# A quantum thin shell surrounding a Black Hole

Cenalo<EMAIL_ADDRESS>University of Cincinnati Blue Ash, Cincinnati,
Ohio 45236.

###### Abstract

In a previous work we obtained exact solutions for the proper time quantum
mechanics of a thin dust shell, collapsing in a vacuum. We extend these
results to the quantum collapse of a dust shell surrounding a pre-existing
black hole. In lieu of exact solutions, which have so far proved difficult to
obtain for this system, we establish the essential features of the quantum
shell through a Wentzel–Kramers–Brillouin (WKB) approximation, which is valid
only when the mass of the shell is much greater than the Planck mass. There
are many similarities with the vacuum collapse: only bound states exist and
the proper energy spectrum of the shell is unaffected by the presence of the
central black hole to this order. There are no peculiar or distinguishing
features of the wave function near the black hole horizon. It vanishes at the
center and oscillates between the origin and the classically forbidden region,
beyond which it decays exponentially.

###### pacs:

04.60.Ds, 04.60.-m

##  I Introduction

Thin shells of matter, collapsing in a variety of environments, have been used
extensively as simplified systems with which to model the final stages of
gravitational collapse under many different conditions [1, 2, 3, 4, 5, 6, 7,
8, 9, 10]. This is because thin shell collapse captures many of the features
of more realistic collapse models while avoiding some of their technical
difficulties. Moreover, the quantum mechanics of the shell is exactly solvable
in some cases, which helps to shine a light on some of the problems of quantum
gravity. For example, “time” has different meanings in classical general
relativity and in the quantum theory. All choices of the time function yield
the same local geometries, but quantum theories built on different time
parameters are not unitarily equivalent. In [11], we showed that exact
quantizations, based on different time variables, of a shell that is
collapsing in a vacuum yield incompatible descrptions. When the shell
quantization is based on coordinate time, solutions exist only when its mass
is less than the Planck mass [12], but when it is based on proper time,
solutions exist only when its mass is greater than the Planck mass, which is
more in keeping with what is observed.

Among other important issues that one would like to understand from the point
of view of quantum gravity is the Hawking effect [13] and the information loss
paradox in black hole physics. Most discussions of the Hawking effect examine
particle production in a scalar field propagating in the classical background
geometry of a collapsing body from the point of view of the asymptotic
observer. This is an effective field theory approach from which black hole
evaporation and information loss are inferred. The geometry excites scalar
field quanta, which then propagate to infinity as thermal, nearly thermal or
unitary radiation (recently, it was argued in [14, 15] from this point of view
that the radiation from a thin shell during its collapse is unitary). The
result is that the black hole appears to evaporate over time as energy is
drawn from it by the excited field quanta. Reasoning that this effect should
have a counterpart in a time dependent quantum gravitational collapse and from
the point of view of a comoving observer (the black hole either evaporates or
it does not), we constructed a midisuperspace quantization of a non-rotating
dust ball (the simplest form of collapse) [16, 17] using the equivalent of
Kuchař variables [27] in the LeMaître-Tolman-Bondi (LTB) frame [19]. We were
able to build exact diffeomorphism invariant states on a lattice, thereby
treating the dust ball as a series of shells labeled by their LTB radial
coordinate, and showed that matching the shell wave functions across the
apparent horizon requires ingoing modes in the exterior to be matched to
outgoing modes in the interior and, vice versa, ingoing modes in the interior
are matched to outgoing modes in the exterior [20, 21]. In each case, the
relative amplitude of the outgoing wave is suppressed by the square root of
the Boltzman factor at the Hawking temperature determined by the total Misner-
Sharp mass contained within the shell. There are two independent solutions. In
one, exterior, infalling waves representing the collapsing shells of dust are
accompanied by interior, outgoing waves. These interior waves, which are of
quantum origin, represent an interior “Unruh” radiation. In the other
solution, waves move away from the apparent horizon on both sides of it.
Interior, infalling waves representing the continued collapse of the dust
shells across the apparent horizon are accompanied by exterior, outgoing
waves. These latter ougoing waves represent the exterior Unruh radiation,
which is thermal. Continued collapse across the apparent horizon from an
initial diffuse state can be achieved by combining the two solutions and
requiring the net flux to vanish at the apparent horizon. The effect is that
the collapse ends in a central singularity and is accompanied by thermal Unruh
radiation in the exterior. The net effect in the proper time quantum theory is
therefore a quasi-classical tunneling of particles as described in [22, 23],
but only if a continued collapse beyond the apparent horizon is assumed.

