On leave of absence from: ]National Research Center “Kurchatov Institute”,
Petersburg Nuclear Physics Institute, St. Petersburg 188300, Russia

#  Moving mirrors and event horizons in asymptotically non-flat spacetimes

Evgenii Ievlev<EMAIL_ADDRESS>[  William I. Fine Theoretical Physics
Institute, School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA  Physics Department &
Energetic Cosmos Laboratory, Nazarbayev University,
Astana 010000, Kazakhstan

###### Abstract

Moving mirrors have been used for a long time as simple models for studying
various properties of black hole radiation, such as the thermal spectrum and
entanglement entropy. These models are typically constructed to mimic the
collapse of a spherically symmetric distribution of matter in the
asymptotically Minkowski background. We generalize this correspondence to the
case of non-trivial background geometry (e.g. a black hole in the AdS
spacetime) and somewhat relax the assumption of spherical symmetry.

††preprint: FTPI-MINN-23-21

###### Contents

  1. I Introduction
  2. II Particle production in dynamical models
    1. II.1 Quantum mechanical analogue
    2. II.2 Moving mirror in 1+1d
    3. II.3 Null shell in 3+1d
  3. III Mirror-gravity correspondence in flat background
    1. III.1 Null shell collapse
    2. III.2 Statement of the correspondence
    3. III.3 Possible generalizations
  4. IV The mirror and the non-trivial background geometry
  5. V Applications
    1. V.1 Schwarzschild – de Sitter
      1. V.1.1 Mirror’s trajectory
      2. V.1.2 Analysis of the dynamics
      3. V.1.3 Discussion of the results
    2. V.2 BTZ
      1. V.2.1 Mirror’s trajectory
      2. V.2.2 AdS peculiarities in the mirror
      3. V.2.3 Radiation and temperature
      4. V.2.4 Note on a spinning black hole
  6. VI Conclusions
  7. A The time coordinate inside the shell

##  I Introduction

It has been known for a long time that the vacuum in a quantum system can
change its structure when the system undergoes a transition, either smooth or
discontinuous. This implies that even if the system has started as a state
with lowest possible energy and no particles, at late times particles may
appear.

In the Heisenberg picture of quantum mechanics this is seen from the evolution
of the Hamiltonian and its spectrum. This evolution means that, although the
wave function $\ket{\Psi}$ is not changing, its decomposition into the energy
eigenstates is not the same at early and late times. Even if $\ket{\Psi}$ was
a lowest energy eigenstate if the early-times Hamiltonian, at late times it
may have contributions from excited eigenstates.

In quantum field theory (QFT), one can analyze vacuum fluctuation of the
quantum field. Under the influence of, for example, a black hole horizon in
3+1d [1] or a dynamical boundary condition in 1+1d [2, 3, 4], these
fluctuations can be amplified, which leads to a detectable energy/particle
flux being produced.

The moving mirror model, or dynamical Casimir effect (DCE), is a QFT of a
massless scalar field111In the simplest case; other conformal field theories
(CFTs) can also be studied. with a Dirichlet boundary condition (b.c.): the
field is set to zero at the point where the mirror is located. The non-trivial
dynamics occurs if the mirror is accelerated [5, 2, 3, 4, 6, 7, 8, 9, 10, 11,
12], see also e.g. the textbooks [13, 14].

It was quickly realized that the phenomena of particle production near a black
hole horizon and in moving mirror model bear striking resemblance, and that
this resemblance can be used for understanding certain aspects of the QFT in
3+1-dimensional curved spacetime by studying a simpler 1+1 flat spacetime
model [2, 3, 4]. Later a specific recipe was constructed [15] that relates a
collapsing null shell to a specific moving mirror trajectory. However, this
recipe is valid as-is only in asymptotically flat spacetimes with spherical
symmetry. Two such examples have been considered in the literature, the
Schwarzschild [16] and Reissner–Nordström [17] black hole mirror analogies.

There have also been attempts to apply this recipe to spacetimes that are not
asymptotically flat [18, 19, 20] or have only axial symmetry [21, 22].
However, as these examples violate the initial prescription assumptions
(asymptotic flatness and spherical symmetry), there is no guarantee that the
results will be sensible; and indeed, inconsistencies may arise [23].

In this work we generalize the black hole – mirror correspondence to the case
when the spacetime is not asymptotically flat. More specifically, we consider
a black hole that is formed from a collapsing shell of null dust immersed in a
spacetime that had non-vanishing curvature to begin with. The thin shell
approximation allows to capture the essential features of the process while
keeping full tractability. We were also able to demonstrate consistency of
relaxing the spherical symmetry assumption: instead of requiring that the
spacetime itself should be spherically symmetric, we only require that the
tortoise coordinate for such a metric does not depend on the angles; this
corroborates the results on the Kerr(-Newman) black holes [21, 22, 24].

The paper is organized as follows. In Sec. II we review basic results on
particle and energy production. We start with a simple quantum mechanical
problem, and then briefly review the moving mirrors and black hole radiation.
Readers familiar with the subject can safely skip this Section. In Sec. III we
discuss the details of the black hole – mirror correspondence in a flat
background, while in Sec. IV we generalize it to a curved background geometry.
Sec. V presents two examples, the Schwarzschild – de Sitter and the BTZ black
holes ((non-)spherical symmetry is discussed in subsection V.2.4). In Sec. VI
we outline the main results and possible future directions.

##  II Particle production in dynamical models

In this section we will briefly review the concept of particle production in
some models with externally varied conditions. We will start with an
undergraduate-level quantum mechanical problem, and then progress to moving
mirrors and black holes. This Section focuses on the key features that help to
understand the connection between moving mirrors and collapsing geometries.

###  II.1 Quantum mechanical analogue

Consider a simple harmonic oscillator with a linear perturbation,

$\hat{H}=\frac{\hat{p}^{2}}{2m}+\frac{\omega^{2}\hat{x}^{2}}{2}+F(t)\hat{x},$
(1)

where $F(t)$ is some predefined function of time (its physical meaning is that
$-F(t)$ is a homogeneous external force applied to the oscillator). Suppose
that $F(t)$ vanishes at $t\to\pm\infty$, and that at early times the harmonic
oscillator was in the vacuum state $\ket{0}_{\text{in}}$ (no particles). We
want to understand the particle content at late times.

At a first glance, it would seem that since the Hamiltonian in Eq. (1) is the
same at early and late times, the final vacuum state $\ket{0}_{\text{out}}$
would be the same as $\ket{0}_{\text{in}}$, and there will not be any excited
states. However, this turns out not to be the case.

To see this, let us pass to the creation-annihilation operator formalism in
the Heisenberg picture. We introduce

$\displaystyle\hat{a}(t)$
$\displaystyle=\sqrt{\frac{m\omega}{2\hbar}}\hat{x}+\frac{i}{\sqrt{2\hbar
m\omega}}\hat{p},$ (2) $\displaystyle\hat{a}^{\dagger}(t)$
$\displaystyle=\sqrt{\frac{m\omega}{2\hbar}}\hat{x}-\frac{i}{\sqrt{2\hbar
m\omega}}\hat{p},$

and

$[\hat{a}(t),\hat{a}^{\dagger}(t)]=1.$ (3)

In terms of these operators, the Hamiltonian Eq. (1) is rewritten as

$\hat{H}(t)=\hbar\omega\left(\hat{a}^{\dagger}(t)\hat{a}(t)+\frac{1}{2}\right)+\sqrt{\frac{\hbar}{2m\omega}}F(t)\left(\hat{a}^{\dagger}(t)+\hat{a}(t)\right)\,.$
(4)

The dynamics is governed by the equation
$\hbar\mathop{}\\!\mathrm{d}\hat{a}=i[\hat{H},\hat{a}(t)]$ and its conjugate.
Solving this linear equation we arrive at

$\hat{a}(t)=e^{-i\omega
t}\left(\hat{a}_{\text{in}}-\frac{i}{\sqrt{2m\hbar\omega}}\int\limits_{-\infty}^{t}F(\tau)e^{i\omega\tau}\mathop{}\\!\mathrm{d}\tau\right),$
(5)

where $\hat{a}_{\text{in}}$ does not depend on time. From the solution (5) we
immediately infer

$\displaystyle\hat{a}(t)$ $\displaystyle=e^{-i\omega
t}\hat{a}_{\text{in}},\quad$ $\displaystyle\text{at }t\to-\infty,$ (6)
$\displaystyle\hat{a}(t)$ $\displaystyle=e^{-i\omega
t}\hat{a}_{\text{out}}\,,\quad\hat{a}_{\text{out}}=\hat{a}_{\text{in}}+\alpha,\quad$
$\displaystyle\text{at }t\to+\infty,$

where the $c$-number $\alpha$ is given by the integral

$\alpha=-\frac{i}{\sqrt{2m\hbar\omega}}\int\limits_{-\infty}^{\infty}F(\tau)e^{i\omega\tau}\mathop{}\\!\mathrm{d}\tau.$
(7)

