# Testing wormhole solutions in extended gravity through the Poynting-
Robertson effect

Vittorio De Falco1<EMAIL_ADDRESS>Emmanuele Battista2,3
<EMAIL_ADDRESS>Salvatore Capozziello4,5,6<EMAIL_ADDRESS>Mariafelicia De Laurentis4,5,7<EMAIL_ADDRESS>1Department
of Mathematics and Applications “R. Caccioppoli”, University of Naples
Federico II, Via Cintia, 80126 Naples, Italy,
$2$ Institute for Theoretical Physics, Karlsruhe Institute of Technology
(KIT), 76128 Karlsruhe, Germany
$3$ Institute for Nuclear Physics, Karlsruhe Institute of Technology (KIT),
Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
4 Università degli studi di Napoli “Federico II”, Dipartimento di Fisica
“Ettore Pancini”, Complesso Universitario di Monte S. Angelo, Via Cintia
Edificio 6, 80126 Napoli, Italy
5 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso
Universitario di Monte S. Angelo, Via Cintia Edificio 6, 80126 Napoli, Italy
6Scuola Superiore Meridionale, Università di Napoli “Federico II”, Largo San
Marcellino 10, 80138 Napoli, Italy
7 Lab.Theor.Cosmology,Tomsk State University of Control Systems and
Radioelectronics(TUSUR), 634050 Tomsk, Russia

###### Abstract

We develop a model-independent procedure to single out static and spherically
symmetric wormhole solutions based on the general relativistic Poynting-
Robertson effect and the extension of the ray-tracing formalism in generic
static and spherically symmetric wormhole metrics. Simulating the flux emitted
by the Poynting-Robertson critical hypersurface (i.e., a stable structure
where gravitational and radiation forces attain equilibrium) or also from
another X-ray source in these general geometrical environments toward a
distant observer, we are able to reconstruct, only locally to the emission
region, the wormhole solutions which are in agreement with the high-energy
astrophysical observational data. This machinery works only if wormhole
evidences have been detected. Indeed, in our previous paper we showed how the
Poynting-Robertson critical hypersurfaces can be located in regions of strong
gravitational field and become valuable astrophysical probe to observationally
search for wormholes’ existence. As examples, we apply our method to selected
wormhole solutions in different extended theories of gravity by producing
lightcurves, spectra, and images of an accretion disk. In addition, the
present approach may constitute a procedure to also test the theories of
gravity. Finally, we discuss the obtained results and draw the conclusions.

Physics of black holes, singularities, wormholes.

###### pacs:

04.20.Dw, 04.70as, 04.25.dg

##  I Introduction

The recent and future great amount of high-energy astrophysical observational
data (e.g., International Gamma-Ray Astrophysics Laboratory (INTEGRAL) Winkler
_et al._ (2003), Swift Burrows  _et al._ (2005), XMM-Newton Beckwith and Done
(2004); Tomsick  _et al._ (2014), Event Horizon Telescope (EHT) Chael  _et
al._ (2016); Akiyama  _et al._ (2019), Advanced Telescope for High-ENergy
Astrophysics (ATHENA) Barcons  _et al._ (2017), enhanced X-ray Timing and
Polarimetry mission (eXTP) Zhang  _et al._ , Imaging x-ray Polarimetry
Explorer (IXPE) Soffitta  _et al._ (2013), Laser Interferometer Gravitational-
Wave Observatory (LIGO) and Virgo Gourgoulhon  _et al._ (2019); Abuter  _et
al._ (2020)) is triggering the attention of all scientific community in
developing more and more advanced models and in proposing new strategies to
test gravity in strong field regimes. A particular attention is reserved to
wormholes (WHs), which are exotic astrophysical objects endowed with no
horizons and singularities, and a characteristic traversable bridge (WH neck)
connecting two different universes or extremely far regions of spacetime
Visser (1995).

The study of these objects permits to have more insight into the fundamental
physics and entails important implications, like: understanding how gravity
shapes spacetime topology, learning more about the exotic matter and how
quantum mechanics effects couple with gravity in strong field regimes, having
natural laboratories for testing quantum gravity models and the intriguing
possibility for undertaking interstellar travels.

