inaccurate as the carrier density is lowered toward the MIT, but the theory
never becomes invalid until there is an actual phase transition (e.g. at the
MIT). However, the theory should be quantitatively accurate deep into the
metallic phase, and indeed we get quantitative agreement with the experimental
resistivity at high carrier density, but our estimate of the critical density
for the MIT is off by a factor of 2. Our screening of the Coulomb disorder
uses the random phase approximation, which is also exact only at high
densities, becoming inaccurate at lower densities. Going beyond these
approximations is difficult, if not impossible, and therefore, it is
satisfying that we get a reasonable agreement with the experimental transport
data over a broad density range with our theory becoming quantitatively
inaccurate at low densities as expected. We also find that associating the
very low density (i.e. large $r_{s}$) observation of the MIT does not
necessarily imply the observation of WC since disorder-induced localization
effects are also strongly enhanced at lower densities– this point has already
been emphasized earlier in the literature focusing on 2D semiconductor systems
Ahn and Das Sarma (2023). In fact, the reported observation of the lowest
critical density (highest critical $r_{s}$) MIT was interpreted as a
percolation transition rather than a WC transition, although the critical
$r_{s}$ for the MIT in that 2D hole p-GaAs system was $\gtrsim 50$, much
higher than the putative theoretically accepted value ($\sim 30$) for the 2D
WC transition Manfra  _et al._ (2007). We believe that low-density 2D systems
manifest a complex interplay of disorder and correlation effects, which cannot
be easily described as a simple WC transition, even when the critical $r_{s}$
is as large as 30 (or higher), provided that the AIR condition predicts a
localization MIT also near this high critical $r_{s}$. This becomes obvious
once we take into account the fact that the mean free path for the
localization MIT according to the AIR criterion is $l\sim 1/k_{F}$, which, by
definition, is comparable to the WC lattice constant, implying that any such
WC necessarily manifests periodic coherence only over one (or at most, a few)
lattice constants. Such a strongly disordered WC could also be construed
equivalently as an Anderson insulator with short-range correlations induced by
interactions.

###### Acknowledgements.

The authors thank C.R. Dean, P. Kim, H. Park, A.Y. Joe, E. Demler, I.
Esterlis, J.D. Sau, Q. Shi, J. Pack, and J. Sung for useful discussions. This
work is supported by the Laboratory for Physical Sciences.

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Observational and theoretical aspects of Superspinars

Ramón Torres111E-mail<EMAIL_ADDRESS>

Dept. de Física, Universitat Politècnica de Catalunya, Barcelona, Spain.

###### Abstract

This article delves into the observational signatures and theoretical
underpinnings of rotating astrophysical objects, with a particular focus on
superspinars —exotic objects characterized by preventing the formation of
event horizons due to their high angular momentum.

While solutions within General Relativity (Kerr superspinars) predict such
objects, their classical forms harbor naked singularities, violate causality,
and exhibit problematic repulsive gravitational effects. These characteristics
render classical superspinars theoretically objectionable, leading to the
consideration of them as physically implausible. On the other hand, the
incompatibility between General Relativity and Quantum Mechanics suggests the
exploration of alternative models, particularly those in which Quantum Gravity
dominates the core, yielding _regular superspinars_.

This work demonstrates that regular superspinars avoid all the complications
associated with Kerr superspinars. From a phenomenological standpoint, it is
shown that the silhouettes of regular superspinars are markedly distinct from
those of black holes and classical Kerr superspinars. To substantiate these
differences, we perform a comprehensive analysis of inner null geodesics and
investigate the structure of the Planckian region within regular superspinars.
Our study reveals that only regular superspinars provide the potential for
distant observers to directly observe the extremely high curvature regions
within their interiors.

KEYWORDS: Rotating black holes; Kerr solutions; Black hole phenomenology;
Black hole shadows, Superspinars, Regular black holes; Quantum black holes.

##  1 Introduction

Recent observational advancements are enabling us to test our theoretical
predictions about rotating black holes. Notably, one confirmed prediction is
that observed black holes exhibit high angular momentum, consistent with our
theoretical and numerical models. This confirmation prompts us to consider the
existence of another theoretical prediction: superspinars. Superspinars are
objects with extremely high angular momentum, such that if their angular
momentum were below a certain threshold (which varies depending on the type of
superspinar), they would transition into black holes.

Archetypal superspinars emerge as solutions within the framework of General
Relativity, characterized by angular momentum ($a$) and mass ($M$) that
satisfy $a^{2}>M^{2}$. These solutions, known as hyperextreme Kerr solutions
or Kerr superspinars, face significant theoretical challenges. They are
associated with naked inner singularities, violations of the cosmic censorship
conjecture, causality issues, and problematic repulsive gravitational effects.
Consequently, they are often dismissed as physically unrealistic.

Conventionally, Kerr’s solution is not expected to be valid in the Planckian
regime. It is postulated that Quantum Gravity effects would modify such
objects, potentially eliminating their inner singularities and resulting in
regular superspinars. As one of the goals in this article, we aim to
demonstrate that, if regular superspinars indeed exist, they would be free
from the issues that plague Kerr superspinars.

Other significant issues include the origin and stability of superspinars. Two
mechanisms have been proposed by which superspinars might exist. First, they
could just be born as primordial superspinars. This possibility has been
predicted in both the classical framework (see, for instance, [1]) and Quantum
Gravity frameworks (see, for example, [2]). After their creation, collapsing
matter might reduce the angular momentum of the superspinars, potentially
transforming them into black holes. However, it has been demonstrated that
this possibility is limited to cases where sufficient surrounding matter
exists; otherwise, no event horizon would form, and they would remain as
superspinars [3].

It has been shown through various methods that it is impossible to overspin a
classical Kerr black hole to transform it into a classical Kerr superspinar.
The third law of black hole thermodynamics asserts that no physical process
can convert a black hole with $a^{2}<M^{2}$ into an extremal one with
$a^{2}=M^{2}$ in a finite number of steps [4]. Moreover, such a process would
violate the Weak Cosmic Censorship Hypothesis [5], which stipulates that the
singularity should remain concealed from distant observers.

However, within various approaches to quantum gravity, it may be possible to
bypass the third law of black hole thermodynamics by overspinning a quantum-
enhanced black hole. Some examples of this phenomenon can be found, for
instance, in [6, 7, 8]. Furthermore, in such cases, the Weak Cosmic Censorship
Hypothesis does not apply because the resulting superspinar would be
nonsingular (and would lie outside the framework of General Relativity). As a
result, this procedure could represent a second possible scenario for the
existence of regular superspinars.

In this article, we explore various phenomenological and theoretical aspects
of regular superspinars in contrast to classical superspinars, building upon
existing literature. Previous studies on the phenomenology of classical
superspinars can be found, for example, in [9, 10, 11, 12, 13], while some
instances of regular superspinars have been addressed in [14]. Our objective
is to enhance these investigations by conducting a meticulous analysis of the
interior of the superspinars. This will utilize recent advancements in
understanding the interiors of rotating astrophysical objects [15], a detailed
examination of the behaviours of null geodesics, and an assessment of the
expected shape of the Planckian regions in regular superspinars.

Through these efforts, we aim to address several key questions: Do regular
superspinars exhibit the same issues as classical superspinars? How do regular
superspinars appear to distant observers compared to rotating black holes or
Kerr superspinars? Given the absence of horizons in superspinars, is it true
that distant observers will necessarily see the high-curvature region inside
superspinars?

These questions will be addressed as follows: In Section 2, we examine the
general metric of superspinars with a focus on conditions for regularity, the
absence of extensions through their disk in the regular case, the global
spacetime structure and non-causality violations in the regular scenario.
Section 3 is dedicated to identifying the Planckian region of the regular
models, where curvature, effective densities and effective pressures reach
Planckian values. Section 4 provides a detailed analysis of null geodesics
that reach and/or traverse the disk or ring. The shape of superspinars as
perceived by distant observers is investigated in Section 5. This includes an
examination of the images produced by their unstable circular orbits and the
portrayal of the Planckian region as observed from a distance, both in
classical and regular cases. Finally, Section 6 summarizes the findings of
this article.

##  2 Metric of Regular Superspinars

Presently, the absence of a fully developed and reliable quantum theory of
gravity poses a significant challenge in accurately describing quantum black
holes and superspinars. Therefore, it is crucial to explore phenomenological
approaches to assess potential models of regular rotating objects and their
implications, including the potential for observable astrophysical
predictions.

There are numerous proposals for _regular_ rotating black holes spacetimes
with their corresponding metrics in the literature (see, for example, [16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]). These proposals
share many common properties, including a critical feature akin to Kerr’s
solution: if the angular momentum exceeds a specific threshold, the model
lacks an event horizon and therefore does not represent a regular rotating
_black hole_ , but rather what we term a regular superspinar.

