#  Exploring Black Hole Mimickers:
Electromagnetic and Gravitational Signatures of AdS Black Shells

Suvendu Giri  Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA  Princeton Gravity Initiative, Princeton University,
Princeton, NJ 08544, USA  Institutionen för fysik och astronomi, Uppsala
Universitet, Box 803, SE-751 08 Uppsala, Sweden  Ulf Danielsson  Institutionen
för fysik och astronomi, Uppsala Universitet, Box 803, SE-751 08 Uppsala,
Sweden  Luis Lehner  Perimeter Institute for Theoretical Physics, 31 Caroline
St., Waterloo, ON, N2L 2Y5, Canada  Frans Pretorius  Department of Physics,
Princeton University, Princeton, New Jersey 08544, USA  Princeton Gravity
Initiative, Princeton University, Princeton, NJ 08544, USA

###### Abstract

We study electromagnetic and gravitational properties of AdS black shells
(also referred to as AdS black bubbles)—a class of quantum gravity motivated
black hole mimickers, that in the classical limit are described as ultra
compact shells of matter. We find that their electromagnetic properties are
remarkably similar to black holes. We then discuss the extent to which these
objects are distinguishable from black holes, both for intrinsic interest
within the black shell model, and as a guide for similar efforts in other sub-
classes of exotic compact objects (ECOs). We study photon rings and lensing
band characteristics, relevant for very large baseline inteferometry (VLBI)
observations, as well as gravitational wave observables—quasinormal modes in
the eikonal limit and the static tidal Love number for non-spinning
shells—relevant for ongoing and upcoming gravitational wave observations.

††preprint: UUITP-12/24

###### Contents

  1. I Black shells: an overview
  2. II Electromagnetic properties of the black shell
    1. II.1 The black shell is black
    2. II.2 The black shell has no hair
    3. II.3 The black shell as a conductor
  3. III The black shell as an electric generator producing jets
    1. III.1 Magnetic field inside the shell
    2. III.2 The black shell as a generator
  4. IV Electromagnetic signatures of black shells
    1. IV.1 Geodesic equations and initial conditions
    2. IV.2 Lensing bands and critical curves
    3. IV.3 Ray tracing: results
  5. V Gravitational wave signatures
    1. V.1 Light ring frequency and QNMs
    2. V.2 Tidal Love number & Gravitational Waves
  6. VI Summary, Discussion and Outlook

## Introduction and overview

The ability to scrutinize black holes has recently kicked into high gear. The
detection of gravitational waves has opened a window to directly examine
dynamical, strong field gravity, in particular from systems describing stellar
mass black hole mergers [1, 2]. Electromagnetic signals detected through very
large baseline interferometry are providing a glimpse into horizon-scale
emission regions of supermassive black holes [3]. These exquisite detection
efforts, which will be followed by increasingly more sensitive campaigns,
herald unprecedented opportunities to test fundamental ideas about the nature
of black holes, and potential deviations from their simple structure predicted
by (classical) General Relativity.

Multiple lines of arguments rooted in quantum theory call into question the
classical picture. Indeed, the so-called black hole information paradox, and
the connection of black hole entropy to its area, have motivated diverse
models attempting to interpret and reconcile such puzzling aspects of black
holes. These, and connected efforts to address the presence of singularities
inside black holes, have motivated “extensions” to black holes, altering their
structure in the vicinity of the classical horizon and its interior. Proposals
for objects like fuzzballs, gravastars, black shells, and other alternatives
have been presented, collectively referred to as exotic compact objects (ECOs)
(not all necessarily motivated by quantum gravity considerations).

Here, we focus on a subclass of ECOs, “AdS black bubbles” which, for their
characteristics described in this work, will be referred to as shells. Such an
object is modeled as an ultra-compact thin-shell: a 2-sphere surface layer of
matter close to the would-be horizon of the analogous black hole, but still at
a macroscopic distance outside, that separates a nonsingular interior
spacetime from the exterior, asymptotically flat spacetime [4, 5, 6, 7, 8]. In
previous works we have focused on understanding the dynamics of black shells
in spherical symmetry, metric properties of slowly rotating shells including
its quadrupole moment, and discussed prospects and challenges to be addressed
for testing the black shell paradigm [6, 8]. In this paper, short of
considering binary black shell systems, we study a set of observable
consequences tied to black shells in both electromagnetic and gravitational
observables. We discuss the extent to which such objects are distinguishable
from black holes, both for intrinsic interest within the black shell model,
and also as a guide for similar efforts in other subclasses of ECOs.

In section I, we begin with an overview highlighting key aspects of black
shells, which will serve as the concrete model that we will study in the rest
of this paper.

In sections II and III, we focus on the electromagnetic properties of black
shells. By assigning sufficiently large values to parameters determining the
permittivity, permeability and conductivity of the shell, we find that the
black shells satisfy a “no-hair” property, akin to the electromagnetic no-hair
properties of Kerr-Newman black holes. We also find that they are fully
absorbing, and that currents flowing within the thin shells have the same
properties as the effective currents on black hole horizons. We argue that
black shells can serve as electric generators producing jets in a similar way
as black holes.

In section IV, we study observational signatures of black shells in the
electromagnetic spectrum, relevant for very large baseline interferometry
(VLBI) observations made by the event horizon telescope (EHT). Since black
shells are macroscopically bigger than black holes, this changes the size and
shape of the inner shadow. Owing to the difference in quadrupole moment of a
rotating black shell as compared to Kerr, the critical curve (photon ring) has
a shape distinct from that of a Kerr black hole with identical mass and spin.
Both of these are potentially observable effects that could distinguish a
black shell from a black hole.

In section V, we study observables that are relevant for gravitational wave
observations of binary mergers by the LIGO-Virgo-Kagra (LVK) collaboration, as
well as upcoming ground-based and space-based missions. We begin, in section
V.1, by examining the characteristic frequencies associated with the light
ring—the orbital frequency and the instability exponent—that can be used to
estimate the quasi-normal mode frequencies in the large harmonic number $\ell$
limit. These frequencies are different from those of a Kerr black hole because
of a difference in multipole moments. Unsurprisingly, the difference is small,
owing to only percent level differences in the multipole moments. We then
compute the tidal Love number of stationary black shells in section V.2, where
we find it to be small and positive. A non-zero Love number would be
observable in the inspiral phase of a gravitational wave signal prior to
merger, imprinting itself on the signal as an overall phase difference
compared to black hole waveforms (for which the Love number vanishes).

Finally, in section VI we comment on what to expect in the gravitational wave
signal from a black hole binary merger event, a discussion on other pertinent
aspects of the black shell model which generalizes to ECOs, and conclude with
an outlook on future directions.

##  I Black shells: an overview

Introduced in [4], a black shell is a bubble of four-dimensional anti-de
Sitter (AdS) spacetime enclosed by a 2+1 dimensional thin material shell,
outside of which there is an asymptotically flat Minkowski vacuum. This model
is inspired by string theory, giving an alternative endpoint to gravitational
collapse than a black hole. The construction assumes that our four-dimensional
Minkowski spacetime is metastable and may transition to an AdS vacuum through
a first-order phase transition, via quantum tunneling. Such AdS vacua are
ubiquitous in string theory, and the decay proceeds through the nucleation of
a spherical brane bubble in four dimensions. This tunneling process is highly
suppressed, making the four-dimensional Minkowski vacuum extremely long lived.
However, during gravitational collapse, a new possibility emerges. Should a
bubble form atop a collapsing shell of matter, the matter transforms into a
gas of open strings on the nucleating brane. Assuming that this gas
thermalizes at the Unruh temperature, it acquires an entropy comparable to
that of the corresponding black hole, significantly increasing the phase space
for nucleation and rendering its creation inevitable.

It was argued in [4] that the natural radius for converting infalling matter
into radiation is the _Buchdahl radius_ ($=9M/4$), making it the natural
radius for the formation of a black shell as suggested by the entropy argument
above. To ensure stability of the shell at this radius, it is necessary to
allow a transfer of energy between the matter components. This was studied in
detail in [6], where dynamical, non-linear stability in response to radial
perturbations and accretion was shown to be possible for a range of parameters
describing the internal flux between the matter components of the shell.

The construction was extended in [5, 7, 8] to include rotating black shells
with a moderately large spin (up to $a^{6}$ perturbatively). The external
spacetime of these rotating shells shows deviations from the Kerr metric,
including multipole moments that vary from those of Kerr at a percent level.
Additionally, the gas on top of the rotating shell exhibits significant
viscous properties, such as non-zero heat conductivity and shear viscosity.
Despite its stringy origins, astrophysical black shells can largely be modeled
using classical physics, provided that the matter is described by a suitable
equation of state. The electromagnetic and gravitational characteristics of
these shells will be further examined in the remainder of this article.

##  II Electromagnetic properties of the black shell

###  II.1 The black shell is black

The defining property of a black hole is that it is black. Any black hole
mimicker, such as a black shell, should reproduce this property at least to a
reasonable extent. In this section we will discover that blackness can be
achieved if the shell is made of a material with relative permittivity
$\epsilon_{r}$ and relative permeability $\mu_{r}$, such that
$\epsilon=\epsilon_{r}\epsilon_{0}$ and $\mu=\mu_{r}\mu_{0}$, with
$\epsilon_{r}=\mu_{r}\gg 1$. Given that the shell is carrying a large number
of degrees of freedom, on order of $R^{2}/l_{4}^{2}$, it is reasonable to
expect that their response to electric and magnetic fields can be strong, if
these degrees of freedom are associated with electric and magnetic dipoles. We
will not make any attempt to estimate these values, but plan to return to this
question in the future. As we will see, as long as they are sufficiently big,
physically measurable quantities will not depend on their exact values. The
speed of light within the material that makes up the thin shell will then be
much smaller than the speed of light in vacuum:

$c_{s}\coloneqq\frac{1}{\sqrt{\mu\epsilon}}\ll c$ (1)

For this reason, we will refer to the region within the shell as the black
domain.111This phrase is borrowed from “Death’s end” by Cixin Liu, the third
book in the trilogy starting with “The three body problem”. In the book,
_black domain_ is a region of space where light slows to a standstill.

