# Inversion points of the accretion flows onto super-spinning Kerr attractors

D. Pugliese&Z. Stuchlík<EMAIL_ADDRESS>Research Centre for
Theoretical Physics and Astrophysics, Institute of Physics, Silesian
University in Opava, Bezručovo náměstí 13, CZ-74601 Opava, Czech Republic

###### Abstract

We study the accretion flows towards a central Kerr super-spinning attractor,
discussing the formation of the flow inversion points, defined by condition
$u^{\phi}=0$ on the particles flow axial velocity. We locate two closed
surfaces, defining _inversion coronas_ (spherical shells), surrounding the
central attractor. The coronas analysis highlights observational aspects
distinguishing the central attractors and providing indications on their spin
and the orbiting fluids. The inversion corona is a closed region, generally of
small extension and thickness, which is for the counter-rotating flows of the
order of $\lesssim 1.4M$ (central attractor mass) on the vertical rotational
axis. There are no co-rotating inversion points (from co-rotating flows). The
results point to strong signatures of the Kerr super-spinars, provided in both
accretion and jet flows. With very narrow thickness, and varying little with
the fluid initial conditions and the emission process details, inversion
coronas can have remarkable observational significance for primordial Kerr
super-spinars predicted by string theory. The corona region closest to the
central attractor is the most observably recognizable and active part,
distinguishing black holes solutions from super-spinars. Our analysis expounds
the Lense–Thirring effects and repulsive gravity effects in the super-spinning
ergoregions.

Singularities in general relativity–Fluids in curved spacetime–Accretion
disk–Classical black holes

##  I Introduction

This work explores astrophysical tracers distinguishing Kerr super-spinning
solutions, with dimensionless spin $a>1$ from Kerr black hole (BH) solutions
with $a<1$.

We focus on the (accretion) flows towards a central super-spinning Kerr
attractor, exploring the formation of azimuthal velocity inversion points in
the accretion flows, defined by the condition $u^{\phi}=0$ on the particles
axial velocity. The analysis follows paper [1], where the case of Kerr BH
attractors has been considered and [2], and where the case of Kerr Naked
singularity (NS) has been detailed. The inversion points in the super-spinar
geometries are related to a combination of the repulsive gravity effects and
the frame dragging effects, typical of the ergoregion.

The possibility to distinguish between BHs and super-spinars is still a major
challenge of the present day Astrophysics. A conclusive proof of the Cosmic
Censorship hypothesis (CCh) is still lacking at present., on the other hand,
super-spinars may appear in a variety of contexts, implying, in some cases,
effects of quantum gravity or string theory [3, 4, 5]. Super-spinars can
appear for example in the frame of string theory. The region of causality
violations and ring—like Kerr singularity defining the Kerr naked-singularity
(NS) can be removed and substituted by a spacetime region represented by a
string solution. These "primordial" super-spinars would be distinguished by an
exterior geometry of a Kerr NS. Then, Kerr NSs could exist, without
contradicting with the CCh, as remnants of a primordial high-energy phase of
the early Universe [3].

Kerr super-spinars are devoid of the physical ring singularity and the
causality-violation region, substituted by an interior regular solution. (The
removed pathological causality violation region is determined from the
condition $g_{\phi\phi}<0$, occurring at $r<0$.). Consequently, the minimal
condition for the boundary surface of Kerr super-spinars is $r(\theta)=R=0$
(in the Boyer–Lindquist coordinates) although, in many analyses, the Kerr
super-spinars are considered at the boundary $R=0.1M$ [4, 5, 6, 7]. As
consequence of this, the Kerr super-spinars do not contradict the CCh (on the
other hand, CCh counterexamples, where existence of spherically symmetric NSs
or evidences of a violation of the weak CCh in five-dimensional spacetimes are
often debated [8]). The region of causality violation (at $r<R$=constant)
would be covered, for example, by a stringy solution, expected to be limited
to the region $r<R\leq M$ [3, 4, 5, 6, 7]. The observation of such Kerr super-
spinars could be therefore interpreted in the framework of the string theory.
Outside a Kerr super-spinar ($r>R$), the Kerr NS geometry is assumed. The Kerr
super-spinars surface (at $r=R>0$) has to be assumed, and in some studies the
solution of a one-way membrane (similarly to the BH horizon) is considered [6,
5, 4] (the surfaces have to be provided by the exact interior solution to be
joined at the boundary to the standard Kerr NS geometry). However, contrary to
the BHs, there is no uniqueness theorem for the (Kerr) NSs (and super-
spinars). Kerr super-spinars are considered as a reinterpretation of NSs
possibly modeling quasars of the early evolution period of the Universe. In
this context therefore observations of a hypothetical Kerr super-spinar could
be expected at high redshift AGN and quasars[9]. In theories different from
General Relativity, such as braneworlds. the Kerr BH spin limit $a=1$ can be
exceeded [10, 11, 12, 13, 14].

For all these reasons, the study of possible astrophysical signatures for Kerr
super-spinars constitutes an extensive and debated literature on a huge
variety of astrophysical phenomena–see for example [5, 6, 15, 16, 17, 18, 19,
20, 21, 22, 4, 23, 24].

In the present analysis, we discuss some relevant astrophysical phenomena
exposing distinctions between Kerr NSs and BHs, related to flows from
accretion tori and jets in the super-spinar gravitational field. Discussing
the differences emerging from this analysis, between the BH and NS scenarios,
we provide distinguishable physical characteristics of jet and accretion flows
in NS geometries, which possibly could provide an observational signature of
the Kerr NSs.

Accretion has been studied for possible mechanisms to convert NSs into extreme
BH states111There are strong indications that the inverse process, consisting
in the creation of a NS through a conversion process from an extreme BH, is
forbidden [25]. It has been shown that, following accretion from orbiting
disks, Kerr super-spinars may be efficiently converted to extreme Kerr (or
near extreme Kerr) BHs[21, 22]. The conversion into a Kerr BH, following
accretion from counter-rotating Keplerian discs, could be very quick.
(However, crucially NSs from a primordial era could survive to the era of
high–redshift quasars or even in later times $z\sim 2$ for a sub Eddington
accretion process)222 The efficiency of the counter-rotating machine is by
orders smaller of the co-rotating accretion disk conversion efficiency.) In
this respect, Kerr super-spinars can also be efficient accelerators for
extremely high–energy collisions–[7, 26]..

Kerr NS spacetimes are characterized by an articulated combination of Lense-
Thirring and the repulsive gravity effects333It should be stressed that here
the gravitational repulsion is not related to the vacuum energy effect as in
inflationary models, but only to special configurations of attractive
gravitational fields as demonstrated by F. de Felice [27]., peculiar of the NS
ergoregion. In the inversion point analysis, the counter-rotating flows and
the NS ergoregion are a particular focus. We here distinguish counter-rotating
and co-rotating flows from accretion disks (_"accretion driven flows"_) or
from structures constituted by open funnels of matter along the central
attractor rotational axis (these structures are well known to be associated to
accreting toroids, see for example [29, 30, 31, 32, 33, 34, 35, 28], and are
here called, for brevity, _proto-jets_)–("_proto-jets driven flows_ "). We
also considered the formation of double toroidal structures with equal
specific angular momentum $\ell$, and ZAMOS (zero angular momentum observers)
tori (with fluid specific angular momentum $\ell=0$), typical of NS
geometries. Two main scenarios are therefore addressed: the accretion driven
inversion points and the proto-jets driven inversion points, where the flux of
particles and photons is related to an open cusped toroidal configuration with
high magnitude of specific fluid angular momentum and variously associated to
jets emission [28, 36, 37, 38, 40, 39, 41]. This setup fixes particle
trajectories initial data and constants of motion.

Although the inversion points do not depend on the details of the accretion
processes, or the precise location of the tori inner edge, in our analysis the
flow of free-falling particles is driven from an inner edge of the accretion
torus or proto-jet (open, cusped configuration). As for the BH case, analyzed
in [1], the flow inversion points in the NS spacetimes, define a closed and
regular surface at fixed $\ell$, _inversion surface_ , surrounding the central
attractor, being influenced by the range of location of the torus inner edge
or the proto-jet cusps.

More in details, article plan is as follows:

In Sec. (II) we introduce the spacetime metric and the constants of motion. In
Sec. (II.1) we explore further the notion of co-rotation and counter-rotation
in NSs spacetimes. Spacetime geodesic structure and tori constraints are
addressed in Sec. (II.2). In Sec. (III), we define the flow inversion points
and detail general properties, as inversion points limits and extremes, for
co-rotating and counter-rotating flows. In Sec. (IV) inversion points of the
flows are constrained according to the NS spin, the fluid specific angular
momentum and the particles energy. Sec. (IV.1) focuses on the counter-rotating
flows, discussing inversion radius $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ in Sec.
(IV.1.1) and the inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}$ in Sec.
(IV.1.2). Constants of motion and inversion points in counter-rotating flows
are the subject of Sec. (IV.1.3). Inversion points of the co-rotating flows
are studied in Sec. (IV.2). Finally we explore the limiting case of tori with
$\ell=0$ in Sec. (IV.3). Sec. (V) contains discussion and final remarks. In
Appedix (A) we explicit Kerr Naked singularities spacetime circular geodesic
radii. We also introduced Table (1), as a look-up table we refer the reader to
for reminding of notation and definitions used throughout this paper with
links to associated sections, equations or figures.

##  II The spacetime metric and constants of motion

The Kerr spacetime metric reads

$\displaystyle
ds^{2}=-\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\left[(r^{2}+a^{2})+\frac{2Mra^{2}}{\Sigma}\sin^{2}\theta\right]\sin^{2}\theta
d\phi^{2}-\frac{4rMa}{\Sigma}\sin^{2}\theta dtd\phi,$ (1)

in the Boyer-Lindquist (BL) coordinates $\\{t,r,\theta,\phi\\}$444We adopt the
geometrical units $c=1=G$, Latin indices run in $\\{0,1,2,3\\}$. The radius
$r$ has unit of mass $[M]$, and the angular momentum units of $[M]^{2}$, the
velocities $[u^{t}]=[u^{r}]=1$ and $[u^{\phi}]=[u^{\theta}]=[M]^{-1}$ with
$[u^{\phi}/u^{t}]=[M]^{-1}$ and $[u_{\phi}/u_{t}]=[M]$. For the seek of
convenience, we always consider the dimensionless energy and an angular
momentum per unit of mass $[L]/[M]=[M]$., where $\Delta\equiv
a^{2}+r^{2}-2rM\quad\mbox{and}\quad\Sigma\equiv a^{2}cos^{2}\theta+r^{2}$. The
Kerr NSs have $a>M$. Parameter $a=J/M\geq 0$ is the metric spin, where total
angular momentum is $J$ and the gravitational mass parameter is $M$. A Kerr BH
is defined by the condition $a\in\,[\,0,M\,]$ with Killing horizons horizons
$r_{-}\leq r_{+}$ where $r_{\pm}\equiv M\pm\sqrt{M^{2}-a^{2}}$). The extreme
Kerr BH has dimensionless spin $a/M=1$ and the non-rotating case $a=0$ is the
Schwarzschild BH solution.

In the following we will use the angular coordinate
$\sigma\equiv\sin^{2}\theta$. The equatorial plane, $\sigma=1$, is a metric
symmetry plane and the equatorial circular geodesics are confined on the
equatorial plane as a consequence of the metric tensor symmetry under
reflection through the plane $\theta=\pi/2$. In this work we address the NSs
case, focusing in particular on the Lense-Thirring effects induced from the
frame-dragging, particularly relevant in the ergoregion, whose boundaries are
the outer and inner stationary limits $r_{\epsilon}^{\pm}$ (ergosurfaces),
given by

$\displaystyle r_{\epsilon}^{\pm}\equiv
M\pm\sqrt{M^{2}-a^{2}(1-\sigma)}\quad\mbox{or,
equivalently}\quad\sigma_{erg}\equiv\frac{(r-2M)r}{a^{2}}+1,$ (2)

respectively—Figs (1)–where $r_{\epsilon}^{-}=0$ and $r_{\epsilon}^{+}=2M$ in
the equatorial plane $\theta=\pi/2$ ($\sigma=1$), and $r_{+}<r_{\epsilon}^{+}$
on $\theta\neq 0$.

More precisely there is

$\displaystyle
g_{tt}>0\;\mbox{for}:a>M,\quad\sigma\in\,\,]\,\sigma_{\epsilon},1\,]\,,r\in\,]\,r_{\epsilon}^{-},r_{\epsilon}^{+}\,[\,,\quad\mbox{where}\;\sigma_{\epsilon}\equiv\frac{a^{2}-M^{2}}{a^{2}}.$
(3) $\displaystyle
g_{tt}<0\;\mbox{for}:a>M,\left(\sigma\in\,[\,0,\sigma_{\epsilon}\,[\,,r\geq
0\right),\quad\left(\sigma=\sigma_{\epsilon},\quad r\neq
r_{\epsilon}^{-}\right),\quad\left(\sigma\in\,]\,\sigma_{\epsilon},1\,]\,,\quad
r\in\,[\,0,r_{\epsilon}^{-}\,[\,\cup\,]\,r_{\epsilon}^{+},+\infty\,[\,\right),$

(see Figs (1)). NS poles $\sigma=0$ are associated to $g_{tt}<0$ for $a\geq 0$
and $r\geq 0$.

