de Broglie wavelength
λμ p m p/μ, and λμ RH.
A first rough approximation for the potential energy UN
is obtained by considering a square well for r < λμ,
( − r ),
p
m p RH
where is the Heaviside step function and the coupling
constant α = 2p/λ2μ = μ2/m2p. The energy balance (3.1)
then leads to N α = 1 and, with λμ RH,
A better approximation for the potential energy was
employed in Ref. [10], which takes the harmonic form
V = 21 μ ω2 (r 2 − d2) (d − r )
≡ V0(r ) (d − r ),
where the parameters d and ω 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
yields the wellknown eigenfunctions
ψnjm (r, θ , φ; λμ)
r2
= N rl e− 2 λ2μ 1 F1(−n, l + 3/2, r 2/λ2μ) Ylm (θ , φ), (3.7)
where N is a normalization constant, 1 F1 the Kummer
confluent hypergeometric function of the first kind and Ylm (θ , φ)
are the usual spherical harmonics. The corresponding
eigenvalues are given by
where 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 Enl must not be confused
with the ADM energy Ea j of that section, here equal to N μ
by construction.
If we denote with n0 and l0 the quantum numbers of the
highest “energy” state, and include the graviton effective
mass μ in the constant V0(0), the condition (3.1) becomes
0, or
which yields
We now further assume that d λμ RH(+) and use the
Compton relation for μ, so that the above relation fully
determines
finite size of the well, the Schrödinger equation in spherical
coordinates,
The potential can be finally written as
and the eigenvalues as
−h¯ ω[2 (n0 − n) + (l0 − l)]
3
−2 μ 2 n0 + l0 + 2
≡ −2 μ0[2 (n0 − n) + (l0 − l)],
[2 (n0 − n) + (l0 − l)]
which of course holds only for n ≤ n0 and l ≤ l0. Let us
remark that the fact the above “energy” is negative for the
allowed values of n and j is indeed in agreement with the
postNewtonian 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 F = H⊗N , where H is a suitable 1particle Hilbert
space. However, both the Hamiltonian of the system Hˆ and
the gravitational radius RˆH are global observables and act as
N body operators on F .
3.1.1 Single eigenstates
The simplest configuration corresponds to all toy gravitons
in the same mode, and the quantum state of the system is
therefore given by
≡  M J =
where  g represents the wave function of a single
component. In particular, this  is a Hamiltonian eigenstate, for
which the total ADM energy is simply given by
and each graviton is taken in one of the modes (3.7). For the
sake of simplicity, we shall set n = n0 = 0, l = l0 = 2 and
m = ±2, that is
where the normalization constant N = 4/(√15 π 1/4 λ7μ/2).
The total angular momentum is thus given by
≡ Jˆ2
= 4 (N+ − N−)(N+ − N− + 1/2) m2p
≡ 4 L2 N 2 m2p,
8 It becomes positive if we consider the effective mass μ < 0 for virtual
gravitons.
where N+ ≥ N− = N − N+ is the number of spin up
constituents (with m = +2). We also introduced the constant
L2 =
NN+ − NN− NN+ − NN− + 21N
1
≡ (2 n+ − 1) 2 n+ − 1 + 2 N
(2 n+ − 1)2,
where the approximate expression holds for N 1. Note
that L2 = 0 for n+ ≡ N+/N = 1/2 (the nonrotating case
with N+ = N−) and grows to a maximum L2 1+O(1/N )
for the maximally rotating case n+ = 1 (or N+ = N ).
Since we are considering an eigenstate of both the
Hamiltonian Hˆ and the total angular momentum Jˆ2, the wave
functions (2.37) for the two horizons will reduce to single
eigenstates of the respective gravitational radii as well. In
particular, replacing the above values into (2.30) and (2.31)
yields
The classical condition for the existence of these horizons is
that the square root be real, which implies
The above bound vanishes for N+ = N− = N /2, as expected
for a spherical black hole, and is maximum for N+ = N , in
which case it yields
again for N 1.
Since we are modeling black holes, it is particularly
interesting to study in detail the consequences of assuming that
all the constituents of our system lie inside the outer horizon.
In other words, we next require that the Compton length of
gravitons, λμ = p m p/μ, is such that the modes (3.16)
are mostly found inside the outer horizon radius Rˆ H(+) .