There are some ambiguities involved in the midisuperspace quantization program
that are avoided in the quantization of a thin shell. Therefore, in this
paper, we address the quantum mechanics of single thin dust shell that is
collapsing in the background of a pre-existing black hole from the comoving
observer’s point of view. The purpose is to understand what differences the
background brings about in the shell wave function, in particular in its
behavior at the event horizon of the black hole and at infinity, i.e., is
there a similar tunneling effect as described above for the dust ball and if
so does the shell evaporate thermally as suggested by Hawking?

On the classical level, the shell has just one degree of freedom and is
completely described by its radius, $R(t)$ and its conjugate momentum, $P(t)$.
We are unable to find exact wave functions, as we did in the case of a shell
collapsing in a vacuum, but a WKB approximation is sufficient to extract many
of their key features. We find several differences between the behavior of
shells in a dust ball and the single shell, all of them traceable to the fact
that, unlike the shells of a dust ball, the single shell possesses a self-
interaction that is inversely proportional to its area radius. As a
consequence the single shell is classically bound. We will show that there is
no tunneling across the horizon and determine the energy spectrum of the
shell. Its WKB wave function extends from the origin (where it vanishes), is
well behaved at the black hole horizon and falls off exponentially beyond the
classically allowed region.

In Sec. II we derive the proper time dynamics of a classical thin shell
collapsing onto a pre-existing mass. The classical dynamics are obtained by an
application of the Israel-Darmois-Lanczos (IDL) formalism [24, 25, 26], which
yields a first integral of the motion involving the pre-existing mass at the
center, $M^{-}$, the Arnowitt-Deser-Misner (ADM) mass at infinity, $M^{+}$,
and the proper mass, $m$, of the shell. Of these, the mass at the center and
the proper mass of the shell are non-dynamical parameters, constant over the
entire phase space. The ADM mass is a dynamical variable that represents the
energy of the system. There are three time variables, viz., the coordinate
times in the interior and the exterior, and the shell proper time. What is not
clear is the time variable in which the ADM mass generates the evolution. We
follow Hajiček, Kay and Kuchař [12] and assume that it generates the evolution
in the time coordinate inside the shell. This allows us to construct
Lagrangians and Hamiltonian evolutions for the shell in the other time
variables, in particular in the shell proper time. Here we also show that the
Hamiltonian obtained in this way is structurally similar to the proper time
Hamiltonian derived for the full Einstein-dust system in the LTB dust ball
models, which lends confidence in the choice of [12]. In section III, we
quantize the classical model. The Wheeler-DeWitt equation is elliptic with a
positive semi-definite inner product for energies less than the shell’s proper
mass. We find the WKB approximation of the wave function and, in Sec. IV,
analyze its $U(1)$ current. Requiring that a lowest energy state exists and
that the $U(1)$ current is finite and well behaved everywhere, a complete set
of bound states exists and we determine its spectrum. By comparing the
solutions with the shell collapsing in a vacuum, we conclude that the
approximation is valid only so long as the proper mass of the shell is much
greater than the Planck mass. We conclude in Sec. V with a brief summary of
our results and tie them in with a previously suggested model for quantum
black holes.

##  II The Classical Shell Model

The equation of motion of a spherical, thin, massive shell is obtained by
applying the Israel-Darmois-Lanczos conditions on the timelike surface
$\Sigma=\mathbb{R}\times\mathbb{S}^{2}$ that represents its world sheet. The
world sheet forms the three dimensional boundary between an internal
spacetime, ${\mathcal{M}}^{-}$, and an external spacetime,
${\mathcal{M}}^{+}$. ${\mathcal{M}}^{\mp}$ are described in coordinates
$x_{\mp}^{\mu}$ by metrics $g_{\mu\nu}^{\mp}$ that solve Einstein’s equations.
Let $\xi^{a}$ be a set of intrinsic coordinates on the surface of the shell
and differentiable functions of $x^{\mu}_{\mp}$, then
${e^{\mp}}^{\mu}_{a}=\partial x_{\mp}^{\mu}/\partial\xi^{a}$ are the
components of the three basis vectors on this surface and
$h^{\mp}_{ab}=g^{\mp}_{\mu\nu}{e^{\mp}}^{\mu}_{a}{e^{\mp}}^{\nu}_{b}$ is the
induced metric on the shell on the two sides of it. The first junction
condition requires the shell to have a well defined metric, i.e.,
$h^{-}_{ab}=h^{+}_{ab}$ or $[h_{ab}]=0$.