Figure 1: Left panel: Plane $\sigma_{\epsilon}$ in Eq. (3) as function of the
NS spin $a>1$. The ergoregion is defined for
$\sigma\in[\,\sigma_{\epsilon},1\,]$. There is
$\sigma\equiv\sin^{2}\theta\in[\,0,1\,]$, where $\sigma=1$ is the equatorial
plane. Radius $r=r_{\epsilon}^{+}=2$ is the outer-ergosurface on the
equatorial plane. Right panel: the curves (ergosurfaces) $r_{\epsilon}^{\pm}$
such that $g_{tt}(r_{\epsilon}^{\pm})=0$ are plotted for selected spin, signed
on the panel in different colors on the plane $(r,\sigma)$. The limiting
values $\sigma_{\epsilon}$ for each spin are also shown as vertical colored
dashed lines. There is $g_{tt}>0$ in the region bounded by the curves
$r_{\epsilon}^{\pm}$. See analysis in Eq. (3). All quantities are
dimensionless.

The role of the limit $\sigma_{\epsilon}$ in the NS geometries is a main
difference with the BH cases—see Figs (1). For very fast super-spinning NSs
$(a\gg M)$, there is $\sigma_{\epsilon}\to 1$ and the range for the existence
of the ergoregion narrows /at $a=M$ there is $\sigma_{\epsilon}=0)$–Figs (1).
In the following, where appropriate, to easy the reading of complex
expressions, we will use the units with $M=1$ (where $r\rightarrow r/M$ and
$a\rightarrow a/M$).

Constants of motion

We consider the following constant of geodesic motions

$\displaystyle\mathcal{E}=-(g_{t\phi}\dot{\phi}+g_{tt}\dot{t}),\quad\mathcal{L}=g_{\phi\phi}\dot{\phi}+g_{t\phi}\dot{t},\quad
g_{ab}u^{a}u^{b}=\kappa,$ (4)

defined from the Kerr geometry rotational Killing field
$\xi_{\phi}=\partial_{\phi}$, and the Killing field $\xi_{t}=\partial_{t}$
representing the stationarity of the background, with
$u^{a}\equiv\\{\dot{t},\dot{r},\dot{\theta},\dot{\phi}\\}$ where $\dot{q}$
indicates the derivative of any quantity $q$ with respect the proper time or a
properly defined affine parameter for the light–like orbits (for $\kappa=0$),
and $\kappa=(\pm 1,0)$ is a normalization constant ($\kappa=-1$ for test
particles).

The constant $\mathcal{L}$ in Eq. (4) may be interpreted as the axial
component of the angular momentum of a test particle following timelike
geodesics and $\mathcal{E}$ represents the total energy of the test particle
related to the radial infinity, as measured by a static observer at infinity.
The velocity components $(u^{t},u^{\phi})$ read

$\displaystyle
u^{t}=\frac{g_{\phi\phi}\mathcal{E}+g_{t\phi}\mathcal{L}}{g_{t\phi}^{2}-g_{\phi\phi}g_{tt}},\quad
u^{\phi}=-\frac{g_{t\phi}\mathcal{E}+g_{tt}\mathcal{L}}{g_{t\phi}^{2}-g_{\phi\phi}g_{tt}}.$
(5)

We introduce also the specific angular momentum $\ell$ and the relativistic
angular velocity $\Omega\equiv{u^{\phi}}/{u^{t}}$

$\displaystyle\ell\equiv\frac{\mathcal{L}}{\mathcal{E}}=-\frac{g_{\phi\phi}u^{\phi}+g_{\phi
t}u^{t}}{g_{tt}u^{t}+g_{\phi
t}u^{\phi}}=-\frac{g_{t\phi}+g_{\phi\phi}\Omega}{g_{tt}+g_{t\phi}\Omega},\quad\Omega(\ell)=-\frac{g_{t\phi}+g_{tt}\ell}{g_{\phi\phi}+g_{t\phi}\ell}.$
(6)

By considering definition of constants of geodesic motion $\mathcal{L}$ and
$\ell$ in Eq. (4) and Eq. (6) we discuss in Sec. (II.1) the notion of co-
rotation and counter-rotation in the Kerr NS spacetime. These definitions will
constitute a primary template of analysis of the flows we investigate in this
work and the inversion surfaces as (newly introduced) geometric properties of
the Kerr NSs. Finally, spacetime geodesic structure and tori constraints are
the focus of Sec. (II.2). In this section we will explore the circular
geodesic co-rotation and counter-rotation motion of the Kerr NSs. These radii
constrain the circular test particle geodesics and toroidal extended matter
orbiting around the central spinning attractor as, for example, accretion
disks. The flows we study are made of matter (test particles) and photons,
freely moving towards the central attractor or from the central attractors (we
address these different aspects more closely also in Sec. (IV)). The motion is
regulated by the geodesic equations and can be parametrized by the constants
of motion in Eq. (4). We will use the constant $\ell$, as it is the parameter
defining the properties focused in this analysis. It is useful to classify the
flows and the characteristics of the accreting structures which could possibly
engine the flows, according to the $\ell$ parameter ranges, as studied in Sec.
(II.2), where we discuss the flows and accretion tori properties, providing
constraints for our set-up.

###  II.1 On co-rotation and counter-rotation in NSs spacetimes

In this section we introduce some of the constraints and parameters used in
this analysis, discussing the specific angular momentum $\ell$ parametrizing
the flows, orbiting tori and inversion points. We shall examine both
co–rotating and counter—rotating motion with respect to the central attractor,
investigating their definitions properties in the Kerr NS spacetimes.

The specific angular momentum $\ell=$constant parametrizes the GRHD barotropic
geometrically thick disks– [28]. These toroids are constant pressure surfaces,
whose construction in the axes–symmetric spacetimes is based on the
application of the von Zeipel theorem. The surfaces of constant angular
velocity $\Omega$ and of constant specific angular momentum $\ell$ coincide.
The toroids rotation law, $\ell=\ell(\Omega)$, is therefore independent of the
details of the equation of state, providing the integrability condition of the
Euler equation in the case of barotropic fluids. Consequently, in the
geometrically thick disks, the functional form of the angular momentum and
entropy distribution, during the evolution of dynamical processes, depends on
the initial conditions of the system and not on the details of the dissipative
processes [28].

We analyze two families of flows, with fluid specific angular momentum
$\ell=\ell^{+}<0$ (counter-rotating) and $\ell=\ell^{-}\lesseqgtr 0$ (co-
rotating and counter-rotating) respectively555We also used notation
$\ell^{\pm}$ both for the function $\ell^{\pm}$ and referring to the constant
values (and constant of motions) $\ell^{\pm}=$constant characterizing orbits,
tori or inversion surfaces. Saying differently function $\ell^{\pm}$ of Eq.
(7) provides the (radial) distribution of the constant of motion $\ell$ for
each orbit. Figs (2) show the distribution $\ell^{\pm}$ as function of $r$ for
different spin $a$, for NSs and BHs.:

$\displaystyle\ell^{\mp}\equiv\frac{a^{3}\mp
r^{3/2}\sqrt{\Delta^{2}}-a(4-3r)r}{a^{2}-(r-2)^{2}r},$ (7)

[28, 42, 45]. Note that tori are governed by the distribution of specific
angular momentum on the equatorial plane, therefore $\ell^{\pm}$ in Eqs (7)
are independent of the $\sigma$-coordinate.

Figure 2: Specific angular momentum $\ell^{+}$ (right column) and $\ell^{-}$
(left column) of Eqs (7) as functions of radius $r$ for different Kerr BH spin
$a\in[\,0,1\,]$ (upper line) and BH and NS spin $a>1$ (bottom line) signed on
the curves. Right column panels follow the color code explicit in the right
panels. (All quantities are dimensionless).

Figs (2) show fluid specific angular momentum $\ell^{\pm}$ as function of $r$
for different spin $a$, for NSs and BHs.

In the NS geometries especially for small values of the spin–mass ratio
($a\in]\,1,a_{2}\,]$ where $a_{2}\equiv 1.29904$) it is necessary to discuss
further the notion of co-rotating and counter-rotating motion. Considering
quantities of $\\{\mathcal{L},\ell\\}$, with $a>0$, we introduce the following
definitions for fluid and test particle motion:

–

For a circularly orbiting test particle, particle counter–rotation (co-
rotation) is _defined_ by $\mathcal{L}a<0$ ($\mathcal{L}a>0$).

–

For fluids, the counter-rotation (co-rotation) is _defined_ by $\ell a<0$
($\ell a>0$).

We can read these definitions666The choice to adopt the co-rotation and
counter-rotation classification for fluids and particles in accordance with
the quantities $\ell$ or $\mathcal{L}$ respectively is a common choice in
literature. A particle or fluid orbiting in the NS spacetimes can satisfy
$\mathcal{L}a<0$ but also $\ell=\mathcal{L}/\mathcal{E}>0$, as there can be
$\mathcal{L}a<0$ and $\mathcal{E}a<0$. It is therefore important, in the NS
spacetimes, to differentiate the two definitions. (In the Kerr BH spacetime
the two definitions are equivalent, as the rotation orientation definition
according $\ell$ or $\mathcal{L}$ are coincident.). In this regard, many
aspects of the particles dynamics are governed by (and thus parametrized in
accordance with) with the constants $\mathcal{L}$ and $\mathcal{E}$, and it is
therefore useful in this case to use the co-rotation and counter-rotation
classification according to particle momentum $\mathcal{L}$ for test
particles. Viceversa, many aspects of fluids and, more generally, of the
orbiting extended matter, including those considered in this article, are
essentially regulated by (and thus parametrized in accordance with) the
momentum $\ell$ (for example, there are accretion disks defined by the
parameter $\ell$–see Sec.(II.2)). It is therefore natural to adopt for these
systems, differently from particles, the rotation orientation classification
according to $\ell$. in terms of $\ell$, $\\{\mathcal{E},\mathcal{L}\\}$ and
$\Omega$. (Note the velocity $u^{\phi}$ sign coincides with $\Omega$, assuming
$u^{t}>0$). )

For this purpose, it is convenient to introduce radii
$\\{r_{0}^{\pm},r_{\delta}^{\pm}\\}$ defined by the conditions that on the
equatorial circular orbits $r_{0}^{\pm}<r_{\epsilon}^{+}=2$, there is
$\mathcal{L}=\ell=0$, while on the orbits
$r_{\delta}^{\pm}:r_{0}^{-}<r_{\delta}^{-}<r_{\delta}^{+}<r_{0}^{+}$, there is
$\mathcal{E}=0$ and $\mathcal{L}<0$ see Figs (3) and [44, 43, 42, 22]. That is
radii $r_{0}^{\pm}$ can be found from the condition $\mathcal{L}=\ell=0$ on
the geodesic motion and $r_{\delta}^{\pm}$ can be found from the condition
$\mathcal{E}=0$ on the geodesic motion. Their explicit form is in Appendix
(A). From the analysis of $r_{\delta}^{\pm}$ and $r_{0}^{\pm}$ we introduce
the spins,

$\displaystyle
a_{0}:r_{\delta}^{+}=r_{\delta}^{-}\equiv\frac{4}{3}\sqrt{\frac{2}{3}}=1.08866,\quad
a_{2}:r_{0}^{+}=r_{0}^{-}\equiv\frac{3\sqrt{3}}{4}=1.29904.$ (8)

–see Figs (3). Hence spins $a_{0}$ and $a_{2}$ can be found assuming
conditions $r_{\delta}^{+}=r_{\delta}^{-}$ and $r_{0}^{+}=r_{0}^{-}$
respectively, and represent a limiting case, where in these NS spacetimes
there is only one limiting orbit where $\mathcal{E}=0$ and $\mathcal{L}=0$
respectively (Figs (3)). As detailed below $a_{0}$ and $a_{2}$ are maximum
spin where radii $r_{\delta}^{\pm}$ and $r_{0}^{\pm}$ exist. Therefore, we can
summarize the situation for counter-rotating flows and co-rotating flows as
follows:

Counter-rotating tori $(\ell<0)$

This case includes fluids with $\ell=\ell^{+}<0$ and $\ell=\ell^{-}<0$ in the
ergoregion for NSs with $a\in\,[\,1,a_{2}\,]$, more precisely there are:

  * (I)

Tori with $\ell=\ell^{+}<0$, that is with $(\mathcal{L}<0,\mathcal{E}>0)$
(located far from the ergoregion)

  * (II)

Tori with momentum $\ell=\ell^{-}<0$, that is with
$(\mathcal{L}<0,\mathcal{E}>0)$, in
$]\,r_{0}^{-},r_{\delta}^{-}\,[\,\cup\,]\,r_{\delta}^{+},r_{0}^{+}\,[$, for
spacetimes $a\in\,[\,1,a_{0}\,]$, and in the orbital region
$]\,r_{0}^{-},r_{0}^{+}\,[$, for NSs with spin $a\in\,]\,a_{0},a_{2}\,]$–Figs
(3).

Co-rotating tori ($\ell>0$)

Co-rotating tori, i.e. with $\ell=\ell^{-}>0$, are defined for
$(r>0,a>a_{2})$,
$(r\in\,]\,0,r_{0}^{-}\,[\,\cup\,]\,r_{0}^{+},+\infty\,[,a\in\,]\,a_{0},a_{2}\,[\,)$
and
$(r\in\,]\,0,r_{0}^{-}\,[\,\cup\,]\,r_{\delta}^{-},r_{\delta}^{+}\,[\,\cup\,r>r_{0}^{+},a\in\,]\,1,a_{0}\,[\,)$–see
Figs (3).