Maxwell’s equations with $\vec{D}=\epsilon\vec{E}$ and $\vec{B}=\mu\vec{H}$,
are given by

$\vec{\nabla}\times\vec{H}=\vec{J}+\frac{d\vec{D}}{dt},$ (2)

where $\vec{J}=\sigma\vec{E}$, and

$\vec{\nabla}\times\vec{E}=-\frac{d\vec{B}}{dt}.$ (3)

The solution for an electromagnetic wave travelling within the thin shell is
proportional to $e^{-\kappa z-ikz+i\omega t}$, where

$k=\omega\sqrt{\frac{\epsilon\mu}{2}}\left(\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^{2}}+1\right)^{1/2}\xrightarrow{\lim{\sigma\ll\epsilon\omega}}\omega\sqrt{\mu\epsilon},$
(4)

verifying the expected expression for the speed of light in the medium.

We also find an exponential decay governed by

$\kappa=\omega\sqrt{\frac{\epsilon\mu}{2}}\left(\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^{2}}-1\right)^{1/2}\xrightarrow{\lim{\sigma\ll\epsilon\omega}}\sqrt{\frac{\mu}{\epsilon}}\frac{\sigma}{2}\,.$
(5)

The length scale associated to this decay can be identified with a
characteristic _skin depth_

$\delta_{s}\coloneqq\frac{2}{\sigma}\sqrt{\frac{\epsilon}{\mu}},$ (6)

to which fields can penetrate into the shell before being damped by an e-fold.
The effective resistance associated with currents flowing within this slice is
proportional to
$\sigma\delta_{s}\sim\sqrt{\epsilon/\mu}=\sqrt{\epsilon_{0}/\mu_{0}}$, which
is independent of $\sigma$ and identical to the impedance of the vacuum. This
is exactly the same value as in the case of the black hole horizon. Provided
that the skin depth is much smaller than the thickness of the shell,
$\delta_{s}\ll\delta a$, the waves will be absorbed quickly and fully, long
before they reach the other side of the thin shell.

Using the Fresnel equations, ignoring conductivity, one notes that any
incident wave exactly normal to the surface will be perfectly transmitted
without any reflection if $\epsilon_{r}=\mu_{r}$. Taking conductivity into
account, there will only be a very small, imaginary, reflection coefficient
proportional to $\sigma/(\epsilon\omega)\ll 1$. This is promising for a black
shell.

In the case of a wave coming in at an angle, Snell’s law tells us that the
wave will be refracted towards the normal, and, given the high refractive
index within the shell, continue on the inside almost exactly along the
normal. According to the Fresnel equations, there will now also exist a
reflected wave without any suppression. However, the Fresnel equations assume
a sharp boundary with $\epsilon$ and $\mu$ behaving as step functions. In any
physical system, this cannot be true. There must be a gradient in the
permittivity and permeability such that they increase from the vacuum value to
the extreme values of the black domain over a finite width. In the presence of
a finite gradient, the refracted wave will not take a sharp turn, but instead
smoothly change its direction along a geodesic in the medium. Furthermore, any
reflection will be suppressed if the wavelength of the light is much smaller
than the width of the step. In our case the wavelength of light will decrease
the further into the medium the wave penetrates. Given that
$c_{s}=1/\sqrt{\epsilon\mu}$ becomes very small, there will be an enormous
pile up of waves across the step. For simplicity, we assume that the width of
the step is much smaller than the skin depth, $\delta_{w}\ll\delta_{s}$. In
order for reflection to be suppressed we then need

$c_{s}/\omega\ll\delta_{w}\ll\delta_{s}\,.$ (7)

This is also compatible with our assumption that $\sigma/(\epsilon\omega)\ll
1$. Hence, any reflection due to either the step or conductivity is
suppressed. One may note that suppression is strictly speaking needed only for
wavelengths $\sim c/\omega\ll a$, where $a$ is the radius of the shell. This
implies that $\sigma a/\left(\epsilon c\right)$ can be of order one. We
conclude that all waves with a wavelength much smaller than the Schwarzschild
radius will be fully absorbed, and the black shell will look perfectly black.
_We expect these considerations to be relevant for any black hole mimicker._

Let us see how this works in more detail. For the purpose of this problem, it
suffices to zoom in on the shell and look at a tiny section that is
effectively a flat slab of material with a graded refractive index. We will
work in Cartesian coordinates, and put the boundary at $z=0$, with the
positive $z$-axis pointing towards the black domain. For definiteness, we
choose the polarization of the incoming light such that $E_{y}(x,z)$,
$H_{x}(x,z)$ and $H_{z}(x,z)$ are nonzero. For normal incidence, $H_{z}=0$. We
put $\mu=\mu_{0}\alpha(z)$ and $\epsilon=\epsilon_{0}\alpha(z)$, with
$\alpha(z)$ representing a step of width $\delta_{w}$ around $z=0$, such that
$\alpha\rightarrow 1$ as $z\rightarrow-\infty$, and
$\alpha\rightarrow\epsilon_{r}=\mu_{r}$ as $z\rightarrow+\infty$. Taking the
electric and magnetic fields to be $\vec{E}=E_{y}(x,z)e^{i\omega
t}\widehat{y}$, $\vec{H}=H_{x}(x,z)e^{i\omega
t}\widehat{x}+H_{z}(x,z)e^{i\omega t}\widehat{z}$, Maxwell’s equations can be
written as a set of coupled, driven wave equations:

$\displaystyle\nabla^{2}E_{y}+\omega^{2}\mu\epsilon
E_{y}-i\omega\frac{d\mu}{dz}H_{x}$ $\displaystyle=0,$
$\displaystyle\nabla^{2}H_{x}+\omega^{2}\mu\epsilon
H_{x}-i\omega\frac{d\epsilon}{dz}E_{y}+\frac{1}{\mu}\frac{d\mu}{dz}\frac{\partial
H_{z}}{\partial x}$ $\displaystyle=0,$ (8)
$\displaystyle\nabla^{2}H_{z}+\omega^{2}\mu\epsilon
H_{z}+\frac{\partial}{\partial
z}\left(\frac{1}{\mu}\frac{d\mu}{dz}H_{z}\right)$ $\displaystyle=0,$

where $\nabla^{2}\coloneqq\left(\partial_{x}^{2}+\partial_{z}^{2}\right)$. In
case of normal incidence, where $k_{x}=0$, the third equation is trivially
solved by $H_{z}=0$, while the first two equations decouple and are solved by
$E_{y}\sim e^{-ik_{z}\int^{z}\alpha dz}$, with $k_{z}=\omega/c$ and
$H_{x}=-\sqrt{\epsilon_{0}/\mu_{0}}E_{y}$. This is a purely ingoing wave with
varying momentum but no reflection, in agreement with Fresnel. If the light
comes in at an angle, with $k_{x}\neq 0$, it is easiest to solve the third
equation. Writing $H_{z}(x,z)=e^{-ik_{x}x}f(z)/\sqrt{\mu(z)}$, the last
equation above becomes (here and below, primes denote derivatives with respect
to $z$)

$f^{\prime\prime}+\left(\left(\frac{\omega}{c}\right)^{2}\alpha^{2}-k_{x}^{2}+\frac{\alpha^{\prime\prime}}{2\alpha}-\frac{3\left(\alpha^{\prime}\right)^{2}}{4\alpha^{2}}\right)f=0$
(9)

$\mu=\mu_{r}\mu_{0},\epsilon=\epsilon_{r}\epsilon_{0}$
---
$\mu=\mu_{0},\epsilon=\epsilon_{0}$
---
Figure 1: A sketch showing that a ray of light incident at an angle to the
black shell bends smoothly towards the normal and continues along the normal
thereafter, until it gets completely absorbed. The refractive index of the
medium, which increases smoothly from its vacuum value outside the shell to a
large value in the shell, is shown as a graded background. The dimming of the
light ray as it passes through the shell represents a reduction in its
intensity until it is absorbed into the shell.

Through Maxwell’s equations, the solutions for $H_{z}$ will also generate
solutions for the other components $E_{y}$ and $H_{x}$. The solution is
uniquely specified by the condition that it is an incoming wave at an angle.
In the limit of normal incidence, $k_{x}=0$, the exact solution is

$f=\frac{1}{\sqrt{\alpha}}e^{-ik_{z}\int^{z}\alpha dz}\,,$ (10)

which implies zero reflection compatible with the Fresnel result. The limit
is, however, degenerate since the amplitude of $H_{z}$ needs to be put to zero
for normal incidence. Turning on $k_{x}$, we can no longer easily find an
exact solution. (Note that $k_{x}^{\rm max}=\omega/c$, corresponding to a
grazing ray of light). To better understand the physics, we change coordinates
to $d\zeta=\alpha(z)dz$. In this new coordinate, the wavelength of light
becomes essentially constant within the black domain. We then introduce
$g(z)=\sqrt{\alpha(z)}f(z)$, to find the very simple equation

$-\frac{d^{2}g}{d\zeta^{2}}+V(\zeta)g=0\,,$ (11)

with the potential

$V(z)=\frac{k_{x}^{2}}{\alpha(z(\zeta))^{2}}-\left(\frac{\omega}{c}\right)^{2}$
(12)

such that we can think of the problem as a one-dimensional Schrödinger
equation for a particle passing over a barrier. We note that for large
$|\zeta|$ (i.e. large $|z|$, where $\alpha(z)$ becomes effectively constant)
the potential goes to a constant according to

$\displaystyle V\rightarrow
k_{x}^{2}-\left(\frac{\omega}{c}\right)^{2}\,\,\,{\rm
for}\,\,\,\zeta\rightarrow-\infty$ (13) $\displaystyle
V\rightarrow\frac{k_{x}^{2}}{\alpha_{r}^{2}}-\left(\frac{\omega}{c}\right)^{2}\,\,\,{\rm
for}\,\,\,\zeta\rightarrow+\infty$ (14)

where $\alpha_{r}=\epsilon_{r}=\mu_{r}$ is large. Since $\alpha$ is monotonic
in $z$ (and $\zeta$), the potential $V$ is a smooth and monotonic function
that goes from the negative value $k_{x}^{2}-\left(\omega/c\right)^{2}$ far
from the black domain, to the even more negative value
$(k_{x}/\alpha_{r})^{2}-(\omega/c)^{2}$ deep inside the black domain.