In order to impose this condition, we compute the
singleparticle probability density
−1
= N 2 r 6 exp
where we used ψ02+22 = ψ02−22. From Fig. 1, we then
see that this probability is peaked well inside Rˆ H(+) for λμ =
Rˆ H(+) /4, whereas λμ = Rˆ H(+) /2 is already borderline and
λμ = Rˆ H(+) is clearly unacceptable.
and the angular momentum
4 N 2 L2 m4p
for all values of L ≥ 0. This seems to suggest that N
constituents of effective mass μ ∼ m p/√N cannot exceed the
classical bound for black holes, or that naked singularities
cannot be associated with such multiparticle states.
However, a naked singularity has no horizon and we lose the
condition (3.1) from which the effective mass μ 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
Let us now plug the effective mass (3.27) into Eq. (3.19),
Fig. 1 Plots of P02(ρ; λμ) as a function of ρ = r/ RˆH(+) for λμ =
RˆH(+) /4 (solid line), λμ = RˆH(+) /2 (dashed line) and λμ = RˆH(+)
(dotted line)
We find it in general convenient to introduce the variable
which should be at least 1 according to the above estimate,
so that Eq. (3.19) reads
1 −
4 L2 m4p
(x − 1)2
1 −
with the condition x ≡ (L/γ ) x ≤ 1 to ensure the existence
of the square root. The only positive solution is given by
for which the existence condition reads ( − 1)2 ≥ 0 and is
identically satisfied. The effective mass is then given by
Amsasas ifnutnecrptioolnateosf aNlm/2ost≤linNea+rly≤betNw,eethneμa20bo=ve γsqmu2pa/reNd
for N+ = N− = N /2 (so that L2 = 0) and μ¯ 2 (1 +
γ 2) m2p/(γ N ) for the maximally rotating case N+ = N 1
(for which L2 1). The Compton length reads
the ADM mass is
2 − L2).
n+ = nc ≡
2 N − 1 +
Since 1/2 ≤ n+ ≤ 1, the critical value nc becomes relevant
only for γ N 1. For N 1 and γ 1, the horizon
radii are thus given by
Rˆ H(−)
2 L2 p
and 2 γ λμ Rˆ H(+) , as we required. The above horizon
structure for 1/2 ≤ n+ ≤ 1 is displayed for γ = 2 and
N = 100 in Fig. 2, where we also recall that λμ = Rˆ H(+) /4.
It is particularly interesting to note that the extremal Kerr
geometry can only be realized in our model if γ is sufficiently
small. In fact, Rˆ H(−) Rˆ H(+) requires
For γ = 1 and N = 100, the horizon structure is displayed
in Fig. 3, where we see that the two horizons meet at L2 1,
that is, the configuration with n+ 1 in which (almost)
all constituents are aligned. Note also that, technically, for
N 1 and γ small, there would be a finite range nc <
n+ ≤ 1 in which the expressions of the two horizon radii
switch. However, this result is clearly more dubious as one
would be dealing with a truly quantum black hole made of a
few constituents just loosely confined. Such configurations
could play a role in the formation of black holes, or in the
9 See Refs. [6,7] for spherically symmetric charged sources.
Fig. 2 Horizon radius Rˆ H(+) (solid line) and Rˆ H(−) (dashed line) in
Planck length units for N = 100 and γ = 2
Fig. 4 Black hole probability (3.39) as a function of γ = Rˆ H(+) /2 λμ
for N = 1 (solid line), N = 102 (dashed line) and N = 106 (dotted
line)
where r ≡ (r, θ , φ), the joint probability density in position
space is simply given by
= N 2N r16 · · · r N6 exp
Fig. 3 Horizon radius Rˆ H(+) (solid line) and Rˆ H(−) (dashed line) in
Planck length units for N = 100 and γ = 1
final stages of their evaporation, but we shall not consider
this possibility any further here.
Finally, let us apply the HQM and compute the
probability (2.15) that the system discussed above is indeed a
black hole. We first note that, since we are considering
eigenstates of the gravitational radii, the wave function (2.37) for
the outer horizon will just contribute a Dirac delta peaked
on the outer expectation value (3.19) to the general
expression (2.14), that is,
PBH(n+, N ) = P (+)(r1 <
<
Moreover, since
where we used Eq. (3.22). It immediately follows that
PBH(n+, N ) =
Rˆ H(+) ) =
×(64 γ 4 + 40 γ 2 + 15)e−4 γ 2
where we recall γ was defined in Eq. (3.23), and depends on
N and n+.