$\mathcal{L}$ test particles angular momentum (constant of motion) | Eq.(4)
---|---
$\mathcal{E}$ test particles energy (constant of motion) | Eq.(4)
$\ell\equiv\mathcal{L}/\mathcal{E}$ specific angular momentum (constant of motion) | Eq. (6)
$\ell^{+}$ counter-rotating specific angular momentum | Eq. (7)–Sec. (II.1)
$\ell^{-}$ co-rotating or counter-rotating specific angular momentum | Eq. (7)–Sec. (II.1)
$\sigma_{\epsilon}:r_{\epsilon}^{+}=r_{\epsilon}^{-}$ | Eq. (3)
$r_{\delta}^{\pm}:\mathcal{E}(r_{\delta}^{\pm})=0,\mathcal{L}(r_{\delta}^{\pm})<0$ | Figs (3)– Appendix (A)
$r_{0}^{\pm}:\mathcal{L}(r_{0}^{\pm})=\ell(r_{0}^{\pm})=0$ | Figs (3)–Sec. (II.1)–Appendix (A)
$a_{0}:r_{\delta}^{+}=r_{\delta}^{-}\equiv\frac{4}{3}\sqrt{\frac{2}{3}}=1.08866$ | Eqs (8)– Figs (3)
$a_{2}:r_{0}^{+}=r_{0}^{-}\equiv\frac{3\sqrt{3}}{4}=1.29904$ | Eqs (8)– Figs (3)
${r}_{[mb]}^{\pm}:\;\ell^{\pm}(r_{mb}^{\pm})=\ell^{\pm}({r}_{[mb]}^{\pm})\equiv{\ell_{mb}^{\pm}}$ | Eq. (9)–Figs (3)
$r_{[\gamma]}^{\pm}:\ell^{\pm}(r_{\gamma}^{\pm})=\ell^{+}(r_{[\gamma]}^{+})\equiv\ell_{\gamma}^{+}$ | Eq. (9)–Figs (3)
$a_{1}\equiv 1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ | Sec. (II.2)
$r_{\mbox{\scriptsize{{T}}}}^{\mp}(a,\sigma;\ell):u^{\phi}=0$ inversion radius | Eq. (12)– Figs. (7)– Figs. (8)– Figs. (4)
$\sigma_{\mbox{\scriptsize{{T}}}}(a,r;\ell):u^{\phi}=0$ inversion plane | Eq. (12)
$()_{\mbox{\scriptsize{{T}}}}$ any quantitiy $()$ evaluated on the inversion surface | Sec. (III)
$r_{cr}:r_{\mbox{\scriptsize{{T}}}}^{-}=r_{\mbox{\scriptsize{{T}}}}^{+}$ | Eq. (13)– –Figs (5),(6),(7),(8)
$r_{{\mbox{\scriptsize{{T}}}}\max}^{0}:\partial_{a}\sigma_{\mbox{\scriptsize{{T}}}}=0$ | Eq. (17)–Figs (11)
$r_{{\mbox{\scriptsize{{T}}}}\max}^{\pm}:\partial_{r_{\mbox{\scriptsize{{T}}}}}\sigma_{\mbox{\scriptsize{{T}}}}=0$ | Eq. (17)–Figs (11)
$\ell_{e}^{-}:\partial_{a}r_{\mbox{\scriptsize{{T}}}}^{\pm}=0$ | Eqs. (18), Eqs. (19)–Figs (12).

Table 1: Lookup table containing the main notation and quantities used
throughout this paper. Links to associated sections, definitions or figures
are also listed.

There is therefore $\ell^{-}<0$ with $(\mathcal{L}<0,\mathcal{E}>0)$ in the NS
ergoregion, together with co-rotating solutions $\ell^{-}>0$ with
$(\mathcal{L}<0,\mathcal{E}<0)$ with negative energy $\mathcal{E}$. However,
these solutions correspond to the relativistic angular velocity, i.e. the
Keplerian velocity with respect to static observers at infinity, $\Omega>0$.
In this sense, they are all co-rotating with respect to the static observers
at infinity.

###  II.2 Extended geodesic structure and tori constraints

Test particle motion and many aspects of the physics of accretion around
compact objects is constrained by the spacetime geodesic structure. The flow
dynamics we consider in this analysis is regulated by the set of co-rotating
and counter-rotating Kerr NSs circular equatorial geodesics radii. This set of
radii defines and constrains different extended orbiting surfaces, as
accretion tori having different topology and their instability points. It is
therefore useful to consider here in details the spacetime co–rotating and
counter–rotating geodesic structure.

The Kerr NS background geodesic structure is constituted by the radii
$\\{r_{\gamma}^{\pm},r_{mb}^{\pm},r_{ms}^{\pm}\\}$, marginally circular
(photon) orbit, marginally bounded orbit, and marginally stable orbit
respectively, regulating the location of the tori and proto-jets cusps. (For
convenience we report in Appedix (A) the explicit expression of the radii of
the Kerr NS spacetime geodesc structure). The geodesic structure is extended
to include radii $\\{r_{[\gamma]}^{\pm},r_{[mb]}^{\pm},r_{[ms]}^{\pm}\\}$
regulating the location of the toroidal configurations centers :

$\displaystyle{r}_{[mb]}^{\pm}:\;\ell^{\pm}(r_{mb}^{\pm})=\ell^{\pm}({r}_{[mb]}^{\pm})\equiv{\ell_{mb}^{\pm}},\quad
r_{[\gamma]}^{\pm}:\ell^{\pm}(r_{\gamma}^{\pm})=\ell^{+}(r_{[\gamma]}^{+})\equiv\ell_{\gamma}^{+},$
(9) $\displaystyle\mbox{where}\quad
r_{\gamma}^{\pm}<r_{mb}^{\pm}<r_{ms}^{\pm}<{r}_{[mb]}^{\pm}\quad\mbox{and}\quad{r}_{[mb]}^{+}<r_{[\gamma]}^{+}<r_{\gamma}^{-}\equiv
0,$ (10)

respectively where there is $r_{\gamma}^{-}\equiv 0$. Increasing the NS spin,
radii of geodesic structure and tori depart from the central singularity, for
both co-rotating and counter-rotating fluids, constituting a major difference
with the BH case. Note there is $r_{[ms]}^{+}=r_{ms}^{+}$ where
$r_{[ms]}^{+}:\ell_{ms}^{+}\equiv\ell(r_{ms}^{+})=\ell^{+}(r)$, analogously a
similar relation holds for fluids with $\ell^{-}$ at NSs spins $a>a_{2}$.

Figure 3: Analysis of co-rotating and counter-rotating tori. Curves
$\ell^{-}=$constant (right and center panel) and $\ell^{+}$=constant (left
panel) ($\ell^{+}<0$ and $\ell^{-}\lesseqgtr 0$ are the fluid specific angular
momentum) in the plane $r-a$. (All quantities are dimensionless). Right panel
is a zoom of the center panel. $r_{\gamma}^{\pm}$ are the last circular orbit
(there is $r_{\gamma}=0$). There is $r_{0}^{\pm}:\mathcal{L}=\ell=0$. Radii
$r_{mb}^{\pm}$ are the marginally bounded orbits. Spins $\\{a_{0},a_{2}\\}$
are defined in Eqs (8), $a_{1}\equiv
1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec. (II.2). Radii
$r_{ms}^{-}=\\{\tilde{r}_{ms}^{-},\bar{r}_{ms}^{-}\\}$ and $r_{ms}^{+}$ are
the marginally stable orbits for the fluid with specific angular momentum
$\ell^{-}$ and $\ell^{+}$ respectively defined in Eqs (7). Radii
$r_{[mb]}^{\pm}$ are in Eqs (9). Radii of the geodesic structures are in
Appendix (A).

The situation for fluid with momentum $\ell^{-}$ is however more complex.
There is $\ell^{-}\lesseqgtr 0$, constrained by the radii
$\\{r_{mb}^{-},\tilde{r}_{ms}^{-},\bar{r}_{ms}^{-},r_{\gamma}^{-}\\}$, with
$\\{\mathcal{L}\lesseqgtr 0,\mathcal{E}\lesseqgtr 0\\}$. Radii
$r_{ms}^{-}=\\{\bar{r}_{ms}^{-},\tilde{r}_{ms}^{-}\\}$, are the marginally
stable orbits related to the motion $\ell^{-}\gtrless 0$, where
$r_{ms}^{-}=\bar{r}_{ms}^{-}$ for $a>a_{1}$ and
$r_{ms}^{-}=\tilde{r}_{ms}^{-}$ for $a\in\,]\,1,a_{1}\,]$ and
$\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ on $a_{1}\equiv 1.28112$–Figs (3).
However, for simplicity of notation, when it is not necessary to specify, we
will use the abbreviated notation $r_{ms}^{-}$ for
$\\{\bar{r}_{ms}^{-},\tilde{r}_{ms}^{-}\\}$.

Barotropic counter-rotating closed quiescent (not cusped tori, regular
topology), cusped or open configurations have centers coincident with maximum
of pressure and density point in the disk, the tori cusp is the minimum point
of pressure and density.

Tori with specific angular momentum $\ell=\ell^{+}<0$ are constrained as
follows [44]

–

For
$\ell^{+}\in\,\,\mathbf{L_{1}^{+}}\equiv\,]\,\ell^{+}_{mb},\ell^{+}_{ms}\,[$
there are quiescent and cusped tori. The cusp is
$r_{\times}\in\,]\,r_{mb}^{+},r_{ms}^{+}\,]$, and the torus center with
maximum pressure is $r_{center}\in\,]\,r_{ms}^{+},r_{[mb]}^{+}\,]$;

–

For
$\ell^{+}\in\,\mathbf{L_{2}^{+}}\equiv\,[\,\ell^{+}_{\gamma},\ell^{+}_{\mathrm{mb}}\,[$
there are quiescent tori and proto-jets. Proto-jets are open cusped
equipotential surfaces. Unstable point, cusp, is on
$r_{j}\in\,]\,r_{\gamma}^{+},r_{mb}^{+}\,]\,$ and center with maximum pressure
$r_{center}\in\,]\,r_{[mb]}^{+},r_{[\gamma]}^{+}\,]\,$;

–

For $\ell^{+}\in\,\mathbf{L_{3}^{+}}:\ell^{+}<\ell^{+}_{\gamma}$, there are
quiescent tori, with center $r_{center}>r_{[\gamma]}^{+}$.

A very large $\ell$ in magnitude is typical of proto-jets or quiescent toroids
orbiting far from the central attractor for counter-rotating tori (with
$\ell=\ell^{+}<0$) or some co-rotating tori with $\ell=\ell^{-}>0$) in a class
of slow rotating NSs.

A more articulated structure characterizes NSs with spin $a\in\,]\,1,a_{2}\,]$
and fluids with $\ell^{-}$ on the equatorial plane ergoregion. For $a>a_{2}$,
where $\mathcal{E}>0$ and $\mathcal{L}>0$ $(\ell^{-}>0)$, the geodetical
orbital structure for $\ell^{-}$ is similar to the counter-rotating tori with
$\ell=\ell^{+}$, where $r_{\gamma}^{-}=0$. For $a\in\,[\,1,a_{0}\,[$, the
$\ell^{-}$ geodesic structure is more articulated. For $a\in\,[\,1,a_{2}\,]$,
there can be tori with $\ell^{-}<0$ and ($\mathcal{E}>0,\mathcal{L}<0$) in the
ergoregion and tori with $\ell^{-}>0$ and ($\mathcal{E}<0,\mathcal{L}<0$).
There can be double tori system with $\ell=\ell^{+}=\ell^{-}<0$ or
$\ell=\ell^{-}>0$. For tori with centers and cusps located on $r_{0}^{\pm}$
there is $\ell=\mathcal{L}=0$, and tori in the limiting case of $\ell=0$ are
considered in Sec. (IV.3). On $r_{\delta}^{\pm}$ ("center" and "cusp"
respectively) there is $\mathcal{E}=0$ ($\ell$ is not well defined).

##  III Flow inversion points

In this section the inversion points and inversion surfaces definitions are
introduced and we start with the analysis of the inversion surfaces extremes
and limits focusing, in particular, on the location of the inversion points
with the respect to the central singularity.

The flow inversion points are defined by the condition $u^{\phi}=0$ on the
flow particle velocities, i.e. as points of vanishing axial velocity of the
flow motion as related to distant static observers.

From the definition of constant $\ell$ and $\mathcal{E}$, Eqs (4) and Eq. (6),
fixed by the orbiting torus, and the inversion point definition we obtain:

$\displaystyle\ell=-\left.\frac{g_{t\phi}}{g_{tt}}\right|_{\mbox{\scriptsize{{T}}}}=-\frac{2ar_{{\mbox{\scriptsize{{T}}}}}\sigma_{{\mbox{\scriptsize{{T}}}}}}{\Sigma_{\mbox{\scriptsize{{T}}}}-2r_{\mbox{\scriptsize{{T}}}}},\quad\mbox{where}\quad\mathcal{E}_{\mbox{\scriptsize{{T}}}}\equiv-
g_{tt}({\mbox{\scriptsize{{T}}}})\dot{t}_{\mbox{\scriptsize{{T}}}},\quad\mathcal{L}_{\mbox{\scriptsize{{T}}}}\equiv
g_{t\phi}({\mbox{\scriptsize{{T}}}})\dot{t}_{\mbox{\scriptsize{{T}}}},$ (11)

where $\mathcal{E}_{\mbox{\scriptsize{{T}}}}$ and
$\mathcal{L}_{\mbox{\scriptsize{{T}}}}$ are the energy and momentum of the
flow particles at the inversion point. More in general we adopt the notation
$q_{\bullet}\equiv q(r_{\bullet})$ for any quantity $q$ evaluated on a radius
$r_{\bullet}$. Therefore, notation $q_{\mbox{\scriptsize{{T}}}}$ or
$q({\mbox{\scriptsize{{T}}}})$ is for any quantity $q$ considered at the
inversion point, and $q_{0}=q(0)$ for any quantity $q$ evaluated at the
initial point of the (free-falling) flow trajectories.