As we have seen, reflection is identically zero for $k_{x}=0$. This is obvious
from (12), since the potential is constant everywhere. For $k_{x}\neq 0$ this
is no longer true, but any reflection will still be heavily suppressed. In the
original coordinate $z$, the step has width $\delta_{w}$, while in the new
coordinate $\zeta$ the width is rescaled to $\alpha_{r}\delta_{w}$. Although
$\delta_{w}$ is supposed to be much smaller than any realistic wave length,
$\alpha_{r}\delta_{w}$ is assumed to be much larger. This implies that we are
in an essentially classical regime, and the reflection can be neglected for
all values of $k_{x}$. This is sketched in fig. 1.

What happens is that the graded boundary of the black domain mimics space-time
just outside of the horizon of a black hole. That is, light rays are bent so
that they travel along the normal. This is compatible with the well-known
electromagnetic boundary condition on the horizon [9, 10, 11], which says that
the normal components $\vec{E}_{n}$ and $\vec{B}_{n}$, as well as the parallel
components, $\vec{E}_{H}$ and $\vec{B}_{H}$, as measured by a freely falling
observer are finite. In addition

$\vec{E}_{H}=\vec{n}\times\vec{B}_{H}\,\,\,\vec{B}_{H}=-\vec{n}\times\vec{E}_{H}\,,$
(15)

which implies that all electromagnetic fields near the horizon, look like
incoming waves along the normal. This can be understood as a consequence of a
diverging Lorentzboost.222The similarity between Maxwell’s equations with
nontrivial permittivity and permeability, keeping the impedance
$\sqrt{\epsilon/\mu}$ constant, and electromagnetism in a curved spacetime (in
particularly close to a horizon) has previously been observed in [12]. The
fully developed black domain works more like the horizon itself— including its
electric properties as we will see later on.

###  II.2 The black shell has no hair

As an attempt to distinguish a black shell from a black hole, one may wonder
about the fate of a charge that falls onto the stationary black shell. Had it
been a black hole, the information about the angular coordinates where the
point charge is dropped in will eventually be lost to an external observer
after the charge passes the horizon, and any measurement performed will then
see electric field lines emanating radially from the spherical horizon of the
Schwarzschild black hole. This is a consequence of the black hole _no-hair_
theorem. Since the black shell lacks a horizon, one does not naively expect
such a no-hair effect. On the other hand, the black shell has a large number
of degrees of freedom by construction, and one may expect the charge to
somehow “dissolve” into these degrees of freedom, with the whole black shell
carrying the charge. But since charge is quantized, if the no-hair effect is
to be redeemed, dissolving can not be the correct solution.

Another possibility is that the shell simply has a non-zero conductivity, and
that charges redistribute themselves so that the entire shell is at the same
electric potential. This is a simple way to satisfy the no-hair theorem. As we
have already seen, we do need a non-zero conductivity for other reasons, but
this cannot on its own explain no hair. Due to the large permittivity, that we
needed for the black shell to be black, the electric field sourced by a charge
within the black domain will be extremely weak (as we will show in detail
below), and any current much to weak to realize the no-hair theorem fast
enough.

Luckily, as we will now show, the large permittivity in itself implies no
hair. But this is not achieved through free charges that redistribute due to
currents, but through bound charges that appear at the surface. To figure out
what is going on, we consider a point charge embedded within the shell and
compute the electric field that results. Since the shell sits at $r=9M/4$, and
we are interested in the effect of the point charge within a stellar mass or
larger shell; we will, for the moment, disregard the spacetime curvature and
work in flat space. The qualitative picture will be the same. In this limit,
this is simply a problem in classical electrostatics. The general case of a
point charge embedded both within and inside a dielectric shell was discussed
in [13]. In the following, we will reproduce their results for the case that
is of interest to us.

To solve for the electric field, we need to solve the Poisson equation for the
scalar potential in three regions: inside, within, and outside the shell.
While there are no sources inside and outside the shell, the point charge
within the shell provides a source term proportional to the charge.

(a) $\vec{E}$

(b) $\vec{D}$

Figure 2: Plots showing the electric field $\vec{E}$ and the displacement
field $\vec{D}$ for a point charge (in red) embedded within a shell (shown in
black) of infinitely large relative permittivity $\epsilon\to\infty$. The
location of the charged particle within the shell is completely shielded from
an external observer, giving a no-hair property for the black shell.

Let us consider a shell of inner radius $a$, outer radius $b$ containing a
point charge $q$ at a radial distance $s$ (and $\theta=0$), where $a<s<b$. We
now make the following ansatz for the scalar potential $\Phi$ (assuming
regularity at the origin and at infinity):

$\displaystyle\Phi_{\textrm{in}}$
$\displaystyle=\sum_{n=0}^{\infty}A_{n}r^{n}P_{n},$
$\displaystyle\Phi_{\textrm{shell}}$
$\displaystyle=\frac{q}{\epsilon\left|\vec{r}-\vec{s}\right|}+\sum_{n=0}^{\infty}\left(B_{n}r^{n}+C_{n}r^{-n-1}\right)P_{n},$
(16) $\displaystyle\Phi_{\textrm{out}}$
$\displaystyle=\sum_{n=0}^{\infty}D_{n}r^{-n-1}P_{n}.$

${\left|\vec{r}-\vec{s}\right|}^{-1}$ accounts for the point charge source
within the shell, while the other terms are Green’s functions for the Laplace
equation. $P_{n}=P_{n}(\theta)$ are the Legendre polynomials. The scalar
potential on the shell can be rewritten using the standard expansion from
[14],

$\frac{1}{\left|\vec{r}-\vec{s}\right|}=\left\\{\begin{array}[]{lr}\displaystyle\sum_{n=0}^{\infty}r^{n}s^{-n-1}P_{n}&(a<r<s)\,,\\\
\displaystyle\sum_{n=0}^{\infty}s^{n}r^{-n-1}P_{n}&(s<r<b)\,.\end{array}\right.$
(17)

Boundary conditions are given by:

$\displaystyle\left(\frac{\partial\Phi_{\textrm{in}}}{\partial
r}=\epsilon\frac{\partial\Phi_{\textrm{shell}}}{\partial
r}\right)\bigg{|}_{r=a},$
$\displaystyle\left(\frac{\partial\Phi_{\textrm{in}}}{\partial\theta}=\frac{\partial\Phi_{\textrm{shell}}}{\partial\theta}\right)\bigg{|}_{r=a}\,,$
(18) $\displaystyle\left(\frac{\partial\Phi_{\textrm{out}}}{\partial
r}=\epsilon\frac{\partial\Phi_{\textrm{shell}}}{\partial
r}\right)\bigg{|}_{r=b},$
$\displaystyle\left(\frac{\partial\Phi_{\textrm{out}}}{\partial\theta}=\frac{\partial\Phi_{\textrm{shell}}}{\partial\theta}\right)\bigg{|}_{r=b}.$
(19)

A solution to this system of equations is given by (after correcting a minor
misprint in [13])

$\displaystyle A_{n}$
$\displaystyle=(2n+1)\frac{q}{E_{n}s^{n+1}}\left[(n+1)(\epsilon-1)\left(\frac{s}{b}\right)^{2n+1}+n(\epsilon+1)+1\right],$
$\displaystyle B_{n}$
$\displaystyle=\left(1-\frac{1}{\epsilon}\right)(n+1)\frac{q}{E_{n}s^{n+1}}\left[n(\epsilon-1)\left(\frac{a}{b}\right)^{2n+1}+(\epsilon+\epsilon
n+n)\left(\frac{s}{b}\right)^{2n+1}\right],$ $\displaystyle C_{n}$
$\displaystyle=\left(1-\frac{1}{\epsilon}\right)n\frac{q~{}a^{2n+1}}{E_{n}s^{n+1}}\left[(\epsilon
n+n+1)+(n+1)(\epsilon-1)\left(\frac{s}{b}\right)^{2n+1}\right],$ (20)
$\displaystyle D_{n}$
$\displaystyle=(2n+1)\frac{q~{}a^{2n+1}}{E_{n}s^{n+1}}\left[n(\epsilon-1)+(\epsilon+\epsilon
n+n)\left(\frac{s}{a}\right)^{2n+1}\right],$ $\displaystyle E_{n}$
$\displaystyle=(\epsilon n+n+1)(\epsilon+\epsilon
n+n)-n(n+1)(\epsilon-1)^{2}\left(\frac{a}{b}\right)^{2n+1}.$

Consider the limit in which the shell is thin, $\delta a/a\ll 1$, and the
shell has a large relative permittivity $\epsilon_{r}\to\infty$. If the
permittivity goes to infinity faster than the thickness goes to zero, i.e.
$\Delta\coloneqq\epsilon_{r}\,\delta a/a\gg 1$, the electric field
($\vec{E}=-\vec{\nabla}\Phi$) inside, and within the shell vanishes, while it
points radially outward outside the shell:

$\lim_{\Delta\to\infty}E_{\textrm{in}}=\lim_{\Delta\to\infty}E_{\textrm{shell}}=0,\lim_{\Delta\to\infty}E_{\textrm{out}}=\frac{q}{r^{2}}.$
(21)

Remarkably, this hides the position of the charge within the shell from an
external observer, recovering a no-hair property for the shell.333For the
shell to have no electromagnetic hair, it needs to have $\epsilon_{r}\,\delta
a/a\gg 1$. For the classical analysis in this article, this is a well-
motivated assumption, and we would like to understand its stringy origin in a
future work. Within the shell, the displacement vector
($\vec{D}=-\epsilon\vec{\nabla}\Phi$) is nontrivial in the limit
$\epsilon\to\infty$ and shows that the field lines from the charge bend along
the shell. The fields $\vec{E}$ and $\vec{D}$ are shown in fig. 2. It is
intriguing that the same properties of the shell that lead to no-hair also
account for why the shell is black. As explained at the end of the last
section, this is presumably due to the outer layer of black domain (which
refracts the field lines in both cases) acting in a way that is very similar
to curved spacetime just outside of an horizon.