Both in general relativity (GR) and in extended/alternative theories of
gravity Capozziello and De Laurentis (2011) there exists a restricted class of
WH solutions which have the peculiar propriety to be _black hole (BH)
mimickers_ , meaning that they possess geometrical proprieties, which make
them observationally similar to BHs Damour and Solodukhin (2007); Bambi (2012,
2013). This can be one of the reasons why no observational evidence of their
existence has been found so far. To this end, there are international efforts
in providing feasible observational procedures to detect their signature,
like: analysing the quasinormal-mode spectrum in the gravitational wave
emission Cardoso  _et al._ (2016); Konoplya and Zhidenko (2016); producing
numerical images of a thin accretion disk forming around WHs Paul  _et al._
(2019); the influence of gravitational fluxes, propagating from one universe
to the other one through the WH neck, and influencing the accelerations of
stellar objects Dai and Stojkovic (2019); the presence of strong tidal effects
close to the WH neck, strongly depending on their geometries Banerjee  _et
al._ (2019); the absence of chaotic motions near strong gravitational field
regions due to the lack of an horizon Hashimoto and Tanahashi (2017); Dalui
_et al._ (2019). Recently, we have proposed another strategy for the detection
of WHs’ existence through the general relativistic Poynting-Robertson (PR)
effect. This method is based on the analysis of possible metric changes in the
vicinity of a BH event horizon through the fit of observational data using the
flux emitted by the PR critical hypersurface, located in such strong field
regime regions, toward a distant observer De Falco  _et al._ (2020).

The general relativistic PR effect is an important phenomenon involving the
motion of relatively small test particles (e.g., dust grains, accreting matter
elements, meteors, comets) around compact objects (like neutron stars or BHs),
which are affected not only by the gravitational pull but also by a radiation
field from an emitting source (e.g., a boundary layer around an neutron star,
a hot corona around a BH, or a type-I X-ray burst on the neutron star
surface). Besides having the radiation pressure opposite in direction to the
gravitational attraction, there is also the radiation drag force, arising
during the process of absorption and re-emission of radiation from the test
particle Ballantyne and Strohmayer (2004); Ballantyne and Everett (2005);
Worpel  _et al._ (2013); Ji  _et al._ (2014); Keek  _et al._ (2014); Worpel
_et al._ (2015). The latter force, known in the literature as PR effect
Poynting (1903); Robertson (1937), removes very efficiently energy and angular
momentum from the test particle, being therefore a full-fledged dissipative
effect in GR.

The PR effect in GR is steadily acquiring a prominent role in high-energy
astrophysics, as it can be seen from the increasing numbers of theoretical and
applicative works on such topic. Among the recent works, it is worth citing:
its description in two-dimensional (2D) Bini  _et al._ (2009, 2011) and three-
dimensional (3D) spaces De Falco  _et al._ (2019); Bakala  _et al._ (2019);
Wielgus (2019); De Falco  _et al._ (2020), its treatment under a Lagrangian
point of view De Falco  _et al._ (2018); De Falco and Battista (2019); De
Falco and Battista (2020); De Falco (2020), the proof that the critical
hypersurfaces are stable configurations, different models of accretion disk
dynamics affected by intense thermonuclear explosions Walker and Meszaros
(1989); Walker (1992); Lančová  _et al._ (2017); Fragile  _et al._ (2020).

In this work, we want to complement our previous analysis De Falco  _et al._
(2020), where we developed an astrophysical strategy to search for the WH
existence. Indeed, we proved that if metric changes occur in the transition
surface layer (located between the BH event horizon and the photon sphere),
then a WH could exist. In light of this hypothesis, we aim at distinguishing
among the various WH solutions the most suitable for fitting the observational
data. To this end, we extend the ray-tracing treatment to generic Morris-
Thorne-like metrics, which represent the new geometrical framework of our
current analysis. On the other hand, since the WH metric parameters strictly
depend on the underlying gravity theory (see, e.g. Visser (1989); Barcelo and
Visser (1999); Böhmer  _et al._ (2012); Anchordoqui  _et al._ (2000);
Bahamonde  _et al._ (2016); Capozziello  _et al._ (2020)), this approach could
be also considered as a further method to test extended/alternative gravity
models.