These quantities are independent from the normalization condition, being a
consequence of the definition of constant $\\{\ell,\mathcal{E},\mathcal{L}\\}$
only.

Using Eq. (11), we can discuss the existence of the inversion points radii
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ and planes
$\sigma_{\mbox{\scriptsize{{T}}}}$, for co-rotating $\ell>0$ or counter-
rotating $\ell<0$ fluids, with constant $\ell$, introducing the quantities

$\displaystyle
r_{\mbox{\scriptsize{{T}}}}^{\mp}=\mp\sqrt{a^{2}\left(\sigma_{\mbox{\scriptsize{{T}}}}+\frac{\sigma_{\mbox{\scriptsize{{T}}}}^{2}}{\ell^{2}}-1\right)-\frac{2a\sigma_{\mbox{\scriptsize{{T}}}}}{\ell}+1}-\frac{a\sigma_{\mbox{\scriptsize{{T}}}}}{\ell}+1,\quad\mbox{and}\quad\sigma_{\mbox{\scriptsize{{T}}}}\equiv\frac{\ell\Delta_{\mbox{\scriptsize{{T}}}}}{a(a\ell-2r_{\mbox{\scriptsize{{T}}}})}.$
(12)

where $\Delta_{\mbox{\scriptsize{{T}}}}$ is $\Delta$ defined at the inversion
point–see Fig. (4).

Figure 4: Inversion surfaces $r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell)$ in the
Kerr NS spacetime with spin-mass ratio $a=1.028$, at different counter-
rotating specific angular momentum $\ell$ for
$\ell\in\\{\ell_{\gamma}^{+},\ell_{mb}^{+},\ell_{ms}^{+},-3\\}$ as signed on
the curves. For $a=1.028$ there is $\ell_{\gamma}^{+}=-7.04659$ (yellow
curve), $\ell_{mb}^{+}=-4.84816$ (red curve), $\ell_{ms}^{+}=-4.4173$ (black
curve). Hence $\ell_{\gamma}^{+}<\ell_{mb}^{+}<\ell_{ms}^{+}<-3$ and
$r_{\epsilon}^{\pm}<r_{\mbox{\scriptsize{{T}}}}(\ell_{\gamma}^{+})<r_{\mbox{\scriptsize{{T}}}}(\ell_{mb}^{+})<r_{\mbox{\scriptsize{{T}}}}(\ell_{ms}^{+})<r_{\mbox{\scriptsize{{T}}}}(-3)$.
Gray curves are for radii $r_{\epsilon}^{+}>r_{\epsilon}^{-}$, i.e. the outer
and inner ergosurfaces respectively. All the quantities are dimensionless.
Notation $ms$ refers the marginally stable circular orbit, $mb$ to marginally
bounded orbit and $(\gamma)$ to the marginally circular orbit. There is
$r=\sqrt{z^{2}+y^{2}}$ and $\sigma=y^{2}/(z^{2}+y^{2})$, for
$\sigma\equiv\sin^{2}\theta$ (where $\sigma=1$ is the equatorial plane).

There is

$\displaystyle
r_{\mbox{\scriptsize{{T}}}}^{-}=r_{\mbox{\scriptsize{{T}}}}^{+}=r_{cr}\equiv
a\sqrt{1-\sigma_{\mbox{\scriptsize{{T}}}}}\quad\mbox{on}\quad\ell_{cr}=\frac{a\sigma_{\mbox{\scriptsize{{T}}}}[r_{cr}+1]}{1-r_{cr}^{2}}.$
(13)

There are no timelike or photon-like inversion points for $\ell>0$, as clear
from the study of the conditions $(\mathcal{E},\mathcal{L})$ constant and
normalization condition. Radius $r_{cr}$ is shown in Figs (5),(6),(7),(8).

Figure 5:  Panels show radius
$r_{cr}:r_{{\mbox{\scriptsize{{T}}}}}^{-}=r_{\mbox{\scriptsize{{T}}}}^{+}$
(flow inversion point radius) defined in Eq. (13) on the plane
$\sigma_{a}(\ell)$ for different fluid specific angular momenta $(\ell)$, as
function of the NS spin $a>1$. Left panel shows the analysis for the counter-
rotating fluid specific angular momentum $\ell^{+}<0$. Center and right panels
show the analysis for the momentum $\ell^{-}$. Central panel is a close-up
view of the right panel. Plane $\sigma_{\epsilon}$ is defined in Eq. (3), the
plane $\sigma_{a}$ is defined in Eq. (25). The ergoregion is defined for
$\sigma\in[\,\sigma_{\epsilon},1\,]$. There is
$\sigma\equiv\sin^{2}\theta\in[\,0,1\,]$, where $\sigma=1$ is the equatorial
plane. Notation $(mb)$ is for quantities evaluated on the marginally bounded
orbit, $(ms)$ for marginally stable orbit, $(\gamma)$ for marginally circular
orbit. Radius $r=r_{\epsilon}^{+}=2$ is the outer-ergosurface on the
equatorial plane. Spins $\\{a_{0},a_{2}\\}$, vertical dotted lines, are
defined in Eqs (8) and $a_{1}\equiv
1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec. (II.2). There
is $r_{ms}^{-}\equiv\\{\tilde{r}_{ms}^{-},\bar{r}_{ms}^{-}\\}$, respectively
there is $\\{\tilde{\ell}_{ms}^{-},\bar{\ell}_{ms}^{-}\\}$–see Sec. (II.2).
Here the most general solutions $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs
(12) are shown. (All quantities are dimensionless).

Figure 6: Analysis of counter-rotating flow inversion points
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ with fluid specific angular momentum
$\ell^{+}<0$. (Most general solutions $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ are
shown, discussion on further constraints on the constants of motion is in Sec.
(IV).) Left panel shows in light-gray dashed curves the geodesic structure of
the NSs as in Figs (3). Dotted vertical lines are the spins
$a_{0}<a_{1}<a_{2}$. Spins $\\{a_{0},a_{2}\\}$ are defined in Eqs (8) and
$a_{1}\equiv 1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec.
(II.2). Radius $r_{cr}=r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eq. (13) is shown
as function of the NS spin-mass ratio $a$, at different planes
$\sigma\equiv\sin^{2}\theta$ signed on the curves (left and center panels),
where $\sigma=1$ is the equatorial plane, and as function of the plane
$\sigma\in[\,0,1\,]$, for different spins, signed on the curves (right panel).
(All quantities are dimensionless).

Figure 7: Inversion points $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs (12) are
shown as functions of the NS spin-mass ratio $a$, for different planes
$\sigma\equiv\sin^{2}\theta$ signed on the panels, where $\sigma=1$ is the
equatorial plane. (Here the most general solutions
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ are shown, while discussion on the
constants of motion is in Sec. (IV).) Dashed gray curves are the outer and the
inner ergosurfaces $r_{\epsilon}^{+}\geq r_{\epsilon}^{-}$ respectively. Plain
curves are $r_{\mbox{\scriptsize{{T}}}}^{+}$, while
$r_{\mbox{\scriptsize{{T}}}}^{-}$ are dashed curves. Curves
$r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell)$ for $\sigma=1$ are generally non
linear functions of the spin $a$ when evaluated on the function of the spin
$\ell=\ell^{+}(a)\in\,\\{\ell_{ms}^{+},\ell_{mb}^{+},\ell_{\gamma}^{+}\\}$.
The radii are evaluated for the counter-rotating fluid specific angular
momentum $\ell=\ell^{+}<0$, on the marginally stable orbit $(ms)$-(darker-blue
curve), marginally bounded orbit $(mb)$–(blue-curves) and marginally circular
orbit $r_{\gamma}^{+}$ (light-blue curves). Radius
$r_{cr}=r_{\mbox{\scriptsize{{T}}}}^{\pm}$ (plain gray line) is defined in Eq.
(13). Radius $r_{ms}^{-}$ is the marginally stable orbit for fluid with
$\ell=\ell^{-}$. (All quantities are dimensionless).

Figure 8: Inversion points $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs (12) of
the counter-rotating flows with specific angular momentum $\ell^{+}<0$ as
functions of the planes $\sigma\equiv\sin^{2}\theta$, for different NS spin-
mass ratio $a$ signed on the panels, where $\sigma=1$ is the equatorial plane.
(All quantities are dimensionless). Plain curves are
$r_{\mbox{\scriptsize{{T}}}}^{+}$, while $r_{\mbox{\scriptsize{{T}}}}^{-}$ are
dashed curves. Spins $\\{a_{0},a_{2}\\}$ are defined in Eqs (8) and
$a_{1}\equiv 1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec.
(II.2). Limiting plane $\sigma_{cr}$ is function $\sigma_{a}$ of Eqs (25)
evaluated on the selected fluid specific angular momentum. Dashed gray curves
are the outer and the inner ergosurfaces $r_{\epsilon}^{+}\geq
r_{\epsilon}^{-}$ respectively. Inversion points are evaluated at specific
angular momentum on the marginally stable orbit $(ms)$-(darker-blue curve),
marginally bounded orbit $(mb)$–(blue-curves) and marginally circular orbit
$r_{\gamma}^{+}$ (light-blue curves). See also Figs (7). (The most general
solutions $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ are shown not considering the
constants of motion analysis of Sec. (IV).)

Quantities
$(\sigma_{\mbox{\scriptsize{{T}}}}(r_{\mbox{\scriptsize{{T}}}}),r_{{\mbox{\scriptsize{{T}}}}}(\sigma_{{\mbox{\scriptsize{{T}}}}}))$
depend on the constant of motion $\ell$ only777Inversion radius
$r_{\mbox{\scriptsize{{T}}}}$ and plane $\sigma_{\mbox{\scriptsize{{T}}}}$ of
Eqs (11) are not independent variables, and they can be found solving the
equations of motion or using further assumptions at any other points of the
fluid trajectory., describing both matter and photons, are independent from
the initial particles velocity
($\\{\dot{\sigma}_{{\mbox{\scriptsize{{T}}}}},\dot{r}\\}$, therefore their
dependence on the tori models and accretion process is limited to the
dependence on the fluid specific angular momentum $\ell$, and the results
considered here are adaptable to a variety of different general relativistic
accretion models. Function
$r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{\pm}(\sigma)$
(equivalently $\sigma_{\mbox{\scriptsize{{T}}}}(r_{\mbox{\scriptsize{{T}}}})$)
defines an almost continuous locus of inversion points, surrounding the
central singularity, _inversion surface_ , where the condition $u^{\phi}=0$ is
satisfied. For flows from the orbiting tori there is a maximum and a minimum
boundary
$r_{\mbox{\scriptsize{{T}}}}(a;\ell,\sigma_{\mbox{\scriptsize{{T}}}})$,
associated to a maximum and minimum value of $\ell$. This region, as well as
its boundaries, will be called the _inversion corona_. The extremes of $\ell$
parameters are determined by the tori parametrized with $\ell$. Therefore, the
accretion driven inversion corona has boundaries defined by
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ (or $\sigma_{\mbox{\scriptsize{{T}}}}$),
evaluated on $\ell_{ms}^{\pm}$ and $\ell_{mb}^{\pm}$, while the proto-jets
driven coronas has (in general) boundaries defined by
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ (or $\sigma_{\mbox{\scriptsize{{T}}}}$),
evaluated on $\ell_{mb}^{\pm}$ and $\ell_{\gamma}^{\pm}$–Fig. (4). In Sec.
(III.0.1), the inversion radii and inversion plane extremes and limits are
discussed. We focus on the necessary conditions for the existence of the
inversion points of the counter-rotating and co-rotating flows in Sec. (IV),
considering first the condition $\ell=$constant and then
$\\{\ell,\mathcal{E},\mathcal{L}\\}$ constant.

####  III.0.1 Extremes and limits of the flows inversion points

We consider functions $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ in all generality,
considering $\ell$ constant, but not taking into account the condition of
normalization or the conditions on the signs of $(\mathcal{E},\mathcal{L})$.
There is

$\displaystyle\lim_{\sigma_{\mbox{\scriptsize{{T}}}}\rightarrow
0}r_{{\mbox{\scriptsize{{T}}}}}^{\pm}=r_{\pm},\quad\lim_{\ell\rightarrow(\pm\infty)}r_{{\mbox{\scriptsize{{T}}}}}^{\pm}=r^{\pm}_{\epsilon}\quad\mbox{respectively
and}\quad\lim_{a\rightarrow+\infty}\sigma_{\mbox{\scriptsize{{T}}}}=1,\quad\lim_{\ell\rightarrow\pm\infty}\sigma_{\mbox{\scriptsize{{T}}}}=\sigma_{erg}.$
(14)

The last limit is well defined (i.e.
$\sigma_{\mbox{\scriptsize{{T}}}}\in[\,0,1\,]$), only for
$r_{\mbox{\scriptsize{{T}}}}\leq 2$–Figs (9).

Figure 9: Analysis of the inversion point plane
$\sigma_{\mbox{\scriptsize{{T}}}}$ considered in Sec. (IV.1.2). Here the most
general solution $\sigma_{\mbox{\scriptsize{{T}}}}$ is shown, not considering
the further constraints of Sec. (IV)). Limit for large specific angular
momentum magnitude, defined in Eq. (14), coincident with the ergosurface
$\sigma_{erg}$ defined in Eqs. (2), for different NSs spin-mass ratios signed
on the curves. Spins $\\{a_{0},a_{2}\\}$ are defined in Eqs (8) and
$a_{1}\equiv 1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec.
(II.2). (All quantities are dimensionless).