The Hamiltonian in Eq. (4) is then represented as

$\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{\text{in}}=\hbar\omega\left(\hat{a}_{\text{in}}^{\dagger}\hat{a}_{\text{in}}+\frac{1}{2}\right),\quad$
$\displaystyle\text{at }t\to-\infty,$ (8) $\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{\text{out}}=\hbar\omega\left(\hat{a}_{\text{out}}^{\dagger}\hat{a}_{\text{out}}+\frac{1}{2}\right),\quad$
$\displaystyle\text{at }t\to-\infty.$

Now, having solved for the time evolution of the operators, let’s discuss the
particle production. Recall that at early times $t\to-\infty$ we had started
with the lowest energy eigenstate $\ket{0}_{\text{in}}$ defined as

$\hat{a}_{\text{in}}\ket{0}_{\text{in}}=0,$ (9)

so that

$\hat{H}_{\text{in}}\ket{0}_{\text{in}}=\frac{1}{2}\hbar\omega\ket{0}_{\text{in}}.$
(10)

The normalized eigenstates of the final Hamiltonian $\hat{H}_{\text{out}}$ are
given by

$\ket{n}_{\text{out}}=\frac{(\hat{a}_{\text{out}}^{\dagger})^{n}}{\sqrt{n!}}\ket{0}_{\text{out}}\,,\quad\hat{a}_{\text{out}}\ket{0}_{\text{out}}=0.$
(11)

Using $\hat{a}_{\text{out}}=\hat{a}_{\text{in}}+\alpha$ from Eq. (6) one can
rewrite the in-vacuum Eq. (9) as

$\hat{a}_{\text{out}}\ket{0}_{\text{in}}=\alpha\ket{0}_{\text{in}}.$ (12)

This is just the equation for a coherent state. We conclude that the in-vacuum
becomes a coherent state when expressed in eigenvectors of the final
Hamiltonian:

$\ket{0}_{\text{in}}=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}_{\text{out}}.$
(13)

This result shows that the final state is a superposition of multiparticle
states. The particle number expectation value

${}_{\text{in}}\expectationvalue{\hat{a}_{\text{out}}^{\dagger}\hat{a}_{\text{out}}}{0}_{\text{in}}=|\alpha|^{2},$
(14)

is now non-zero. The energy expectation value

${}_{\text{in}}\expectationvalue{\hat{H}_{\text{out}}}{0}_{\text{in}}=\hbar\omega\left(|\alpha|^{2}+\frac{1}{2}\right),$
(15)

is larger than the initial zero-point energy Eq. (10).

This simple quantum mechanical problem demonstrates an important effect. Even
though the Hamiltonian, Eq. (1), seems to be the same at $t\to\pm\infty$, the
intermediate time evolution makes the relationship between the initial and
final creation-annihilation operators non-trivial. This leads to an energy
increase and a non-zero final particle count.

###  II.2 Moving mirror in 1+1d

(a) Timelike mirror has no horizon.

(b) Asymptotically lightlike mirror develops a horizon $\mathcal{H}$.

Figure 1:  Two Penrose diagrams for 1+1 mirrors with and without horizons. An
incoming light ray starts at the past null infinity $\mathcal{I}^{R}_{-}$ and
propagates at $v=v_{-}=\text{const}$, then gets reflected at the mirror and
propagates to the future null infinity $\mathcal{I}^{R}_{+}$ with
$u=u_{+}=\text{const}$. The relationship between $u_{+}$ and $v_{-}$ is given
by the mirror trajectory as $u_{+}=f_{m}(v_{-})$ or, equivalently,
$v_{-}=p_{m}(u_{+})$. For an asymptotically lightlike mirror rays that start
with $v>v_{\mathcal{H}}$ never get reflected; in a sense, they are lost to
$\mathcal{I}^{L}_{+}$.

The moving mirror model is a 1+1d conformal field theory with a boundary
(BCFT); a boundary condition is imposed along a timelike or a lightlike curve.
In this paper we focus on the simplest case of the free boson CFT with a
scalar field $\Phi$ and a boundary condition $\Phi=0$ at the location of the
mirror [2, 3, 4].

The mirror location is taken to be preset externally. Its trajectory can be
specified in a variety of ways. In terms of the usual Minkowski coordinates
$(t,z)$ we can specify the mirror position with the help of a function
$t_{m}(z)$ or the inverse function $z_{m}(t)$ as

$t=t_{m}(z)\quad\text{or}\quad z=z_{m}(t)\,.$ (16)

Equivalently, in terms of the light cone coordinates $(v,u)$ defined as

$v=t+z\,,\quad u=t-z\,,$ (17)

the mirror position can be set in terms of a function $f_{m}(v)$ or in terms
of the inverse function $p_{m}(u)$ as

$u=f_{m}(v)\quad\text{or}\quad v=p_{m}(u)\,.$ (18)

We will employ the usual convention that at late times the mirror goes to the
left, i.e. to the negative $z$.

There are two qualitatively different cases that depend on whether or not the
mirror asymptotically approaches the speed of light. If the mirror is always
timelike, then all incident rays get reflected, see Fig. 1a. If the mirror is
lightlike at late times, then it develops a horizon, as late rays never reach
the mirror and do not get reflected, see Fig. 1b. A comprehensive discussion
of the moving mirror classification can be found in [25].

Consider a ray that starts at the past null infinity $\mathcal{I}^{R}_{-}$
with $v=v_{-}$ with a frequency $\omega_{-}$. If this ray does get reflected
from the mirror, then it travels to the future null infinity
$\mathcal{I}^{R}_{+}$ with $u=u_{+}$ (see Fig. 1) and a red shifted frequency
$\omega_{+}$. The red shift is determined by a standard Doppler factor

$\frac{\omega_{+}}{\omega_{-}}=\frac{1-|\dot{z}_{m}|}{1+|\dot{z}_{m}|}=\derivative{p_{m}}{u}=\left(\derivative{f_{m}}{v}\right)^{-1}\,,$
(19)

where the dot denotes a derivative with respect to time $t$.

If the mirror is stationary or moves at a constant speed, we can always do a
Lorenz boost to a frame where the mirror is at $z=z_{m}(t)\equiv 0$ at all
times. Such a boost does not change the scalar field energy-momentum tensor.
If we start in the vacuum state, we will always stay in the vacuum state. We
can reformulate this in the following way: if the red shift in Eq. (19) is
time-independent, there is no particle production.

However, if the mirror moves non-uniformly (i.e. the speed is not constant in
time), the situation drastically changes. In the CFT language, we can still
perform a conformal map so that the mirror is at the origin, but the stress-
energy tensor of the scalar field $\Phi$ become non-trivial [3]. In terms of
the trajectory $v=p_{m}(u)$, one finds the energy flux at
$\mathcal{I}^{R}_{+}$

$\left\langle
T_{uu}\right\rangle=\frac{1}{24\pi}\left[\frac{3}{2}\left(\frac{p^{\prime\prime}}{p^{\prime}}\right)^{2}-\frac{p^{\prime\prime\prime}}{p^{\prime}}\right]\,,$
(20)

where primes denote derivatives with respect to $u$.

In the usual field theory language, while we start in the trivial vacuum
$\ket{0}_{\text{in}}$, after the mirror’s acceleration the state
$\ket{0}_{\text{in}}$ is no longer a vacuum. Rather, it is a non-trivial
linear combination of a new vacuum and excited states. This is reflected by
non-trivial Bogolyubov coefficients. The relevant equation is (see e.g. [14,
13, 26, 27])

$\beta^{R}_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int\limits_{-\infty}^{v_{\mathcal{H}}}dve^{-i\omega^{\prime}v-i\omega
f_{m}(v)}\,.$ (21)

Here, $v_{\mathcal{H}}$ is the position of the horizon
($v_{\mathcal{H}}=+\infty$ for a timelike mirror). Using these Bogolyubov
coefficients one can calculate the particle number registered at
$\mathcal{I}^{R}_{+}$

$\expectationvalue{N_{\omega}}=\int\limits_{0}^{\infty}\mathop{}\\!\mathrm{d}\omega^{\prime}\,|\beta^{R}_{\omega\omega^{\prime}}|^{2}$
(22)

as well as the corresponding energy

$\expectationvalue{E}=\int\limits_{0}^{\infty}\mathop{}\\!\mathrm{d}\omega\,\omega\expectationvalue{N_{\omega}}=\int\limits_{0}^{\infty}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\omega^{\prime}\,\omega|\beta^{R}_{\omega\omega^{\prime}}|^{2}\,.$
(23)

###  II.3 Null shell in 3+1d

(a) Timelike shell that does not create a horizon.

(b) Lightlike (null) shell forms a horizon $\mathcal{H}$.

Figure 2:  Penrose diagrams for 3+1 spacetime with simple shells. An incoming
light ray starts at the past null infinity $\mathcal{I}^{R}_{-}$ and
propagates at $v=v_{-}=\text{const}$ (superscript “in” our “out” depending on
the picture), then gets “reflected” at the origin $r=0$ and propagates to the
future null infinity $\mathcal{I}^{R}_{+}$ with
$u^{\text{out}}=u_{+}^{\text{out}}=\text{const}$. The relationship between
$u_{+}$ and $v_{-}$ is determined by the spacetime geometry. For a lightlike
shell rays that start with $v>v_{\mathcal{H}}$ are lost to the singularity
(analog of $\mathcal{I}^{L}_{+}$ in the mirror case).