The general metric for such rotating models was discovered by Gürses and
Gürsey [33] as a specific rotating instance of the algebraically special Kerr-
Schild metric:

$ds^{2}=(\eta_{\alpha\beta}+2Hk_{\alpha}k_{\beta})dx^{\alpha}dx^{\beta},$
(2.1)

where $\eta$ is the metric of Minkowski, $H$ is a scalar function and
$\vec{k}$ is a light-like vector both with respect to the spacetime metric and
to Minkowski’s metric. Specifically, in Kerr-Schild coordinates
$\\{\tilde{t},x,y,z\\}$ the Gürses-Gürsey metric (2.1) corresponds with the
choices

$H=\frac{\mathcal{M}(r)r^{3}}{r^{4}+a^{2}z^{2}}$

and

$k_{\alpha}dx^{\alpha}=-\frac{r(xdx+ydy)-a(xdy-
ydx)}{r^{2}+a^{2}}-\frac{zdz}{r}-d\tilde{t},$

where $r$ is a function of the Kerr-Schild coordinates implicitly defined by

$r^{4}-r^{2}(x^{2}+y^{2}+z^{2}-a^{2})-a^{2}z^{2}=0,$ (2.2)

$\mathcal{M}(r)$ is known as the mass function and the constant $a$ is a
rotation parameter.

This metric can be written in Boyer-Lindquist-like coordinates by using the
coordinate change defined by

$\displaystyle x+iy$ $\displaystyle=$
$\displaystyle(r+ia)\sin\theta\exp\left[i\int(d\phi+\frac{a}{\Delta}dr)\right]$
(2.3) $\displaystyle z$ $\displaystyle=$ $\displaystyle r\cos\theta$ (2.4)
$\displaystyle\tilde{t}$ $\displaystyle=$ $\displaystyle
t+\int\frac{r^{2}+a^{2}}{\Delta}dr-r.$ (2.5)

where now $\Delta=r^{2}-2\mathcal{M}(r)r+a^{2}$. The resulting metric takes
the form

$ds^{2}=-\frac{\Delta}{\Sigma}(dt-a\sin^{2}\theta
d\phi)^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}(adt-(r^{2}+a^{2})d\phi)^{2},$ (2.6)

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$. This metric reduces to the well-
known Kerr’s solution in B-L coordinates if $\mathcal{M}(r)=M$=constant. In
this case, there is a curvature singularity at $(r=0,\ \theta=\pi/2)$, as can
be shown by the divergence of its Kretschmann scalar.

Interestingly, for $a\neq 0$ and $\theta\neq\pi/2$, a surface defined by $t=$
constant and $r=0$ is free from singularities and has a metric

$ds_{2}^{2}=a^{2}\cos^{2}\theta\,d\theta^{2}+a^{2}\sin^{2}\theta\,d\phi^{2}=dx^{2}+dy^{2},$

where the coordinate change $x\equiv a\sin\theta\cos\phi$, $y\equiv
a\sin\theta\sin\phi$ explicitly shows that the surface is flat. This flat
surface corresponds to $x^{2}+y^{2}<a^{2}$, conventionally referred to as the
disk. The boundary $x^{2}+y^{2}=a^{2}$ is termed the ring.

Note also the symmetry $\\{a,\phi\\}\leftrightarrow\\{-a,-\phi\\}$ in this
metric. This allows us, for the sake of simplicity, to assume $a>0$ in this
article, since the negative case is covered by using the trivial coordinate
change $\phi\rightarrow-\phi$.

In order to analyze the general properties of the superspinar we will use the
following null tetrad-frame:

$\displaystyle\mathbf{l}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\left((r^{2}+a^{2})\frac{\partial}{\partial
t}+\Delta\frac{\partial}{\partial r}+a\frac{\partial}{\partial\phi}\right),$
$\displaystyle\mathbf{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2\Sigma}\left((r^{2}+a^{2})\frac{\partial}{\partial
t}-\Delta\frac{\partial}{\partial r}+a\frac{\partial}{\partial\phi}\right),$
$\displaystyle\mathbf{m}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\varrho}\left(ia\sin\theta\frac{\partial}{\partial
t}+\frac{\partial}{\partial\theta}+i\csc\theta\frac{\partial}{\partial\phi}\right),$
$\displaystyle\mathbf{\bar{m}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\bar{\varrho}}\left(-ia\sin\theta\frac{\partial}{\partial
t}+\frac{\partial}{\partial\theta}-i\csc\theta\frac{\partial}{\partial\phi}\right),$

where $\varrho\equiv r+ia\cos\theta$, $\bar{\varrho}\equiv r-ia\cos\theta$ and
the tetrad is normalized as follows
$\mathbf{l}^{2}=\mathbf{n}^{2}=\mathbf{m}^{2}=\mathbf{\bar{m}}^{2}=0$ and
$\mathbf{l}\cdot\mathbf{n}=-1=-\mathbf{m}\cdot\mathbf{\bar{m}}$. The metric
(2.6) with $\mathcal{M}(r)\neq 0$ is Petrov type D and the two double
principal null directions are $\mathbf{l}$ and $\mathbf{n}$ [34].

We can also define a real orthonormal basis
$\\{\mathbf{t},\mathbf{x},\mathbf{y},\mathbf{z}\\}$ formed by a timelike
vector $\mathbf{t}\equiv(\mathbf{l}+\mathbf{n})/\sqrt{2}$ and three spacelike
vectors: $\mathbf{z}\equiv(\mathbf{l}-\mathbf{n})/\sqrt{2}$,
$\mathbf{x}=(\mathbf{m}+\bar{\mathbf{m}})/\sqrt{2}$ and
$\mathbf{y}=(\mathbf{m}-\bar{\mathbf{m}})i/\sqrt{2}$. Then, $\mathbf{t}$ and
$\mathbf{z}$ are two eigenvectors of the Ricci tensor with eigenvalue [34]

$\lambda_{1}=\frac{2a^{2}\cos^{2}{\theta}\mathcal{M}^{\prime}+r\Sigma\mathcal{M}^{\prime\prime}}{\Sigma^{2}}.$
(2.7)

$\mathbf{x}$ and $\mathbf{y}$ are two eigenvectors of the Ricci tensor with
eigenvalue

$\lambda_{2}=\frac{2r^{2}\mathcal{M}^{\prime}}{\Sigma^{2}}.$ (2.8)

The metric has a coordinate singularity in those cases where there are values
of $r$ such that $\Delta=0$, which can be removed by a coordinate change (see
[35]). At $\Delta=0$ the hypersurface $r=$constant becomes light-like and no
observer can remain at the specific value for $r$, thus the hypersurface is
called a null horizon.

The simplest example is Kerr’s solution: if $M^{2}>a^{2}$ there are two roots:
$r^{Kerr}_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$, where the null horizon $r^{Kerr}_{+}$
is an event horizon, while the null horizon $r^{Kerr}_{-}$ is a Cauchy
horizon. The limiting case $M^{2}=a^{2}$ has a degenerate null horizon and the
spacetime is called the extreme Kerr black hole. If $M^{2}<a^{2}$ there are no
roots for $\Delta=0$ and, therefore, no horizons. This is the so called
hyperextreme case that provides the model of a classical superspinar.

###  2.1 Regularity of Superspinars

For a finite $\mathcal{M}(r)$, scalar curvature singularities may exist only
where $\Sigma=0$ or, in other words, in $(r=0,\theta=\pi/2)$ (as previously
noted for Kerr’s solution). Nevertheless, while $\Sigma=0$ is necessary, it is
not a sufficient condition to guarantee the existence of curvature
singularities. Conversely, a necessary and sufficient condition for the
_absence_ of scalar curvature singularities is provided by the following

###### Theorem 1

[34] Assuming a metric of type (2.6) possessing a $C^{3}$ function
$\mathcal{M}(r)$, all its second order curvature invariants will be finite at
$(r=0,\theta=\pi/2)$ if, and only if,

$\mathcal{M}(0)=\mathcal{M}^{\prime}(0)=\mathcal{M}^{\prime\prime}(0)=0.$
(2.9)

Note that any function admitting an expansion around $r=0$ in the form
$\mathcal{M}(r)=cr^{n}+O(r^{n+1})$, with $c$ a constant and $n\geq 3$, will
satisfy the conditions. (This will be written here also as
$\mathcal{M}(r)=O(r^{n})$ or $\mathcal{M}(r)\sim r^{n}$). Consider the
examples of the mass functions provided by Bonanno and Reuter (BR)
[36][26][27] and by Hayward [37] (that will be useful later to illustrate
different properties of regular superspinars)

$\mathcal{M}(r)_{BR}=\frac{mr^{3}}{r^{3}+l^{2}(r+L)}\ \ ;\ \
\mathcal{M}(r)_{Hayward}=\frac{mr^{3}}{r^{3}+l^{3}},$ (2.10)

where $l$ and $L$ are constants. These mass functions admit the following
expansions around $r=0$:

$\mathcal{M}(r)_{BR}=\frac{m}{l^{2}L}r^{3}+O(r^{4})\ \ ;\ \
\mathcal{M}(r)_{Hayward}=\frac{m}{l^{3}}r^{3}+O(r^{6}).$

In the specific BR case, computations in the Quantum Einstein Gravity
framework lead the authors to specific values for the constants with $l\sim
l_{Planck}$ and $L\sim$ Schwarzschild‘s radius. In this way, the deviations
from the classical case will be relevant for $r$ of the order of some Planck
lengths. This is not specific of this example, but an usual result and an
usual assumption in case of heuristic models (such as Hayward’s model, $l\sim
l_{Planck}$).