The reason to expect large $\epsilon$ is, as already argued, related to the
huge number of degrees of freedom of the gas on top of the shell. If each
degree of freedom can induce a dipole moment, the relative permittivity and
permeability will be proportional to the number density, which is large. The
same argument applies to permeability $\mu$. Since we expect the black shell
to be democratic with respect to electric and magnetic fields, we expect
$\epsilon_{r}=\mu_{r}$ both large. In other words, the impedance will be the
same as that of the vacuum. While we are making these arguments in the context
of the string inspired black shell model, _they should apply to any model that
attempts to explain the origin of BH entropy._

In the above, we ignored the conductivity of the shell for the simple reason
that the electric field for this particular configuration was more or less
vanishing. In the case of an incoming electromagnetic wave, the electric field
within the black shell is non-vanishing, and the conductivity is crucial for
the wave to be absorbed. In the next sections, we will study currents flowing
through the shell, where again conductivity will be important. We will first
consider a source exterior to the shell, and then consider the shell itself as
a generator.

###  II.3 The black shell as a conductor

Consider a black shell with a wire connected to the north pole and another
wire connected to the south pole; see fig. 3. We then connect the wires to an
external voltage source and expect a current to start flowing through the
circuit, including the black shell. As in the case of a black hole we do not
expect matter to be able to leave the shell. In order for a current to flow,
we therefore need, say, electrons entering the shell through one of the wires
and positrons entering through the other.444A scenario already familiar in
both neutron stars and black holes [15, 16].

$V$
---
Figure 3: A schematic of a black shell as a resistor in a circuit with an
external battery.

Maxwell’s equations for the conducting shell are simply given by

$\displaystyle\nabla\times E$ $\displaystyle=-\partial_{t}B\,,$
$\displaystyle\nabla\times H=\sigma E+\partial_{t}D\,,$ (22)
$\displaystyle\nabla\cdot H$ $\displaystyle=0\,,$ $\displaystyle\nabla\cdot
E=0\,.$

Here we have used the constitutive equations for the shell:

$D=\epsilon E,\quad B=\mu H,\quad J=\sigma E.$ (23)

We note that the electric field is source-free, $\nabla\cdot E=0$, everywhere,
except where the external wires are connected. In the static case, the fields
will penetrate the shell in a way similar to fig. 4(a). However, in a physical
situation there will always be some time dependence, with the importance of
such a time dependence amplified by $\mu$ and $\epsilon$.

Let us examine this in more detail. It is useful to split the fields into
components parallel to the equator, labelled by a subscript $||$, and the
remainder labelled by a subscript $\perp$ (which is normal to the parallel
component):

$\displaystyle\nabla\times
E_{\perp}=-\mu\partial_{t}H_{\parallel},\quad\nabla\times H_{\parallel}=\sigma
E_{\perp}+\epsilon\partial_{t}E_{\perp},$ (24) $\displaystyle\nabla\times
E_{\parallel}=-\mu\partial_{t}H_{\perp},\quad\nabla\times H_{\perp}=\sigma
E_{\parallel}+\epsilon\partial_{t}E_{\parallel}\,.$ (25)

We now consider a situation where $E_{\parallel}=H_{\perp}=0$ so that we only
have a current flowing in the poloidal direction, generating a magnetic field
around the equator. It is convenient to combine the two equations in (24) into

$\nabla\times(\nabla\times
H_{\parallel})+\sigma\mu\partial_{t}H_{\parallel}+\epsilon\mu\partial_{t}^{2}H_{\parallel}=0\,.$
(26)

Using $H_{\parallel}=h(r,\theta,t)\widehat{\phi}$, we find

$-\Delta h+\sigma\mu\partial_{t}h+\epsilon\mu\partial_{t}^{2}h=0\,,$ (27)

where
$\Delta\equiv(1/r)\partial_{r}^{2}(rh)+(1/r^{2})\partial_{\theta}\left((1/\sin\theta)\partial_{\theta}(h\sin\theta)\right)$
is the $\widehat{\phi}$ component of the vector Laplacian. We make the ansatz

$h(r,\theta,t)=\sum_{m}\frac{1}{r}A_{\omega,m}(r)B_{m}(\theta)e^{i\omega
t}\,,$ (28)

where $B_{m}(\theta)$ is an eigenfunction of the angular operator

$\frac{\partial}{\partial\theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}(\sin{\theta}B_{m}(\theta)\right)=-m^{2}B_{m}(\theta)\,,$
(29)

and can be written in terms of hypergeometric functions:

$\displaystyle B_{m}(\theta)$
$\displaystyle={}_{2}F_{1}\left[-\frac{1}{4}-\lambda,-\frac{1}{4}+\lambda;\frac{1}{2};\cos^{2}\theta\right]\csc\theta\,c_{1}$
(30) $\displaystyle\hskip
11.0pt+{}_{2}F_{1}\left[\frac{1}{4}-\lambda,\frac{1}{4}+\lambda;\frac{3}{2};\cos^{2}\theta\right]\cot\theta\,c_{2}\,,$
(31)

with $\lambda\coloneqq\sqrt{1+4m^{2}}/4$. The leads to the radial equation:

$\frac{\partial^{2}A_{n,m}}{\partial
r^{2}}+\left(k^{2}-\frac{m^{2}}{r^{2}}\right)A_{\omega,m}=0\,.$ (32)

where (in the limit $\sigma\ll\epsilon\omega$)

$k=\sqrt{\epsilon\mu\omega^{2}-\sigma\mu\omega
i}\sim\sqrt{\epsilon\mu}\omega-i\frac{\sigma}{2}\sqrt{\frac{\mu}{\epsilon}}\,.$
(33)

The physically relevant solution is the one that decays as we go into the
black domain. This is proportional to a modified Bessel function of the second
kind:

$A_{\omega,m}=\sqrt{r}K_{2\lambda}(-ikr)\,,$ (34)

where the imaginary part of $k$ gives the expected exponential fall off. In
summary, we find our solution

$h(r,\theta,t)=\sum_{m}\frac{1}{\sqrt{r}}K_{2\lambda}(-ikr)e^{i\omega
t}B_{m}(\theta)\,.$ (35)

We now use $K_{\nu}(z)\sim\sqrt{\frac{\pi}{2z}}e^{-z}$ for large $|z|$ (and
$\arg z<3\pi/2$) to get 555 We assume that the skin depth in (6) is small
compared to the thickness of the shell. This requires a large $\sigma$, which
implies large $|z|$.

$h(r,\theta,t)\sim\frac{1}{r}e^{r\frac{\sigma}{2}\sqrt{\mu/\epsilon}}e^{i\sqrt{\epsilon\mu}\omega
r+i\omega t}\,.$ (36)

The radial component decays exponentially as $r$ decreases from its maximal
value at the boundary of the shell. The length scale associated with this
decay is again given by the skin depth $\delta_{s}$ defined in (6). Just as in
the case of the EM wave, we note that $\sigma$ cancels out in the effective
resistance, and we reproduce (at large $\omega$) the value for a black hole.
The requirements for this to happen are:

$\omega>\frac{\sigma}{\epsilon},\quad\delta_{s}\ll\delta a\,,$ (37)

implying

$\omega\gg\frac{c_{s}}{\delta a}\,,$ (38)

where, as before, $\delta a$ is the thickness of the shell. This means that
any variation faster than it takes for the slowed down light to cross the thin
shell will experience a resistance equal to that of a black hole. By choosing
$\epsilon_{r}=\mu_{r}$ sufficiently large, this will not lead to any practical
difference. In principle, one would expect the resistance to decrease for
processes sustained over a very long time.

The above is the familiar phenomenon of resistance depending on the skin
depth. At high frequencies the skin depth is small, and the current is only
flowing in a thin slice of the shell. The limiting resistance at high
frequencies is finite. At low frequencies, the field is penetrating the entire
width and the shell functions as an almost perfect conductor.

The actual solution for a specific configuration depends on the details of how
the wires are attached to the shell. The simplest possibility is when the thin
wires have vanishing resistance, and extend into the black domain. The
relevant solution then has only $m=0$ turned on, which leads to $E_{r}=0$ and
$E_{\theta}\neq 0$. The electric field (and the current) is then sourced by
the inserted wires all the way down to the skin depth. We then have exactly
$K_{1/2}(z)=\sqrt{\frac{\pi}{2z}}$, and the exact solution is given by

$h(r,\theta,t)=\frac{1}{r}e^{r/\delta_{s}}e^{i\sqrt{\epsilon\mu}\omega
r+i\omega t}\frac{1}{\sin\theta}\,.$ (39)

It is also possible to construct solutions where the wires end at the surface,
where they appear as localized sources. We will not discuss this here.

##  III The black shell as an electric generator producing jets

Let us now examine the impact of introducing a rotating hollow shell with
relative permeability $\mu$ and relative permittivity $\epsilon$ into a
uniform magnetic field. It is reasonable to assume that the external magnetic
field will penetrate the shell, resulting in a magnetic field within the black
domain. Note, though, that the interior AdS is effectively shielded from any
magnetic field. As the shell rotates, the bound charges within it will
encounter a Lorenz force. Consequently, this will give rise to a non-uniform
distribution of charges within the black domain, leading to a non-trivial
electric field.

One key difference between a shell and a solid sphere composed of the same
material is that the magnetic field within a solid sphere is directly
proportional to the external field, resulting in a constant magnetic field.
However, this relationship does not hold true for a shell. As a result, we
will approach the problem in two stages.

  1. (i)

we will first compute the magnetic field within the black domain

  2. (ii)

using this magnetic field, we will compute the electric field generated by the
rotating shell.

(a) $\vec{B}$

$\Omega$
---
$\vec{B}$
---

Figure 4: (a) Plot showing the magnetic field outside and within the cross-
section of a static shell with large $\mu,\epsilon$ placed in a uniform
constant external magnetic field pointing vertically upwards. (b) Schematic of
a rotating black shell placed within a circuit in an external magnetic field.
It acts like a generator in a closed electric circuit with current entering at
the poles and leaving through a conductor in the form of an accretion disc at
the equator. There is also an external load.