The second junction condition, which follows from Einstein’s equations, says
that the surface stress energy tensor, $S_{ab}$, of the shell is given by

$S_{ab}=-\frac{\varepsilon}{8\pi}\left([K_{ab}]-[K]h_{ab}\right)$ (1)

where $K_{ab}$ is the extrinsic curvature of the boundary, $K=K^{a}_{a}$ and
$\varepsilon=+1$ for a timelike shell. If ${\mathcal{M}}^{\mp}$ are taken to
be vacuum spacetimes, then spherical symmetry implies that $g_{\mu\nu}^{\mp}$
are Schwarzschild metrics, with mass parameters $M^{\mp}$ respectively, and
$M^{+}$ represents the total mass of the system. We may write the respective
line elements as

$ds^{2}_{\mp}=-g^{\mp}_{\mu\nu}dx_{\mp}^{\mu}dx_{\mp}^{\nu}=B^{\mp}dt_{\mp}^{2}-\frac{1}{B^{\mp}}dr^{2}_{\mp}-r_{\mp}^{2}d\Omega^{2}$
(2)

where $B^{\mp}=1-2GM^{\mp}/r_{\mp}$ and we have assumed that the interior and
exterior share the same spherical coordinates, $\theta$ and $\phi$. The shell
is described by the parametric equations $r_{\mp}=r=R(\tau)$,
$t_{\mp}=t_{\mp}(\tau)$, where $\tau$ is the proper time for comoving
observers and the interior and exterior time coordinates are related to the
shell proper time (and indirectly to each other) by

$\frac{dt_{\mp}}{d\tau}=\frac{\sqrt{B^{\mp}+R_{\tau}^{2}}}{B^{\mp}},$ (3)

where the subscript indicates a derivative with respect to $\tau$. Choosing
the intrinsic coordinates of the shell to be $\xi^{a}=\\{\tau,\theta,\phi\\}$,
the induced metric is

$ds_{\Sigma}^{2}=d\tau^{2}-R^{2}(\tau)d\Omega^{2}.$ (4)

while the non-vanishing components of the extrinsic curvature are

${K_{\mp}^{\theta}}_{\theta}={K_{\mp}^{\phi}}_{\phi}=\frac{\beta^{\mp}}{R},~{}~{}{K_{\mp}^{\tau}}_{\tau}=\frac{\beta_{\tau}^{\mp}}{\dot{R}_{\tau}},$
(5)

where

$\beta^{\mp}=\sqrt{B^{\mp}+\dot{R}_{\tau}^{2}}.$ (6)

Therefore, according to (1),

$\displaystyle{S^{\tau}}_{\tau}$ $\displaystyle=$
$\displaystyle\frac{\beta^{+}-\beta^{-}}{4\pi GR}=-\sigma$ (7)
$\displaystyle{S^{\theta}}_{\theta}={S^{\phi}}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\beta^{+}-\beta^{-}}{8\pi
GR}+\frac{\beta_{\tau}^{+}-\beta_{\tau}^{-}}{8\pi GR_{\tau}}=p$ (9)

where we have set ${S^{a}}_{b}=\text{diag}(-\sigma,p,p)$.

The mass density of the shell is “$\sigma$” and “$p$” is its tangential
pressure, which, for dust shells, we take to be zero. Integrating the second
equation in (9),

$\beta^{+}-\beta^{-}=-\frac{Gm}{R},$ (10)

where $m$ is a constant of the integration, which represents the rest mass of
the shell, as is seen by inserting this solution into the first. Equation (10)
may be put in the form

$M^{+}-M^{-}=\Delta M=m\sqrt{B^{-}+R_{\tau}^{2}}-\frac{Gm^{2}}{2R}.$ (11)

It is reasonable think of the above as a first integral of the motion and
associate $\Delta M$ with the total energy, $E$, of the shell. When expressed
in terms of the momentum conjugate to $R(\tau)$, (11) will represent the
Hamiltonian of the system. It is, however, given in terms of the velocities,
which are dependent variables in the canonical theory and, to determine the
momentum, it becomes necessary to know in which of the three time coordinates
the Hamiltonian is evolving the system.