The reader can consult other mass functions in the literature with the
following approaches: In the framework of Conformal Gravity [28], in the
framework of Shape Dynamics in [29], inspired by Supergravity in [30], by Loop
Quantum Gravity in [31] and by non-commutative gravity in [32].

###  2.2 Absence of extensions through the disk

The curves that reach the disk of a superspinar are reaching a regular point.
In Kerr’s case, and due to a lack of differentiability at the disk, an
analytic extension of the spacetime has to be made through $r=0$ in order to
continue the curves. This is accomplished by letting the coordinate $r$ to
take negative values [38]. However, this analytic continuation also introduces
some of the issues associated with Kerr’s solutions: The $r<0$ extended
spacetime can be seen as a negative mass spacetime and causality violations
manifest in the extended spacetime [39]. In contrast, the following result has
been demonstrated for the _regular_ case [15] (summarized in the appendix):

Regular rotating models with metric 2.6 (without an extension through $r=0$)
have a high degree of differentiability at the disk (at least $C^{n}$, with
$n\geq 3$).

In this way, regular rotating objects do not have differentiability problems
at the disk and an extension through $r=0$ with $r<0$ is not needed. As a
consequence, regular superspinars with non-negative mass functions will not be
able to act repulsively.

###  2.3 Causality

In general, it is reasonable to require a time-orientable spacetime to be free
of closed causal curves, as their existence would lead to logical paradoxes. A
spacetime devoid of closed causal curves is referred to as causal [38].
Furthermore, if no closed causal curves appear even under any small
perturbation of the metric, the spacetime is termed stably causal. As
mentioned, it is well-known that the standard analytical extension of the Kerr
metric is non-causal. However, for regular rotating objects it can be shown
the following

###### Theorem 2

[20][35] If $r\mathcal{M}(r)\geq 0$ for all $r$, then the rotating model with
metric (2.1) [or (2.6)] will be stably causal.

Note that, as a consequence, for a regular superspinar (unextended through
$r=0$), a non-negative mass function suffices to guarantee a stably causal
spacetime.

###  2.4 No horizon condition and global structure of the spacetime

A superspinar is characterized by the absence of horizons (for all $r$), which
necessitates the condition

$\Delta=r^{2}-2\mathcal{M}(r)r+a^{2}>0,$ (2.11)

implying that $r$ is a spacelike coordinate throughout the entire spacetime.
This simplifies obtaining the global structure of the spacetime starting from
BL coordinates. It is important to note that the causal structure of a Kerr
superspinar includes an extended region with negative values for $r$. In
contrast, for a regular superspinar, there is no need for an extension through
$r=0$. The Penrose diagrams for both cases are shown in Figure 1.

Figure 1:  _Left_ : Penrose diagram for Kerr’s superspinar. The spacetime is
extended through $r=0$ to an asymptotically flat region with negative values
for $r$. (Note that the diagram is valid for $\theta\neq\pi/2$. The diagram
with $\theta=\pi/2$ will require to draw the ring singularity). _Right_ :
Penrose diagram for a regular superspinar. An extension through $r=0$ is not
required. (Note that, since there are not singularities, the diagram is valid
for all $\theta$).

As suggested by the figure, the absence of an event horizon in superspinars is
significant, as it seems to imply that a distant observer could receive
information from the inner high-curvature regions near $r=0$. In principle,
this could be used to observationally test various approaches to Quantum
Gravity. This possibility will be explored in Section 5.

##  3 Effective energy-momentum and Planckian region

Even if here we are not confined to General Relativity, we can consider the
existence of an effective energy-momentum tensor defined through

$T_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}\mathcal{R}g_{\mu\nu}.$

One can explicit $\mathbf{T}$ for a superspinar as [34]

$T_{\mu\nu}=-\lambda_{2}(-t_{\mu}t_{\nu}+z_{\mu}z_{\nu})-\lambda_{1}(x_{\mu}x_{\nu}+y_{\mu}y_{\nu}).$

The (effective) density being $\mu=\lambda_{2}$ and the (effective) pressures
being $p_{x}=p_{y}=-\lambda_{1}$ and $p_{z}=-\lambda_{2}$ (see (2.7) and
(2.8)).

It seems natural that any deviations of a realistic regular superspinar from
its classical counterpart should be concentrated in a region around the ring.
To rigorously analyze this intuition, let us study the ring in greater depth
by introducing a set of coordinates better adapted to its structure. These
coordinates will help us examine the directional behavior of physical
quantities around the ring: the toroidal coordinates $\\{\rho,\varphi,\psi\\}$
(see figure 2). These coordinates are related to the Kerr-Schild coordinates
through

$\displaystyle x$ $\displaystyle=$ $\displaystyle(a+\rho\cos\psi)\cos\varphi,$
$\displaystyle y$ $\displaystyle=$ $\displaystyle(a+\rho\cos\psi)\sin\varphi,$
(3.1) $\displaystyle z$ $\displaystyle=$ $\displaystyle\rho\sin\psi.$

Note that in these coordinates the ring is defined by $\rho=0$.

Figure 2:  A plot of the behaviour of the toroidal coordinates with the
(regular) ring ($\rho=0$) highlighted in black.

From the relationship between K-S and B-L coordinates, it follows that the
coordinates $\\{\rho,\psi\\}$ are related to the Boyer-Lindquist coordinates
$\\{r,\theta\\}$ through

$\displaystyle r\cos\theta$ $\displaystyle=$ $\displaystyle\rho\sin\psi,$
(3.2) $\displaystyle(r^{2}+a^{2})\sin^{2}\theta$ $\displaystyle=$
$\displaystyle(a+\rho\cos\psi)^{2}.$

This implies that around the ring

$\displaystyle r$ $\displaystyle\simeq$
$\displaystyle\sqrt{a(1+\cos\psi)\rho},$ (3.3) $\displaystyle\cos\theta$
$\displaystyle\simeq$
$\displaystyle\frac{\rho^{1/2}\sin\psi}{\sqrt{a(1+\cos\psi)}}.$

Let us now analyze the dependence of the effective density and pressures on
$\rho$ in a region around the ring for a mass function $\mathcal{M}(r)\sim
r^{n},\ (n\geq 3)$ around $r=0$. With the help of (3.3):

$\displaystyle\mu=-p_{z}$ $\displaystyle=$
$\displaystyle\lambda_{2}=\frac{2r^{2}\mathcal{M}^{\prime}}{\Sigma^{2}}\sim\rho^{\frac{n-3}{2}}$
$\displaystyle p_{x}=p_{y}$ $\displaystyle=$
$\displaystyle-\lambda_{1}=-\frac{2a^{2}\cos^{2}{\theta}\mathcal{M}^{\prime}+r\Sigma\mathcal{M}^{\prime\prime}}{\Sigma^{2}}\sim\rho^{\frac{n-3}{2}}.$

In this way, the effective density and pressures vanish at the ring for $n>3$.
On the other hand, for $n=3$ the densities and pressures are finite functions
of the direction of approach to the ring $\psi$. Only in this case the density
and pressures are not continuous at the ring.

Specifically, for $\mathcal{M}(r)\equiv m_{3}r^{3}+O(r^{4})$ with $m_{3}>0$ a
constant,

$\displaystyle\mu=-p_{z}$ $\displaystyle=$
$\displaystyle\lambda_{2}=6m_{3}\cos^{4}(\psi/2),$ (3.4) $\displaystyle
p_{x}=p_{y}$ $\displaystyle=$
$\displaystyle-\lambda_{1}=3m_{3}\cos^{2}(\psi/2)(\cos\psi-3).$

For both the effective density and pressures, the value zero is reached in the
direction of the disk $\psi=\pi/2$ and the maximum absolute value is reached
in the opposite direction $\psi=0$. Polar plots of the effective density and
pressures for this particular case are shown in figure 3.

Figure 3:  Polar plot of $\tilde{\mu}\equiv\mu/(6m_{3})$ and (minus)
$\tilde{p}_{x}\equiv p_{x}/(3m_{3})$ around the ring for the case
$\mathcal{M}(r)\sim r^{3}$ around $r=0$.