Our main model of gravitational collapse is going to be the dust shell [28,
15, 29], see also [14]. We consider a shell of zero electrical charge and
uniform surface energy density, with total energy $M$. The shell is immersed
in a 3+1 dimensional space with some underlying geometry (this background
might be flat, or there might be e.g. a cosmological constant). Upon this, we
consider a neutral massless scalar field $\Phi$ minimally coupled to gravity.

The shell particles and the theory of gravity are taken as classical, while
the field $\Phi$ will be considered as a quantum field propagating in the
background of classical geometry induced by the shell; incorporation of the
back reaction is left for a future work. The Hawking-like radiation that we
want to study is the radiation of $\Phi$ quanta happening in the time-varying
gravity background.

Generally, there are two qualitatively different possibilities. In one
situation there is no horizon; this happens when the shell is timelike and
does not collapse, i.e. does not reach its Schwarzschild radius, see Fig. 2a.
In the other a horizon develops, as the shell collapses. This happens e.g. in
the famous case of the null shell and corresponding Vaidya spacetime, see Fig.
2b. In all of these cases one should differentiate between inside and outside
of the shell, as the metric tensors are different in these regions. Therefore,
we denote the light cone coordinates inside the shell as $v^{\text{in}}$,
$u^{\text{in}}$, and outside the shell $v^{\text{out}}$, $u^{\text{out}}$.
Precise relation of these coordinates to each other and to the usual spherical
coordinates will be specified below.

In this paper we will focus on the case of null dust collapsing in an already
non-trivial background geometry. This case turns out to be much simpler than
the timelike shell (the latter will be treated in a subsequent publication).
Indeed, consider a ray that starts at the past null infinity
$\mathcal{I}^{R}_{-}$ with $v^{\text{out}}=v^{\text{out}}_{-}$ with a
frequency $\omega_{-}$ (see Fig. 1). If it does not hit the horizon, then it
travels to the future null infinity $\mathcal{I}^{R}_{+}$ with
$u=u_{+}^{\text{out}}$ and a red shifted222Red shifted for a contracting
shell. If the shell is expanding, then, generally speaking, there can be a
blue shift. frequency $\omega_{+}$. The simplicity of the null shell case
stems from the fact that such a ray crosses the shell exactly once; in the
timelike case there would be at least two crossings, with the corresponding
non-local effects severely complicating the analysis.

We define a ray tracing function $G(u)$ via

$v^{\text{out}}_{-}=G(u_{+}^{\text{out}})\,.$ (24)

As soon as the function $G(u)$ is known, the Doppler shift factor can be
easily found

$\frac{\omega_{+}}{\omega_{-}}=\derivative{G}{u}\,.$ (25)

Hawking radiation in this setup is the production of the scalar field $\Phi$
quanta in such a time-varying background. As was shown in [30], the power
radiated across a large sphere and registered at the future null infinity
$\mathcal{I}^{R}_{+}$ is given by

$P=\frac{1}{24\pi}\left[\frac{3}{2}\left(\frac{G^{\prime\prime}}{G^{\prime}}\right)^{2}-\frac{G^{\prime\prime\prime}}{G^{\prime}}\right]\,.$
(26)

In certain cases, the spectrum of Hawking radiation can also be computed;
however, the calculations usually involve QFT on a curved spacetime background
and often present challenges.

##  III Mirror-gravity correspondence in flat background

In this section we are going to discuss the correspondence between the
collapsing 3+1d geometry on one hand, and the moving mirror in 1+1d on the
other hand.

We are going to start with a review of some known results on the mirror-
gravity correspondence. We will focus on the case when the black hole is
modeled as a collapse of a thin shell made of null dust immersed in flat
spacetime. For such a shell one can construct a moving mirror that has (under
approximations to be discussed below) the same radiation spectrum as that of
the black hole under consideration.

The discussion presented here focuses on the key details that will be needed
for further generalizations, which are also discussed below.

Figure 3:  Spacetime is curved in the presence of a massive object. The thick
red line is a schematic representation of the distance to the origin.

###  III.1 Null shell collapse

We take the bulk as the 3+1 Minkowski spacetime and introduce in it a
collapsing null shell. The corresponding mirror was worked out in [15]. In
this setup, the space inside the shell is flat, while outside it is given by
the Schwarzschild solution:

$\displaystyle ds_{\text{in}}^{2}$
$\displaystyle=dt_{\text{in}}^{2}-dr^{2}-r^{2}d\Omega^{2}\,,\quad
r<r_{\text{sh}}(t)\,,$ (27) $\displaystyle ds_{\text{out}}^{2}$
$\displaystyle=F(r)dt^{2}-\frac{1}{F(r)}dr^{2}-r^{2}d\Omega^{2}\,,\quad
r>r_{\text{sh}}(t)\,.$

Here, $r_{\text{sh}}(t)$ is the radius of the collapsing shell, $M$ is the
shell’s energy, while the metric function $F$ is given by the standard
expression

$F(r)=1-\frac{2M}{r}\,.$ (28)

Note that the time coordinate is different on the inside and on the outside;
see Appendix A for the details.

We introduce the tortoise coordinate $r_{*}(r)$ defined by

$\derivative{r_{*}}{r}=\frac{1}{F(r)}\,.$ (29)

The light cone coordinates outside and inside the shell are then defined,
respectfully, as

$\displaystyle v^{\text{out}}=t+r_{*}(r)\,,$ $\displaystyle
u^{\text{out}}=t+r_{*}(r)\,;$ (30) $\displaystyle
v^{\text{in}}=t_{\text{in}}+r\,,$ $\displaystyle
u^{\text{in}}=t_{\text{in}}+r\,.$

Note that in terms of $r_{*}$, the $(t,r)$ components of the metric outside
can be rewritten as $ds_{\text{out}}^{2}=F(r)\left(dt^{2}-dr_{*}^{2}\right)$
One can see that it is conformally flat, which makes it easier to solve the
Klein-Gordon wave equation in these coordinates.

The correspondence with the moving mirror is made through the light cone
coordinates. In order to do that, we need to find the map that related the
inner and outer regions of the shell.

First of all, one identifies $v^{\text{in}}=v^{\text{out}}\equiv v$. This can
be done because our shell is comprised of the incoming null dust. Indeed,
following e.g. [31, Sec. 29.2], we write down the kinematic equation for such
a shell as $v^{\text{in}}=v_{\text{sh}}^{\text{in}}$ or, equivalently, as
$v^{\text{out}}=v_{\text{sh}}^{\text{out}}$ for some constants
$v_{\text{sh}}^{\text{in,out}}$. By making appropriate shifts one can arrange
for $v_{\text{sh}}^{\text{in}}=v_{\text{sh}}^{\text{out}}$, so that the
identification $v^{\text{in}}=v^{\text{out}}$ is consistent on the collapsing
shell. There could still be a proportionality factor between them, but it
turns out to be 1.

To determine the transformation function $u^{\text{in}}(u^{\text{out}})$, we
study what has been called “the trajectory of the origin”. From Eq. (30) it
follows that the point $r=0$ is determined by the equation
$v^{\text{in}}=u^{\text{in}}$. Requiring the shell’s area to be a gauge
invariant we can write

$r_{*}\left(\frac{v_{\text{sh}}-u^{\text{in}}(u^{\text{out}})}{2}\right)=\frac{v_{\text{sh}}-u^{\text{out}}}{2}\,,$
(31)

where $v_{\text{sh}}=t+r_{\text{sh}}(t)$ is the shell’s location in spacetime.
Since for a given metric $F(r)$ the tortoise coordinate $r_{*}(r)$ is known,
Eq. (31) determines the transformation function
$u^{\text{in}}(u^{\text{out}})$.

###  III.2 Statement of the correspondence

Next, one identifies this 3+1d collapsing geometry with a 1+1d moving mirror.
The mirror’s trajectory is taken to be the trajectory of the origin; in a
certain sense [15], the distance from a one-dimensional observer to the mirror
is taken to be the same as the distance to the origin as “seen” by the three-
dimensional observer near a massive object, see Fig. 3 for a schematic
representation. In the spherical coordinates a ray sent to the origin is seen
to be “reflected”, see Fig. 2, which looks very similar to one side of the
mirror Fig. 1.

To pass from the qualitative to the quantitative, we note that the ray tracing
function $G(u^{\text{out}})$ (see Sec. II.3) has the same meaning as the
mirror’s trajectory function $p(u)$. They both map the incoming ray
$u$-coordinate to the reflected outgoing ray $v$-coordinate. The Doppler shift
is given by the derivative of this function, see Eq. (19) and Eq. (25); the
radiated energy flux is given by the Schwarzian derivative of this function,
see Eq. (20) and Eq. (26). The beta Bogolyubov coefficients, which are easily
constructed for the mirror (see Eq. (21)), can also be constructed (to some
extent) on the gravity side [1, 14]; the results are also in agreement with
each other.