Within the thin shell construction, there is no à priori way to determine a
canonical Hamiltonian because the constraint equation has been derived from
the IDL conditions and not a fundamental action principle. One approach would
be to compare the thin shell Hamiltonians with a similar system for which a
canonical theory has been derived from an action principle. Our goal will be
to recover a proper time Hamiltonian that is compatible with the midi-
superspace Hamiltonian [16, 17] obtained for the spherically symmetric
Einstein-dust action by an application of a canonical chart analogous to that
employed by Kuchař in [27, 28]. Because a dust ball can be thought of as a
sequence of non-interacting shells, the proper time Hamiltonian for a single
shell should be of the same form apart from any self-interaction terms
peculiar to the thin shell itself. Thus, for example, if the evolution is
taken to be in the shell proper time and the energy is taken to be $\Delta M$,
we have

$R_{\tau}=\frac{\partial H_{I}}{\partial p}$ (12)

and the Hamiltonian is [29, 30]

$H_{I}=m\sqrt{B}\cosh\frac{p}{m}-\frac{Gm^{2}}{2R}.$ (13)

It does not have the same form as the Hamiltonian derived for the dust ball in
[16, 17].

As mentioned in the introduction, we will show that the choice of [12], taking
$\Delta M$ to evolve the system in the coordinate time of the interior, yields
a compatible proper time Hamiltonian. Because the right hand side of (11)
involves only the interior we drop the superscripts $\pm$ and, employing (3),
we can rewrite it as

$\Delta M=\frac{mB^{3/2}}{\sqrt{B^{2}-R_{t}^{2}}}-\frac{Gm^{2}}{2R}.$ (14)

Then $R_{t}=\partial H_{II}/\partial p$ gives

$H_{II}=-P_{(t)}=\sqrt{m^{2}B+B^{2}p^{2}}-\frac{Gm^{2}}{2R},$ (15)

and

$p=\frac{mR_{t}}{\sqrt{B}\sqrt{B^{2}-R_{t}^{2}}}.$ (16)

The action for the shell may now be given as a Legendre transform of $H_{II}$,

$S=\int dt\left[-m\sqrt{B-\frac{R_{t}^{2}}{B}}+\frac{Gm^{2}}{2R}\right]$ (17)

and then transformed into an action in proper time, once again with the help
of (3). One finds

$S=\int d\tau\left[-m+\frac{Gm^{2}}{2R}\frac{\sqrt{B+R_{\tau}^{2}}}{B}\right]$
(18)

and the proper time Hamiltonian

${\mathcal{H}}=-P_{(\tau)}=m-\sqrt{\frac{f^{2}}{B}-BP^{2}},$ (19)

where we have set $f(R)=Gm^{2}/2R$ and the momentum, $P$, conjugate to $R$, is
now given by

$P=\frac{fR_{\tau}}{B\sqrt{B+R_{\tau}^{2}}}.$ (20)

This proper time Hamiltonian is bounded from above by the mass of the shell
and the shell momentum is bounded from above by $f/B$. From the comoving
observer’s point of view, the shell is always bound to the center.

The equations of motion that follow from (19) are derivable from the super-
Hamiltonian

$h_{(\tau)}=(P_{(\tau)}+m)^{2}+BP^{2}-\frac{f^{2}}{B}=0.$ (21)

In this form, the Hamiltonian structure of the shell is identical to that of
the dust ball, as derived in the Einstein-dust system, with one important
exception: the shells in a dust ball do not possess a self-interaction that
depends on their area radius: for the shells in a dust ball, $f(R)$ gets
replaced by the Misner-Sharp mass density, $F^{\prime}(r)$, where $r$ is the
LTB shell label coordinate and $F(r)$ represents the mass of the dust ball up
to $r$. The dependence of the self-interaction term, $f(R)$, on the area
radius is responsible for the fact that the shell is classically bound in the
proper time description. It will also play an important role in the matching
conditions at the horizon.

##  III The Quantum Shell

The structure of the super-Hamiltonian in (21) indicates that the DeWitt
metric is

$\gamma_{ab}=\left(\begin{matrix}1&0\cr 0&1/B\end{matrix}\right),$ (22)

so we choose a factor ordering that is symmetric with respect to the measure
“$dR/\sqrt{B}$”. Raising the momenta to operator status following Dirac we get
the Wheeler-DeWitt equation

${\widehat{h}}_{(\tau)}\Psi(\tau,R)=\left[\left(-i\hbar\frac{\partial}{\partial\tau}+m\right)^{2}-\hbar^{2}\sqrt{B}\frac{\partial}{\partial_{R}}\sqrt{B}\frac{\partial}{\partial_{R}}-\frac{f^{2}}{B}\right]\Psi(\tau,R)=0.$
(23)

We have been unable to find exact solutions to this equation, but the WKB
approximation suffices to yield a general picture of the quantum shell. With
$\Psi(\tau,R)=e^{iW(\tau,R)/\hbar}$, (23) reads