Note that the disk (and the ring when approached with $\psi=\pi$) can be
described as minkowskian ($\mu=p_{x}=p_{y}=p_{z}=0$).

We call planckian region the region in which the curvature, effective density
and effective pressures reach their corresponding high Planckian values. In
the literature, this region is sometimes assumed to be an ellipsoid containing
the disk [10]. Here, we will locate it using the assumptions in subsection
2.1. To accomplish this, it suffices to compute the values of the effective
density and pressures from the values of $\lambda_{1}$ and $\lambda_{2}$ in BL
coordinates. (See figure 4). From this, we can deduce that the planckian
region is not ellipsoidal. Moreover, it is not even isotropic around the ring,
when projected in a plane perpendicular to the x-y plane and containing the
z-axis. In other words, the planckian region does not have the shape of a
torus, but rather what we will call a pseudotorus. The planckian pseudotorus
has a major radius corresponding to the ring of size $|a|$, and it has a mean
minor radius of the order of a few $l_{P}$ [with the smallest radius (0) in
the direction of the disk ($\psi=\pi$) and the largest radius in the direction
of the equatorial plane ($\psi=0$)]. As a consequence, the ratio between the
major and the mean minor radius will satisfy

$\frac{|a|}{l_{p}}\gtrsim\frac{M}{M_{P}},$

where $M_{P}$ is Planck’s mass. Thus, the major radius of the pseudotorus will
be immensely large compared to the mean minor radius, except when dealing with
superspinars of Planckian mass. For example, a superspinar with a mass twice
that of our Sun will have a major radius greater than 3 km, which is
approximately $10^{38}$ times larger than the mean minor radius of the
Planckian region. This implies that, under these assumptions, an observer
could pass through the (Minkowskian) disk of an astrophysical superspinar
without ever traversing a true planckian region.

Figure 4:  An example of the region with planckian densities
($\mu\sim\mu_{Planck}$) in BL coordinates (dark region). Note that the region
has a maximum minor radius of the order of a few $l_{P}$ and that it is
concentrated around $\theta=\pi/2$ (i.e., the equatorial plane). In this
example it has not been necessary to plot the region with $0\leq\theta\leq
1.53$ since it does not have planckian densities. We have specifically used
the BR model. However, the results are qualitatively similar for all regular
models under the assumptions in subsection 2.1.

##  4 Null geodesics reaching $r=0$

The trajectory of the photons in the spacetime of a regular superspinar are
described by an action $S=S(x^{\alpha})$. The momentum of the photons is

$p_{\mu}\equiv\frac{\partial S}{\partial x^{\mu}}$

and satisfy

$g^{\alpha\beta}p_{\alpha}p_{\beta}=0.$ (4.1)

The stationary and axisymmetric of the spacetime described by the metric (2.6)
imply two conserved quantities in the trajectory of the photon: the energy
$E\equiv-p_{t}$ and the angular momentum $L\equiv p_{\phi}$. If there is a
separable solution for $S$, by using the definition of the momentum, we could
rewrite it as

$S=-Et+L\phi+S_{r}+S_{\theta},$

where we have introduced the new functions $S_{r}=S_{r}(r)$ and
$S_{\theta}=S_{\theta}(\theta)$. In this way, (4.1) can now be written as

$-\Delta\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{[(r^{2}+a^{2})E-aL]^{2}}{\Delta}=\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+\frac{(L-aE\sin^{2}\theta)^{2}}{\sin^{2}\theta}.$
(4.2)

In this equation, the left-hand side depends only on $r$, while the right-hand
side depends only on $\theta$. In this way, they define a constant which we
will denote by

$K=\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+\frac{(L-aE\sin^{2}\theta)^{2}}{\sin^{2}\theta}.$
(4.3)

From $dx^{\mu}/d\lambda=p^{\mu}=g^{\mu\nu}p_{\nu}$ and using (4.2) and (4.3)
one gets (see, for instance, [40])

$\displaystyle\Sigma\frac{dt}{d\lambda}$ $\displaystyle=$
$\displaystyle-a(aE\sin^{2}\theta-L)+\frac{(r^{2}+a^{2})P(r)}{\Delta},$ (4.4)
$\displaystyle\Sigma\frac{dr}{d\lambda}$ $\displaystyle=$
$\displaystyle\sigma_{r}\sqrt{R(r)},$ (4.5)
$\displaystyle\Sigma\frac{d\theta}{d\lambda}$ $\displaystyle=$
$\displaystyle\sigma_{\theta}\sqrt{\Theta(\theta)},$ (4.6)
$\displaystyle\Sigma\frac{d\phi}{d\lambda}$ $\displaystyle=$
$\displaystyle-\left(aE-\frac{L}{\sin^{2}\theta}\right)+\frac{aP(r)}{\Delta}$
(4.7)

where

$\displaystyle P(r)$ $\displaystyle\equiv$ $\displaystyle E(r^{2}+a^{2})-aL,$
(4.8) $\displaystyle R(r)$ $\displaystyle\equiv$ $\displaystyle
P(r)^{2}-\Delta[(L-aE)^{2}+\mathcal{Q}],$ (4.9) $\displaystyle\Theta(\theta)$
$\displaystyle\equiv$
$\displaystyle\mathcal{Q}+\cos^{2}\theta\left(a^{2}E^{2}-\frac{L^{2}}{\sin^{2}\theta}\right).$
(4.10)

with $\mathcal{Q}\equiv K-(L-aE)^{2}$ the Carter constant and $\sigma_{r}$ and
$\sigma_{\theta}$ can take the values $\pm 1$ independently.

$R$ can also be rewritten as

$R/E^{2}=r^{4}+(a^{2}-\xi^{2}-\eta)r^{2}+2\mathcal{M}(r)[(\xi-a)^{2}+\eta]r-a^{2}\eta,$
(4.11)

where $\eta\equiv\mathcal{Q}/E^{2}$ and $\xi\equiv L/E$.

For the null geodesics traversing the disk, $r=0$ must be reached for some
value of their affine parameter. For a finite mass function, $R$ will take the
value $R(r=0)=-a^{2}E^{2}\eta$. According to (4.5), $R\geq 0$, what implies as
condition for the null geodesics reaching the disk or the ring: $\eta\leq 0$.

####  4.0.1 The case $\eta<0$

Due to the symmetry, we will consider here that null geodesics approaching
$r=0$ from $0\leq\theta\leq\pi/2$.

The function $\Theta$ can be written as

$\frac{\Theta}{E^{2}}=\eta+(a-\xi)^{2}-\left(a\sin\theta-\frac{\xi}{\sin\theta}\right)^{2}$

which implies that a necessary condition for the existence of the solutions is

$|\eta|\leq(a-\xi)^{2}.$ (4.12)

$\Theta(\theta=\pi/2)=E^{2}\eta<0$ and this is not possible according to
(4.6). Thus, there will always be a maximum value for $\theta=\theta_{max}$
which cannot be traversed. As a consequence, these geodesics can not reach the
ring ($\theta=\pi/2$).

Two possibilities can be considered according to the value of $\xi$:

  * •

If $\xi\neq 0$, $\Theta$ is necessarily negative around $\theta=0$. Therefore
the null geodesics with $\xi\neq 0$ will not cross the disk near the z-axis as
there will also be a minimum value for $\theta=\theta_{min}>0$. See figure 5.
The limits allowed for $\theta$ will be the solutions of $\Theta=0$ that,
writing $\mu\equiv\cos\theta$, are simply the solutions of

$\eta-(\xi^{2}+\eta-a^{2})\mu^{2}-a^{2}\mu^{4}=0.$

The existence of the solutions (non-negative discriminant and
$0\leq\mu^{2}<1$) imposes that the geodesics crossing the disk must also
satisfy $|\eta|\leq a^{2}$ and $|a|-\sqrt{|\eta|}\geq|\xi|$.

  * •

If $\xi=0$, according to (4.12) the geodesic should satisfy $|\eta|\leq
a^{2}$. The requirement that $\Theta\geq 0$ imposes a limit (as discussed
above) that in this case takes the explicit form
$\cos^{2}\theta_{max}=|\eta|/a^{2}$. However, it does not impose a minimum
value for $\theta$, implying that in this case the geodesics can cross the
disk near or at the $z$ axis.

Figure 5:  The disk of a superspinar bounded by the ring (in the plane $x-y$
of K-S coordinates) with the allowed regions (green) for the null geodesics
crossing it in the case $\xi\neq 0$ and $\xi=0$.