###  III.1 Magnetic field inside the shell

Consider a spherical shell of inner radius $a$, and outer radius $b$ of
relative permeability $\mu$ and relative permittivity $\epsilon$ placed in a
uniform external magnetic field $\vec{H_{0}}=H_{0}\widehat{z}$. Outside the
shell (vacuum with $\mu_{0}=1$), $\vec{B}=\vec{H_{0}}=H_{0}\widehat{z}$. The
scalar field $\Phi$ corresponding to a magnetic field $\vec{H}$ is given by
$\vec{H}=-\vec{\nabla}\Phi$. Outside the shell,
$\vec{\nabla}\Phi=-H_{0}\widehat{z}$, which gives (in radial coordinates
centered at the origin of the shell)

$\Phi_{0}=-H_{0}r\cos\theta=-H_{0}rP_{1}\,,$ (40)

where $P_{n}$ is the $n$-th Legendre polynomial. To solve Maxwell’s equation
$\vec{\nabla}\cdot\vec{B}=0=\nabla^{2}\Phi$, we make the following ansatz for
$\Phi$ (assuming regularity at the origin and at infinity):

$\Phi=\left\\{\begin{array}[]{lr}ArP_{1}&(r<a)\,,\\\
BrP_{1}+CP_{1}/r^{2}&(a<r<b)\,,\\\
-H_{0}rP_{1}+DP_{1}/r^{2}&(r>b)\,.\end{array}\right.$ (41)

Boundary conditions are given by continuity of the potential $\Phi$ and
continuity of the radial component of the magnetic field $\vec{B}=\mu\vec{H}$
(which follows from $\vec{\nabla}\cdot\vec{B}$) across the shell at $r=a,b$:

$\displaystyle A$ $\displaystyle=B+C/a^{3}\,,$ $\displaystyle B+C/b^{3}$
$\displaystyle=-H_{0}+D/b^{3}\,,$ $\displaystyle A$
$\displaystyle=\mu(B-2C/a^{3})\,,$ (42) $\displaystyle\mu(B-2C/b^{3})$
$\displaystyle=-H_{0}-2D/b^{3}\,.$

Solving (41) with the above boundary conditions gives

$\displaystyle A$ $\displaystyle=-9\mu b^{3}\frac{H_{0}}{E}\,,$ $\displaystyle
B$ $\displaystyle=-3(1+2\mu)b^{3}\frac{H_{0}}{E}\,,$ $\displaystyle C$
$\displaystyle=3(1-\mu)a^{3}b^{3}\frac{H_{0}}{E}\,,$ (43) $\displaystyle D$
$\displaystyle=(1+\mu-2\mu^{2})(b^{3}-a^{3})b^{3}\frac{H_{0}}{E}\,,$

with $E\coloneqq-2a^{3}(1-\mu)^{2}+b^{3}(2+5\mu+2\mu^{2})$. In the limit of
large $\mu,\epsilon$, this results in the following magnetic field
$\vec{B}=\mu\vec{H}=-\mu\vec{\nabla}\Phi$:

$\displaystyle\vec{B}_{\textrm{in}}=\mathcal{O}\left(1/\mu\right)\,,$
$\displaystyle\vec{B}_{\textrm{shell}}=-\frac{3b^{3}H_{0}}{2r^{3}\left(b^{3}-a^{3}\right)}\left[-\left(r^{3}-a^{3}\right)\cos\theta~{}\widehat{r}\right.$
$\displaystyle\hskip
45.0pt\left.+\left(a^{3}+2r^{3}\right)\sin\theta~{}\widehat{\theta}\right]+\mathcal{O}\left(1/\mu\right)\,,$
(44)
$\displaystyle\vec{B}_{\textrm{out}}=\frac{H_{0}}{r^{3}}\left[\left(2b^{3}+r^{3}\right)\cos\theta~{}\widehat{r}-\left(r^{3}-b^{3}\right)\sin\theta~{}\widehat{\theta}\right]$
$\displaystyle\hskip 45.0pt+\mathcal{O}\left(1/\mu\right)\,.$

It should be noted that when the value of $\mu$ is large, the interior of the
shell is completely protected from the magnetic shield. The solution for this
scenario is illustrated in fig. 4(a). As expected, the magnetic field within
the shell does not remain constant, unlike in the case of a solid sphere.
Instead, it exhibits a non-trivial dependence on both $r$ and $\theta$.

###  III.2 The black shell as a generator

The black shell acts like a generator in a way very similar to a black hole.
Let us assume an external magnetic field similar to the one in the previous
section, with a radial component of the magnetic field pointing outwards in
the northern hemisphere and inwards in the southern hemisphere. We let the
black shell spin with angular velocity $\Omega$ with its axis pointing upwards
and connect to it a very long wire that extends upwards from the north pole as
illustrated in fig. 4. The wire then turns around very far away from the shell
and connects back to the equator via sliding contacts mounted on an equatorial
disc such that they maintain constant contact with the shell, closing the
circuit. This induces an emf in the closed circuit given by

${\cal E}=\frac{\Omega\Phi}{2\pi}$ (45)

where $\Phi$ is the total magnetic flux entering through the southern
hemisphere (and exiting through the northern). This is the same result as for
a black hole. A current will flow from the north pole to the equator,
generating a magnetic field along the equator. As argued in section II.3, the
current will be flowing in a very thin sheet of the shell. This current
shields the interior from the magnetic field pointing in the $\phi$-direction
on the outside of the shell.

Let us figure out the structure of the electric and magnetic fields within the
thin shell. As discussed above, the radial component of the magnetic field
will give rise to a Lorentz force driving a current along the surface of the
shell in the direction of $\widehat{\theta}$. A bit deeper into the black
domain, the magnetic field must turn in the angular direction in order for the
fields in the southern and northern hemispheres to be connected. This will
give rise to a Lorentz force pushing the current towards the surface. Let us
now work out the details.

To proceed we need to relate the $E$, $D$, $B$ and $H$ fields in the inertial
system to the corresponding fields, $E^{*}$, $D^{*}$, $B^{*}$ and $H^{*}$ in a
frame co-rotating with the shell. We have the constitutive relations:

$D^{*}=\epsilon E^{*},\quad B^{*}=\mu H^{*},\quad J^{*}=\sigma E^{*}.$ (46)

These fields are related to the inertial frame through the Lorentz
transformations:

$\displaystyle E_{\perp}^{*}=\gamma(E_{\perp}+v\times B)\,,$ $\displaystyle
B_{\perp}^{*}=\gamma(B_{\perp}-\frac{v}{c^{2}}\times E)\,,$ (47)
$\displaystyle H_{\perp}^{*}=\gamma(H_{\perp}-v\times D)\,,$ $\displaystyle
D_{\perp}^{*}=\gamma(D_{\perp}+\frac{v}{c^{2}}\times H)\,.$

Note that the component of the current transverse to $v$, has
$J_{\perp}=J_{\perp}^{*}$, while the parallel component has
$J_{\parallel}=\gamma J_{\parallel}^{*}$. We also note that if there is no
charge density in the rest frame of the shell, $\rho^{*}=0$, there will be one
generated in the inertial frame given by $\rho=-\gamma
vJ_{\parallel}^{*}/c^{2}$.

Maxwell’s equations in the frame of the shell are identical to those we used
in the previous section:

$\displaystyle\nabla\times E^{*}=-\mu\partial_{t}H^{*}\,,$
$\displaystyle\nabla\times H^{*}=\sigma E^{*}+\epsilon\partial_{t}E^{*}\,,$
(48) $\displaystyle\nabla\cdot H^{*}=0\,,$ $\displaystyle\nabla\cdot
E^{*}=0\,,$

where the last relation is due to $J^{*}=\sigma E^{*}$ being conserved
everywhere, except where the external wires are connected. If one wants to
compare with Maxwell’s equations in the inertial frame it is useful to note
that $\nabla\times(v\times H^{*})=\nabla\times(v\times E^{*})=0$. We note,
though, that $\nabla\cdot E_{\perp}=-\nabla\cdot(v\times H^{*})\neq 0$,
reflecting the cancellation of the Lorentz force pushing the current away from
the rotational axis.

Since the equations for $E^{*}_{\parallel}=0$ and $H^{*}_{\perp}$ (determining
the magnetic field) decouple from $E^{*}_{\perp}$ and $H^{*}_{\parallel}$
(determining the current), except through boundary conditions, we can use the
results from the previous sections. The only difference is how the wires are
connected to the shell. For a black shell with an accretion disc, generating
jets, we expect something like fig. 4. We have thin wires connected at the
north and the south pole together with a thin, disc like wire connected to the
equator. With the magnetic field as well as the rotation vector pointing
upwards, we will obtain a current flowing from the poles to the equator.

The detailed profile of the solution depends, as discussed previously, on the
anatomy of the attached wires. The simplest case is again $m=0$, but now the
currents flow along the longitudes from both poles (with $E_{r}=0$), and are
intercepted by the disc shaped conductor that makes sliding contact along the
entire equator.

##  IV Electromagnetic signatures of black shells

Having discussed electromagnetic properties of the black shell, let us now
turn our attention to their observational signatures in the electromagnetic
spectrum i.e. images obtained in VLBI experiments like the EHT.

Photons that travel from the object to the observer’s camera make up the image
on the observer’s screen. Given the limited field of view of the camera, most
photons emitted near the object never make it to the screen. An efficient way
to reconstruct the image on the screen is, therefore, to trace photon
trajectories back in time from the screen back towards the source. In the case
of a black shell (or a black hole), given enough time, the backward traced
photons will either end up at the surface of the black shell (or at the event
horizon in the case of a black hole) or escape out of the camera sphere, to
infinity. Computing these photon trajectories goes by the name of _ray
tracing_ and can be efficiently performed by integrating null geodesics in the
given spacetime. While this can be done analytically for the Kerr spacetime,
the geodesic equations need to be integrated numerically for general
spacetimes. Let us summarize the equations to solve, and initial conditions
below.

$d(\varphi)$
---
$\varphi$
---
inner shadow
---
critical curve
---
Figure 5: Snapshot of simulated synchrotron emission at $230\textrm{GHz}$
(obtained from a GRMHD simulation) of a Kerr black hole with $M=6.5\times
10^{9}M_{\odot},a=0.45$, observed by a camera placed at a distance of
$16.9\textrm{Mpc}$ from the source, and inclined at $17^{\circ}$ to the spin
axis. Superimposed on the image are the critical curve (blue curve), boundary
of the inner shadow (white curve), and the projected diameter $d(\varphi)$.
The axes are labelled in units of $M$.