The limit of the inversion radii on the poles is well defined in the BH case
only. The limit for large specific angular momentum in magnitude is
independent from the co-rotation or counter-rotation flow direction. As noted
also for the BH case, in the limit for large magnitude of the specific angular
momentum, the inversion point approaches the ergosurface, from above or from
below, if the flow is counter-rotating or co-rotating (this issue is detailed
in Secs (IV)). A very large $\ell$ in magnitude is typical of proto-jets
emission or quiescent toroids orbiting far from the central attractor for
counter-rotating tori (with $\ell=\ell^{+}<0$) or some co-rotating tori with
$\ell=\ell^{-}>0$) in a class of slow rotating NSs. The limit of
$r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{\pm}$ for
$a\rightarrow+\infty$ (a very faster spinning NS) is not well defined.
Increasing the spin $a>0$, the inversion point approaches the equatorial plane
$(\sigma_{\mbox{\scriptsize{{T}}}}\leq 1)$–Figs (9,10,7,8). It is interesting
to note that, contrary to the BH case, in the NS there can be two inversion
points radii $r_{\mbox{\scriptsize{{T}}}}^{\pm}$.

Figure 10: Inversion point plane $\sigma_{\mbox{\scriptsize{{T}}}}$ considered
in Sec. (IV.1.2) evaluated on the specific angular momentum $\ell^{+}$ (left
panel) and $\ell^{-}$ (right panel) as function of the inversion radius
$r_{\mbox{\scriptsize{{T}}}}$. Here the most general solution
$\sigma_{\mbox{\scriptsize{{T}}}}$ is shown, not considering the further
constraints of Sec. (IV)). (All quantities are dimensionless). There is
$\sigma\equiv\sin^{2}\theta\in[\,0,1\,]$, and $\sigma=1$ is the equatorial
plane. Left panel: counter-rotating inversion planes for counter-rotating
flows $\ell=\ell^{+}<0$ evaluated at specific angular momentum on the
marginally stable orbit $(ms)$, marginally bounded orbit $(mb)$ and marginally
circular orbit $r_{\gamma}^{+}$, for different spin-mass ratios, as functions
of the radius inversion point. Radius
$r_{ms}^{-}=\\{\bar{r}_{ms}^{-},\tilde{r}_{ms}^{-}\\}$ is the marginally
stable orbit, $r_{mb}^{-}$ is the marginally bounded orbit, for
$\ell=\ell^{-}$. Right panel: Analysis of the inversion point radius
$r_{\mbox{\scriptsize{{T}}}}$ and plane $\sigma_{\mbox{\scriptsize{{T}}}}$
from accretion flows with specific angular momentum $\ell=\ell^{-}$. Dashed
curve is for $a=1.01$; dotted - dashed curve is for $a=a_{0}$, dotted curve is
for $a=a_{2}$, plain curve for $a=2$. Spins $\\{a_{0},a_{2}\\}$ are defined in
Eqs (8).

We proceed below as follows: we first consider the inversion surfaces
morphology analysing the critical radius $r_{cr}$, defined as
$r_{cr}=r_{\mbox{\scriptsize{{T}}}}^{+}=r_{\mbox{\scriptsize{{T}}}}^{-}$,
which provides indication on the inversion surface radial structure, as the
maximum radial distance from the central spinning attractor, according to the
NS spin and flow parameters. Secondly, we concentrate our attention on the
inversion points in the regions close to the singularity poles. The inversion
surfaces is in fact in some cases not well defined in the region. On the other
hand, in other cases the particular structure of the surfaces close the poles
singles out this special region of the spacetime by the effects associated to
the presence of the inversion points–see Fig. (4. In this framework, it could
be expected that this region may be of interest for jet emission (having
funnels along the rotational axis) or other flows coming towards the
singularity poles on the vertical direction. However, most of the accretion
flows of the common scenarios currently under scrutiny concerns the
singularity equatorial plane. For example, accretion from a disk inner edge is
usually considered from the inner part of the disk laid on the attractor (and
disk) equatorial plane. Hence, we will give particular attention to the
inversion surfaces on the equatorial plane. This part of our analysis is then
finalized by the discussion of the surfaces extreme points. By representing
the inversion surfaces in the plane $(r,\sigma)$, we start examining the
extremes of the inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}$ according
to the flow parameter $\ell$, or the NS spin $a$ and finally to the inversion
radius $r_{\mbox{\scriptsize{{T}}}}$. In this way we closely relate the
inversion surfaces to the flows properties, considering variation of the
inversion plane with the respect to the flow specific angular momentum at
different $a$ and radius $r_{\mbox{\scriptsize{{T}}}}$. We will show also that
there is in fact an extreme of the poloidal angle at the inversion point as
function of the central singularity spin, showing that the inversion surfaces
can show strongly different features for different attractors. Fixing $a$ and
flow momentum $\ell$, we proceed considering a general inversion surface
expressed as the function
$\sigma_{\mbox{\scriptsize{{T}}}}(r_{\mbox{\scriptsize{{T}}}})$, tracing its
profiles examining the zeros of the function
$\partial_{r_{\mbox{\scriptsize{{T}}}}}\sigma_{\mbox{\scriptsize{{T}}}}$.
Results of this analysis are also confirmed by the examination of the maxima
and minima of the inversion radius function $r_{\mbox{\scriptsize{{T}}}}$,
with respect to the spin $a$, the flow $\ell$, providing the maximum and
minimum radial distance of the inversion point from the central singularity,
with the respect to the NS spin or the parameter $\ell$. This study ultimately
fixes the vertical and radial extension of the inversion surfaces which are
spacetime geometrical properties defined as toroidal surfaces embedding the
central singularity (ergoregions) presenting a complex structure in the region
close to the attractor poles and with a non–trivial variation with the spin of
the singularity.

On the boundary radius $r_{\mbox{\scriptsize{{T}}}}^{\pm}=r_{cr}$

The inversion radius
$r_{\mbox{\scriptsize{{T}}}}(\sigma_{\mbox{\scriptsize{{T}}}})$ depends on the
geometry and torus parameter $(a,\ell)$, while $r_{cr}$, of Eqs (13) is a
background property, independent explicitly from $\ell$. It represents a
boundary value for $r_{\mbox{\scriptsize{{T}}}}^{\pm}$–see Figs (7,8,6).
Radius $r_{\mbox{\scriptsize{{T}}}}(\sigma_{\mbox{\scriptsize{{T}}}})$
provides a good indication of the maximum extension of the inversion point for
$\sigma$ and for $a$, as shown for the counter-rotating case in Figs (7), Figs
(8). From Figs (6), we can note that the inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}=r_{cr}$ always increases with the NS spin
and decreases with the plane $\sigma$, being larger at the poles–Fig. (4).
Function $r_{\mbox{\scriptsize{{T}}}}^{\pm}=r_{cr}$ has no extremes for plane
$\sigma$. There is however $\partial_{a}r_{cr}=0$ on the equatorial plane.

Below we study the maximum geometric extension of the inversion spheres on the
equatorial plane and on the vertical direction parallel to the central
attractor rotational axis, which is a relevant aspect particularly for proto-
jets flows.

Inversion points close to the poles and on the equatorial plane

We concentrate on the inversion points close to the NS poles $\sigma=0$ and on
the NS equatorial plane $\sigma=1$. (The inversion surfaces on the equatorial
plane are shown in Fig. (4)). There is

$\displaystyle\sigma_{\mbox{\scriptsize{{T}}}}=0:\
\mbox{for}\quad(\ell=0,r_{\mbox{\scriptsize{{T}}}}>0)\quad\mbox{and}\quad\sigma_{\mbox{\scriptsize{{T}}}}=1:\
\mbox{for}\quad\ell<0\,\cup\,\ell\geq\ell_{\gamma}^{-},\quad
r_{\mbox{\scriptsize{{T}}}}=r_{b},\quad\mbox{where}\quad
r_{b}\equiv\frac{2(\ell-a)}{\ell}.$ (15)

In particular there is $\sigma_{\mbox{\scriptsize{{T}}}}=1$ in
$r_{\mbox{\scriptsize{{T}}}}>2$ only for
$a>1,\ell<0,r_{\mbox{\scriptsize{{T}}}}=r_{b}$. In other words, on the
equatorial plane ($\sigma_{\mbox{\scriptsize{{T}}}}=1$) at inversion point,
there is

$\displaystyle\ell_{\beta}\equiv\frac{2a}{2-r},\quad\mbox{or}\quad
r_{\mbox{\scriptsize{{T}}}}=r_{b}=\frac{2(\ell-a)}{\ell},$ (16)

for $\ell<0$ (counter-rotating flows) or for $\ell>\ell_{\gamma}^{-}>0$,
whereas there are no solutions for $\ell_{\gamma}^{-}>\ell>0$.

Extremes of the inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}$

Below we analyze the extremes of the inversion plane
$\sigma_{\mbox{\scriptsize{{T}}}}$ with respect to the inversion point radius
$r_{\mbox{\scriptsize{{T}}}}$, the fluid specific angular momentum $\ell$ and
the dimensionless NS spin $a$.

There is

$\displaystyle\partial_{\ell}\sigma_{\mbox{\scriptsize{{T}}}}=0:\quad\mbox{for}\quad(\ell\neq
0,r_{\mbox{\scriptsize{{T}}}}=0)$
$\displaystyle\partial_{a}\sigma_{\mbox{\scriptsize{{T}}}}=0:\quad\mbox{for}\quad(\ell<0,r_{\mbox{\scriptsize{{T}}}}=\\{0,r_{{\mbox{\scriptsize{{T}}}}\max}^{0}\\}),(\ell=0,r_{\mbox{\scriptsize{{T}}}}>0),(\ell>0,r_{\mbox{\scriptsize{{T}}}}=0),$
$\displaystyle\quad\quad\quad\mbox{where}\quad
r_{{\mbox{\scriptsize{{T}}}}\max}^{0}\equiv\frac{1}{2}\left[a\ell+2\sqrt{a^{2}\left(\ell^{2}+4\right)+4(1-a\ell)}\right].$
$\displaystyle\partial_{r_{\mbox{\scriptsize{{T}}}}}\sigma_{\mbox{\scriptsize{{T}}}}=0:\quad\mbox{for}\quad(\ell<0,r_{\mbox{\scriptsize{{T}}}}=r_{{\mbox{\scriptsize{{T}}}}\max}^{+}),(\ell=0,r_{\mbox{\scriptsize{{T}}}}>0),(\ell=\ell_{\gamma}^{-},r_{\mbox{\scriptsize{{T}}}}=0),(\ell>\ell_{\gamma}^{-},r_{\mbox{\scriptsize{{T}}}}=r_{{\mbox{\scriptsize{{T}}}}\max}^{-}),$
$\displaystyle\quad\mbox{with }\quad
r_{{\mbox{\scriptsize{{T}}}}\max}^{\mp}\equiv\frac{1}{2}\left[a\ell\mp\sqrt{a\left[a\left(\ell^{2}+4\right)-4\ell\right]}\right],\quad\mbox{where}\quad\sigma_{{\mbox{\scriptsize{{T}}}}\max}^{\pm}\equiv\sigma_{{\mbox{\scriptsize{{T}}}}}(r_{{\mbox{\scriptsize{{T}}}}\max}^{\pm})$
(17)

–Figs (11).

Figure 11: Analysis of the extremes of the flow inversion point plane
$r_{{\mbox{\scriptsize{{T}}}}\max}^{\pm}:\partial_{r_{\mbox{\scriptsize{{T}}}}}\sigma_{\mbox{\scriptsize{{T}}}}=0$
of Eqs (18) and maximum inversion plane
$\sigma_{{\mbox{\scriptsize{{T}}}}\max}^{+}$ of Eqs (17). Spins
$\\{a_{0},a_{2}\\}$ are defined in Eqs (8) and $a_{1}\equiv
1.28112:\bar{r}_{ms}^{-}=\tilde{r}_{ms}^{-}$ is defined in Sec. (II.2). See
also analysis of Secs (IV.1.2). (All quantities are dimensionless). Upper line
panels: radius $r_{\max}^{\pm}$ evaluated on the specific angular momenta
$\ell^{+}<0$ and $\ell^{-}$ on the marginally bounded orbit $(mb)$, marginally
stable orbits $(ms)$– Figs (3). Radius $r_{\epsilon}^{+}=2$ is the outer
ergosurface on the equatorial plane. Upper left panel is a close–up view of
the right panel. Bottom left panel: the maximum plane
$\sigma_{{\mbox{\scriptsize{{T}}}}\max}^{+}\equiv\sigma_{{\mbox{\scriptsize{{T}}}}}(r_{{\mbox{\scriptsize{{T}}}}\max}^{+})$
as function of the NS spin–mass ratio, for different fluid specific angular
momenta following the colouring notation of the upper right panel. Inside plot
is a close-up view. Bottom right panel: radius
$r_{{\mbox{\scriptsize{{T}}}}\max}^{-}$ and plane
$\sigma_{{\mbox{\scriptsize{{T}}}}\max}^{-}\equiv\sigma_{{\mbox{\scriptsize{{T}}}}}(r_{{\mbox{\scriptsize{{T}}}}\max}^{-})$
on the fluid specific angular momentum
$\tilde{\ell}_{ms}^{-}\equiv\ell^{-}(\tilde{r}_{ms}^{-})>a>0$. Dashed-gray
curves are the Kerr NS geodesic structure. (In this analysis the most general
solutions $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ are shown not considering the
constants of motion analysis of Sec. (IV).)).

Therefore the inversion surface morphology (plane
$\sigma_{\mbox{\scriptsize{{T}}}}$), for photons and particles, differentiates
the central attractors by their spin, and the extreme points of the spheres
depend on the spin and fluid parameter $\ell$, distinguishing proto-jets from
accretion tori.