The paper is organized as follows: in Sec. II we introduce the Morris-Throne-
like metric and we derive the ray-tracing equations in such generic WH
spacetimes; in Sec. III the general relativistic PR effect and its critical
hypersurface equations are recalled; in Sec. IV we develop the equations to
calculate the flux from an emitting source toward a distant observer; in Sec.
V, applications to some WH solutions in GR and extended/alternative theories
of gravity are considered; in Sec. VI we discuss our results and draw
conclusions.

##  II Ray-tracing in spherically symmetric and static wormhole spacetimes

###  II.1 Morris-Thorne metric structure

A static and spherically symmetric WH can be generally described by a
Morrison-Thorne-like metric Morris and Thorne (1988), which in spherical
coordinates $\left\\{t,r,\theta,\psi\right\\}$ and geometrical units $G=c=1$
reads as (see Fig. 1)

$\displaystyle ds^{2}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}=$ (1)
$\displaystyle=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\psi^{2}),$

where $\Phi(r)$ and $b(r)$ denote the redshift and shape functions,
respectively. _Equation ( 1), representing a family of metrics depending on
the functions $\Phi(r),b(r)$, is valid both in GR and in extended/alternative
theories of gravity._ This class of metrics may also depend on other
parameters, which can be closely determined with the help of specific
constraints in different extended theories of gravity (see Sec. V, for further
details). In the WH case we use the Arnowitt, Deser, Misner (ADM) mass $M$,
which represents the total mass of the system contained in the whole spacetime
(see Ref. Visser (1995), for more details).

Figure 1: Left panel: Ray-tracing geometry. Middle panel: Symmetrization
process. Right panel: periastron geometry.

####  II.1.1 Geometrical proprieties

We briefly recall the relevant geometrical proprieties of the Morris-Thorne-
like metrics Morris and Thorne (1988). The absence of horizons and
singularities requires $\Phi(r)$ and $b(r)$ to be smooth functions in
$\mathbb{R}$ and $\Phi(r)$ to be everywhere finite; furthermore, the condition
$1-b(r)/r\geq 0$ allows to define a finite proper radial distance; the flaring
outward condition requires that $b^{\prime}(r)<b(r)/r$ near and at the throat,
defined as the minimum radius such that $r_{\rm min}=b_{0}$ and $b(r_{\rm
min})=b_{0}$; the asymptotic flatness entails $b(r)/r\to 0$ and $\Phi(r)\to 0$
for $r\to+\infty$. Traversability of WHs is subjected to the fulfilment of the
flaring outward condition Hochberg and Visser (1998a, b); Capozziello  _et
al._ (2015, 2014) which, depending on the underlying theory of gravity, can be
achieved by considering exotic matter Hochberg  _et al._ (1997); Bronnikov
_et al._ (2013); Garattini (2019) or topological defects Lobo and Oliveira
(2009); Harko  _et al._ (2013); Di Grezia  _et al._ (2017).

####  II.1.2 Conserved quantities and geodesic structure

The metric (1) depends on the radial coordinate $r$ only, therefore there
exist the timelike Killing vector $\bm{\partial_{t}}$ and (for closed orbits
only) the spacelike Killing vector $\bm{\partial_{\psi}}$; the associated
conserved quantities along all types of trajectories are, respectively, the
energy $E$ and the angular momentum $L_{d}$ with respect to any direction
$\bm{d}$ Misner  _et al._ (1973).

The spherical symmetry of the background geometry guarantees that the geodesic
motion of particles and photons lies in one single plane termed _invariant
plane_ Misner  _et al._ (1973). Therefore, we can investigate the dynamics of
a photon in the invariant plane $\theta=\pi/2$, i.e., the equatorial plane in
the coordinate system $\left\\{t,r,\theta,\psi\right\\}$ (see Fig. 1).