$-i\hbar\frac{\partial^{2}W}{\partial\tau^{2}}+\left(\frac{\partial
W}{\partial\tau}+m\right)^{2}-i\hbar B\frac{\partial^{2}W}{\partial
R^{2}}-\frac{i\hbar}{2}B^{\prime}\frac{\partial W}{\partial
R}+B\left(\frac{\partial W}{\partial R}\right)^{2}-\frac{f^{2}}{B},=0$ (24)

where the prime indicates a derivative with respect to $R$, and taking

$W=-{\mathcal{E}}\tau+S_{0}(R)+\frac{\hbar}{i}\ln A(R),$ (25)

where ${\mathcal{E}}$ is the shell proper energy, we find up to
${\mathcal{O}}(\hbar)$,

$\displaystyle BS_{0}{{}^{\prime 2}}+(m-{\mathcal{E}})^{2}-\frac{f^{2}}{B}=0$
(26) (27)
$\displaystyle\sqrt{B}(\sqrt{B}S_{0}^{\prime})^{\prime}+\frac{2B}{A}A^{\prime}S_{0}^{\prime}=0.$
(28)

The first equation is solved by the Hamilton-Jacobi function,

$S_{0}(R)=\pm\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B},$ (29)

and the second gives

$A=\frac{C}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}.$ (30)

Let us now show that the classical limit of this solution yields the classical
dynamical equations that follow from (19). To order $\hbar^{0}$, $W(\tau,R)$
is just the Hamilton-Jacobi function, so the function $R(\tau)$ defined by the
principle of constructive interference,

$\frac{\partial
S_{0}}{\partial{\mathcal{E}}}=0=-\tau\pm\int\frac{(m-{\mathcal{E}})dR}{\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}}$
(31)

and

$P(\tau)=S_{0}^{\prime}=\pm\frac{1}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}$
(32)

should satisfy the Hamiltonian equations based on (19). Taking a derivative of
(31),

$1=\pm\frac{R_{\tau}(m-{\mathcal{E}})}{\sqrt{f^{2}-B(m-{\mathcal{E}})^{2}}}$
(33)

and therefore

$m-{\mathcal{E}}=\frac{f}{\sqrt{B+R_{\tau}^{2}}}$ (34)

Inserting this into (32) shows that

$BP=\sqrt{f^{2}-B(m-{\mathcal{E}})^{2}}=\frac{fR_{\tau}}{\sqrt{B+R_{\tau}^{2}}}$
(35)

or

$R_{\tau}=\frac{BP}{\sqrt{\frac{f^{2}}{B}-BP^{2}}}=\left\\{R,{\mathcal{H}}\right\\}$
(36)

Again, taking the derivative of $P$ in (32) and using (36) we find

$P_{\tau}=\frac{2ff^{\prime}-(f^{2}/B+BP^{2})B^{\prime}}{B\sqrt{\frac{f^{2}}{B}-BP^{2}}}=\left\\{P,{\mathcal{H}}\right\\}.$
(37)

It follows that the trajectories implied by the principle of constructive
interference in (31) are identical to those determined by the Hamiltonian
equations of motion that follow from (19).

The WKB solutions may now be given as

$\Psi_{\pm}(\tau,R)=\frac{C_{\pm}e^{-i{\mathcal{E}}\tau}}{|B^{1/4}\sqrt{S_{0}^{\prime}}}\exp\left[\pm
i\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\right].$ (38)

Inside the black hole horizon, because $B<0$, the the WKB wave function is
always oscillatory, but the situation is different outside. The phase $S_{0}$
is real in the classically allowed region, for which
$f^{2}-(m-{\mathcal{E}})^{2}B>0$, and imaginary in the classically forbidden
region. The wave function thus falls off exponentially when
$f^{2}-(m-{\mathcal{E}})^{2}<0$ i.e., when

$R>R_{+}=\frac{1}{2}\left(\kappa+\sqrt{\kappa^{2}+\frac{\mu^{4}}{(m-{\mathcal{E}})^{2}}}\right),$
(39)

where $\kappa=2GM^{-}$ and $\mu^{2}=Gm^{2}=(m/m_{p})^{2}$. We will henceforth
distinguish between the “interior” region ($R<\kappa$) and the “exterior”
region ($R>\kappa$), which itself consists of the classically allowed region
($\kappa<R<R_{+}$) and the classically forbidden region ($R_{+}<R$). In the
interior we write