The explicit 3-dimensional image of a geodesic crossing the disk in Kerr-
Schild coordinates with a parameter $\xi\neq 0$ is illustrated in figure 6,
while the case of a geodesic crossing the disk with a parameter $\xi=0$ is
illustrated in figure 7. In order to obtain these figures, the system of
differential equations (4.5)+(4.6)+(4.7) has been solved with some initial
conditions, then the coordinate change (2.3) has been used to get
$\\{x(\lambda),y(\lambda),z(\lambda)\\}$. These figures emphasize the fact
that, contrary to Kerr’s solutions, in the regular case the null geodesics do
not cross to a new spacetime with negative $r$. Along their trajectory, first
$r$ decreases until reaching the disk ($r=0$), and then $r$ increases as the
null geodesic moves away from the disk.

Figure 6:  A light-like geodesic (red) with non-null angular momentum
($\xi\neq 0$) crossing the disk of a regular superspinar from the region with
$z>0$ to the region with $z<0$. The planckian region is represented by a
pseudotorus. As required in order to cross the disk, Carter’s constant is
negative $\eta<0$ and the parameters have been chosen among the required
conditions for these geodesics. Specifically, $\eta=-10$, $\xi=1$ and $E=1$.
On the other hand, the BR superspinar is defined by the parameters $M=100$,
$a=100$, $l=1$ and $L=100$ that, as required for superspinars, avoid the
formation of horizons since $\Delta\neq 0$. Once the geodesic crosses the
disk, $r$ monotonically increases along it. Figure 7:  A light-like geodesic
(blue) with null angular momentum ($\xi=0$) crossing the disk of a regular
superspinar from the region with $z>0$ to the region with $z<0$. The
parameters have been chosen among the required conditions for these geodesics.
Specifically, $\eta=-10$, $\xi=0$ and $E=1$. On the other hand, the
superspinar is the same as in figure 6. As can be seen, the main difference
with the previous figure is that the region around the center of the disk is
accessible by the null geodesic with $\xi=0$. In this specific case it crosses
the disk at $(r=0,\theta=0)$.

####  4.0.2 The case $\eta=0$

For null geodesics not restricted to the equatorial plane, the fact that
$\Theta\geq 0$ imposes

$\xi^{2}\leq a^{2}$

and force them to be in the range
$\theta_{min}\leq\theta\leq\pi-\theta_{min}$, where
$\theta_{min}=\arcsin|\xi/a|$.

As noted previously, only null geodesics with $\eta=0$ have the potential to
reach the ring. To examine this possibility, observe that
$\Theta(\theta=\pi/2)=0$. The system (4.5)+(4.6) can be rewritten by defining
a new parameter $s$ such that $d\lambda=\Sigma(s)ds$ as

$\displaystyle\frac{dr}{ds}$ $\displaystyle=$
$\displaystyle\sigma_{r}\sqrt{R(r(s))},$ $\displaystyle\frac{d\theta}{ds}$
$\displaystyle=$ $\displaystyle\sigma_{\theta}\sqrt{\Theta(\theta(s))}.$

Therefore, $(r=0,\theta=\pi/2)$ is an isolated critical point of this system.
The behaviour of the critical point is very different in Kerr’s superspinars
or regular superspinars:

  * •

In Kerr’s superspinars as the isolated critical point is approached
$\partial\sqrt{R}/\partial r\sim r^{-1/2}$. Thus, this diverges at the ring
and the system can not be linearized. Practically all null geodesics
approaching $r=0$ reach the disk ($\theta<\pi/2$) (see figure 8), where they
find a turning point ($dr/d\lambda=dr/ds=0$) and they cannot traverse the disk
towards negative $r$, according to the condition $R>0$ and the fact that,
around $r=0$, $R\simeq 2Mr(a-\xi)^{2}$. This leads us to the important result
that only null geodesics approaching the disk from the equator
($\theta(\lambda)=\pi/2=$constant) can reach or emerge from the ring
singularity. As a consequence, the ring singularity is relatively well
concealed: only observers in the equatorial plane can observe it.

Figure 8:  A numerical integration of the system (4.5)+(4.6) for Kerr’s case
($\mathcal{M}(r)=$constant) and null geodesics satisfying $\eta=0$ showing
that only null geodesics approaching the disk from the equator
($\theta(\lambda)=\pi/2=$constant) are able to reach the ring. The other
geodesics reach the disk ($r=0,\theta<\pi/2$), where they find a turning
point. Conversely, null geodesics from the ring of the Kerr superspinar will
only reach faraway observers in the equatorial plane. In this particular plot
$a/M=2$ and $\xi=0$, but other parameters provide the same _qualitative_
figure.

  * •

For regular superspinars one has, at the isolated critical point,
$\partial\sqrt{R}/\partial
r=\partial\sqrt{\Theta}/\partial\theta=\sqrt{a^{2}-\xi^{2}}\ (\in\Re)$. Thus,
the system in linearizable. In this way, it is easy to check that all null
geodesics coming out from $r=0$ do so from the critical point, i.e., the ring
(see figure 9). Thus, contrary to Kerr’s case, there are null geodesics from
the ring to distant observers outside the equatorial plane. We will delve into
its observational consequences in the next section.

Figure 9:  A numerical integration of the system (4.5)+(4.6) for a regular
superspinar showing that all null geodesics with $\eta=0$ approaching $r=0$ go
to the ring (and, conversely, null geodesics from the ring will reach faraway
observers even if they are outside the equatorial plane). The parameters
defining the BR superspinar have been chosen $l/M=10^{-2},L/M=1$ and $a/M=2$.
The null geodesics have $\xi=0$, but other values provide the same
_qualitative_ figure.

##  5 The silhouette of superspinars

Despite the similarities between black holes and superspinars, their images,
as seen by faraway observers, are very different. For black holes, a defining
characteristic is the existence of an event horizon. To a distant observer,
photons coming from the background of a black hole and passing through the
unstable circular orbits around it will eventually cross the event horizon
and, therefore, will not be able to reach the distant observer. This
phenomenon casts a relatively large _shadow_ for distant observers. There are
numerous studies on shadows of rotating black holes (RBH) in different
frameworks. See, for example, [41, 42, 43, 44, 45, 46, 47, 48, 49]. In order
to illustrate the typical results obtained, Figure 10 shows the shadows of two
regular RBHs compared to the shadow of a Kerr RBH222The computations we have
carried out to plot this figure are the same as the ones we are going to use
to obtain the silhouette of superspinars..

Figure 10:  Shadows of rotating black holes (RBH) are shown for three
different cases. We have fixed the rotation parameter to $a/M=0.8$ and the
inclination angle to $\theta_{i}=\pi/2$ (i.e., it is viewed from the
equatorial plane). The dashed green boundary of the shadow corresponds to the
Kerr black hole case. The blue boundary of the shadow corresponds to Hayward’s
RBH with $l/M=0.5$ [15]. The red boundary of tbe shadow corresponds to
Hayward’s RBH with $l/M=0.6$. Note that for smaller values of $l/M$, the
deviations from Kerr’s shadow are smaller. As expected, when $l$ is on the
order of the Planck mass, the deviation of the shadow from the classical case
is negligible.

Superspinars do not have an event horizon and, therefore, there are not
photons crossing it and causing a shadow. Nevertheless, there are related
phenomena associated with superspinars due to the existence of unstable
circular orbits and the potential visibility of their ring, which will produce
a characteristic _silhouette_.

Let us start with the unstable circular orbits. Some null geodesics coming
from the background and passing near an unstable circular orbit will be unable
to reach a distant observer at the other side. Since the retrograde photon
orbits are less affected by the frame dragging of the superspinar, this
phenomenon can cause an open _arc_ structure that opens to the left for $a>0$
(or to the right for $a<0$). Equation (4.5) implies that there would be
unstable circular orbits at a certain $r=r_{0}$ whenever
$R(r_{0})=R^{\prime}(r_{0})=0$ and $R^{\prime\prime}(r_{0})>0$. To exploit
this, one has to follow the expected steps: Consider the equation $R(r_{0})=0$
from eq.(4.11) and take the derivative of $R$ with respect to $r$, what
provides

$R^{\prime}/E^{2}=4r^{3}+2(a^{2}-\xi^{2}-\eta)r+2\mathcal{M}(r)[(\xi-a)^{2}+\eta]f(r),$

where

$f(r)\equiv 1+\frac{r\mathcal{M}^{\prime}}{\mathcal{M}}.$

Using the conditions for an unstable orbit one gets the quadratic equation
with respect to $\xi$:

$\displaystyle a^{2}$
$\displaystyle(r_{0}-f_{0}\mathcal{M}_{0})\xi^{2}-2a\mathcal{M}_{0}[(2-f_{0})r_{0}^{2}-f_{0}a^{2}]\xi-
r_{0}^{5}+$
$\displaystyle+(4-f_{0})\mathcal{M}_{0}r_{0}^{4}-2a^{2}r_{0}^{3}+2a^{2}\mathcal{M}_{0}(2-f_{0})r_{0}^{2}-a^{4}r_{0}-a^{4}\mathcal{M}_{0}f_{0}=0,$

where $\mathcal{M}_{0}\equiv\mathcal{M}(r_{0})$ and $f_{0}\equiv f(r_{0})$. In
order to obtain the unstable orbit, we must choose the solution