###  II.2 Ray-tracing equations

We derive the ray-tracing equations to describe the effects related to photon
geodesics in metric (1), which are: _light bending, travel time delay, and
solid angle_ Misner  _et al._ (1973). We consider the _WH reference frame_
$\left\\{\bm{x},\bm{y},\bm{z}\right\\}$, which is centred at the origin of the
WH location and having the $\bm{x}$\- and $\bm{y}$-axes lying in the
equatorial plane, and the $\bm{z}$-axis orthogonal to the equatorial plane,
see Fig. 1. To describe the photon emission, it is useful to employ spherical
coordinates, where the radius $r$ joins the centre of the coordinates with any
point in the space, the azimuthal angle $\varphi$ measured clockwise from the
$\bm{x}$-axis in the equatorial plane, and the latitudinal angle $\theta$
measured from the $\bm{x}$-axis. We consider also a static and not rotating
observer located at infinity, who is inclined by an angle $i$ with respect to
the $\bm{z}$-axis, and the $\bm{z^{\prime}}$-axis points in the observer’s
direction. The azimuthal angle $\psi$ is measured from the emission point to
the $\bm{z^{\prime}}$-axis, and it is known in the literature as _light
bending angle_ Misner  _et al._ (1973); Beloborodov (2002); De Falco  _et al._
(2016).

The photon four velocity is given by $k^{\alpha}=dx^{\alpha}/ds$, where $s$ is
the affine parameter along the photon trajectory, and $s\to r$ as $r\to\infty$
Beloborodov (2002). From the conservation laws (see Sec. II.1.2), we have

$k_{t}=-E,\quad k_{\theta}=0,\quad k_{\psi}=L_{d},$ (2)

The photon impact parameter $b_{\rm ph}\equiv L_{d}/E$ is naturally conserved
along the photon trajectory. For determining $k^{r}$, we exploit the other
conserved quantity, given by the module of the null geodesics four-velocity,
namely $k^{\alpha}k_{\alpha}=0$. Without loss of generality we can set $E=1$,
because one could consider the photon four-momentum for unity energy (i.e.,
$\tilde{k}^{\alpha}=k^{\alpha}/E$). Therefore, we obtain

$\displaystyle k^{r}$
$\displaystyle=\sqrt{\frac{-g^{tt}k_{t}^{2}-k_{\psi}^{2}g^{\psi\psi}}{g_{rr}}}$
(3)
$\displaystyle=\sqrt{\left[1-\frac{b(r)}{r}\right]\left[e^{-2\Phi(r)}-\frac{b_{\rm
ph}^{2}}{r^{2}}\right]}.$

####  II.2.1 Impact parameter

Equations (2) and (3) permit to explicitly calculate the impact parameter
$b_{\rm ph}$ in terms of the emission angle $\alpha_{E}$, which is formed
between the emitted photon four-velocity and the local radial direction (see
Fig. 1). Using the conditions
$\tan\alpha_{E}=\sqrt{(k^{\psi}k_{\psi})/(k^{r}k_{r})}$ and
$\sin\alpha_{E}=\tan\alpha_{E}/\sqrt{1+\tan^{2}\alpha_{E}}$ Misner  _et al._
(1973); Beloborodov (2002), we obtain

$b_{\rm ph}=\frac{R\sin\alpha_{E}}{e^{\Phi(R)}},$ (4)

where $R$ is the radius at which the photon is emitted.

####  II.2.2 Light bending

We describe how the photon trajectory is deflected when it is emitted in the
WH spacetime. This is described by the parameter $\psi$, see Fig. 1. We set
$\psi=0$ at the emission point $R$, and a generic angle $\psi$ at the observer
location at infinity. The light bending angle $\psi$ is described by the
following elliptic integral Misner  _et al._ (1973); Beloborodov (2002)

$\displaystyle\psi$
$\displaystyle=\int_{0}^{\psi}d\Psi=\int_{R}^{\infty}\frac{d\Psi}{dr}dr=\int_{R}^{\infty}\frac{k^{\psi}}{k^{r}}dr=$
(5) $\displaystyle=\int_{R}^{\infty}\frac{b_{\rm
ph}}{r^{2}}\left\\{\left[1-\frac{b(r)}{r}\right]\left[e^{-2\Phi(r)}-\frac{b_{\rm
ph}^{2}}{r^{2}}\right]\right\\}^{-1/2}dr.$

This equation is valid for $0\leq\alpha_{E}\leq\pi/2$ Beloborodov (2002); De
Falco  _et al._ (2016). For every $\alpha_{E}$ we can determine the
corresponding $\psi$ by solving the integral (5), and viceversa for every
$\psi$ we can determine the corresponding $\alpha_{E}$ by inverting Eq. (5)
through a numerical interpolation method Press  _et al._ (2002).