Extremes of the inversion point radius $r^{\pm}_{\mbox{\scriptsize{{T}}}}$

Radius $r_{cr}$ always decreases with $\sigma\in\,]\,0,1\,[$–see Figs (7,8,6).
Whereas there is $\partial_{a}r_{cr}=0$, for the limiting case $\sigma=1$ (the
equatorial plane), increasing always with the spin for $\sigma\in\,]\,0,1\,[$.
There is $r_{cr}(\sigma\rightarrow 1)=0$.

The extremes of the inversion radii $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ for
the dimensionless NS spin $a$ are as follows

$\displaystyle\partial_{a}r_{\mbox{\scriptsize{{T}}}}^{-}=0:\quad\sigma_{\mbox{\scriptsize{{T}}}}=1,(\ell<0,\ell>\ell_{\gamma}^{-}),\quad\mbox{and}\quad\partial_{a}r_{\mbox{\scriptsize{{T}}}}^{+}=0:(\sigma_{\mbox{\scriptsize{{T}}}}\in\,]\,0,1\,[\,,\ell=\ell_{e}^{-}<0),(\sigma_{\mbox{\scriptsize{{T}}}}=1,\ell\in\,]\,0,\ell_{\gamma}^{-}\,[\,),$
(18)
$\displaystyle\mbox{where}\quad\ell_{e}^{-}\equiv\frac{a\sigma_{\mbox{\scriptsize{{T}}}}\left[1+\sqrt{1+r_{cr}^{2}}\right]}{-r_{cr}^{2}}$
(19)

see–Figs (12) and also Figs (4).

Figure 12: Analysis of the extremes of the flow inversion point radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs (18). (All quantities are
dimensionless). Right panel: fluid specific angular momentum $\ell_{e}^{-}$
defined in Eqs (19), as functions of the spin-mass ratio $a$, for different
planes $\sigma\equiv\sin^{2}\theta$ (where $\sigma=1$ is the equatorial
plane). Left panel: inversion radii $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ on
$\ell_{e}^{-}$ as functions of $a$, for different $\sigma$. Radii
$r_{\epsilon}^{+}>r_{\epsilon}^{-}$ are the outer and inner ergosurfaces
respectively.

(There are no solutions
$\partial_{\sigma_{\mbox{\scriptsize{{T}}}}}r_{\mbox{\scriptsize{{T}}}}^{\pm}=0$,
and $\partial_{a}r_{\mbox{\scriptsize{{T}}}}^{+}=0$ for $\ell=0$–Figs (8))

Considering the inversion point radius as function of the fluid specific
angular momentum $\ell$, there is

$\displaystyle\partial_{\ell}r_{\mbox{\scriptsize{{T}}}}^{-}=0:\quad\sigma_{\mbox{\scriptsize{{T}}}}=1\quad(\ell<0,\ell>\ell_{\gamma}^{-}>1);\quad\partial_{\ell}r_{\mbox{\scriptsize{{T}}}}^{+}=0:\quad\sigma_{\mbox{\scriptsize{{T}}}}=1\quad(\ell\in\,]\,0,1\,]\,,\ell_{\gamma}^{-}>\ell>1),$
(20)

(see also Eqs (15,16)). There are extremes of the inversion point radius, as
function of $\ell$ on the equatorial plane.

For the co-rotating flows inversion points ($\ell>0$), as in Eqs (20), (18),
(17), (15), condition $\ell>\ell_{\gamma}^{-}>1$, has to be evaluated. (There
is $\ell_{\gamma}^{-}=a$, for $\ell^{-}$ evaluated at the last circular co-
rotating orbit $r_{\gamma}^{-}=0$ and on $r_{[\gamma]}^{-}=a^{2}$, bottom
boundary for the $\mathbf{L_{3}^{-}}$ range.). There is:

$\displaystyle\ell^{-}=\ell_{\gamma}^{-}:\quad\mbox{for}\quad\left(r=r_{[\gamma]}^{-},a\neq
a_{3}\right),(r=0,a\geq 1),\quad\mbox{where}\quad a_{3}\equiv\sqrt{3},$ (21)
$\displaystyle\ell^{-}>\ell_{\gamma}^{-}:\quad\mbox{for}\quad
a\in\,[\,1,a_{0}\,[\,:\left(r\in\,]\,r_{\delta}^{-},r_{\delta}^{+}\,[\,,r>r_{[\gamma]}^{-},r\neq
r_{\delta}^{(1)}\right),$ $\displaystyle\;\qquad\qquad\quad\mbox{for}\quad
a\in\,[\,a_{0},a_{3}\,[\,:\left(r>r_{[\gamma]}^{-},r\neq
r_{\delta}^{(1)}\right),\;$ $\displaystyle\;\qquad\qquad\quad\mbox{for}\quad
a\geq a_{3}:r>r_{[\gamma]}^{-},$

where

$\displaystyle
r_{\delta}^{(1)}\equiv\frac{\delta^{+}+\delta^{-}+8}{6},\quad\mbox{and}\quad\delta^{\pm}\equiv
2^{2/3}\sqrt[3]{27a^{2}\pm 3\sqrt{81a^{4}-96a^{2}}-16},$ (22)

there is $\ell_{\gamma}^{-}=a$ and $r_{[\gamma]}^{-}=a^{2}$–Figs (13).

Figure 13: Radius $r_{\delta}^{(1)}$, defined in Eq (22), as function of the
NS spin-mass ratio. It refers to the conditions $\ell^{-}\geq a$ of Eqs (21).
Gray dashed curves are the geodesic structure of the NSs as in Figs (3). There
is $a_{3}=\sqrt{3}$. Spins $\\{a_{0},a_{2}\\}$ are defined in Eqs (8) and
$a_{1}\equiv 1.28112$. (All quantities are dimensionless).

Figure 14: Analysis of the counter-rotating flow inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs (12) with specific angular momentum
$\ell^{+}<0$, as functions of the NS spin-mass ratio $a$, for different planes
$\sigma\equiv\sin^{2}\theta$ signed on the panels, where $\sigma=1$ is the
equatorial plane. (Here most general solutions
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ shown not considering constraints of Sec.
(IV).) Difference $\Delta r_{\mbox{\scriptsize{{T}}}}^{\pm}$ defined on the
curves according to the notations of the left and center panels. Inversion
points are evaluated at specific angular momentum on the marginally bounded
orbit $(mb)$, marginally stable orbit $(ms)$ and marginally circular orbit
$r_{\gamma}^{+}$. Dashed curves are differences
$(r_{\mbox{\scriptsize{{T}}}}^{-}-r_{\mbox{\scriptsize{{T}}}}^{+})_{\ell}$ at
fixed $\ell$. Plain curves are differences
$(r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell_{1})-r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell_{2}))$
for two fixed values of $\ell$. (All quantities are dimensionless).

Figure 15: Analysis of the counter-rotating flow inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of Eqs (12) with specific angular momentum
$\ell^{+}<0$, as function of planes
$\sigma_{\mbox{\scriptsize{{T}}}}=\sigma\equiv\sin^{2}\theta\in[\,0,\,1]$,
where $\sigma=1$ is the equatorial plane, for different NS spin-mass ratios
signed on the panel. (Here most general solutions
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$ shown not considering constraints of Sec.
(IV).) Spins $\\{a_{0},a_{1}\\}$ are defined in Eqs (8). Difference $\Delta
r_{\mbox{\scriptsize{{T}}}}^{\pm}$ defined on the curves according to the
notations of the left and center panels of Figs (14) . Inversion points are
evaluated at specific angular momentum on the marginally bounded orbit $(mb)$,
marginally stable orbit $(ms)$ and marginally circular orbit $r_{\gamma}^{+}$.
Dashed curves are differences
$(r_{\mbox{\scriptsize{{T}}}}^{-}-r_{\mbox{\scriptsize{{T}}}}^{+})_{\ell}$ at
fixed $\ell$. Plain curves are differences
$(r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell_{1})-r_{\mbox{\scriptsize{{T}}}}^{\pm}(\ell_{2}))$
for two fixed values of $\ell$. (All quantities are dimensionless).

There are no inversion points for $\ell=\ell^{-}>0$ for $a>a_{2}$. The
inversion corona therefore is a closed region, generally of small extension
and thickness. The vertical coordinate $z_{\mbox{\scriptsize{{T}}}}$,
elongation on the central rotational axis of the inversion point, for the
counter-rotating flows, is the order of $\lesssim 1.4$ (central attractor
mass, where $r=\sqrt{z^{2}+y^{2}}$ and
$\sigma\equiv\sin^{2}\theta=y^{2}/(z^{2}+y^{2})$.).

##  IV Inversion points of the counter-rotating and co-rotating flows

Inversion point definition, as the set of points where $u^{\phi}=0$ identifies
a surface, inversion surface, surrounding the central singularity, depending
only on $\ell$ parameter of the freely infalling (or outfalling) matter. The
inversion surface is a general property of the orbits in the Kerr NS
spacetime. In this article we study in general function $\ell(u^{\phi}=0)$,
constraining then $\ell_{\mbox{\scriptsize{{T}}}}$ when framed in the
particles flow and tori flows inversion points. We assume: 1. constance of
$(\mathcal{E}_{\mbox{\scriptsize{{T}}}},\mathcal{L}_{\mbox{\scriptsize{{T}}}})$
(implied by $\ell=$constant) evaluated at the inversion point with
$\dot{t}$>0\. 2. We then consider the normalization condition at the inversion
point, distinguishing particles and photons in the flow. 3. The matter flow is
then related to the orbiting structures, constraining the range of values for
$\ell$, to define the inversion corona for proto-jets or accretion driven
flows, as background geometry properties, depending only on the spacetime
spin. (Inversion surface describes also particles with
$\dot{r}_{\mbox{\scriptsize{{T}}}}>0$ (outgoing particles) or particles with
an axial velocity $\dot{\theta}\neq 0$ along the BH rotational axis.).

Sign of $\\{\ell,\mathcal{L}\\}$ at the inversion point for the counter-
rotation case is addressed in Sec. (IV.1.3). In this section we explore in
details the inversion points of the accreting flows. Specifically, counter-
rotating flows $(\ell<0)$ inversion points are detailed in Sec. (IV.1). In
Sec. (IV.1.1) we analyze the inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$, while in Sec. (IV.1.2) we focus on the
inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}$. The co-rotating ($\ell>0$)
flows inversion points are detailed in Sec. (IV.2).

Figure 16: Plane $\sigma_{a}$, defined in Eq. (25), as function of the NS
spin-mass ratio $a>1$ evaluated on the counter-rotating fluid specific angular
momentum $\ell^{+}<0$ (left panel). Center and right panels show the plane
$\sigma_{a}$ for different specific angular momenta $\ell^{-}$, center panel
is a close–up view of the right panel. Color notation follows the closed-up
view panel. Notation $(mb)$ is for quantities evaluated on the marginally
bounded orbit, $(ms)$ for marginally stable orbit, $(\gamma)$ for marginally
circular orbit. Plane $\sigma_{\epsilon}$ is defined in Eq. (3). The
ergoregion is defined for $\sigma\in[\,\sigma_{\epsilon},\,1]$. There is
$\sigma\equiv\sin^{2}\theta\in[\,0,1\,]$, where $\sigma=1$ is the equatorial
plane. Dotted vertical lines are spins $a_{0}<a_{1}<a_{2}$, defined in Eqs
(8). The marginally stable orbit is
$r_{ms}^{-}\equiv\\{\tilde{r}_{ms}^{-},\bar{r}_{ms}^{-}\\}$, respectively
there is $\\{\tilde{\ell}_{ms}^{-},\bar{\ell}_{ms}^{-}\\}$–see Sec. (II.2).
(All quantities are dimensionless).

###  IV.1 Inversion points of counter-rotating flows

In this section we study the inversion points of counter-rotating flows
$(\ell<0)$, considering in Sec. (IV.1.1) the inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}(\sigma)$, while in Sec. (IV.1.2) we explore
the inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}(r)$. In Sec. (IV.1.1)
and Sec. (IV.1.2) we develop our analysis considering definitions in Eq. (11),
derived by the definition of $\ell=$constant and condition $u^{\phi}=0$.
Whereas in Sec. (IV.1.3), we discuss the conditions for $u^{\phi}=0$ and
$\ell<0$, within $(\mathcal{E},\mathcal{L})$ constant, considering definitions
in Eq. (11).

According to the analysis in Sec. (II.1) and Sec. (II.2) we consider (I) tori
with $\ell=\ell^{+}<0$, that is with $(\mathcal{L}<0,\mathcal{E}>0)$, located
(centered) at $r>r_{\gamma}^{+}$– Figs (7,8,14,15). (II) Tori with momentum
$\ell=\ell^{-}<0$, that is with $(\mathcal{L}<0,\mathcal{E}>0)$, in
$]\,r_{0}^{-},r_{\delta}^{-}\,[\,\cup\,]\,r_{\delta}^{+},r_{0}^{+}\,[$, for
spacetimes $a\in[\,1,\,a_{0}]$, and in the orbital region
$]\,r_{0}^{-},r_{0}^{+}\,[$, for NSs with spin $a\in\,]\,a_{0},a_{2}\,]$. The
case $\ell=0$ (that is on $r_{0}^{\pm}:\mathcal{L}=0$) is considered in Sec.
(IV.3). On $r_{\delta}^{\pm}$, there is $\mathcal{E}=K=0$ and $\mathcal{L}<0$.
(There are no inversion points for $\mathcal{L}>0,\mathcal{E}<0$.)