$\xi_{-}\equiv\frac{4\mathcal{M}_{0}r_{0}^{2}-(r_{0}+f_{0}\mathcal{M}_{0})(r_{0}^{2}+a^{2})}{a(r_{0}-f_{0}\mathcal{M}_{0})}.$

that implies

$\eta=\eta_{-}\equiv\frac{r_{0}^{3}[4(2-f_{0})a^{2}\mathcal{M}_{0}-r_{0}[r_{0}-(4-f_{0})\mathcal{M}_{0}]^{2}]}{a^{2}(r_{0}-f_{0}\mathcal{M}_{0})^{2}}.$

Consider now an observer at a large distance from the rotating object. The
observer views the object with an inclination $\theta_{i}$ relative to the
$z$-axis of the superspinar. The curve formed on the sky by the unstable
orbits can be expressed in celestial coordinates $\alpha$ and $\beta$, using
$r_{0}$ as its parameter, as follows [40]

$\alpha=\frac{-\xi_{-}}{\sin\theta_{i}}\hskip 14.22636pt;\hskip
14.22636pt\beta=\pm\sqrt{\eta_{-}+(a-\xi_{-})^{2}-\left(a\sin\theta_{i}-\frac{\xi_{-}}{\sin\theta_{i}}\right)^{2}}.$
(5.1)

###  5.1 Observing the unstable circular orbits: Classical vs regular case

In the case of a classical superspinar there are three defining observational
characteristics [11][9]:

  1. 1.

Using the eqs.(5.1) one finds that the unstable photon orbits with positive
$r_{0}$ form an (open) _arc_ in the sky.

  2. 2.

Using the eqs.(5.1) one finds that the unstable photon orbits with negative
$r_{0}$ form a closed curve with a dark interior, the _dark spot_ , which is
due to the geodesics crossing the disk and never returning to the region with
positive $r$.

  3. 3.

The geodesics with $\theta$ constant ($\neq\pi/2$) along them (which are
principal null geodesics) traverse the disk and never return to the region
with positive $r$. In this way, observers with an inclination $\theta_{i}$ do
no receive the light of the geodesic with $\theta=\pi-\theta_{i}$. This _dark
point_ is rather irrelevant from an observational point of view, even more if
one takes into account that it would be hidden by the dark spot, except if the
observer is precisely situated in the equatorial plane of the superspinar.

Examples of silhouettes of Kerr superspinars are shown in Figure 11 with the
same relationships $a/M$ as in the examples in [9].

Figure 11:  The silhouette of a Kerr supespinar characterized by an arc (blue)
(due to the existence of unstable photon orbits of constant $r$ ($>0$)) and a
dark spot (grey) (due to the unstable photon orbits with negative $r$). The
parameters defining the superspinar have been chosen $a/M=1+10^{-5}$ (left)
$a/M=2$ (right). The inclination angle of observation in both cases is
$\theta_{i}=\pi/3$.

In the case of a regular superspinar there is not a region with negative $r$.
In this way, there cannot be a dark spot or a dark point. In the case of
regular superspinars there are two defining observational characteristics:

  1. 1.

Using the eqs.(5.1) one finds that the unstable photon orbits with constant
$r$ form an _arc_. Under the assumptions in subsection 2.1 this arc will be
open, as in the classical case. (However, If we skip those assumptions and let
the planckian region to have a size $l\sim M$ this arc could also be closed.
See figure 13).

  2. 2.

There are no null geodesics disappearing at a ring singularity, instead there
is a regular ring. As a consequence, the planckian region around the ring
could emit information that would reach distant observers. This possibility is
specifically treated in the next subsection.

Examples of silhouettes of regular superspinars are shown in figure 12 with
the same relationships $a/M$ as in the previous classical cases.

Figure 12:  The silhouette of a regular superspinar is characterized by an
_arc_ (blue) (due to the existence of unstable photon orbits of constant $r\
(>0)$) and a ellipse (red) (due to the existence of a planckian region around
the superspinar _ring_). The parameters defining the BR superspinar have been
chosen $l/M=10^{-2},L/M=1$ and $a/M=1+10^{-5}$ (left) $a/M=2$ (right). The
inclination angle of observation in both cases is $\theta_{i}=\pi/3$. Note
that the arcs are similar to the classical case if $l/M$ is small. Figure 13:
The silhouette of a regular supespinar in which quantum effects reach a radius
of order $M$, and $a/M$ is small enough, has a closed arc (blue) (due to the
existence of unstable photon orbits of constant $r$). Practically the whole
region inside the closed curve would be dominated by quantum gravity effects.
In this figure the ellipse highlighted in red is now just the position of the
ring. The specific parameters defining the BR superspinar have been chosen
$l/M=0.7,L/M=1$ and $a/M=0.2$. The inclination angle of observation is
$\theta_{i}=\pi/3$.

###  5.2 Observing the ring

As previously shown, it suffices to identify the ring of the superspinar in
order to locate the planckian pseudotorus. Only null geodesics with $\eta=0$
can reach the ring, but we have seen that they do so differently in Kerr
superspinars and in regular superspinars.

Let us first assume that the distant observer that wants to identify the ring
is situated outside the equatorial plane of the superspinar. In this case, we
have seen that the condition $\Theta\geq 0$ imposes on $\xi$ the condition
$\xi^{2}\leq a^{2}$.

  * •

For Kerr’s superspinars only null geodesics on the equatorial plane can reach
or exit the ring. Therefore, the observers outside the equatorial plane are
not able to see the naked singularity at the ring.

  * •

For regular superspinars the null geodesics reaching or leaving the ring
(i.e., null geodesics with $\eta=0$) can be seen with celestial coordinates
[see eqs.(5.1)]

$\alpha=\frac{-\xi}{\sin\theta_{i}}\hskip 14.22636pt;\hskip
14.22636pt\beta=\pm\sqrt{(a-\xi)^{2}-\left(a\sin\theta_{i}-\frac{\xi}{\sin\theta_{i}}\right)^{2}}.$

which satify

$\frac{\alpha^{2}}{a^{2}}+\frac{\beta^{2}}{a^{2}\cos^{2}\theta_{i}}=1.$

In this way, the ring is seen as an ellipse with semi-major axis $|a|$ and
semi-minor axis $|a\cos\theta_{i}|$ by the observer. (Note that the null
geodesics coming from this ellipse and reaching the observer have parameters
$\xi$ between $\pm a\sin\theta_{i}$, which are precisely the values allowed
for $\xi$ for geodesics with $\eta=0$ (subsec. 4.0.2) ). The consequences for
the silhouette of a regular superspinar are explicitly shown in Figure 12. The
extreme cases occur when the observer is located along the $z$ axis
($\theta_{i}=0$), where the ring is seen as a circle, and when the observer is
near the equatorial plane ($\theta_{i}=\pi/2$), where the ring approximates an
horizontal segment. Nevertheless, this limit requires a specific study (both
in the classical and regular case) as follows.

Assuming the ring is viewed from the equatorial plane, now the condition
$\Theta\geq 0$ does not place a restriction on $\xi$. However, the condition
that the null geodesics are directed towards the future imposes (see eq.
(4.4)) that $\xi/a\leq 1$ (where $\xi=a$ is a special case in which we are
dealing with a principal null geodesic). There is another limit to $\xi$ for
the geodesics that do reach the ring: If a null geodesic moving towards
smaller values of $r$ does not have a turning point with $r>0$ then
necessarily will reach $r=0$. At the beginning of this section we have already
treated the turning point conditions. In the case that concerns us now, to
identify the turning point in celestial coordinates we have to use eqs.(5.1)
to find the $r_{0}^{*}$ such that $\beta(r_{0}^{*})=0$, what provides with the
limit $\alpha_{max}\equiv\alpha(r_{0}^{*})$. In summary, assuming $a>0$, the
ring as seen from the equatorial plane consists of a segment ($\beta=0$)
between $-a\leq\alpha\leq\alpha_{max}$ (i.e., with right limit precisely on
the arc). We show figures of superspinars as seen from the equatorial plane
(both their arc and their ring) for Kerr superspinars and regular superspinars
in figures 14 and 15, respectively.

Figure 14:  The silhouette of a Kerr superspinar as seen from the equatorial
plane. The thick black line at $\alpha=0$ is caused by the ring singularity.
Figure 15:  The silhouette of a regular BR superspinar as seen from the
equatorial plane. The thick red line at $\alpha=0$ is caused by the null
geodesics coming from the planckian region and reaching the distant observer.