###  IV.1 Geodesic equations and initial conditions

For a stationary, axisymmetric metric in Boyer-Lindqvist coordinates

$\mathrm{d}s^{2}=g_{tt}\mathrm{d}t^{2}+g_{rr}\mathrm{d}r^{2}+g_{\theta\theta}\mathrm{d}\theta^{2}+g_{\phi\phi}\mathrm{d}\phi^{2}+2g_{t\phi}\mathrm{d}t\mathrm{d}\phi\,,$
(49)

null geodesics are solutions to a set of second order ODEs:

$\ddot{x}^{\mu}+\Gamma^{\mu}{}_{\nu\rho}\dot{x}^{\nu}\dot{x}^{\rho}=0\,,$ (50)

where $\Gamma^{\mu}{}_{\nu\rho}$ is the torsion-free affine connection, and
dots represent derivatives with respect to the affine parameter. These are
four second order equations for the four components of the position vector of
the photon, which can be integrated given the initial position and velocity of
the photon. However, stationarity and axisymmetry imply that energy ($E$) and
angular momentum ($L$) are constant along the geodesic. Using these constants
of motion, we can integrate the $t$ and $\phi$ components of the geodesic
equation once, to reduce them to two first order equations:

$\dot{t}=\frac{Eg_{\phi\phi}+Lg_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}},\quad\dot{\phi}=-\frac{Eg_{t\phi}+Lg_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,.$
(51)

We now have a system of $2$ first order and $2$ second order ODEs, which
require $6$ inital conditions to solve. In addition, we have the constraint
for a null geodesic, $\dot{x}^{\mu}\dot{x}_{\mu}=0$, which we use to monitor
the accuracy of the integration.

A useful way to parametrize the initial condition is in terms of position and
velocity of the photon at the observer’s screen. We use the formulation in
[17], and place the observer at $r_{\textrm{obs}}=100M$ from the center of the
coordinate system, at an inclination of $17^{\circ}$ and $60^{\circ}$
respectively. To perform the integration, geodesics are evolved from the
observer’s screen until they either (i) arrive at the shell (with a tolerance
of $r=r_{\textrm{shell}}+10^{-5}M$), or (ii) exit the camera sphere
$r>r_{\textrm{obs}}$.

###  IV.2 Lensing bands and critical curves

To understand the image produced on the observer’s screen, it is useful to
label each photon by the number of times it crosses the equatorial
plane.666For a stationary axisymmetric object, the equatorial plane (which is
perpendicular to the spin axis) provides a well defined notion of plane
crossing, and in Boyer-Lindqvist coordinates corresponds to $\theta=\pi/2$,
providing a simple way of tracking photon trajectories. Other notions of $n$
have also been used in the literature, which include counting the number of
full turns made by the photon [18], and the number of times the photon crosses
a plane perpendicular to the observer’s axis [19]. For a discussion, see [20].
Photons which cross the equatorial plane exactly once constitute the _direct
image_ of the object. Each subsequent turn that the photons make around the
equatorial plane give a lensed image of this direct image and are referred to
as the _lensing bands_ denoted by the additional number of times they cross
the equatorial plane – the $n$-th lensing band includes photons that cross the
equatorial plane at least $n+1$ times. Each subsequent lensing band becomes
exponentially thinner, and the $n\to\infty$ image coincides with the _critical
curve_ , which gives the location of the unstable photon orbit. The complete
image contains all the lensing bands superposed on top of one another. By
definition, each lensing band includes all the subsequent lensing bands within
it, and therefore the critical curve is the brightest curve in the image. The
innermost region of the direct image, which corresponds to photons that fall
on to the black shell (or hit the horizon of a black hole), therefore remains
empty and is referred to as the _inner shadow_ [19]. The shape of the critical
curve can be conveniently characterized using the _projected diameter_ and
_centroid_ defined by [21].

###  IV.3 Ray tracing: results

With this, we can now study the lensing bands and critical curve for a black
shell rotating moderately fast. We will use the rotating black shell metric
obtained in [8] to order $a^{6}$ in Boyer-Lindqvist coordinates and perform
the computation at $a=0.45$ (corresponding to $a^{6}\sim 0.45^{6}<1\%$.)

Before presenting the results of the computation, let us first discuss our
expectations. The black shell has two important differences from a Kerr black
hole with the same mass and spin: (i) it has a surface which lies outside the
outer horizon for a Kerr black hole, and (ii) it has a multipolar structure
that differs from Kerr at a percent level. We therefore expect to see two
types of differences between the curves of a black shell and a black hole:

  1. (i)

The inner boundary of the direct image (equivalently, boundary of the inner
shadow) consists of photons that barely escape the black shell (or the horizon
of a black hole). A surface that is larger than the horizon should lead to a
larger inner shadow. The lensing bands are higher-order lensed images of the
direct image – the inner boundary in particular is a lensed image of the
boundary of the inner shadow. We would therefore expect the image of the black
shell to have larger inner boundaries for each band, as compared to a black
hole.

  2. (ii)

The shape of the critical curve and the outer boundaries of the lensing bands
are a function of the spacetime metric outside the object. Since the black
shell has a different multipolar structure compared to a Kerr black hole, we
expect the shape of the curves to be different. The leading multipolar
deviations are small: $\sim 1\%$ for the quadrupole, and $\sim 6\%$ for the
octupole and we expect the critical curve to have a small difference in shape
compared to Kerr.

(a) Kerr, $\theta_{\textrm{obs}}=17^{\circ}$

(b) Black shell, $\theta_{\textrm{obs}}=17^{\circ}$

(c) figs. (a) and (b) superimposed

(d) Kerr, $\theta_{\textrm{obs}}=60^{\circ}$

(e) Black shell, $\theta_{\textrm{obs}}=60^{\circ}$

(f) figs. (d) and (e) superimposed

Figure 6: The direct image ($n=0$) and the $n=1$ lensing band (light and dark
gray regions, respectively) for a Kerr black hole and a black shell as
observed from an inclination angle of $17^{\circ}$ and $60^{\circ}$
respectively, relative to the spin axis (dimensionless spin of $a=0.45$) are
shown in figs (a-b), (d-e). The axes are labeled in units of $M$. Critical
curves for the Kerr black hole and the black shell (corresponding to the
$n\to\infty$ lensing band) are shown with solid blue and dotted red lines,
respectively. Figs (a-b) are superimposed in fig (c), and figs (d-e) are
superimposed in fig (f). The edges of the lensing bands for the Kerr black
hole and the black shell are shown with solid and dotted lines respectively.
Figure 7: Shape of the critical curve for a Kerr black hole and a black shell
(BSh) of the same mass and spin ($M=1,a=0.45$), as seen by observers at two
different inclinations with the spin axis, characterized using quantities
defined by [21]. The projected position $f(\varphi)$ as a function of the
normal angle $\varphi$ fully parametrizes the shape of the curve. Even and odd
parity components of $f(\varphi)$ give the projected diameter $d(\varphi)$ and
centroid $C(\varphi)$ respectively.

Finally, we can integrate the geodesic equations numerically from the
observer’s screen with initial conditions given by [17]. Since we are only
interested in the inner and outer boundaries of the lensing bands, we use a
binary search algorithm to find each boundary (_method of bisection_). For the
direct image ($n=0$), and the $n=1$ lensing band, we only need to track
photons that cross the equatorial plane once and twice respectively. Having
obtained those, we directly compute the critical curve (corresponding to the
$n\to\infty$ photon ring), also using a binary search (since the higher order
lensing bands approach the critical curve exponentially fast).

Before looking at the results, it is useful to look at fig. 5 to orient
ourselves. There, we have produced a simulated 230 GHz synchrotron emission
image from an accretion disk made of magnetized plasma around a rotating Kerr
black hole with $a=0.45$ using force-free GRMHD and radiative transfer [22,
23]. We have labelled the critical curve, inner shadow and an instance of the
projected diameter on the image.

We are now ready to take a look at our results in figs. 6 and 7. From fig. 6,
we see that both of our expectations are realized. In the right most panels,
we see that the inner boundaries of the $n=0$ and $n=1$ bands for the black
shell are larger than those for a Kerr black hole with the same mass and spin;
the difference is big enough to see by eye. The outer boundaries and the
critical curve are also not identical, but the difference is much smaller and
harder to make out by eye, just as expected.

In fig. 7, we have computed the projected position, diameter and centroid of
the critical curve as a function of the angle. The largest difference is in
the shape, given by the projected diameter $d(\varphi)$, shown in the center
panel just as expected.

While the critical curve for a black shell has a shape distinct from a black
hole with identical mass and spin, it is imperative to keep in mind that
current uncertainties in mass and spin estimation from astrophysical
observations can conflate the shape of the curves, making them
indistinguishable [24, 25].

The size of the inner shadow, and the inner boundary of the $n=1$ lensing
band, on the other hand, are the biggest differences between a black shell and
a black hole. The next generation EHT (ngEHT) is expected to be able to
measure the $n=1$ photon ring and will have an improved dynamic range to map
the inner shadow [26]. This then seems to be the most promising difference
that one might hope to observe in the near future, e.g. [27].

There is, however, an additional effect that could affect the inner shadow for
a black shell owing to how accreting magnetized plasma interacts with the
shell. While a black hole horizon provides purely ingoing boundary conditions
for the plasma, the surface of a black shell will interact with the plasma in
a much more nontrivial way. Although the black shell is expected to be a very
efficient absorber similar to a black hole, the microscopics of how the
absorption and energy transfer takes place within the black shell could have
an effect on the accretion physics. These effects are beyond the scope of the
present work and are being investigated separately using GRHMD simulations. We
will report on the results in an upcoming work.

##  V Gravitational wave signatures

Having looked at electromagnetic signatures of black shells, let us now turn
our attention to gravitational observables. We will focus on two in
particular: (a) Light ring characteristics and quasinormal modes, and (b) the
tidal Love number.