The singleparticle (N = 1) black hole probability
P(+)(γ ) is represented by the solid line in Fig. 4, from which
it is clear that it practically saturates to 1 for γ 2. The
same graph shows that the minimum value of γ for which
PBH(n+, N ) = [ P(+)(γ )]N approaches 1 increases with N
(albeit very slowly). For instance, if we define γc as the value
at which PBH(n+, N ) 0.99, we obtain the values of γc
plotted in Fig. 5. It is also interesting to note that, for γ = 1,
which we saw can realize the extremal Kerr geometry, we
i ai 2 = 1 and
 Mi Ji =
Fig. 5 Value of γc such that the black hole probability (3.39) is given
by PBH(n+, N ) = [PBH(γ ≥ γc)]N ≥ 99% for N = 102 to N = 1011
PBH(N )
(0.67)N ,
and the system is most likely not a black hole for N 1,
in agreement with the probability density shown in Fig. 1.
One might indeed argue this probability is always too small
for a (semi)classical black hole, and that the extremal Kerr
configuration is therefore more difficult to achieve.
Analogously, we can compute the probability PIH that the
inner horizon is realized. Instead of Eq. (3.35), we now have
P(−)(RH) = δ(RH − Rˆ H(−) ),
which analogously leads to
PIH(n+, N ) =
P<(−)(r < Rˆ H(−) )
It is then fairly obvious that, for any fixed value of γ ,
PIH(n+, N ) ≤ PBH(n+, N ) and that equality is reached
at the extremal geometry with Rˆ H(−) Rˆ H(+) .
Moreover, from 0 ≤ L2 ≤ 1 and Eq. (3.33), we find Rˆ H(−)
Rˆ H(+) /γ 2, so that for γ = 2, the probability PIH (0.04)N
is totally negligible for N 1. This suggest that the inner
horizon can remain extremely unlikely even in configurations
that should represent large (semi)classical black holes.
3.1.2 Superpositions The next step is investigating general superpositions of the states considered above,
so that Mi = Ni μi and Ji = (2 Ni+ − Ni ) ji ≡ Ni (2 ni+ −
1) ji . One can repeat the same analysis as the one performed
for the singlemode case, except that the two HWFs will now
be superpositions of ADM values as well.
In practice, this means that Eqs. (3.35) and (3.42) are now
replaced by
P(±)(RH) =
where, from Eqs. (2.30) and (2.31), the horizon radii are given
by
1 −
and the expectation values of the horizon radii are
correspondingly given by
Rˆ H(±) =
As usual, we obtain the probability that the system is a black
hole by considering the outer horizon, for which
PBH =
ai 2 P<(+)(r1 < R(+), . . . , rNi < RH(+i ))
Hi
The explicit calculation of the above probability
immediately becomes very cumbersome. For the purpose of
exemplifying the kind of results one should expect, let us just
consider a state
where the two modes in superposition are given by N
constituents with quantum numbers n1 = 0, l1 = 2 and m = ±2
in the state (3.16), here denoted with  g1 ; the same number
N of gravitons with quantum numbers n2 = 1, l2 = 2 and
m = ±2 in the state
P<(+)(r < RH(+i )) =
−1
where we further assumed that all constituents have the same
Compton/de Broglie wavelength λμ. It then follows that
M1 = M2 ≡ M , so that
hole provided the Compton/de Broglie length is sufficiently
shorter than the possible outer horizon radius (that is, for
sufficiently large γ1 and γ2).
1 −
4 Conclusions and
+ b2 1 −
with each of the Ji depending both on the numbers of spin up
and the total number of constituents of each type, as defined
in the beginning of this section. We also notice that when
both J1 and J2 go to zero the expression simplifies to
while Rˆ H(−) = 0, as expected for a Schwarzschild black
hole.
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
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 [1–3], this extension is not based on
(quasi)local quantities, but rather on the asymptotic mass
and angular momentum of the Kerr class of spacetimes. As
long as we have no access to local measurements on black
hole spacetimes, 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 selfgravitating sources and
black holes. For example, it would be quite natural to apply
the HQM to regular configurations of the kinds reviewed in
Refs. [23–25].
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.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
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