$\displaystyle\Psi(\tau,R)$ $\displaystyle=$
$\displaystyle\frac{e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}\left\\{F_{1}\exp\left[+i\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\right]\right.$
(42) $\displaystyle\hskip
56.9055pt\left.+F_{2}\exp\left[-i\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\right]\right\\},~{}~{}0<R<\kappa$

and in the exterior,

$\Psi(\tau,R)=\left\\{\begin{matrix}\frac{e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}\left\\{D_{1}\exp\left[+i\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\right]\right.\cr\cr\left.\hskip
56.9055pt+D_{2}\exp\left[-i\int\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\right]\right\\},&\kappa<R<R_{+}\cr\cr\frac{e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}\left\\{D_{3}\exp\left[-\int\frac{dR}{B}\sqrt{(m-{\mathcal{E}})^{2}B-f^{2}}\right]\right.\cr\cr\left.\hskip
56.9055pt+D_{4}\exp\left[+\int\frac{dR}{B}\sqrt{(m-{\mathcal{E}})^{2}B-f^{2}}\right]\right\\},&R>R_{+}\end{matrix}\right.$
(43)

where $D_{j}$ and $F_{j}$ are constants. The wave functions in the exterior
i.e., in the classically allowed and forbidden regions, can be matched in the
standard way by invoking the asymptotic forms of the Airy functions far from
the boundary between the regions. One readily finds the connection rules,

$D_{1}=\left(D_{3}-\frac{i}{2}D_{4}\right)e^{i\pi/4},~{}~{}D_{2}=\left(D_{3}+\frac{i}{2}D_{4}\right)e^{-i\pi/4}.$
(44)

Since the classically forbidden region extends to infinity we take $D_{4}=0$,
which implies that $D_{1}e^{-i\pi/4}=D_{2}e^{i\pi/4}=D_{3}$. In remains to
match the interior and exterior solutions at the horizon, where the integral
defining the phase has an essential singularity of order one.

We define the integral by analytically continuing to the complex plane,
deforming the path so as to go around the pole at $R=\kappa$ in an
infinitesimal circle of radius $\varepsilon$. Let $C_{\varepsilon}$ denote the
deformed path, $S_{\varepsilon}$ the semi-circle of radius $\varepsilon$ about
$R=\kappa$ in the complex $R$ plane, then we define

$\int^{R}\frac{dR}{\sqrt{B}}\sqrt{\frac{f^{2}}{B}-(m-{\mathcal{E}})^{2}}~{}\stackrel{{\scriptstyle\text{def}}}{{=}}~{}\lim_{\varepsilon\rightarrow
0}\underset{(C_{\varepsilon})}{\int^{R}}\frac{dR}{\sqrt{B}}\sqrt{\frac{f^{2}}{B}-(m-{\mathcal{E}})^{2}}$
(45)

and choose the orientation of the semi-circle as a boundary condition.
Performing the integration from left to right for $R=\kappa+\varepsilon$

$\displaystyle\underset{(C_{\varepsilon})}{\int^{\kappa+\varepsilon}}\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}$
$\displaystyle=$
$\displaystyle\underset{(C_{\varepsilon})}{\int^{\kappa-\varepsilon}}\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}$
(48) $\displaystyle\hskip
28.45274pt+\underset{(S_{\varepsilon})}{\int}\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}$

Since $\varepsilon$ is small, we perform a near horizon approximation of the
integrand in the second integral,

$\underset{(S_{\varepsilon})}{\int}\frac{dR}{B}\sqrt{f^{2}-(m-{\mathcal{E}})^{2}B}\approx\int_{S_{\varepsilon}}\frac{dRRf}{(R-\kappa)}=\pm\frac{i\pi\mu^{2}}{2}$
(49)

where the positive sign occurs if the path is deformed in the lower half
complex plane, the negative sign occurs when the path is deformed in the upper
half complex plane. In the present situation, there appears to be no good
reason to choose one over the other. Each choice amounts to the
identifications

$F_{1}=D_{1}e^{\mp\pi\mu^{2}/2}=D_{3}e^{i\pi/4}e^{\mp\pi\mu^{2}/2},~{}~{}F_{2}=D_{2}e^{\pm\pi\mu^{2}/2}=D_{3}e^{-i\pi/4}e^{\pm\pi\mu^{2}/2}.$
(50)

Owing to the sign change in $B$ across the horizon, outgoing waves in the
exterior are matched to infalling waves in the interior and, vice-versa,
infalling waves in the exterior are matched to outgoing waves in the interior,
and the complete wave function is

$\Psi(\tau,R)=\left\\{\begin{matrix}\frac{D_{3}e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}\left[e^{\mp\pi\mu^{2}/2}e^{\frac{i\pi}{4}}e^{iS_{0}}+e^{\pm\pi\mu^{2}/2}e^{-\frac{i\pi}{4}}e^{-iS_{0}}\right]&0<R<\kappa\cr\cr\frac{2D_{3}e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}\cos\left[S_{0}+\frac{\pi}{4}\right],&\kappa<R<R_{+}\cr\cr\frac{D_{3}e^{-i{\mathcal{E}}\tau}}{|B|^{1/4}\sqrt{|S_{0}^{\prime}|}}e^{-\int|S_{0}^{\prime}|dR},&R>R_{+}\end{matrix}\right.$
(51)

where $S_{0}(R)$ is defined in (29).