As a next step we note that the transformation function
$u^{\text{in}}(u^{\text{out}})$ is actually the same as the ray tracing
function $G$ defined in Eq. 24. This is evident from Fig. (2b): the ray starts
inside the shell at $v^{\text{in}}_{-}\equiv v^{\text{out}}_{-}$, propagates
at constant $v$, then reaches the origin where it starts propagating at
constant $u^{\text{in}}=v^{\text{in}}_{+}$ (recall that
$u^{\text{in}}=v^{\text{in}}$ for $r=0$) until it hits the shell. After the
shell it propagates at constant $u^{\text{out}}=u^{\text{out}}_{+}$ such that
$u^{\text{in}}=u^{\text{in}}(u^{\text{out}})$, see Eq. (31). Tracing this ray
back we obtain

$v^{\text{out}}_{-}=u^{\text{in}}(u^{\text{out}}_{+})\,.$ (32)

One can compare Eq. (32) with the definition of the $G$-function Eq. (24).

All this allows to present the corresponding mirror’s trajectory. Given the
transformation function $u^{\text{in}}(u^{\text{out}})$, one takes the
mirror’s trajectory in the form $v=p_{m}(u)$ (see Sec. II.2) as

$p_{m}(u)=u^{\text{in}}(u^{\text{out}}=u)\,.$ (33)

Identification (33) gives a mirror that produces radiation with the same
spectrum as the corresponding black hole, assuming the geometric optics
approximation on the gravity side (see e.g. [1, 30, 14]). Moreover, it
captures only the $s$-wave radiation, hence the assumption of the spherical
symmetry.

###  III.3 Possible generalizations

Let us list a few key assumptions of the above analysis.

  1. 1.

The spacetime under consideration is spherically symmetric.

  2. 2.

The collapsing shell is null, i.e. it collapses at the speed of light.

  3. 3.

The spacetime inside the shell is flat.

  4. 4.

The spacetime outside the shell is given by the Schwarzschild solution.

One can immediately generalize assumption 4 to a charged black hole [17], but
beyond that one has to proceed with care. For example, the cases of Kerr [21]
and Schwarzschild – de Sitter [19] spacetimes require relaxing of the
assumptions 1 and 3-4, respectfully.

Below we are going to completely relax assumptions 3 and 4 and consider the
shell collapsing in an already non-trivial background. We will also comment a
possible way to relax assumption 1 and consider an example of a spacetime that
is only axially symmetric. Generalization to timelike shells in connection to
the mirror is going to be presented in a subsequent paper.

##  IV The mirror and the non-trivial background geometry

Now, let us implement the generalizations announced in the previous section.
First of all, when we introduce a dust shell in an already non-trivial
spacetime (for example, with a cosmological constant present), the metric
inside the shell is no longer flat. Instead of Eq. (27) we will now have for
the metric inside and outside the shell

$\displaystyle ds_{\text{in}}^{2}$
$\displaystyle=F_{\text{in}}(r)dt_{\text{in}}^{2}-\frac{1}{F_{\text{in}}(r)}dr^{2}-r^{2}d\Omega^{2}\,,\quad
r<r_{\text{sh}}(t)\,;$ (34) $\displaystyle ds_{\text{out}}^{2}$
$\displaystyle=F_{\text{out}}(r)dt^{2}-\frac{1}{F_{\text{out}}(r)}dr^{2}-r^{2}d\Omega^{2}\,,\quad
r>r_{\text{sh}}(t)\,.$

Note that the outside metric $F_{\text{out}}(r)$ is also no longer
Schwarzschild but rather some fairly arbitrary function (of course, we assume
it to be smooth and positive).

Since the inside is no longer flat, we need to introduce two corresponding
tortoise coordinates instead of one:

$\derivative{r_{*\text{in}}}{r}=\frac{1}{F_{\text{in}}(r)}\,,\quad\derivative{r_{*\text{out}}}{r}=\frac{1}{F_{\text{out}}(r)}\,.$
(35)

The spacetime in coordinates $(t_{\text{in}},r_{*\text{in}})$ on the inside,
and $(t,r_{*\text{out}})$ on the outside is conformally flat, which motivates
introduction of the null coordinates

$\displaystyle v^{\text{in}}$
$\displaystyle=t_{\text{in}}+r_{*\text{in}}\,,\quad u^{\text{in}}$
$\displaystyle=t_{\text{in}}-r_{*\text{in}}\,;$ (36) $\displaystyle
v^{\text{out}}$ $\displaystyle=t+r_{*\text{out}}\,,\quad u^{\text{out}}$
$\displaystyle=t-r_{*\text{out}}\,.$

We still can identify $v^{\text{in}}=v^{\text{out}}\equiv v$ and describe the
ingoing null shell by $v=v_{\text{sh}}=\text{const}$.

The tortoise coordinates defined via Eq. (35) are determined up to an
integration constant. While for the outer coordinate $r_{*\text{out}}$ this
constant does not matter (it just effectively shifts $u^{\text{out}}$), the
inner coordinate is fixed by requiring that $v^{\text{in}}=u^{\text{in}}$ at
the origin $r=0$, which implies

$r_{*\text{in}}(r=0)=0\,.$ (37)

Following the flat-background prescription, we still want to match the $r$
coordinate at the shell. From Eq. (36) we infer that on the shell
$r=r_{\text{sh}}$

$\begin{cases}r_{*\text{in}}&=\frac{v_{\text{sh}}-u^{\text{in}}}{2}\,,\\\
r_{*\text{out}}&=\frac{v_{\text{sh}}-u^{\text{out}}}{2}\,,\end{cases}\quad\text{on
the shell}$ (38)

from which follows a generalization of Eq. (31)

$r_{*\text{out}}\left[r_{*\text{in}}^{-1}\left(\frac{v_{\text{sh}}-u^{\text{in}}(u^{\text{out}})}{2}\right)\right]=\frac{v_{\text{sh}}-u^{\text{out}}}{2}\,.$
(39)

Note the appearance of the inverse function $r_{*\text{in}}^{-1}$. The
argument of Sec. III.2 can be repeated, and we identify the mirror’s
trajectory $p_{m}(u)=u^{\text{in}}(u)$.

Therefore, the generalized prescription can be formulated as

$r_{*\text{out}}\left[r_{*\text{in}}^{-1}\left(\frac{v_{\text{sh}}-p_{m}(u)}{2}\right)\right]=\frac{v_{\text{sh}}-u}{2}\,.$
(40)

An alternative prescription for the same trajectory but in the form
$u=f_{m}(v)$ would

$r_{*\text{out}}\left[r_{*\text{in}}^{-1}\left(\frac{v_{\text{sh}}-v}{2}\right)\right]=\frac{v_{\text{sh}}-f_{m}(v)}{2}\,.$
(41)

The mirror’s horizon is then located at
$v_{\mathcal{H}}=v_{\text{sh}}-2r_{*\text{in}}(r_{\mathcal{H}})$, where
$r=r_{\mathcal{H}}$ is the position of the collapsed BH horizon in this
geometry (in the simple Schwarzschild case $r_{\mathcal{H}}=2M$). Note that if
the inside is actually flat ($r_{*\text{in}}(r)=r$) then Eq. (40) reduces to
the known prescription Eq. (31).

What we have done here is nothing but the generalization for the ray-tracing
function $G(u)$. The fundamental principle — identification of the ray-tracing
functions on the mirror side and on the gravity side — remains unchanged.
Formula (40) gives the trajectory of the mirror corresponding to a given
collapsing geometry in the sense, that the radiation coming from this mirror
is the same as the Hawking radiation from the given collapsing geometry (in
the geometric optics approximation).

##  V Applications

Now let us apply the prescription derived in the previous section to some
interesting cases.

###  V.1 Schwarzschild – de Sitter

The case of a black hole in de Sitter universe with a cosmological constant
$\Lambda>0$ is interesting because of several reasons. Our Universe is also
expanding, and it was particularly important during the inflation [32, 33].
Moreover, the cosmological horizon by itself also produces radiation with
thermal characteristics [34]. Such spacetimes are also interesting from the
point of view of holography, see e.g. [35] and references therein.

From the QFT perspective, thermodynamics of the Schwarzschild – de Sitter
(SdS) spacetime was first studied in [34], followed by many others (see e.g.
[36, 37] and references therein), and not only in 3+1 dimensions [38, 39, 40].
The analysis of quantum fields in curved spacetimes such as SdS presents
difficulties which are sometimes hard to overcome. From this perspective the
duality between spacetime geometries and moving mirrors seems attractive, as
it allows one to employ QFT in flat 1+1d spacetime with a boundary to study
properties (such us the radiation and its thermality) of higher-dimensional
theories with gravity.

In [19, 20] the authors studied a mirror that corresponds to the following
geometry: collapsing shell of null dust, positive cosmological constant
$\Lambda>0$ outside of the shell, and $\Lambda=0$ inside the shell (in that
setup the metric inside the shell was taken to be flat). Here we want to
consider a more realistic scenario with the cosmological constant positive and
the same everywhere, both inside and outside the shell.