##  6 Summary

Recent advances have significantly improved our ability to test theoretical
predictions related to high-curvature regions in the Universe. On one hand,
direct observations of gravitational waves from astrophysical sources have
been achieved by the LIGO-VIRGO-KAGRA collaborations [50]. On the other hand,
the Event Horizon Telescope (EHT) has provided us with images of black holes
[51][52]. Additionally, substantial improvements are anticipated in the near
future with the launch of the LISA project [53] and the development of new
ground-based observatories [54]. As a result, the study of physics in strong
gravitational fields is becoming increasingly significant, not only in
theoretical physics but also in astrophysical phenomenology.

Classical superspinars are often viewed as speculative and problematic objects
as they have naked singularities, which would violate the cosmic censorship
conjecture [5][55]. Moreover, they also feature regions with effective
negative mass (resulting in repulsive gravity in the $r<0$ region) where
causality violations are unavoidable. In contrast, we have shown that regular
superspinars do not violate the cosmic censorship conjecture, do not exhibit
effective negative mass, and do not lead to causality violations, provided
they are endowed with a non-negative mass function. This suggests that
incorporating Quantum Gravity effects could render superspinars objects with
plausible properties, warranting serious consideration of their existence.

In this way, the phenomenology of regular superspinars becomes an interesting
subject of study. We have demonstrated that, in general, the silhouette of
regular superspinars will possess an arc. The arc in the silhouette should
closely resemble the arc predicted in the silhouette of a Kerr superspinar if
deviations from General Relativity are due to Quantum Gravity effects (and
they are important at Planck’s scale). The regular case will have slightly
smaller arcs than its classical counterpart, but this effect would be
practically impossible to observe. On the other hand, a major difference
between classical and regular superspinars is that the former are predicted to
have a visible dark spot due to the existence of the $r<0$ region beyond the
disk. We have shown that if Quantum Gravity prevents the formation of
singularities, this region becomes superfluous. Consequently, regular
superspinars should not have silhouettes with dark spots.

Another significant difference between classical and regular superspinars is
related to the observation of their rings. We have shown that the singular
ring of classical superspinars would be practically impossible to observe in
practice, as it would only be visible to distant observers positioned
precisely in the equatorial plane of the superspinar. In contrast, regular
superspinars feature a Planckian region shaped like a pseudotorus, with the
ring as part of its boundary. This region is observable by all distant
observers, making it possible to study the Planckian region of superspinars
and potentially gain extraordinary insights into Quantum Gravity.

## Appendix: Differentiability at the disk

In order to analyze the differentiability at the disk for regular superspinars
with metric (2.1), let us proceed with an analysis similar to the one usually
carried out for the classical rotating case. Consider an observer crossing
$r=0$ while moving along the $z$-axis ($x=y=0$). If we choose $r$ to be non-
negative, then equation (2.2) implies that $r=|z|$ along the observer’s
trajectory, so that along this path

$g_{tt}=-1+\frac{2\mathcal{M}(|z|)|z|}{z^{2}+a^{2}}.$ (.1)

The numerator in the fraction indicates that the derivative of this metric
component along the axis, as well as the Christoffel symbols and the extrinsic
curvature of the surface, can be discontinuous across the disk depending on
the chosen mass function. A well-known instance of this discontinuity occurs
when the mass function is constant: Kerr’s solution. As previously mentioned,
the differentiability issues in the Kerr solution can be resolved by
analytically extending the spacetime through $r=0$, allowing for negative
values of $r$.

At first glance, the situation for general regular rotating objects appears
more favorable. If the regular object has a mass function $\mathcal{M}(r)\sim
r^{n}$ with $n\geq 3$ around $r=0$, then the metric component along the
trajectory (.1) will not have differentiability issues at $z=0$
($\partial_{z}g_{tt}(z=0)=0$). In fact, it will be at least
$C^{n}$,333Specifically, it will be $C^{n}$ if $n$ is even and $C^{\infty}$ if
$n$ is odd. suggesting that an extension through the disk may not be necessary
for regular rotating objects. To prove this conjecture, one must examine more
than just a specific trajectory intersecting the disk and consider more than a
single metric component. Notably, as we approach a point on the disk
($x^{2}+y^{2}<a^{2}$), according to equation (2.2), the function $r$
approaches zero whenever $z$ approaches zero, and vice versa. If one chooses
to avoid an extension with $r<0$, we get, solving for $r$ in (2.2), that
around $z=0$444Note that we assumed $a>0$. (If not, here we should replace
$a\rightarrow|a|$).

$r\simeq\frac{a}{\sqrt{a^{2}-(x^{2}+y^{2})}}|z|$

If we introduce this into the metric component (.1) and consider a mass
function $\mathcal{M}(r)\sim r^{n}$ with $n\geq 3$ around $r=0$, we see that
the metric component takes the form

$g_{tt}\simeq-1+\frac{f(x,y)|z|^{n+1}}{g(x,y)z^{2}+a^{2}},$

where $f$ and $g$ are finite differentiable functions in the disk. In this
way, $g_{tt}$ is differentiable at the disk. (Specifically, again
($\partial_{z}g_{tt}(z=0)=0$) and, in fact, $g_{tt}$ is at least $C^{n}$ at
the disk). The reader can easily verify that a similar situation arises for
the other metric components. However, it is important to note that the metric
will not be analytic at the disk, regardless of the value of $n$. This is
because not all metric components will be infinitely differentiable. For
example, even if the particular metric component (.1) for odd $n$ is
$C^{\infty}$ in the disk, other metric components like

$g_{xz}=\frac{2\mathcal{M}(r)r^{2}z(ay+xr)}{(a^{2}+r^{2})(a^{2}z^{2}+r^{4})}$

are not. ($g_{xz}$ is $C^{n}$ for odd $n$). Nevertheless, since usually the
metric is required to be at least $C^{2}$ [38], a $C^{n}$ metric with $n\geq
3$ at the disk is more than enough.

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11institutetext: College of Science and Engineering, Hamad Bin Khalifa
University, Qatar Foundation, Doha, Qatar 22institutetext: Manchester
University NHS Foundation Trust, Manchester Royal Infirmary, Oxford Road,
Manchester, M13 9PL, United Kingdom
22email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Spot the fake lungs: Generating Synthetic Medical Images using Neural
Diffusion Models††thanks: This preprint has not undergone peer review (when
applicable) or any post-submission improvements or corrections. The Version of
Record of this contribution is published in [insert volume title], and is
available online at https://doi.org/doi to be updated . Upon publishing, final
version will be available from Springer Nature Publishers (link to be
updated).

Hazrat Ali 11 Shafaq Murad 22 Zubair Shah 11

###### Abstract

Generative models are becoming popular for the synthesis of medical images.
Recently, neural diffusion models have demonstrated the potential to generate
photo-realistic images of objects. However, their potential to generate
medical images is not explored yet. In this work, we explore the possibilities
of synthesis of medical images using neural diffusion models. First, we use a
pre-trained DALLE2 model to generate lungs X-Ray and CT images from an input
text prompt. Second, we train a stable diffusion model with 3165 X-Ray images
and generate synthetic images. We evaluate the synthetic image data through a
qualitative analysis where two independent radiologists label randomly chosen
samples from the generated data as real, fake, or unsure. Results demonstrate
that images generated with the diffusion model can translate characteristics
that are otherwise very specific to certain medical conditions in chest X-Ray
or CT images. Careful tuning of the model can be very promising. To the best
of our knowledge, this is the first attempt to generate lungs X-Ray and CT
images using neural diffusion models. This work aims to introduce a new
dimension in artificial intelligence for medical imaging. Given that this is a
new topic, the paper will serve as an introduction and motivation for the
research community to explore the potential of diffusion models for medical
image synthesis. We have released the synthetic images on
https://www.kaggle.com/datasets/hazrat/awesomelungs.

###### Keywords:

Diffusion models Generative models Artificial Intelligence Medical imaging
lungs CT X-Ray

##  1 Introduction

During the last decade, there has been a surge in studies on generative models
for medical image synthesis [1, 2]. Generative Adversarial Networks (GANs) and
deep autoencoders are two primary examples of deep generative models that have
shown remarkable advancements in synthesis, denoising, and super-resolution of
medical images [1, 3]. Many studies have shown the great potential of GANs to
generate realistic magnetic resonance imaging (MRI), Computed Tomography (CT),
or X-Ray images that can help in training artificial intelligence (AI) models
[1, 4, 5, 6].

Figure 1: Forward pass and reverse pass in diffusion model training. Figure
modified from [7].

With the recent success of neural diffusion models for the synthesis of
natural images [8, 9], there is now an increasing interest in exploring the
potential of neural diffusion models to generate medical images. For
generating natural images such as art images, objects, models such as
DALLE2111https://openai.com/, Mid-Journey222https://www.midjourney.com/home/,
and Stable Diffusion333https://github.com/CompVis/stable-diffusion have pushed
the state-of-art. Amongst the three, only the latter is available with open-
source code. Compared to GANs, diffusion models are becoming popular for their
training stability.