####  II.2.3 Turning points and symmetrization processes

The photon trajectories reach the observer at infinity only if their impact
parameters $b_{\rm ph}$ are greater than the critical impact parameter
$b_{c}$, i.e., $b_{\rm ph}\geq b_{c}$. However, photons endowed with an impact
parameter $b_{\rm ph}<b_{c}$ fall inside the WH throat, and end up in the
opposite universe of their origins. In this work, we do not analyse the
possible destiny of a photon crossing the WH throat, and we assume that it is
not visible to the observer at infinity. In our numerical simulations we would
like to give more emphasis to the WH geometrical background. Nevertheless, if
an observer at infinity detected something coming from inside the WH throat,
this would be a clear and distinctive signature of a WH existence, and can be
better worked it out through the gathered observational data.

We consider the null geodesic equation

$g_{\alpha\beta}\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=0,\ \Rightarrow\
\frac{E^{2}}{e^{2\Phi(r)}}-\frac{\dot{r}^{2}}{1-\frac{b(r)}{r}}-\frac{L_{d}^{2}}{r^{2}}=0.$
(6)

The effective potential $V(r)$ is given by Chandrasekhar (1983)

$E^{2}\equiv
V(r)=\frac{L_{d}^{2}}{r^{2}}e^{2\Phi(r)}+\left[\frac{e^{2\Phi(r)}}{1-b(r)/r}\right]\dot{r}^{2}.$
(7)

The photon innermost stable circular orbit (ISCO), or also known as _photon
sphere_ $r_{ps}$, is determined by imposing both that $\dot{r}=0$ (circular
orbit) Misner  _et al._ (1973); Chandrasekhar (1983),

$V(r)=\frac{L_{d}^{2}}{r^{2}}e^{2\Phi(r)},$ (8)

and that $[dV(r)/dr]_{r=r_{ps}}=0$ (stable and innermost)

$\displaystyle r_{ps}\Phi^{\prime}(r_{ps})-1=0,$ (9)

where $\Phi^{\prime}(r)=\Phi^{\prime}(r)$. Therefore, the critical impact
parameter $b_{c}$ is determined by substituting the value of $r_{ps}$ in Eq.
(4) and imposing $\alpha_{E}=\pi/2$,

$b_{c}=\frac{r_{ps}}{e^{\Phi(r_{ps})}}.$ (10)

To calculate the maximum emission angle $\alpha_{\rm max}$ for which the
photons do not fall inside the WH throat, we substitute Eq. (10) in Eq. (4),
obtaining thus De Falco  _et al._ (2016)

$\alpha_{\rm max}=\pi-\arcsin\left[b_{c}\frac{e^{\Phi(R)}}{R}\right],$ (11)

which is valid for $R>r_{ps}$. Instead for emission radii $R\leq r_{ps}$, the
maximum emission angle is De Falco  _et al._ (2020)

$\alpha_{\rm max}=\arcsin\left[b_{c}\frac{e^{\Phi(R)}}{R}\right].$ (12)

We note that in the BH case, the gravitational pull is so strong under the
photon sphere that the emission of light is restricted to the so-called _cone
of avoidance_ , which shrinks more and more toward the event horizon
Chandrasekhar (1983); De Falco  _et al._ (2020). In the WH case, it is not
known a-priori where and how the cone of avoidance extends through the WH neck
up to the WH throat, because this behaviour strongly depends on the explicit
expression of the function $\Phi(r)$. However, if $\Phi(r)$ is a _monotone
increasing function_ , as it occurs in the BH case, we can apply Eq. (12) all
the way down to the WH throat $b_{0}$, and the cone shrinks more and more,
until it reaches its maximum at $b_{0}$.