####  V.1.1 Mirror’s trajectory

The metric of the 3+1-dimensional SdS spacetime is of the form Eq. (34) with
the metric functions

$F_{\text{in}}(r)=1-\frac{r^{2}}{L^{2}}\,,\quad
F_{\text{out}}(r)=1-\frac{r^{2}}{L^{2}}-\frac{2M}{r}\,.$ (42)

The cosmological constant is related to the parameter $L$ as
$L=\sqrt{3/\Lambda}$. From Eq. (35) and Eq. (42) we find the inner tortoise
coordinate and it’s inverse:

$\displaystyle r_{*\text{in}}(r)$
$\displaystyle=\frac{1}{2}L\ln(\frac{L+r}{L-r})=L\tanh[-1](\frac{r}{L})\,,$
(43) $\displaystyle r_{*\text{in}}^{-1}(R)$
$\displaystyle=L\tanh(\frac{R}{L})\,.$

Note the normalization $r_{*\text{in}}(0)=0$, which ensures
$v^{\text{in}}=u^{\text{in}}$ at the origin $r=0$, see Sec. IV. The outer
tortoise coordinate reads

$\displaystyle r_{*\text{out}}(r)$
$\displaystyle=\frac{1}{2\kappa_{B}}\ln\left(\frac{r}{r_{B}}-1\right)$ (44)
$\displaystyle-\frac{1}{2\kappa_{C}}\ln\left(1-\frac{r}{r_{C}}\right)$
$\displaystyle+\left(\frac{1}{2\kappa_{C}}-\frac{1}{2\kappa_{B}}\right)\ln\left(1+\frac{r}{r_{B}+r_{C}}\right)\,.$

In Eq. (44), the $r_{B}$, $r_{C}$ are the two Killing horizons related to the
black hole and de Sitter cosmology respectively,

$\displaystyle r_{B}$
$\displaystyle=\frac{2L}{\sqrt{3}}\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{3\sqrt{3}M}{L}\right)+\frac{\pi}{3}\right]\,,$
(45) $\displaystyle r_{C}$
$\displaystyle=\frac{2L}{\sqrt{3}}\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{3\sqrt{3}M}{L}\right)-\frac{\pi}{3}\right]\,.$

The corresponding surface gravities are given by

$\displaystyle\kappa_{B}$
$\displaystyle=\frac{(r_{C}-r_{B})(2r_{B}+r_{C})}{2L^{2}r_{B}}\,,$ (46)
$\displaystyle-\kappa_{C}$
$\displaystyle=-\frac{(r_{C}-r_{B})(2r_{C}+r_{B})}{2L^{2}r_{C}}\,.$

Here $\kappa_{C}$, while the cosmological horizon surface gravity
$-\kappa_{C}$ is negative.

Now, applying the recipe Eq. (41) (and setting $v_{\text{sh}}=0$ for
convenience) we obtain the SdS trajectory:

$\displaystyle f_{m}$
$\displaystyle(v)=-\frac{1}{\kappa_{B}}\ln\left(-\frac{L}{r_{B}}\tanh(\frac{v}{2L})-1\right)$
(47)
$\displaystyle+\frac{1}{\kappa_{C}}\ln\left(1+\frac{L}{r_{C}}\tanh(\frac{v}{2L})\right)$
$\displaystyle+\left(\frac{1}{\kappa_{B}}-\frac{1}{\kappa_{C}}\right)\ln\left(1-\frac{L}{r_{B}+r_{C}}\tanh(\frac{v}{2L})\right)\,.$

Formula (47) gives the mirror corresponding to the SdS black hole.

####  V.1.2 Analysis of the dynamics

Our trajectory in Eq. (47) can be compared to the trajectory derived in [19]
(see Eq. (20-22) in that paper). There are several distinct features. We start
by checking the limiting cases.

For a vanishing cosmological constant $\Lambda\to 0$, i.e. $L\to\infty$, we
have $r_{B}\approx 2M$, $\kappa_{B}\approx 1/(4M)$ and $r_{C}\approx L$,
$\kappa_{C}\approx 1/L$. Decomposing the logs in Eq. (47) we obtain

$f_{m}(v)\approx v-4M\ln(-v/2M)+\text{const}\,,$ (48)

which is the Schwarzschild mirror and agrees with [19, 15].

However, the limit of empty dS space turns out to be different from [19]. In
this limit we take $M\to 0$, then $r_{B}\approx 2M\to 0$, $\kappa_{B}\approx
1/(4M)\to\infty$ and $r_{C}\approx L$, $\kappa_{C}\approx 1/L$. The trajectory
(47) reduces to

$f_{m}(v)=v+\text{const}\,,$ (49)

which is nothing but the stationary mirror, and there is no radiation in this
case. The reasons for such behavior will be discussed below.

Now let us turn to the radiation spectrum. In principle, the spectrum of the
Hawking radiation can be obtained by plugging the trajectory (47) into the
integral (21). This is already a strong result, as it presents the spectrum as
a one-dimensional integral of an analytic function easily tractable by
numerical analysis.

The most important contributions to the spectrum can be easily derived
analytically in the near-horizon approximation. The trajectory equation (47)
diverges at the values of $v=v_{\mathcal{H},B}$ and $v=v_{\mathcal{H},C}$
determined by the equations

$\tanh(\frac{v_{\mathcal{H},i}}{2L})=-\frac{r_{i}}{L}\,,\quad i=B,C\,.$ (50)

We assume $3\sqrt{3}M/L<1$, so that $r_{B}<r_{C}$, and we are below the Nariai
limit. In this case, simple analysis of Eq. (45) shows that
$0<r_{B}<L/\sqrt{3}$ and $L/\sqrt{3}<r_{C}<L$, and
$v_{\mathcal{H},C}<v_{\mathcal{H},B}<0$. (There is also a non-physical third
“horizon”).

When $v$ approaches one of the horizon values, the mirror trajectory (47)
simplifies to a single log:333There are also unimportant constants which make
the logs dimensionless.

$\displaystyle v\to v_{\mathcal{H},B}$ $\displaystyle\Rightarrow
f_{m}(v)\approx-\frac{1}{\kappa_{B}}\ln(-(v-v_{\mathcal{H},B}))\,,$ (51)
$\displaystyle v\to v_{\mathcal{H},C}$ $\displaystyle\Rightarrow
f_{m}(v)\approx-\frac{1}{\kappa_{C}}\ln(v-v_{\mathcal{H},C})\,.$

These are nothing but examples of the Carlitz-Willey mirror [8]. For this
mirror the radiated particle spectrum is known exactly: it is the 1+1 Planck
distribution with a well-defined temperature $T$,

$\displaystyle N_{\omega\omega^{\prime},i}$
$\displaystyle=|\beta_{\omega\omega^{\prime}}|^{2}=\frac{1}{2\pi\kappa_{i}\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa_{i}}-1}\,,$
(52) $\displaystyle T_{i}$ $\displaystyle=\frac{\kappa_{i}}{2\pi}\,,\quad
i=B,C\,.$

Thus we obtain the thermal radiation with the temperature that agrees with
known results. The total radiation spectrum is a sum of the two found
temperature curves plus some residual non-thermal radiation. The latter piece
can be prominent only during the intermediate times, but is negligible at the
late times, when the black hole horizon forms.

####  V.1.3 Discussion of the results

In this example we have seen that the mirror approach allows one to easily
analyze the spectrum of black hole radiation, without having to do the
computation using QFT in curved spacetime. The main results, namely, the two-
temperature spectrum and the expressions for the temperatures, agree with the
literature.

However, there is one seeming peculiarity: the fact that in the limit of empty
$dS$ spacetime, $M\to 0$, we get a mirror that does not radiate. This happened
because in our dynamical null shell collapse actually models only the black
hole radiation. The asymptotically de Sitter space is static in this model,
and it is not captured by the mirror, which is inherently dynamical. The
mirror corresponding to the eternally empty dS space is a static mirror and
does not radiate.

The same is actually true of mirrors that correspond to black holes. All known
constructions of this sort do not deal with eternal black holes; instead, they
involve some dynamical models for the black hole collapse, for example, the
shell of null dust. This inherent property of the mirror-gravity
correspondence is not a serious drawback, in our view, as realistic
astrophysical black holes are not eternal, but rather form in a collapse,
either in the process of the stellar evolution or in fluctuations shortly
after the Big Bang.

From the argument in the two paragraphs above it might seem that in our moving
mirror we should not see any radiation with the cosmological temperature
$T_{C}$. Why then have we observed the two-temperature behavior of the black
hole radiation? The reason seems be that the curved background still affects
the black hole formation in such a way, that it actually starts to radiate in
two modes simultaneously. This seems to be similar to the superradiance.

One can also consider a model that dynamically incorporates the dS
cosmological horizon as well; this might shed more light on this peculiarity.
One such construction was proposed in [18], where a bubble of de Sitter vacuum
was expanding into the flat Minkowski exterior. Incorporation of another
collapsing null shell into this spacetime is left for a future work.