##  IV The Energy Spectrum

For any two solutions of the wave equation in (23) there is a conserved
bilinear current density given by

$J_{i}=-\frac{i}{2}\Phi^{*}\overleftrightarrow{\nabla}_{i}\Psi+m\delta_{i\tau}\Phi^{*}\Psi$
(52)

the time component of which determines a physical inner product

$\langle\Phi,\Psi\rangle=\int\frac{dR}{\sqrt{B}}\left[-\frac{i}{2}\Phi^{*}\overleftrightarrow{\nabla}_{\tau}\Psi+m\Phi^{*}\Psi\right].$
(53)

Consider two stationary states,

$\Psi_{\mathcal{E}}(\tau,R)=e^{-i{\mathcal{E}}\tau}\psi_{\mathcal{E}}(R),~{}~{}\Phi_{{\mathcal{E}}^{\prime}}(\tau,R)=e^{-i{\mathcal{E}}^{\prime}\tau}\psi_{{\mathcal{E}}^{\prime}}(R)$
(54)

of energies ${\mathcal{E}}$ and ${\mathcal{E}}^{\prime}$, then

$\langle\Phi,\Psi\rangle=\left[m-\frac{1}{2}({\mathcal{E}}+{\mathcal{E}}^{\prime})\right]e^{-i({\mathcal{E}}+{\mathcal{E}}^{\prime})\tau}\int\frac{dR}{\sqrt{B}}\phi^{*}_{{\mathcal{E}}^{\prime}}\psi_{\mathcal{E}}$
(55)

is positive semi-definite so long as ${\mathcal{E}}<m$. By the wave equation
we get

$\displaystyle\phi^{*}_{{\mathcal{E}}^{\prime}}\sqrt{B}\partial_{R}\sqrt{B}\partial_{R}\psi_{\mathcal{E}}$
$\displaystyle=$
$\displaystyle\left[(m-{\mathcal{E}})^{2}-\frac{f^{2}}{B}\right]\phi^{*}_{{\mathcal{E}}^{\prime}}\psi_{\mathcal{E}}$
$\displaystyle\psi_{\mathcal{E}}\sqrt{B}\partial_{R}\sqrt{B}\partial_{R}\phi^{*}_{{\mathcal{E}}^{\prime}}$
$\displaystyle=$
$\displaystyle\left[(m-{\mathcal{E}}^{\prime})^{2}-\frac{f^{2}}{B}\right]\phi^{*}_{{\mathcal{E}}^{\prime}}\psi_{\mathcal{E}}$
(57)

Subtracting the second from the first,

$\sqrt{B}\partial_{R}\left(\sqrt{B}\phi^{*}_{{\mathcal{E}}^{\prime}}\overleftrightarrow{\partial_{R}}\psi_{\mathcal{E}}\right)=({\mathcal{E}}-{\mathcal{E}}^{\prime})\left({\mathcal{E}}+{\mathcal{E}}^{\prime}-2m\right)\phi^{*}_{{\mathcal{E}}^{\prime}}\psi_{\mathcal{E}}$
(58)

showing that the inner product in (55) is just a surface term which, mindful
of the three regions, we give as

$\langle\Phi,\Psi\rangle=-\frac{i}{({\mathcal{E}}-{\mathcal{E}}^{\prime})}\left[\int_{0}^{\kappa}dR\partial_{R}\sqrt{B}J_{R}+\int_{\kappa}^{R_{+}}dR\partial_{R}\sqrt{B}J_{R}+\int_{R_{+}}^{\infty}dR\partial_{R}\sqrt{B}J_{R}\right].$
(59)

To guarantee orthonormality of the wave functions, we must require that the
inner product vanishes whenever ${\mathcal{E}}^{\prime}\neq{\mathcal{E}}$.
Therefore, calling $\Omega=\sqrt{B}J_{R}$, we seek the conditions under which