We distinguish between _direct photons_ , where $0\leq\alpha_{E}\leq\pi/2$,
and _photons endowed with one turning point_ , where
$\pi/2\leq\alpha_{E}\leq\alpha_{\rm max}$ De Falco  _et al._ (2016, 2020). For
photons showing a turning point, the bending angle $\psi_{E}$, associated to
the emission angle $\alpha_{E}$, cannot be straightforwardly calculated
through Eq. (5), because it is valid only for $0\leq\alpha_{E}\leq\pi/2$.
However, by employing a _symmetrization process_ with respect to the emission
angle $\alpha_{p}=\pi/2$, we can determine $0\leq\alpha_{S}\leq\pi/2$, given
by $\alpha_{S}=\pi-\alpha_{E}$. The angle $\alpha_{p}$ is associated to the
periastron $p$, which is the minimum distance between the photon trajectory
and the WH, see Fig. 1. This value depends both on the emission radius $R$,
and the emission angle $\alpha_{E}$, and can be calculated by imposing
$dr/d\psi=0$ (i.e., considering the integrating function of Eq. (5) equal to
zero), Misner  _et al._ (1973); Chandrasekhar (1983); De Falco  _et al._
(2016)

$p^{2}-b_{\rm ph}^{2}e^{2\Phi(p)}=0,$ (13)

with the further condition that $p\geq r_{ps}$. Therefore, we can calculate
the periastron bending angle $\psi_{p}=\psi_{p}(p,\alpha_{p})$222It is
important to stress that $\psi_{p}$ is an angle that depends on the selected
photon trajectory, because the value of $p$ depends on the impact parameter
$b_{\rm ph}$ (or equivalently on the emission angle $\alpha_{E}$) and the
emission radius $R$, see Fig. 1. . Using Eq. (5), we calculate $\psi_{S}$, and
employing another symmetrization process with respect to the periastron
bending angle $\psi_{p}$, we finally determine the desired bending angle
$\psi_{E}=2\psi_{p}-\psi_{S}$, associated to $\alpha_{E}$.

####  II.2.4 Time delay

A photon following its null geodesic in the metric (1) from a generic emission
point at distance $R$ to the observer location at infinity will take an
infinite travel time. To avoid such singularity, we consider the relative time
delay between a photon emitted at radius $R$ with a generic impact parameter
$b_{\rm ph}$, $T(b_{\rm ph})$, and another photon emitted always at distance
$R$ (but of course at different azimuthal position $\varphi$) with radial
impact parameter, i.e., $b_{\rm ph}=0$, see Fig. 1. Therefore, the non-
singular photon travel time delay reads as Pechenick  _et al._ (1983);
Poutanen and Beloborodov (2006)

$\displaystyle\Delta t(b_{\rm ph})$
$\displaystyle=\int_{t_{0}}^{\bar{t}}dT(b_{\rm ph})-dT(0)$
$\displaystyle=\int_{R}^{\infty}\left[\frac{dT(b_{\rm
ph})}{dr}-\frac{dT(0)}{dr}\right]dr$
$\displaystyle=\int_{R}^{\infty}\left[\frac{k^{t}}{k^{r}}\Big{|}_{b_{\rm
ph}}-\frac{k^{t}}{k^{r}}\Big{|}_{b_{\rm ph}=0}\right]dr=$
$\displaystyle=\int_{R}^{\infty}\frac{e^{-\Phi(r)}}{\sqrt{1-b(r)/r}}\left\\{\left[1-\frac{b_{\rm
ph}^{2}}{r^{2}}e^{2\Phi(r)}\right]^{-1/2}-1\right\\}dr.$

This formula is valid for $0\leq\alpha_{E}\leq\pi/2$. However, if we have
photons endowed with turning points $\pi/2\leq\alpha_{E}\leq\alpha_{\rm max}$
and an impact parameter $b_{\rm ph}\geq b_{c}$, we make use of the periastron
$p$, see Eq. (13). Indeed, we split the calculation of the total time delay in
the following three parts along the photon trajectory (see Fig. 1):

  * •

time delay between $[\pi/2,\alpha_{E}]$, given by333We prefer to use the
module notation to calculate this time, because a-priori we do not know
whether $p$ is greater or lower than $R$, and a time cannot be negative.