###  V.2 BTZ

The Bañados–Teitelboim–Zanelli (BTZ) black hole [41] is an important solution
of the (topological) gravity in 2+1-dimensional AdS spacetime. Since it is
possible to model this black hole as a collapse of a null dust shell, the
mirror dual can be easily constructed.

This example also uncovers some new advantages of the moving mirror approach.
Quantum theory in the BTZ spacetime has several distinct features.

  1. 1.

AdS space, if taken as-is, admits closed timelike curves. To rectify the
situation one should pass to the universal covering space (CAdS).

  2. 2.

AdS space has a timelike boundary, though which information can escape. In
other words, if a light ray is emitted from the origin, it reaches the
infinity in a finite coordinate time.

  3. 3.

BTZ black hole is topological in the sense that it is not a compact object
curving the spacetime.444This feature actually does not present problems in
QFT, but we list it for completeness. Rather, BTZ is just a discrete quotient
of the empty AdS3. It was first noted in 2+1 dimensions [41, 42], then
generalized to 3+1 [43] and to higher-dimensional spaces [44], see also [45]
for a detailed explanation.

Below we will see that all these features are easily addressed in the moving
mirror approach.

Figure 4:  Null shell in AdS background. Coordinates $t_{\text{in}}$,
$r_{*\text{in}}$ are used. Due to the presence of $AdS$ boundary, rays that
are emitted too early at $v<v_{\text{sh}}-\pi\ell$ never encounter the shell.

####  V.2.1 Mirror’s trajectory

In our setup we consider a null shell of effective mass $M+1$ collapsing on a
pre-existing AdS3 space555The AdS3 space can be viewed as a BTZ black hole
with mass $M_{0}=-1$ [41]. We were not able to construct a mirror
corresponding to a shell that collapses on the $M_{0}=0$ configuration, as in
this case the inner tortoise coordinate would diverge at the origin, and the
normalization discussed in Sec. IV cannot be performed.. Similar setup — AdS
inside, BTZ outside — has been also considered in the gravity side [46]; the
difference is that the authors of the latter paper consider a quantum scalar
field with the conformal coupling to gravity, while here we consider the
minimal coupling. This however does not affect the radiation temperature.

The BTZ black hole can be modeled by a null shell collapse [47, 48, 49]. In
the Eddington-Finkelstein coordinates the Vaidya-like metric in this case is

$ds^{2}=-\left(m(v)+\frac{r^{2}}{\ell^{2}}\right)dv^{2}-2dvdr-r^{2}d\phi^{2}\,,$
(53)

where the mass function is given by

$m(v)=\begin{cases}-1,v<v_{\text{sh}}\\\ M,v>v_{\text{sh}}\,.\end{cases}$ (54)

In spherical coordinates, the metric inside and outside of the shell thus is
the 2+1D analog of (34),

$\displaystyle ds_{\text{in}}^{2}$
$\displaystyle=F_{\text{in}}(r)dt_{\text{in}}^{2}-\frac{1}{F_{\text{in}}(r)}dr^{2}+r^{2}d\phi^{2}\,,$
(55) $\displaystyle ds_{\text{out}}^{2}$
$\displaystyle=F_{\text{out}}(r)dt^{2}-\frac{1}{F_{\text{out}}(r)}dr^{2}+r^{2}d\phi^{2}\,.$

with the metric functions

$F_{\text{in}}(r)=1+\frac{r^{2}}{\ell^{2}}\,,\quad
F_{\text{out}}(r)=-M+\frac{r^{2}}{\ell^{2}}\,.$ (56)

The inner tortoise coordinate is then given by

$\displaystyle r_{*\text{in}}(r)$
$\displaystyle=\ell\tan[-1](\frac{r}{\ell})\,,$ (57) $\displaystyle
r_{*\text{in}}^{-1}(R)$ $\displaystyle=\ell\tan(\frac{R}{\ell})\,.$

The outer tortoise coordinate is also simply found

$\displaystyle r_{*\text{out}}(r)$
$\displaystyle=-\frac{\ell}{\sqrt{M}}\tanh[-1](\frac{\ell\sqrt{M}}{r})\,,$
(58) $\displaystyle r_{*\text{out}}^{-1}(R)$
$\displaystyle=\frac{\ell\sqrt{M}}{\tanh(\frac{-R\sqrt{M}}{\ell})}\,.$

Plugging Eq. (57) and Eq. (58) into the mirror-gravity recipe Eq. (40) or Eq.
(41), after some algebra we get the mirror trajectory:

$f_{m}(v)=v_{\text{sh}}+\frac{2\ell}{\sqrt{M}}\tanh[-1](\frac{\sqrt{M}}{\tan(\frac{v_{\text{sh}}-v}{2\ell})})\,,$
(59)

Here, $v_{\text{sh}}$ is the constant specifying the shell’s position in the
$(u,v)$ frame.

####  V.2.2 AdS peculiarities in the mirror

Now, let us see how the difficulties outlined at the beginning of this section
are resolved in the moving mirror formalism.

  1. 1.

Presence of closed timelike curved in AdS manifested itself in the fact that
the expression (59) is a periodic function in $v$. To rectify this, it
suffices to simply consider any one particular branch of this function.

  2. 2.

AdS boundary manifested itself in the fact that, in addition to the expected
black hole future horizon, the expression (59) has a past horizon. It turns
out that, in order to rectify this, we should take into account only a half of
the trajectory. Let us consider this in detail.

From the AdS tortoise coordinate in Eq. (57) one can see that it takes
$\pi\ell$ amount of time for a light ray to cross the whole of AdS. Therefore,
a light ray emitted too early, i.e. at
$v<v_{\text{crit}}=v_{\text{sh}}-\pi\ell$, never actually encounters the null
shell propagating at $v=v_{\text{sh}}$, see Fig. 4. Therefore such rays should
not be taken into account, and in the moving mirror one should only consider
$v>v_{\text{crit}}$.

The future and the past horizons are defined by the equations

$f_{m}(v_{\mathcal{H},\text{past}})=-\infty\,,\quad
f_{m}(v_{\mathcal{H},\text{fut}})=+\infty$ (60)

that give (choosing a particular branch)

$\displaystyle v_{\mathcal{H},\text{past}}$
$\displaystyle=v_{\text{sh}}+2\ell\tan[-1](\sqrt{M})-2\pi\ell\,,$ (61)
$\displaystyle v_{\mathcal{H},\text{fut}}$
$\displaystyle=v_{\text{sh}}-2\ell\tan[-1](\sqrt{M})\,.$

As it turns out, using only $v>v_{\text{crit}}=v_{\text{sh}}-\pi\ell$ cuts out
exactly a half of this trajectory. The other half at $v<v_{\text{crit}}$ is
just a fictitious reflection of it.

Plugging the resulting trajectory in the beta Bogolyubov integral with the
appropriate limits, we finally arrive at the formula for the BTZ radiation
spectrum,

$\beta^{R}_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int\limits_{v_{\text{sh}}-\pi\ell}^{v_{\mathcal{H},\text{fut}}}dv\,e^{-i\omega^{\prime}v-i\omega
f_{m}(v)}\,.$ (62)

####  V.2.3 Radiation and temperature

Now let us actually analyze the BTZ mirror trajectory and investigate the
radiation spectrum.

First of all we note that, just as it was in the previous example, in the
limit of empty AdS space (i.e. with a shell of vanishing mass $(M+1)\to 0$ we
obtain a stationary mirror:

$f_{m}(v)=-2\ell\tan[-1](\cot(\frac{v}{2\ell}))=v-\pi\ell\,.$ (63)

There is no radiation in this case. This is emphasizes the fact that our
current model describes dynamically only the black hole. The AdS bulk is
static, and possible radiation from the AdS boundary is not captured in this
model.

Next, we want to extract the temperature of the thermal radiation spectrum.
Again, in principle the spectrum is given by the exact formula (62). The
temperature can be easily extracted in the near-horizon approximation, when
the trajectory Eq. (59) simplifies to

$f_{m}(v)\approx-\frac{\ell}{\sqrt{M}}\ln(-(v-v_{\mathcal{H},\text{fut}}))+\text{const}\,.$
(64)

This gives the particle spectrum

$N_{\omega\omega^{\prime}}=|\beta^{R}_{\omega\omega^{\prime}}|^{2}\approx\frac{1}{2\pi\kappa\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa}-1}\,,\quad\kappa=\frac{\sqrt{M}}{\ell}\,.$
(65)

Thus the spectrum is Planck with the temperature

$T_{\text{BTZ}}=\frac{\sqrt{M}}{2\pi\ell}\,.$ (66)

The spectrum and the temperature just obtained coincide with known results
from the gravity side [41, 46]. We want to emphasize that the moving mirror
approach reduces the essential part of the Hawking radiation problem to a QFT
in 1+1d flat spacetime, encoding all the necessary information in the boundary
of that 1+1d. Thus certain difficulties are bypassed; one can, in principle,
write down and study the formula for the exact spectrum. At late times the
spectrum is of course dominated by the near-horizon approximation, and the
computation reduces to just a few lines.