###  V.1 Light ring frequency and QNMs

Quasinormal modes (QNMs) are defined as solutions of the spacetime
perturbation equations (with complex characteristic frequencies) with boundary
conditions that are purely incoming at the horizon (or the surface of a
horizonless compact object) and purely outgoing at infinity. A tantalizing
relation between the QNM frequencies in the eikonal limit and unstable null
circular orbits for any static, spherically symmetric, asymptotically flat
spacetime has been noted [28, 29] (though, for a cautionary note see [30]):

$\omega_{\textrm{QNM}}=\Omega_{c}l-i\left(n+1/2\right)|\lambda|,$ (52)

where $\Omega_{c}$ is the orbital angular velocity at the null circular
geodesic, $\lambda$ is the principal Lyapunov exponent that characterizes the
deviation of nearby photon orbits around the unstable circular orbit, $n$ is
the overtone number, and $l$ is the angular momentum quantum number of the
perturbation. Beyond spherical symmetry, it is known for Kerr [31] and Kerr-
Newmann [32] black holes that equatorial null geodesics can account for
$l=|m|$ modes.

Nevertheless, characteristics of unstable null circular orbits in the
equatorial plane carry information about the compact object, and the deviation
from their Kerr values could serve as a test of how closely they mimic or
differ from a Kerr black hole.

Figure 8: Characteristics of unstable null circular geodesics associated with
the black shell (BSh and a Kerr black hole with the same mass and spin, shown
as a function of the dimensionless spin $a$.

Following [33], let us write down the geodesic equation in the equatorial
plane ($\theta=\pi/2,\dot{\theta}=0$) and find the radial potential for a
stationary axisymmetric metric. The appropriate Lagrangian for the system is

$2\mathcal{L}=g_{tt}\dot{t}^{2}+g_{rr}\dot{r}^{2}+g_{\varphi\varphi}\dot{\varphi}^{2}+2g_{t\varphi}\dot{t}\dot{\varphi}.$
(53)

Next, we write down momenta canonically conjugate to the coordinates
$p_{\mu}=\partial\mathcal{L}/\partial x^{\mu}$. Stationarity and axisymmetry
imply that energy and angular momentum are constants of motion
($p_{t}=-E,p_{\varphi}=L$), from which it follows that $\dot{t},\dot{\varphi}$
are given by

$\dot{t}=\frac{Eg_{\varphi\varphi}+Lg_{t\varphi}}{g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi}},\quad\dot{\varphi}=-\frac{Eg_{t\varphi}+Lg_{tt}}{g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi}}.$
(54)

The Hamiltonian corresponding to the system
($2\mathcal{H}=2\left(p_{\mu}\dot{x}^{\mu}-\mathcal{L}\right)\coloneqq\delta\equiv\textrm{constant}$)
can be written as

$\dot{r}^{2}=\frac{L^{2}g_{tt}+2ELg_{t\varphi}+E^{2}g_{\varphi\varphi}}{g_{rr}\left(g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi}\right)}+\frac{\delta}{g_{rr}}\coloneqq
V_{r}.$ (55)

For null ($\delta=0$) circular geodesics to exist at $r=r_{c}$,
$V_{r}(r=r_{c})=V_{r}^{\prime}(r=r_{c})=0$. The orbital angular frequency on
these circular geodesics follows from (54)

$\Omega_{c}\coloneqq\frac{\dot{\varphi}}{\dot{t}}\bigg{|}_{r=r_{c}}=-\frac{Lg_{tt}+Eg_{\varphi\varphi}}{Lg_{t\varphi}+Eg_{\varphi\varphi}}\bigg{|}_{r=r_{c}}.$
(56)

The Lyapunov exponent $\lambda$ is a measure of the instability of the orbit
and is given by

$\lambda=\sqrt{\frac{V_{r}^{\prime\prime}}{2\dot{t}^{2}}}\bigg{|}_{r=r_{c}}.$
(57)

To each of these quantities, we can attach an associated orbital time scale
$T_{\Omega}\coloneqq 2\pi/\Omega_{c}$, an instability timescale
$T_{\lambda}\coloneqq 2\pi/\lambda$, and define a critical exponent $\gamma$
following [34]

$\gamma\coloneqq\frac{\Omega_{c}}{2\pi\lambda}=\frac{T_{\lambda}}{T_{\Omega}},$
(58)

or simply a dimensionless instability exponent $\lambda/\Omega_{c}$.

We can now use the metric of the rotating black shell to order $a^{6}$ in [8]
to compute the orbital frequency and the dimensionless instability exponent
for a slowly rotating black shell. It is convenient to introduce the impact
parameter $b\coloneqq L/E$. In fig. 8 we present the orbital frequency as a
function of spin $a$ for both the black shell and a Kerr black hole for the
same mass and spin. It is interesting to note that

$\Omega_{c}=\frac{1}{b}.$ (59)

This special relationship between the angular frequency of null circular
geodesics and the impact parameter is known to hold for the Kerr solution
[33]. It was discovered in [29] that it holds for a $d$ dimensional Myers-
Perry black hole [35] (which is a rotating black hole solution in dimensions
$d>4$) as well, where it was further speculated that it holds for any
stationary spacetime. That it holds for the black shell provides evidence in
support of this speculation.

The dimensionless instability exponent of a rotating black shell compared to a
Kerr black hole of the same mass and spin, as a function of dimensional spin
parameter $a$ is shown in fig. 8. Since the rotating black shell metric is
known perturbatively to order $a^{6}$, useful results can only be obtained for
spin up to $a\sim 0.45$. The difference from a Kerr black hole is extremely
small, at sub-percent to a percent level, for this range of spin.

###  V.2 Tidal Love number & Gravitational Waves

Gravitational waves from inspiraling compact objects, prior to the merger
stage, can be obtained through a Post-Newtonian approach. The system is then
described by the dynamics of two effective “point particles” where details of
its structure are captured, to leading order, for non-spinning objects at the
5th Post-Newtonian level via a tidal Love number [36]. Such a number helps
describe the deformability of a self-gravitating object immersed in an
external tidal field. In an inspiraling binary, such a deformation leaves a
potentially detectable imprint in the gravitational-wave (GW) signal emitted
by the binary in the late stages of its orbital evolution (e.g. [37]). We here
undertake the computation of the tidal Love number for non-spinning black
shells. (For simplicity, we do not concern ourselves in this work with
potential modulations introduced by spin-induced quadrupole moments [38].)

We follow [39] and compute the (static) tidal Love numbers for non-rotating
black shells. To this end, we expand the (static) metric as
$g_{ab}=g_{ab}^{(0)}+\delta g_{ab}$, using $(t,r,\theta,\phi)$ coordinates,

$\begin{split}\delta
g_{ab}=\mbox{diag}&\left(fH_{0}^{l}(r),\frac{H_{2}^{l}(r)}{h},r^{2}K^{l}(r),r^{2}K^{l}(r)\sin^{2}\theta\right)\times\\\
&\times Y_{l0}(\theta,\phi),\end{split}$

with $Y_{lm}$ spherical harmonics, and we focus on the spherical symmetric
case background so $m=0$ can be chosen. The function $f$ is given by

$f=1-\frac{2\widehat{M}}{r}+\widehat{\Lambda}\frac{r^{2}}{3},$ (60)

with $\widehat{\Lambda}=\Lambda\Theta(R-r)$ and $\widehat{M}=M\Theta(r-R)$
with $\Theta$ the Heaviside step function and $R$ the shell’s radius. In [39,
40] the case for a gravastar was studied, from which most results can be
derived, as the sign of $\Lambda$ was not crucial. The solution for the fields
in each region, at linearized order imply all functions can be determined from
$H_{0}^{l}(r)$. This, in turn, obeys

$\frac{\mathrm{d}^{2}H_{0}^{l}}{\mathrm{d}r_{*}^{2}}-VH_{0}^{l}=0,$ (61)

with the potential

$V=\frac{9r^{2}}{R^{4}}\left(l(l+1)f+4\frac{\widehat{M}^{2}}{r^{2}}-\frac{2}{9}r^{2}\widehat{\Lambda}(9+r^{2}\widehat{\Lambda})\right),$
(62)

with the tortoise coordinate defined as $\mathrm{d}r/\mathrm{d}r_{*}\equiv
g(r)=rR^{-2}(6\widehat{M}-\widehat{\Lambda}r^{3}-3r)$.

Equation (61) can be solved independently inside/outside the shell requiring
regularity and asymptotic decay:

$\displaystyle H_{0}^{i,l}$ $\displaystyle=$
$\displaystyle\alpha_{1}\frac{r^{l}\sqrt{\Lambda}}{3+r^{2}\Lambda}{}_{2}F_{1}\left[\frac{l-1}{2},\frac{l}{2},l+\frac{3}{2},-r^{2}\Lambda/3\right],$
$\displaystyle H_{0}^{o,l}$ $\displaystyle=$
$\displaystyle\alpha_{2}P^{2}_{l}\left(\frac{r}{M}-1\right)+\beta_{2}Q^{2}_{l}\left(\frac{r}{M}-1\right),$
(63)

where $P^{2}_{l},Q^{2}_{l}$ are associated Legendre functions and
${}_{2}F_{1}$ the hypergeometric function. The constants
$\\{\alpha_{i},\beta_{i}\\}$ are fixed (up to an overall value) by imposing
Israel’s junction conditions [41] (2 equations) together with coordinate
continuity (1 equation) and the equation of state describing the shell that
modulates the jump at linear order.777Reference [39] works out explicitly for
$l=2$ the junction conditions together with the coordinate condition
transformation with respect to time scalings so that the boundary is at the
same coordinate value $r$. For concreteness and simplicity we take $\delta
p=-\delta\sigma$ (with $p,\sigma$ the pressure and energy density of the
shell). For simplicity we focus on the $l=2$ mode, asymptotically,

$H_{0}^{2}\rightarrow
3c_{1}\frac{r^{2}}{M^{2}}-6c_{1}\frac{r}{M}+\frac{8}{5}\frac{M^{3}}{r^{3}}c_{2}+O(r^{-4}),$
(64)

where we have introduced the constants $c_{i}$ which are functions of the
solutions obtained for $\\{\alpha_{2},\beta_{2}\\}$. The quadrupole tidal love
number is given by [42]

$\lambda_{\textrm{BSh}}=\frac{8}{45}\frac{c_{2}}{c_{1}}M^{5}.$ (65)

The solution depends on $\Lambda$, taking its value to be (negative) and
large, we find to leading order $\lambda_{\textrm{BSh}}\approx 0.27M^{5}$.