####  IV.1.1 Inversion radius $r_{\mbox{\scriptsize{{T}}}}^{\pm}$ of the
counter-rotating $(\ell<0)$ flows

We concentrate on the existence of a inversion radius
$r_{\mbox{\scriptsize{{T}}}}^{\pm}$, using the definitions Eq. (11). For
counter-rotating fluids $(\ell<0)$ there is

$\displaystyle\ell<0\quad\mbox{for}:$
$\displaystyle\sigma=\sigma_{\mbox{\scriptsize{{T}}}}:\quad(r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,2\,]\,,\ell<0),(r_{\mbox{\scriptsize{{T}}}}>2,\ell\in\,[\,\ell_{\beta},0\,[\,)).$
(23)

Conditions (23) can be expressed also as

$\displaystyle\sigma_{\mbox{\scriptsize{{T}}}}\in\,]\,0,\sigma_{\epsilon}\,[\,:(\ell=\ell_{a\sigma},r=r_{\mbox{\scriptsize{{T}}}}^{-}),(\ell\in\,]\,\ell_{a\sigma},0\,[\,,r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{\pm});\quad\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,\sigma_{\epsilon},1\,[\,:(r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{\pm});\quad\sigma_{\mbox{\scriptsize{{T}}}}=1:(r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{+}).$

where

$\displaystyle\ell_{a\sigma}^{\mp}\equiv\frac{a\sigma_{\mbox{\scriptsize{{T}}}}}{1-r_{cr}^{2}}\mp\sqrt{\frac{a^{2}r_{cr}^{2}\sigma_{\mbox{\scriptsize{{T}}}}^{2}}{\left[1-r_{cr}^{2}\right]^{2}}},$
(24)

or equivalently it is possible to express this condition also as

$\displaystyle\sigma_{\mbox{\scriptsize{{T}}}}=\sigma_{a}:(r=r_{\mbox{\scriptsize{{T}}}}^{-});\quad\sigma_{\mbox{\scriptsize{{T}}}}\in\,]\,\sigma_{a},1\,[\,:(r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{\pm});\quad\sigma_{\mbox{\scriptsize{{T}}}}=1:(r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{+}),$

where

$\displaystyle\sigma_{a}\equiv\frac{1}{2}\left[\sqrt{\ell^{2}\left(-\frac{4\ell}{a}+\ell^{2}+4\right)}-\ell^{2}\right]+\frac{\ell}{a},$
(25)

(see Figs (16) and Figs (5)). There are no solutions on the poles i.e.
$\sigma=0$ –Figs (10). There are two inversion points
$r_{\mbox{\scriptsize{{T}}}}^{\pm}(\sigma)$ for fixed
$(\sigma_{\mbox{\scriptsize{{T}}}},\ell)$, while on the equatorial plane there
is always one only radius $r_{\mbox{\scriptsize{{T}}}}^{+}$. This implies
that, at plane different from the equatorial, there are two radii, where
$\ell<0$ and $u^{\phi}=0$. This fact, analyzed in details in Sec. (IV.1.3),
constitutes a major difference with the BH case.

The inversion point radius $r_{\mbox{\scriptsize{{T}}}}^{+}$ on the equatorial
plane is always _out_ of the ergoregion and there is

$\displaystyle\sigma_{\mbox{\scriptsize{{T}}}}=1:\ell<0,r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{+}\quad\mbox{equivalently}\quad
r_{\mbox{\scriptsize{{T}}}}>2,\ell=\ell_{\beta},$ (26)
$\displaystyle\mbox{there are \emph{no}
solutions}\quad\ell<0\quad\mbox{for}\quad
r_{\mbox{\scriptsize{{T}}}}\in[\,r_{\epsilon}^{-},r_{\epsilon}^{+}\,]$ (27)

–see also Figs (7,8,12,4). More specifically, there are solutions

$\displaystyle
r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{-}\quad\mbox{for}\quad
r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,r_{\epsilon}^{-}\,[\,,\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,\sigma_{\epsilon}^{+},1\,[$
(28) $\displaystyle\mbox{equivalently
for}\quad\sigma=\sigma_{\mbox{\scriptsize{{T}}}}:\quad
r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,\,1[\,,\ell\leq\ell_{\rho},\quad\mbox{where}\quad\ell_{\rho}\equiv-\frac{2\left(a^{2}-1\right)r}{a(r-1)^{2}},$
$\displaystyle
r_{\mbox{\scriptsize{{T}}}}=r_{\mbox{\scriptsize{{T}}}}^{+}\quad\mbox{for}\quad
r_{\mbox{\scriptsize{{T}}}}>r_{\epsilon}^{+},\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,\sigma_{\epsilon},1\,]\,,$
$\displaystyle\mbox{equivalently for
}\quad\sigma=\sigma_{\mbox{\scriptsize{{T}}}}:\quad(r_{\mbox{\scriptsize{{T}}}}\in\,]\,1,2\,]\,,\ell\leq\ell_{\rho}),(r_{\mbox{\scriptsize{{T}}}}>2,\ell\in\,[\,\ell_{\beta},\ell_{\rho}\,]\,).$

The counter–rotating case includes also tori with $\ell=\ell^{+}<0$ and
$\ell=\ell^{-}<0$ in the ergoregion for NSs with $a\in\,[\,1,a_{2}\,]$.

####  IV.1.2 Inversion plane $\sigma_{\mbox{\scriptsize{{T}}}}$ of the
counter-rotating $(\ell<0)$ flows

We discuss the necessary conditions for the existence of a inversion plane
$\sigma_{\mbox{\scriptsize{{T}}}}\in[\,0,1\,]$ as function of the inversion
radius $r_{\mbox{\scriptsize{{T}}}}$, using the definitions Eq. (11). As seen
in Sec. (IV.1.1), a counter-rotating inversion point _must_ be out of the
ergoregion, as confirmed also from the analysis in Figs (11,10,7,8).
Therefore, for $\ell<0$ ($\ell=\ell^{+}<0$ or $\ell=\ell^{-}<0$, defined in
$a\in\,]\,1,a_{2}\,[$ and
$r\in\,]\,r_{0}^{-},r_{0}^{+}\,[\,-\,]\,r_{\delta}^{-},r_{\delta}^{+}\,[$)
there are _no_ inversion points in $[\,r_{\epsilon}^{-},r_{\epsilon}^{+}\,]$.

There are inversion points
$\sigma=\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,0,1\,]$ in:

$\displaystyle\ell<0,r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,r_{b}\,]\quad\mbox{or,
more
precisely}\quad(\ell<0,r_{\mbox{\scriptsize{{T}}}}>0)\quad\mbox{and}\quad\left(\ell\in\,\left[\,\ell_{\beta},0\,\right[\,,r_{\mbox{\scriptsize{{T}}}}\leq
2\right).$ (29)

Focusing on the ergoregion, there are inversion points
$\sigma=\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,0,1\,]$ for

$\displaystyle
r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,r_{\epsilon}^{-}\,[\,:\quad\ell\leq\ell_{\rho}<0,\quad
0<r_{\mbox{\scriptsize{{T}}}}\leq r_{b}<1;$ (30) $\displaystyle\quad
r_{\mbox{\scriptsize{{T}}}}>r_{\epsilon}^{+}:\quad
r_{\mbox{\scriptsize{{T}}}}\in\,[\,r_{B}^{+},r_{b}\,]\,,\quad\mbox{where}\quad
r_{B}^{\mp}\equiv\frac{1}{a\ell}+1-\frac{a}{\ell}\mp\sqrt{\frac{\left(a^{2}-1\right)\left(a^{2}-2a\ell-1\right)}{a^{2}\ell^{2}}}.$
$\displaystyle\mbox{more
precisely:}\quad\left(r_{\mbox{\scriptsize{{T}}}}\in\,]\,1,2\,]\,,\ell\leq\ell_{\rho}\right),\left(r_{\mbox{\scriptsize{{T}}}}>2,\ell\in\,[\,\ell_{\beta},\ell_{\rho}\,]\,\right).$

In BH case the region $]\,0,r_{\epsilon}^{-}\,[$ is not observable by the
distant observer, constituting a main difference with the super-spinars where
counter-rotating inversion points are possible in this region.

####  IV.1.3 Counter-rotation at the inversion point

We explore the conditions for $u^{\phi}=0$ and $\ell<0$, within
$(\mathcal{E}>0,\mathcal{L}<0)$ constant, using definition Eq. (11). (In this
analysis we do not consider the normalization condition.). This case includes
tori with specific angular momentum $\ell=\ell^{+}<0$ and $\ell=\ell^{-}<0$
(possible for $a<a_{2}$ with $\mathcal{E}>0$ and $\mathcal{L}<0$ in
$r\in\,]\,r_{0}^{-},r_{0}^{+}\,[\,-\,[\,r_{\delta}^{-},r_{\delta}^{+}\,]$).

There is

$\displaystyle(\ell\leq 0):\quad u^{\phi}=0\;(u^{t}>0)\quad\\{\mathcal{E}\geq
0,\mathcal{L}\leq
0\\},\quad(\sigma_{\mbox{\scriptsize{{T}}}}\in\,]\,0,\sigma_{\epsilon}\,[\,,r_{\mbox{\scriptsize{{T}}}}>0),(\sigma_{\mbox{\scriptsize{{T}}}}\in\,[\,\sigma_{\epsilon},1\,]\,,r_{\mbox{\scriptsize{{T}}}}\in\,]\,0,r_{\epsilon}^{-}\,[\,\cup\,r_{\mbox{\scriptsize{{T}}}}>r_{\epsilon}^{+})$
(31)

where the velocity $u^{t}$, the energy $\mathcal{E}$ and angular momentum
$\mathcal{L}$ are

$\displaystyle
u^{t}=\frac{-\mathcal{L}\Sigma_{\mbox{\scriptsize{{T}}}}}{2ar_{\mbox{\scriptsize{{T}}}}\sigma_{\mbox{\scriptsize{{T}}}}}=\frac{-\mathcal{E}\Sigma_{\mbox{\scriptsize{{T}}}}}{2r_{\mbox{\scriptsize{{T}}}}-\Sigma_{\mbox{\scriptsize{{T}}}}},\quad\mathcal{E}=\frac{2r_{\mbox{\scriptsize{{T}}}}+\Sigma_{\mbox{\scriptsize{{T}}}}}{\Sigma_{\mbox{\scriptsize{{T}}}}}u^{t}=\frac{-\mathcal{L}\left[\Sigma_{\mbox{\scriptsize{{T}}}}-2r_{\mbox{\scriptsize{{T}}}}\right]}{2ar_{\mbox{\scriptsize{{T}}}}\sigma_{\mbox{\scriptsize{{T}}}}},\quad\mathcal{L}=-\frac{2ar_{\mbox{\scriptsize{{T}}}}\sigma_{\mbox{\scriptsize{{T}}}}u^{t}}{\Sigma_{\mbox{\scriptsize{{T}}}}}=-\frac{2a\mathcal{E}r_{\mbox{\scriptsize{{T}}}}\sigma_{\mbox{\scriptsize{{T}}}}}{\Sigma_{\mbox{\scriptsize{{T}}}}-2r_{\mbox{\scriptsize{{T}}}}}.$
(32)

The case $\ell=0$ is possible only for $\mathcal{L}=0$ and $\sigma=0$ (on the
BH axis) where $\mathcal{E}={u^{t}\Delta}/({a^{2}+r^{2}})$, which is
considered in Sec. (IV.2) Hence below we focus on the time-like and photon-
like components.

Time-like and photon-like components

For $(\mathcal{E}>0,\mathcal{L}<0)$ the analysis is in Eq. (31). Further
constraints follow from the normalization condition on the flow components.
Let us define :

$\displaystyle\mathcal{E}_{\Gamma}\equiv\frac{\Sigma_{{\mbox{\scriptsize{{T}}}}}-2r_{{\mbox{\scriptsize{{T}}}}}}{\Sigma_{{\mbox{\scriptsize{{T}}}}}}\dot{t_{{\mbox{\scriptsize{{T}}}}}},\quad\mathcal{L}_{\Gamma}=-\frac{2ar_{{\mbox{\scriptsize{{T}}}}}\sigma_{{\mbox{\scriptsize{{T}}}}}\dot{t}_{{\mbox{\scriptsize{{T}}}}}}{\Sigma_{{\mbox{\scriptsize{{T}}}}}},\quad\dot{\theta}^{2}_{\Gamma}\equiv\frac{\tilde{\mathbf{x}}_{\mbox{\scriptsize{{T}}}}-a^{2}(\sigma_{{\mbox{\scriptsize{{T}}}}}-1)\left(\dot{t}_{{\mbox{\scriptsize{{T}}}}}^{2}-1\right)}{\Sigma_{{\mbox{\scriptsize{{T}}}}}^{2}},\quad\tilde{\mathbf{x}}_{\mbox{\scriptsize{{T}}}}\equiv
r_{{\mbox{\scriptsize{{T}}}}}\left[(r_{{\mbox{\scriptsize{{T}}}}}-2)\dot{t}_{{\mbox{\scriptsize{{T}}}}}^{2}-r_{{\mbox{\scriptsize{{T}}}}}\right]$
(33)
$\displaystyle\dot{r}^{2}_{\Gamma}\equiv-\frac{\Delta_{{\mbox{\scriptsize{{T}}}}}\left[a^{4}(\sigma_{{\mbox{\scriptsize{{T}}}}}-1)^{2}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}+a^{2}(\sigma_{{\mbox{\scriptsize{{T}}}}}-1)\left(\dot{t}_{{\mbox{\scriptsize{{T}}}}}^{2}-1-2r_{{\mbox{\scriptsize{{T}}}}}^{2}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\right)+r_{{\mbox{\scriptsize{{T}}}}}^{4}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}-\mathbf{\tilde{x}}_{\mbox{\scriptsize{{T}}}}\right]}{\Sigma_{{\mbox{\scriptsize{{T}}}}}^{2}},\quad\dot{t}_{\Gamma}\equiv\sqrt{\frac{\Sigma_{{\mbox{\scriptsize{{T}}}}}}{\Sigma_{{\mbox{\scriptsize{{T}}}}}-2r_{{\mbox{\scriptsize{{T}}}}}}}.$