####  V.2.4 Note on a spinning black hole

To conclude this section, we want to make a remark concerning the case of BTZ
with angular momentum. Although the prescription described in this paper is
not a priori guaranteed to work in this case, it nevertheless gives the right
temperature.

For a BTZ black hole with mass $M$ and angular momentum $J$, the metric has
the form [41]

$ds_{\text{out}}^{2}=F_{\text{out}}(r)dt^{2}-\frac{1}{F_{\text{out}}(r)}dr^{2}+r^{2}\left(d\phi-\frac{J}{2r^{2}}dt\right)^{2}$
(67)

with

$F_{\text{out}}(r)=-M+\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}},.$ (68)

Introducing

$r_{\pm}=\ell\sqrt{\frac{M}{2}\left(1\pm\sqrt{1-\frac{J}{M\ell}}\right)}\,,$
(69)

one can write the expression for the tortoise coordinate in a convenient form

$r_{*\text{out}}(r)=\frac{\ell^{2}}{r_{+}^{2}-r_{-}^{2}}\left[r_{-}\tanh[-1](\frac{r_{-}}{r})-r_{+}\tanh[-1](\frac{r_{+}}{r})\right]\,.$
(70)

Now let us take a null shell collapsing to form this spinning BTZ black hole.
Assuming AdS metric inside, one can easily write down the mirror trajectory:

$\displaystyle f_{m}(v)=v_{\text{sh}}$
$\displaystyle+\frac{\ell^{2}}{r_{+}^{2}-r_{-}^{2}}\Bigg{[}r_{-}\tanh[-1](\frac{r_{-}}{\tan(\frac{v_{\text{sh}}-v}{2\ell})})$
(71) $\displaystyle-
r_{+}\tanh[-1](\frac{r_{+}}{\tan(\frac{v_{\text{sh}}-v}{2\ell})})\Bigg{]}\,.$

On the gravity side, the surface $r=r_{+}$ is the black hole horizon. The
corresponding mirror horizon $v_{\mathcal{H},\text{fut}}$ is determined by the
equation $\tan(v/2\ell)=r_{+}$; in the near-horizon approximation the mirror
becomes

$f_{m}(v)\approx-\frac{\ell^{2}r_{+}}{r_{+}^{2}-r_{-}^{2}}\ln(-(v-v_{\mathcal{H},\text{fut}}))+\text{const}\,.$
(72)

This is again a Carlitz-Willey mirror, which gives thermal radiation spectrum
with the temperature

$T_{\text{BTZ},J}=\frac{r_{+}^{2}-r_{-}^{2}}{2\pi\ell^{2}r_{+}}\,.$ (73)

This again coincides with known results [41, 46].

When the angular momentum is zero, $J=0$, the two radii become
$r_{+}=\ell\sqrt{M}$, $r_{-}=0$, and we recover the results from the above.

In the extremal limit $J=M\ell$ we have $r_{+}=r_{-}=\ell\sqrt{M/2}$, and the
temperature in Eq. (73) drops to zero. This does not mean that such a black
hole does not radiate, it only means that the radiation spectrum is no longer
thermal. Indeed, the mirror trajectory in this case becomes

$\displaystyle f_{m}(v)=v_{\text{sh}}$
$\displaystyle-\frac{\ell}{\sqrt{2M}}\Bigg{[}\tanh[-1](x)-\frac{1}{x-1/x}\Bigg{]}\,,$
(74) $\displaystyle x$
$\displaystyle\equiv\frac{\ell\sqrt{M/2}}{\tan(\frac{v_{\text{sh}}-v}{2\ell})}\,.$

In the near-horizon limit $x\to(1\,-\\!0)$ the first term in the square
brackets gives the expected log, but the second term has a pole and spoils
thermality. This is consistent with known results on the extremal Kerr black
hole [24].

Now, the important question is, why has our prescription given the correct
results, even though the spacetime in Eq. (67) was not spherically symmetric?
The answer is that, although there is no strict spherical symmetry, the
current setup is still spherically symmetric in a sort of a weak sense,
meaning that the tortoise coordinate in Eq. (70) still does not depend on the
angle variable.

##  VI Conclusions

In this paper we presented a generalization of the correspondence between
black holes and moving mirrors. Previously this correspondence was well-
motivated only in spherically symmetric asymptotically flat cases. Here we
extended it to the non-trivial (asymptotically non-flat) background geometry.

We were also able to relax the symmetry assumption to the weak spherical
symmetry: the spacetime does not have to be spherically symmetric, only the
corresponding tortoise coordinate is required to be angle-independent. It is
plausible that this assumption can also be lifted, at least when studying the
$s$-wave radiation component; however, further investigation is needed to
clarify this issue.

This construction allows one to study certain aspects of black holes, like the
radiation and temperature, by modeling such a black hole by a collapse of a
shell made of null dust, and then constructing a dynamical moving mirror in
1+1d that corresponds to the spacetime of interest. The moving mirror approach
is much simpler as compared to the direct analysis of QFT in the curved
spacetime of interest. We have demonstrated this by studying the spectrum of
Hawking radiation of the Schwarzschild – de Sitter and BTZ black holes.

In this paper we focused on generalizing and testing the moving mirror
approach, before applying it to new problems. This correspondence is promising
from the point of view of the information loss and other entropy-related
questions. It is also possible to include the back reaction into this setup.
We are going to pursue these avenues in the immediate future.

Another direction that will be pursued in the future works is incorporation of
timelike shells and balls. There are indications that such timelike objects
might describe what has been called remnants.

###### Acknowledgements.

The author thanks Michael R.R. Good for very useful discussions and a reading
of the manuscript draft. This work is supported by the William I. Fine
Theoretical Physics Institute at the University of Minnesota, and also by the
FY2021-SGP-1-STMM Faculty Development Competitive Research Grant No.
021220FD3951 at Nazarbayev University.

##  Appendix A The time coordinate inside the shell

In this Appendix we make a brief note on the time coordinate in the flat inner
region, $t_{\text{in}}$, see Eq. (27). As the shell is lightlike, we have
$ds^{2}=0$ on the shell. From the metric Eq. (27) it follows then

$\displaystyle 0$
$\displaystyle=\left(\derivative{t_{\text{in}}}{t}\right)^{2}-\left(\derivative{r_{\text{sh}}(t)}{t}\right)^{2}$
(75) $\displaystyle 0$
$\displaystyle=F(r_{\text{sh}}(t))-\frac{1}{F(r_{\text{sh}}(t))}\left(\derivative{r_{\text{sh}}(t)}{t}\right)^{2}$

from which we immediately obtain

$\derivative{t_{\text{in}}}{t}=F(r_{\text{sh}}(t))$ (76)

Therefore, the metric Eq. (27) can be represented in the same coordinates
inside and outside of the shell as

$\displaystyle ds_{\text{in}}^{2}$
$\displaystyle=h^{2}(t)dt^{2}-dr^{2}-r^{2}d\Omega^{2}\,,\quad
r<r_{\text{sh}}(t)\,,$ (77) $\displaystyle ds_{\text{out}}^{2}$
$\displaystyle=F(r)dt^{2}-\frac{1}{F(r)}dr^{2}-r^{2}d\Omega^{2}\,,\quad
r>r_{\text{sh}}(t)\,,$ $\displaystyle h^{2}(t)$
$\displaystyle=F^{2}(r_{\text{sh}}(t))\,.$

This demonstrates that, although the metric inside the shell is indeed flat,
it is not the naive “$dt^{2}-dr^{2}$”.

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1 1 7<EMAIL_ADDRESS>11institutetext:  Departamento de Astronomía,
Universidad de Concepción, Chile 22institutetext: Departamento de Astronomía,
Universidad de Chile, Chile

# Origin of supermassive black holes: predictions for the black hole
population

M. Liempi 11 L. Almonacid 11 D.R.G. Schleicher & A. Escala 1122

La presencia de agujeros negros super masivos con corrimiento al rojo $z>6$
plantea algunas preguntas sobre su formación y crecimiento en el universo
primitivo. Debido a la construcción de nuevos telescopios como el ELT, el cual
estará capacitado para observar y detectar agujeros negros super masivos, será
útil derivar estimaciones teóricas de la población y comparar observaciones y
predicciones de modelos en el futuro. En consecuencia nuestro principal
objetivo es estimar la población de estos objetos utilizando un código semi-
analítico conocido como Galacticus, el cual es un código para la formación y
evolución de galaxias donde podemos simular diferentes escenarios para la
formación de agujeros negros indicando la masa inicial de estos y las
condiciones de formación y evolución de los distintos componentes de las
galaxias. Nuestros resultados preliminares muestran que el mecanismo principal
del crecimiento de los agujeros negros es la fusión de galaxias y la
acumulación de material. Además, calculamos los radios de influencia de los
agujeros negros para comparar con los datos observacionales en la literatura.