In contrast to a Schwarzschild black hole (whose static Love number vanishes),
a stationary black shell has a positive Love number. A non-zero Love number
will induce a phase shift in the early, low frequency part of a detected
gravitational wave signal from roughly comparable mass mergers (the phase
shift is suppressed by the third power of the mass ratio), and is potentially
a measurable feature of the gravitational waveform (see [37]). For instance,
this has been exploited to scrutinize the neutron star equation of state in
GW170817 [43] and to systematically constrain the Love number for black holes
in the LVK catalog [44].

To put the non-zero Love number for a black shell into perspective, it is
useful to compare it with other compact objects e.g. neutron stars. They have
$\lambda\approx 10^{5}$ times larger, and a proportionately large impact on
observable consequences, e.g. the phase offset in the gravitational wave.
While a comparable (say in terms of masses) binary neutron star system merges
at a lower frequency (as black shells are significantly more compact) the
overall phase difference ($\delta\Phi)$ due to tidal effects, from far
separation until “contact” is much smaller for the black shell binary. For
instance, in the equal mass case, $\delta\Phi_{\textrm{NS}}\approx
10^{3}\delta\Phi_{\textrm{BSh}}.$ Naturally, measuring such small phase
offsets would require next generation (3G) detectors.

##  VI Summary, Discussion and Outlook

Understanding all potential observables related to exotic compact objects is
imperative to scrutinize them and, in particular, to realize their role to
help reveal insights into open problems on the fundamental physics of compact
objects. In this work we have studied electromagnetic and gravitational
properties of black shells, and discussed the extent to which they are
distinguishable from black holes. While we have focused exclusively on black
shells as a sub-class of ECOs, another aim of this paper is to provide
guidance and lessons for similar effects in the study of other ECOs.

We have analyzed the electromagnetic properties of black shells in the
classical limit, and flat spacetime,888A reasonable approach, since the shell
is located at the Buchdahl radius, a point where spacetime curvature effects
are weak for supermassive black holes. A more detailed investigation is
certainly doable, as is the case of axisymmetry. highlighting the lack of
reflection for light with wavelengths commensurate and smaller than the black
shell. This is a natural consequence of such wavelengths being reduced as the
wave propagates into a material with high relative permittivity and
permeability, and the conductivity lies within a wide interval.999We have
argued in section II that a large relative permeability and permittivity,
$\epsilon_{r}=\mu_{r}\gg 1$, is natural to expect from the stringy nature of
black shells. However, we have not yet derived this from first principles, and
for the classical analysis in this article, this has served as a well
motivated _assumption_. We would like to return to this in the future and
attempt to compute/estimate these electromagnetic properties of the shell from
string theory. To avoid contributing to reflection, the conductivity needs to
be small enough, i.e. $\sigma\ll\epsilon\omega$, but large enough to make sure
that the electromagnetic skin depth $\delta_{s}$, defined in (6), is much
smaller than the thickness of the shell ($\delta a$), i.e $\delta_{s}\ll\delta
a$, or $\sigma\gg(2/\delta a)\sqrt{\epsilon/\mu}$. For consistency, this
requires $(\delta a/c_{s})\gg(2/\omega)$, where $c_{s}\coloneqq
1/\sqrt{\epsilon\mu}$ is the speed of light within the shell. This only needs
to apply for $\omega\gg c/a$, that is, light with a wavelength smaller than
the Schwarzschild radius. This implies

$\epsilon_{r}\,\delta a/a\gg 1\,,$ (66)

which is the same condition as we derived in section II.2 around (21). In
summary, $\sigma$ needs to satisfy

$\frac{2}{\delta a}\sqrt{\frac{\epsilon}{\mu}}\ll\sigma\ll\frac{\epsilon
c}{a}\,,$ (67)

Hence, we conclude that all constraints are satisfied if the time it takes for
the slowed down light to cross the thin shell (i.e. the black domain) is
longer than the light crossing time of the Schwarzschild radius. On the other
hand, for processes that are really slow, with time scales orders of magnitude
longer than the light crossing time, we could in principle allow for a
reduction in the resistance due to an increased skin depth. Should precise
future estimates of relevant magnetic fields and other plasma characteristics
become available, an opportunity to calculate this resistance would emerge,
thereby facilitating a meaningful connection with the underlying model.

We have further initiated an effort to connect the black shell model with VLBI
observations of supermassive compact objects, and identify key observables
that can distinguish them from black holes. Our first analysis focused on the
photon ring, where the differences between a Kerr black hole and corresponding
black shell are subtle. Future steps involve investigating astrophysical
accretion flow dynamics to achieve a realistic depiction of the image of a
black shell. While the accreting plasma falls into the horizon of a black
hole, the interaction with a black shell proves significantly more complex.
Upon contact with the shell, the incoming fluid transforms into string degrees
of freedom, resulting in a novel state of matter with a different equation of
state. The nuances of this interaction might reveal characteristics that
better distinguish black shells from black holes than the structure of the
photon ring.101010 In [45, 46], attempts have been made to constrain the size
and nature of ECOs using EHT data from M87* and Sgr A*, assuming that their
surface is in thermal equilibrium with the accreting environment. Since black
shells have a very large heat capacity, they are not expected to thermalize
and are not bounded by the analysis there. Our current efforts are focused on
simulating this interaction through a force-free GRMHD simulation of
magnetized plasma accreting onto a black shell. We anticipate discussing these
findings in forthcoming publications.

We have inferred gravitational wave properties by estimating the QNM
frequencies in the eikonal limit (through the photon orbital frequency) and
computing the tidal Love numbers. This complements our previous study [6]
which unearthed features related to propagation of scalar waves through the
hollow interior of the black shell. Future efforts involve extending this to
the gravitational wave case, computing the spectrum of black shell QNMs, and
contrasting this with that of a black hole.

A typical characteristic of most ECOs is the presence of a _stable light
ring_. In Schwarzschild spacetime, the potential for a radial null geodesic
peaks at $r=3M$ (unstable light ring), and declines to
$V(r)\rightarrow-\infty$ as $r\rightarrow 0$. In contrast, ECOs typically have
non-singular centers, with a potential that rises to $V(r)\rightarrow+\infty$
as $r\rightarrow 0$. This results in a local minimum between $r=0$ and $r=3M$,
which allows for a trapping of null geodesics, hence called the _stable light
ring_. The presence of a stable light ring is often thought to allow for the
possibility of non-linear instabilities, even if the underlying compact object
is linearly stable [47]. For a black shell, the stable light ring is situated
on the shell.

While we have not discussed black shell binaries so far, we can make some
informed comments on what one would reasonably expect in such a situation.
Like a binary black hole system, the evolution of a black shell binary would
go through roughly three stages: (i) inspiral, (ii) merger, (iii) ringdown,
though arguably, the ringdown phase can be further sub-divided into: _early
ringdown_ , _ringdown_ and _late-ringdown_ stages.

  1. (i)

We expect the inspiral to be similar to that of a black hole binary, as tidal
effects are very small. Furthermore, the high compactness means that up until
frequencies $\sim 4/(9m_{T})$, (where $m_{T}$ is the total mass of the system)
the gravitational waves are expected to be very similar. This expectation is
supported by non-linear simulations of highly compact boson star merger [48]
as well as of binary black holes in higher curvature gravity [49].

  2. (ii)

The merger is a highly non-linear process, the details of which we cannot yet
begin to describe. In [6], it was shown how the black shell grows in size
under slow accretion of matter. If the accretion is fast enough, or if two
black shells collide, we expect tunneling events to take place similar to the
nucleation of a black shell when, e.g., a star collapses. The nucleation of a
common shell should be a close parallel to the sudden formation of a common
apparent horizon when two black holes collide.

  3. (iii)

During the _early ringdown_ , we can imagine that the shell formed from the
merger is highly distorted, and will shed its structure on a “viscous”
timescale. Assuming that the spherically symmetric model analysed in [6] can
inform what happens here, bulk viscosity would play a role here. A back-of-
the-envelope estimate gives that the timescale for modes to _smooth out_ is of
order $\rho_{g}/(\xi k^{2})$, where $\rho_{g}$ is given by the black hole
temperature (and commensurate with $m_{T}$), $\xi$ is the bulk viscosity
(which we choose to be $\sim 1$), and $k$ is each wavenumber that can be
supported by the black shell (hence they are $\propto(2\pi/n)$. As a result,
high frequencies would smooth out very quickly. Supported sound modes would
have frequencies quantized by $1/c_{s}$, and the _rotating_ quadrupole would
have a frequency correlated to $\Omega_{\textrm{orbital}}$ at the light ring
frequency.

  4. (iv)

The _ringdown_ phase would be marked by quasinormal modes dominating the
gravitational waveform. These would be longer wavelength/slower modes, and we
would expect their decay rate to be comparable to that of a black hole of
comparable mass. The lowest of them are estimated by the light ring frequency
as discussed in section V.1.

  5. (v)

The _late ringdown_ would be dominated by the long term behaviour of modes
that make it into the shell and leak to the outside. These waves would not be
present in a comparable black hole ringdown.

Obtaining an accurate description of all these stages demand significant new
developments, some of which will be presented in due course. Nevertheless, the
heuristics given in this work provide valuable guidance moving forward.

We conclude by emphasizing our hope that this work inspires similar efforts
involving other exotic compact objects, with the aim to provide as much
information as possible to discover or constrain them through observation.
Beyond the heuristics presented here, these methods are likely to be relevant
and valuable for such an enterprise.

###### Acknowledgements.

We are grateful to Alejandro Cárdenas-Avendaño, Daniel Mayerson, Paolo Pani,
Nils Siemonsen, and George Wong for helpful discussions at various stages of
this work. The work of SG was conducted with funding awarded by the Swedish
Research Council grant VR 2022-06157. LL thanks financial support via the
Carlo Fidani Rainer Weiss Chair at Perimeter Institute. LL receives additional
financial support from the Natural Sciences and Engineering Research Council
of Canada through a Discovery Grant and CIFAR. This work was supported in part
by Perimeter Institute for Theoretical Physics. Research at Perimeter
Institute is supported by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Economic Development, Job Creation and Trade.
FP acknowledges support from the NSF through the grant PHY-220728.

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