$\lim_{R\rightarrow\infty}\Omega-\lim_{R\rightarrow
R_{+}^{+}}\Omega+\lim_{R\rightarrow
R_{+}^{-}}\Omega-\lim_{R\rightarrow\kappa^{+}}\Omega+\lim_{R\rightarrow\kappa^{-}}\Omega-\lim_{R\rightarrow
0}\Omega=0,$ (60)

where the superscripts indicate the left/right limits. The first term vanishes
because the wave function vanishes exponentially at infinity. Direct
computation also shows that the second and third terms cancel and the last
term vanishes but the fourth and fifth terms, which must be evaluated at the
black hole horizon, neither separately vanish nor cancel one another. This
occurs because the black hole horizon is an essential singularity of the phase
integral. One finds

$-\lim_{R\rightarrow\kappa^{+}}\Omega+\lim_{R\rightarrow\kappa^{-}}\Omega\sim\sin\left[\frac{\mu^{2}}{2}\ln\left(\frac{m-{\mathcal{E}}}{m-{\mathcal{E}}^{\prime}}\right)\right],$
(61)

and therefore

$\frac{m-{\mathcal{E}}}{m-{\mathcal{E}}^{\prime}}=e^{\frac{2n\pi}{\mu^{2}}}$
(62)

for integer values of $n$. Assuming the existence of a ground state, it
implies the energy spectrum,

${\mathcal{E}}_{n}=m\left(1-e^{-2n\pi/\mu^{2}}\right),$ (63)

where $n$ is a whole number. This is identical to the spectrum of the shell
collapsing in a vacuum when $\mu\gg 1$, i.e., when the proper mass of the
shell is much larger than the Planck mass. The presence of the external black
hole does not disturb the spectrum to this order.

##  V Conclusions

The purpose of this work was to understand the similarities and differences in
the quantum mechanics of a single thin shell and the midisuperspace
quantization of the shells in a collapsing dust ball. We examined the WKB
approximation to the proper time quantum mechanics of the thin dust shell when
it surrounds a pre-existing black hole. We have shown that although the
construction of the Hamiltonians governing the evolution of the two systems
have very different origins (the dynamics of the thin shell are obtained via
an application of the IDL conditions whereas the dynamics of the dust ball are
fully derived from the Einstein-dust system) the Hamiltonians one ends up are
structurally similar with a crucial exception: the thin shell posseses a self-
interaction that depends on its area radius, but the shells of a dust ball do
not. This self interaction causes the thin shell to always stay bound to the
center, regardless of whether the interior of the shell is a vacuum or a black
hole. On the contrary, the shells of a dust ball may be unbound. Again as a
consequence of the self-interaction, the matching of the wave function at the
horizon of the black hole is accomplished with essentially no information
about the horizon length, only the proper mass of the shell plays a role.
Neither does the black hole play any role in the energy spectrum of the shell,
which we have shown is identical to the spectrum of a shell collapsing in a
vacuum to this order.

What is most surprising is that bound state solutions exist for shell proper
masses greater than the Planck mass and the shell does not collapse into the
central singularity, suggesting that quantum uncertainty plays a role in the
collapse over distance scales determined by the size of the black hole’s event
horizon and larger than previously suspected. The wave function vanishes at
the center, extends out to the turning point, which, depending on the energy
of the shell, may lie close to the horizon but always outside it, and falls
off exponentially beyond this point. This is reminiscent of a gravitational
atom and supports another solution of the quantum dust ball collapse, which
does not involve continued collapse as described in the introduction: if the
collapse does not continue past the apparent horizon, the solution is
described by the first of the two solutions given in the introduction and the
shells will coalesce on the apparent horizon. No event horizon will form and
the collapse will end in an ultra compact star instead of a black hole [31,
32].

In general, proper time quantization seems to enjoy several advantages over
coordinate time quantizations. For one, the proper time quantum theory exists
for shells of mass greater than the Planck mass, unlike the quantum mechanics
that is based on coordinate time. But the most important is that it satisfies
a basic requirement of the quantum theory, i.e., observer independence of the
time parameter. It is therefore “democratic” in regard to all foliations of
spacetime: all coordinate time variables would be functions of the phase space
(in the simple case of the shell these are given by (3)) as are the spatial
coordinates. The same would be true of the metric components. Thus they would
all be operator valued and we would be able to speak of time intervals and
spatial distances only in terms of averages. In the proper time formulation,
these averages can be calculated and fluctuations about them quantified
because the Wheeler-DeWitt equation yields a conserved, positive, semi-
definite inner product.

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