###### Abstract

The presence of supermassive black holes at redshift $z>6$ raises some
questions about their formation and growth in the early universe. Due to the
construction of new telescopes like the ELT to observe and detect SMBHs, it
will be useful to derive theoretical estimates for the population and to
compare observations and model predictions in the future. In consequence our
main goal is to estimate the population of SMBHs using a semi-analytic code
known as Galacticus which is a code for the formation and evolution of
galaxies where we are about to include different scenarios for SMBHs formation
indicating the initial mass of the black hole seed, its formation conditions
and recipes for the evolution of the components of the galaxies. We found that
the principal mechanism of growing SMBHs is is via galaxy mergers and
accretion of matter. For the comparison of our results with observations, we
calculate the radius of influence of the black hole to estimate which part of
the population could be detected, leading to relations similar to the observed
ones.

###### keywords:

black hole physics — galaxies: nuclei — methods: analytical

##  1 Introduction

It is known that the origin of supermassive black holes (SMBHs), whose
presence in the center of most galaxies has been confirmed in the last decades
(Kormendy & Ho, 2013), is still unclear (Volonteri, 2010). Moreover, the
detection of nuclear star clusters (NSCs) (Böker et al. (2001); Côté et al.
(2006)) in the center of some galaxies requires a mechanism to explain the
(co)-existence of SMBHs and/or NSCs in the galactic nuclei.

The best example is our Galaxy, which hosts a SMBH called Sagittarius A*. That
was detected with the EHT telescope (Chael et al., 2023) and a NSC detected
with studies of stellar orbits around the SMBH (Ghez et al. (2008); Genzel et
al. (2010)); Gillessen et al. (2017)).

The detection of SMBHs at high redshifts (e.g., Banados et al. (2018); Onoue
et al. (2019); Tripodi et al. (2022)) further complicates the escenario since
they must have had a short period of time to reach masses often exceeding
$>10^{9}\leavevmode\nobreak\ \mathrm{M}_{\odot}$ (Paliya et al., 2019).

In order to solve the question regarding the origin of SMBHs, some pathways
were proposed to explain how they were formed in the early universe.

The first situation proposes the existence of SMBH seeds in the early
universe. A seed is a black hole with an initial mass in the range from a few
hundreds to about a million solar masses (Volonteri et al., 2021). The
channels currently linked to the origin of black hole seeds are based on the
ideas of Rees (1978), starting from a “gas cloud” and depending on the
physical processes that the cloud goes through (e.g. star formation,
contraction, collapse and/or accretion) it could form a black hole seed. More
recent works use those ideas for the direct collapse of a primordial cloud
onto a SMBH (e.g., Bromm & Loeb (2003); Schleicher et al. (2013)).

Another channel linked to SMBH formation is driven by runaway stellar
collisions in galactic nuclei. The latest results show that it is possible to
form SMBHs via stellar collisions in NSCs. If a central massive object more
massive than $10^{8}\leavevmode\nobreak\ \mathrm{M}_{\odot}$, with a
relaxation time longer than its collision time, is too dense it will collapse
into a SMBH (Escala (2021); Vergara et al. (2022)).

In the first instance, we will focus our investigation on the evolution of
SMBHs seeds scenario in Galacticus.

##  2 Methodology

###  2.1 The code

Galacticus is an open source semi-analytic model to study galaxy formation and
evolution written by Benson (2012). It is designed to be easily expanded and
explored. The advantages of Galacticus are its computationally inexpensive
cost compared to cosmological codes, allowing the fast exploration of the
parameter space. The basic structure in Galacticus is a merger tree, which
consists of a linked tree of nodes. Each node represents a single dark matter
halo or sub-halo (and its content) which have various properties (e.g. black
hole mass, stellar mass, gas mass, etc.). For more details about the
components and the interaction between each other see Fig. 1.

###  2.2 Merger tree sample building

We use the merger tree task of Galacticus to build our merger trees where we
start from a uniformly distributed sample at $\mathrm{z}=0$ with masses
between $10^{6}-10^{15}\leavevmode\nobreak\ \mathrm{M}_{\odot}$. The initial
distribution is specified at $z=0$ for the reason that Galacticus works
reconstructing the history of merger trees following it backward in time. The
internal merger tree builder uses the algorithm of Cole et al. (2000) with a
minor modification and requires some description of the branching
probabilities in trees. We choose the branching probability distribution of
Parkinson et al. (2008) and a sampling rate for the mass evolution of the halo
given by a power-law with the exponent $\alpha=1$. All the details about the
models available can be found on
GITHUB111https://github.com/galacticusorg/galacticus.

###  2.3 Supermassive black holes

In this section we describe briefly the physical meaning of the parameter set
in model A & B. The initial conditions for the SMBH initialization are given
in Table 1. The evolution of the SMBH is given by:

$M_{\bullet}=(1-\epsilon_{\mathrm{radiation}}-\epsilon_{\mathrm{jet}})\dot{M}_{0},$
(1)

where $\epsilon_{\mathrm{radiation}}$ is the radiative efficiency of the
accretion flow feeding the black hole, $\epsilon_{\mathrm{jet}}$ is the
efficiency with which accretion power is converted into jet power and
$\dot{M}_{0}$ is the rest mass accretion rate.Galacticus assumes the rest mass
accretion as a Bondi-Hoyle-Lyttlelon accretion rate from the spheroid gas
reservoir. The black hole is assumed to cause feedback in two ways (see Fig.
1):

  * •

The first way is a radio mode, where any jet power from the black hole-
accretion disk system is included in the hot halo heating rate.

  * •

The other way is given by the addition of the mechanical wind luminosity of
Ostriker et al. (2010) to the gas component of the spheroid. The expression
used is

$L_{\mathrm{wind}}=\epsilon_{\mathrm{wind}}\epsilon_{\mathrm{radiation}}\dot{M}_{0}c^{2},$
(2)

with $\epsilon_{\mathrm{wind}}$ the black hole wind efficiency. The initial
value of $\epsilon_{\mathrm{wind}}$ is fixed to be equal to the best fit model
value described in Galacticus GITHUB website.

##  3 Preliminary results

We calculate the radius of the sphere of influence of the SMBHs using the
following definition

$r_{\mathrm{influence}}=\frac{GM_{\bullet}}{\sigma^{2}},$ (3)

where $G$ is the gravitational constant, $M_{\bullet}$ is the mass of the
black hole and $\sigma$ stellar velocity dispersion of the host bulge
following the definition of Kormendy & Ho (2013).

We define a critical value $r_{\mathrm{influence}}=1\leavevmode\nobreak\
\mathrm{pc}$, calculated with the data provided in Kormendy & Ho (2013), in
order to discriminate SMBHs which could be detected at z=0 via stellar/gas
dynamics. These results are summarized in Fig. 3 where it is possible to see,
how a better resolution can impact in the detection of SMBHs with
$r_{\mathrm{influence}}<1\leavevmode\nobreak\ \mathrm{pc}$.

We also explore the mass evolution for the model A and B (see Fig. 2), where
we found a similar evolution independent of the initial black hole seed mass.

Table 1: Relevant parameters used to initialize the black hole component in Galacticus. Model |  $\epsilon_{\mathrm{wind}}$ | Seed mass$\leavevmode\nobreak\ \mathrm{M}_{\odot}$ | Mass resolution$\leavevmode\nobreak\ \mathrm{M}_{\odot}$
---|---|---|---
A | 0.0024 | $10^{2}$ | $10^{10}$
B | 0.0024 | $10^{5}$ | $10^{10}$
Figure 1: Key interactions between the components in Galacticus. Not all the
components exists in each galaxy. The complete details of the interactions and
the model used can be constructed seeing the manual indicated in subsection
2.2. Figure 2: Mass evolution of models A & B in function of redshift. We show
the mean mass and the maximum mass of the SMBHs at a given z. Model A
represents a light seed while model B represents a heavy seed. Figure 3:
Figure 3: Comparison between the influence radius for simulated data and
observed data. We also show how the population changes when we select SMBHs
with $r_{\mathrm{influence}}>1\leavevmode\nobreak\ \mathrm{pc}$. The
observational data was extracted from Kormendy & Ho (2013), where classical
bulges are defined as an elliptical galaxy, except that they are embedded in
disks, while pseudo-bulges are classical bulges which have properties that are
more disk-like than those of classical bulges.

##  4  Conclusion and future work

Although our work is preliminarily in agreement with Volonteri et al. (2021),
who concludes that accretion is the first source for the SMBHs growth, and
mergers play a secondary role in the mass budge of SMBHs, we need to explore
the dependence on the mass resolution to improve our data analysis and to made
the statistical work for the SMBH population. We found that observations
detected SMBHs with $r_{\mathrm{influence}}\gtrsim 1\leavevmode\nobreak\
\mathrm{pc}$. It is evident that the number of the SMBHs detections could be
higher if the resolution to detect dynamically the local SMBHs increases, and
of course, this is something that we will explore in more detail to predict
the SMBH population. Finally, in the future we plan to implement a black hole
formation scenario based on collisions in NSCs.

We gratefully acknowledge support by the ANID BASAL projects ACE210002 and
FB210003, as well as via the Millenium Nucleus NCN19-058 (TITANs).

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