For time-like particles, inversion points are for
$(\dot{r}_{{\mbox{\scriptsize{{T}}}}}^{2}=\dot{r}^{2}_{\Gamma},\mathcal{L}=\mathcal{L}_{\Gamma},\mathcal{E}=\mathcal{E}_{\Gamma})$
and

$\displaystyle\sigma_{{\mbox{\scriptsize{{T}}}}}\in\,]\,0,\sigma_{\epsilon}\,[\,:\,(\dot{t}_{{\mbox{\scriptsize{{T}}}}}=\dot{t}_{\Gamma},\dot{\theta}^{2}=0);(\dot{t}_{{\mbox{\scriptsize{{T}}}}}>\dot{t}_{\Gamma},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\,\in\,[\,0,\dot{\theta}^{2}_{\Gamma}\,\,]\,);$
(34)
$\displaystyle\sigma_{{\mbox{\scriptsize{{T}}}}}=\sigma_{\epsilon}:(r_{{\mbox{\scriptsize{{T}}}}}\neq
r_{\epsilon}^{-},(\dot{t}_{{\mbox{\scriptsize{{T}}}}}=\dot{t}_{\Gamma},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}=0);(\dot{t}_{{\mbox{\scriptsize{{T}}}}}>\dot{t}_{\Gamma},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\,\in\,[\,0,\dot{\theta}^{2}_{\Gamma}\,\,]\,));$
$\displaystyle\sigma_{{\mbox{\scriptsize{{T}}}}}\in\,]\,\sigma_{\epsilon},1\,]\,:\,(r_{{\mbox{\scriptsize{{T}}}}}\in\,]\,0,r_{\epsilon}^{-}\,[\,\cup\,\,r_{{\mbox{\scriptsize{{T}}}}}>r_{\epsilon}^{+},(\dot{t}_{{\mbox{\scriptsize{{T}}}}}=\dot{t}_{\Gamma},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}=0);(\dot{t}_{{\mbox{\scriptsize{{T}}}}}>\dot{t}_{\Gamma},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\in\,[\,0,\dot{\theta}^{2}_{\Gamma}\,\,]\,).$

Let us introduce:

$\displaystyle\dot{\theta}^{2}_{X}\equiv\frac{\dot{t}_{{\mbox{\scriptsize{{T}}}}}^{2}\left[\Sigma_{{\mbox{\scriptsize{{T}}}}}-2r_{{\mbox{\scriptsize{{T}}}}}\right]}{\Sigma_{{\mbox{\scriptsize{{T}}}}}^{2}};\quad\dot{r}^{2}_{X}=-\frac{\Delta_{{\mbox{\scriptsize{{T}}}}}\left[a^{4}(\sigma_{{\mbox{\scriptsize{{T}}}}}-1)^{2}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}+a^{2}(\sigma_{{\mbox{\scriptsize{{T}}}}}-1)\left(\dot{t}_{{\mbox{\scriptsize{{T}}}}}^{2}-2r_{{\mbox{\scriptsize{{T}}}}}^{2}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\right)+r_{{\mbox{\scriptsize{{T}}}}}^{2}(r_{{\mbox{\scriptsize{{T}}}}}^{2}\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}-1)-\tilde{\mathbf{x}}_{\mbox{\scriptsize{{T}}}}\right]}{\Sigma_{{\mbox{\scriptsize{{T}}}}}^{2}}.$
(35)

For photon–like particles there is
$(\dot{r}_{{\mbox{\scriptsize{{T}}}}}^{2}=\dot{r}^{2}_{X},\mathcal{L}=\mathcal{L}_{\Gamma},\mathcal{E}=\mathcal{E}_{\Gamma})$,
with

$\displaystyle\sigma_{{\mbox{\scriptsize{{T}}}}}\in\,\,]\,0,\sigma_{\epsilon}\,[\,:\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\in\,[\,0,\dot{\theta}^{2}_{X}\,\,]\,;\,\quad\sigma_{{\mbox{\scriptsize{{T}}}}}=\sigma_{\epsilon}:(r_{{\mbox{\scriptsize{{T}}}}}\neq
r_{\epsilon}^{-}\,[\,,\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\,\in\,[\,0,\dot{\theta}^{2}_{X}\,\,]\,);\;\;\sigma_{{\mbox{\scriptsize{{T}}}}}\in\,\,]\,\sigma_{\epsilon},1\,]\,:(r_{{\mbox{\scriptsize{{T}}}}}\in\,]\,0,r_{\epsilon}^{-}\,[\,\cup\,r_{{\mbox{\scriptsize{{T}}}}}>r_{\epsilon}^{+},\dot{\theta}_{{\mbox{\scriptsize{{T}}}}}^{2}\in\,[\,0,\dot{\theta}^{2}_{X}\,\,]\,).$

Although the inversion spheres are similarly defined for photons and matter
components by the $\ell$–parameter range only, the inversion point coordinate
and the velocity components distinguish photon–like and time-like components
differently according to the super-spinars spin.

###  IV.2 Inversion points of the co-rotating ($\ell>0$) flows

There are _no_ inversion points for $\ell>0$ with $\mathcal{L}\geq 0$ and
$\mathcal{E}\geq 0$. A necessary condition for the existence of an inversion
point, from definition Eq. (11) of $\mathcal{L}$ at
$(r_{\mbox{\scriptsize{{T}}}},\sigma_{\mbox{\scriptsize{{T}}}})$, is
$\mathcal{L}<0$ and therefore it occurs in the co-rotating case only for
$\mathcal{L}<0$ and $\mathcal{E}<0$. (However these solutions are only for
spacelike (tachyonic) particles.).

###  IV.3 Inversion points: limiting case $\ell=0$

In the NSs with spin $a\in[1,a_{2}]$ there are orbits with angular momentum
$\mathcal{L}=0$, therefore with specific angular momentum $\ell=0$, (and
$\mathcal{E}>0$) on $r_{0}^{\pm}$ with $r_{0}^{-}\leq
r_{0}^{+}<r_{\epsilon}^{+}=2$ (on the equatorial plane). Within these
conditions inversion points can form in some limiting cases $(\sigma=0,r\geq
0),(\sigma\in\,]\,0,1\,[\,,r=0)$. However within condition $\ell=0$ there
could be $u^{\phi}\neq 0$, i.e., ZAMOs are not (generally) zero angular
velocity observers (ZAVOS).

##  V Discussion and Conclusions

Inversion coronas are spherical shells surrounding the central singularity,
interpreted as a property of the background geometry which can identify the
central attractor, distinguishing BHs and NSs, and the accretion configuration
characteristics defining accretion or proto-jets driven flows. Inversion
coronas are generally closed spherical shells with narrow thickness and small
extension on the equatorial plane and rotational central axis, varying little
with the fluid initial conditions and the details of the emission processes,
having therefore a possible remarkable observational significance and
applicability to various orbiting toroidal models.

We focused particularly on the estimation of the coronas maximum extension
along the central axis and on the equatorial plane, analysing the coronas
thickness, the constraints to the verticality and equatorial extension of the
inversion points corona, for accretion driven and proto-jets driven flows. The
inversion surface extension on the central rotational axis, maximum vertical
height, is of the order of $\lesssim 1.44$ for accretion driven flows.

We studied function $\ell_{\mbox{\scriptsize{{T}}}}$, defining the inversion
surface in all generality, later assuming constraints on the constants
($\mathcal{E}$,$\mathcal{L}$) signs, fixing the normalization constant and
studying the inversion coronas for tori driven or proto-jets driven inversion
points. We discussed the definition of fluids and particles co-rotation and
counter-rotation, in relation to the $\\{\ell,\mathcal{L},\mathcal{E}\\}$ sign
and the NS casual structure. Co-rotating flows ($a\ell>0$) have no (timelike
or photon like particle) inversion points.

The disk inner region is an extremely active part of the disk and, although
the free falling hypothesis we consider is widely used and justified, the flow
in the inner edge region could be exposed to several different factors. The
inner region of the disk can be subjected to Pointing effects and magnetic
fields. Inner edge can also be distorted by local oscillations affecting the
-accretion rate. High-frequency quasi periodic oscillations could also
modulate the accretion disk in oscillatory modes or having stabilizing
effects. All these aspects can be of great importance for the study of the
components of the accretion and emission flux of jets, they are however not
considered in this first work on the analysis of azimuthal inversion points.
On the other hand, the characteristics demonstrated for inversion coronas,
defined as background property, with little internal variation depending on
the particle specific angular momentum assure of their significance and very
broad applicability.

BHs and super-spinning geometries are also distinguished by the presence of
double inversion points for planes different from the equatorial. This
feature, which can be inferred from Figs. (7,8), has profound consequences for
particles motion and the observational properties of the coronas. We argue
that the inner corona region, closest to the central attractor, is the most
active part and observably recognizable.

The results of this analysis therefore individuate strong signatures of Kerr
super-spinars, in both accretion and jet counter-rotating, flows,
distinguishing Kerr NSs an BHs, and a relevance of our findings is in the fact
that, as for the BH case, the inversion corona is a narrow region surrounding
the central attractor, independent by the details of the accretion processes,
the emission mechanisms and tori models which are plagued, generally, by large
arbitrariness. Inversion coronas are distinguished for proto-jets and
accretion driven flows.

Our findings discern BHs and super-spinning objects, depending on the central
object spin $a$ and the different orbiting objects, differentiating co-
rotating and counter–rotating tori and proto-jets, inner and outer tori of an
orbiting couple. The flow is located in restricted orbital range localized in
the orbital cocoon, inversion coronas, surrounding the central singularity,
out of ergoregions. Inversion surfaces are ultimately a geometric property of
the Kerr spacetime emerging, in the conditions analyzed here, for the free
flows of particles and photons changing on these surfaces the toroidal
component $u^{\phi}$, and therefore the surfaces could be observed from the
effects related to the change in toroidal velocity, the condition
$u^{\phi}=0$. While we recognize that more work is required to assert more
than the possibility of observational relevance of our results, we expect that
the inversion coronas might manifest as an active part of the accreting flux
of matter and photons, particularly the region close to the central
singularity and the rotational axis which could be characterized by an
increase of the flow luminosity and temperature, given the particular
structure of the surfaces in that region, see Fig. (4) (or vice versa, the
effects considered here could distinguish the region from the emissions with a
component subject to the inversion sphere by emerging the peculiarity of the
folding of the surface at the poles). On the other hand, the possible
observational properties related to the flows in this region depend strongly
on the processes timescales, i.e. they depend on the time flow reaches the
inversion points, reached at different times $t_{\mbox{\scriptsize{{T}}}}$
depending on the initial data.

###### Acknowledgements.

The authors are grateful to RCTPA, IoP, Silesian University in Opava.

##  Appendix A Kerr Naked singularities spacetime circular geodesic structure

For convenience we provide below an explicit expression for the set of radii
$\\{r_{ms}^{+},r_{mb}^{\pm},r_{\gamma}^{+},\tilde{r}_{ms}^{-},\bar{r}_{ms}^{-}\\}$
of the Kerr NS spacetime geodesic structure, introduced in Sec. (II.2). Hence,
there is

$\displaystyle r_{ms}^{+}\equiv 3+\upsilon^{+}+\Upsilon,\quad r_{mb}^{+}\equiv
2+a+2\sqrt{a+1},\quad r_{\gamma}^{+}\equiv
4\cos^{2}\left(\frac{1}{6}\cos^{-1}\left[2a^{2}-1\right]\right)$ (36)
$\displaystyle\tilde{r}_{ms}^{-}\equiv
3-\upsilon^{-}-\Upsilon,\quad\bar{r}_{ms}^{-}\equiv\upsilon^{-}-\Upsilon+3,\quad
r_{mb}^{-}\equiv 2+a-2\sqrt{a+1},$

where

$\displaystyle\upsilon^{\mp}\equiv\sqrt{B[(H\mp
2\Upsilon)+4]},\quad\Upsilon\equiv\sqrt{3a^{2}+(H+1)^{2}},\quad B\equiv
2-H,\quad H\equiv\sqrt[3]{1-a^{2}}\left(\sqrt[3]{1-a}+\sqrt[3]{a+1}\right)$
(37)

(see Figs (3)). We introduce also radii $\\{r_{0}^{\pm},r_{\delta}^{\pm}\\}$
as:

$\displaystyle r_{0}^{\mp}\equiv\frac{\sqrt{\hat{K}}\mp X}{\sqrt{6}},\quad
r_{\delta}^{-}\equiv\frac{8}{3}\sin^{2}\left[\frac{1}{6}\cos^{-1}\left(1-\frac{27a^{2}}{16}\right)\right],\quad
r_{\delta}^{+}\equiv\frac{4}{3}\left[\sin\left(\frac{1}{3}\sin^{-1}\left[1-\frac{27a^{2}}{16}\right]\right)+1\right],$
(38)

where

$\displaystyle
X\equiv\sqrt{a^{2}\left(\frac{6\sqrt{6}}{\sqrt{\hat{K}}}-6\right)-\hat{K}},\quad\hat{K}\equiv\frac{4a^{4}}{\sqrt[3]{T}}-2a^{2}+\sqrt[3]{T},\quad
T\equiv 3a^{4}\left[\sqrt{81-48a^{2}}+9\right]-8a^{6}.$ (39)

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