A diffusion model, in simple words, is a parameterized Markov chain trained
using variational inference. The transition is learned through a diffusion
that adds noise to the data. In principle, the diffusion model transforms the
input data into noisy data by adding Gaussian noise and then recovers the data
distribution by reversing the noise. Once the model learns the distribution,
it can generate useful data from random noise input. So, diffusion models
transform a latent encoded representation into a more meaningful
representation of image data. In this context, diffusion models can be
compared to denoising autoencoders. As shown in Figure 5, the overall process
can be summarized as a two-step phenomenon, the forward pass, i.e., the
transformation of the data distribution to noise ($X_{i}$ to $X_{T}$), and the
reverse pass, i.e., reversing the noise distribution to data distribution
($X_{T}$ to $X_{i}$). Training a diffusion model implies the learning of the
reversing process i.e., $p(x_{t-1}|x_{t})$. The diffusion model can be
implemented by using a neural network for the forward and reverse training
steps. However, the architecture must have the same input and output
dimensions.

While previously, the generating ability of diffusion models was mostly used
for unconditional generation of data, more recent attempts have shown
conditioned generation by introducing guided-diffusion models [9, 10, 11].
These works have demonstrated the generation of photo-realistic images guided
by the context of the input text or image. The existing use cases of diffusion
models comprise text-to-image applications, i.e., generating images according
to a given text prompt. In addition, Han et al., [12] presented a
classification and regression diffusion model (CARD), and demonstrated the use
of the diffusion model for classification as well as regression tasks. In
CARD, the authors approached the task of supervised learning using generative
modeling conditioned on the class labels. Though the objective was not to
claim state-of-the-art results, the method has shown promising results on the
benchmark dataset. For CIFAR-10 classification, the model reached an accuracy
of 90.9%.

Given the potential of diffusion models to learn the representation, one can
expect their potential to generate a diverse set of medical images.
Furthermore, they can add a new dimension to existing approaches for medical
image applications, such as noise adaptation, noise removal, super-resolution,
domain-to-domain translation, and data augmentation. To the best of our
knowledge, no work other than the recent pre-print [13], exists currently on
the synthesis of medical images using neural diffusion models. Walter et al,
[13] used latent diffusion models to generate T1w MRI images of the brain.
Using 31,740 brain MRI images from the UK Biobank, they have generated a stack
of 100,000 images conditioned on key variables such as age, sex, and brain
volume. In this work, we explore neural diffusion models to generate synthetic
images of lung CT and X-Ray. We use the DALLE2 model and the stable diffusion
model to generate the images and present them to two radiologists for their
feedback. We then summarize the feedback received from the radiologists and
identify some of the challenges in using the neural diffusion model for
medical image synthesis.

The remaining paper is organized as Section 2 explains the methodology of our
work. Section 3 presents the results of generating lung CT and X-Ray images,
while Section 4 provides insights into the results and also highlights the
limitations of the approach. Finally, Section 5 concludes the paper.

##  2 Methodology

In this work, we devised two experiments for generating synthetic images of
lungs X-Ray and CT. In the first experiment, we used the _OpenAI_ DALLE2 API
444https://openai.com/ to generate images based on the input text. The DALLE2
model recently gained much attention for its ability to generate photo-
realistic images of objects given a certain input text. Using the API, we
generated multiple images of lungs CT and X-Ray. We then presented a randomly
selected set of the generated images to two trained radiologists. We asked the
radiologists for two key tasks. First, we asked them to label each image as
real, fake, or uncertain about, as per their perceived understanding. Second,
we asked them to provide a brief description of the possible information
related to lung condition or diagnosis of disease (for example, normal lungs,
severely damaged lungs, pneumonia-affected lungs, etc). The radiologists did
not have prior information on the labels of the images. In fact, all the
images that we presented to the radiologists were synthetic. The radiologists
did not know each other and performed the tasks independently. Of the two
radiologists, one radiologist had prior knowledge of artificial intelligence
and generative models, while the other radiologist was naïve to deep
generative models.

In the second experiment, we used the stable diffusion model [14]. We trained
the stable diffusion model using 3165 X-Ray images from [15]. We resized the
images to 256 by 256 resolution. No other pre-processing was done. Using the
X-Ray images, we trained a stable diffusion model on a server equipped with
NVIDIA Quadro RTX 8000 GPU with a 48 GB memory. We set the batch size equal to
32 and ran the training for 700000 training steps.

##  3 Results

Using the DALLE2 API, we generated a total of 150 images. We have uploaded the
synthetic images to Kaggle
555https://www.kaggle.com/datasets/hazrat/awesomelungs. We believe the number
of generated images is only limited by the tokens available to us. Sample
X-Ray and CT images are shown in Figure 2 and Figure 3, respectively. Out of
40 images that we presented to the radiologists, radiologist $\mathcal{A}$
identified 14 X-Ray images and three CT images as real, while 4 X-Ray and 17
CT images as fake. Radiologist $\mathcal{A}$ labeled two X-Ray images as
unsure. The second radiologist (radiologist $\mathcal{B}$) identified 10 X-Ray
images and only two CT images as real, while all the remaining images as fake.

Agreement between radiologists: Of the 20 CT images, only three images were
labeled as real by both radiologists. Similarly, five X-Ray images were marked
as real by both radiologists. There were 2 X-Ray and 2 CT images for which
both the radiologists were uncertain.

For task 2, where we asked the radiologist to provide a description of what
the images may reveal, the radiologists made some interesting observations.
For example, some descriptions are listed in Table 1. These descriptions
clearly reveal that some of the images carried representations similar to real
X-Ray or CT images, and the model was able to generate features that are
specific lung conditions.

Table 1: Samples of remarks from radiologists (no-specific order) Image modality |  Remarks*
---|---
CT |  Possible effusions
|  Pneumonia
X-Ray |  Left lower lobe effusions
|  Possibility of pneumonia
|  Bilateral infection
*The remarks do not imply a definite decision.

Figure 2: Samples of lungs X-Ray images generated with the diffusion model.

Figure 3: Samples of lungs CT images generated with diffusion model.

Figure 4: Samples of synthetic images for lungs X-Ray (left two images) and CT
(right two images) identified as fake by at least two radiologists. Figure 5:
Evaluation by Radiologist A

##  4 Discussion

Some of the generated images lacked the characteristics of realistic images
and were quickly identified by the radiologists as fake. These images were
termed as having unusual ribs appearance or showing unusual exposure.
Similarly, it was easy to spot big vessels contour and lung fields that
appeared to have been drawn and not imaged. One key observation for fake
images was that the trachea is visible behind the heart shadow, which does not
happen in real X-Ray imaging. A few sample images that were termed fake by at
least two radiologists are shown in Figure 4. Many of the generated images
from the pre-trained model clearly lacked the characteristics of realistic
images and were quickly identified by the radiologists as fake. These images
were termed as having unusual ribs appearance, strange clavicle appearance, or
showing unusual exposure.

###  4.1 Limitations

One challenge identified in diffusion models is the limited ability to produce
details in complex scenes [9]. So, generating complex medical images would
need to be complemented with noise adaptation or super-resolution techniques
[5]. Like many other AI models, diffusion model training is prone to bias in
the dataset; for example, unbalanced representation of medical conditions in
the input X-Ray or CT image or inherent noise in data. Thus, the synthetic
data from such a diffusion model will also carry the bias. Eventually, if the
generated data are made public and used for onward model training, the bias
may turn into a cascade behavior and will be further augmented [16]. The model
has been used pretty much as a black-box model; hence, not much explainability
can be offered on how certain images were generated. Unlike the work reported
in Walter et al., [13], our generated images are not conditioned on additional
variables such as gender, age, etc. Diffusion models are very slow to train as
they require the number of training steps to be in the order of several
hundred thousand. Our training took around one day for 100k training steps.

##  5 Conclusion and Future Work

In this work, we have demonstrated the potential of neural diffusion models
for the synthesis of lungs X-Ray and CT images. Though the radiologists
spotted many images as fake, few images were still labeled as real by them.
The labeling from the radiologists reflects that some of the generated X-Ray
images carried a great resemblance to real images. However, the identification
of fake images was straightforward for the CT images. Through qualitative
analysis of the generated images, we showed that neural diffusion models have
great potential to learn complex representations of medical images. Although
the performance of diffusion models is superior to GANs-based methods for
synthesizing natural images, research efforts on the diffusion model for
medical image synthesis have yet to mature.

##  6 Acknowledgments

The authors are grateful to Surendra Maharjan from Indiana University Purdue
University Indianapolis, USA, for useful comments on this work. The authors
are thankful to Dr. Jens Schneider for facilitating the GPU access.

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