$\Delta t_{E-p}(b_{\rm
ph})=\left|\int_{R}^{p}\frac{e^{-\Phi(r)}}{\sqrt{1-b(r)/r}}\left[1-\frac{b_{\rm
ph}^{2}}{r^{2}}e^{2\Phi(r)}\right]^{-1/2}dr\right|;$ (15)

  * •

then using the symmetrization process we determine
$\alpha_{S}=\pi-\alpha_{E}\in[0,\pi/2]$. Since the integrating function of Eq.
(II.2.4) is symmetric with respect to $\alpha_{p}=\pi/2$, we have that $\Delta
t_{p-S}=\Delta t_{E-p}$;

  * •

finally we consider the last part of the orbit, which gives the time delay
from $\alpha_{S}$ to the observer at infinity, $\Delta t_{S}$ (as that of a
photon with zero-turning points), which can be calculated through Eq.
(II.2.4).

Summing up these three pieces, we obtain the proper formula of the travel time
delay with turning points

$\Delta t(b_{\rm ph})=\Delta t_{S}(b_{\rm ph})+2\Delta t_{E-p}(b_{\rm ph}).$
(16)

####  II.2.5 Solid angle

We consider the emission reference frame
$\left\\{\bm{x},\bm{y},\bm{z}\right\\}$ and the observer reference frame
$\left\\{\bm{x^{\prime}},\bm{y^{\prime}},\bm{z^{\prime}}\right\\}$, where the
two systems are rotated by the observer inclination angle $i$ around
$\bm{y}=\bm{y^{\prime}}$. We use spherical coordinates in both systems, and we
differentiates them by labelling those belonging to the observer reference
frame by primes. In this case, it is more convenient to use the following
spherical coordinates $\left\\{r,\psi,\varphi\right\\}$ in the emission
reference frame $\left\\{\bm{x},\bm{y},\bm{z}\right\\}$, where $r$ is the
radius, $\psi$ is the polar angle measured from the $\bm{z}$-axis, and
$\varphi$ is the azimuthal angle measured clockwise from the positive
$\bm{x}$-axis in the $\bm{x}$ – $\bm{y}$ plane. The same definitions hold also
for the spherical coordinates
$\left\\{r^{\prime},\psi^{\prime},\varphi^{\prime}\right\\}$ in the observer
reference frame
$\left\\{\bm{x^{\prime}},\bm{y^{\prime}},\bm{z^{\prime}}\right\\}$ (see Fig.
1, for a visual representation).

The solid angle, $d\Omega$, in the observer reference frame reads as
$d\Omega=\sin\psi^{\prime}\,d\varphi^{\prime}\,d\psi^{\prime}$. The observer
is located at a distance $D$ far from the emission point $R$, i.e., $R\ll
D<\infty$, which can be considered as at infinity. The solid angle equation
can be also expressed in terms of the impact parameter, $b_{\rm ph}$, by its
first-order approximation for infinitesimally small $\psi^{\prime}$ (i.e.,
$b_{\rm ph}\approx D\ \psi^{\prime}$) as De Falco  _et al._ (2016, 2020)

$d\Omega=\frac{b_{\rm ph}\ db_{\rm ph}\ d\varphi^{\prime}}{D^{2}}.$ (17)

In the emission reference frame, Eq. (17) becomes

$d\Omega=\frac{b_{\rm
ph}}{D^{2}}\frac{\partial\varphi^{\prime}}{\partial\varphi}\frac{\partial
b_{\rm ph}}{\partial r}dR\ d\varphi,$ (18)

where we considered the following dependencies
$\varphi=\varphi(\varphi^{\prime})$ and $b_{\rm ph}=b_{\rm ph}(r,\psi)$. The
Jacobian of the transformation is always
$\frac{\partial\varphi^{\prime}}{\partial\varphi}\frac{\partial b_{\rm
ph}}{\partial r}$ independent of the value of $\frac{\partial b_{\rm
ph}}{\partial\psi}$, since the photon moves in an invariant plane. Therefore,
Eq. (18) is valid for any emission point De Falco  _et al._ (2016). Using the
following coordinates transformation $\cos\psi=\sin i\cos\varphi$ that relates
the angles in the observer and emission reference frames, the term
$\frac{\partial b_{\rm ph}}{\partial r}=-\frac{\partial b_{\rm
ph}}{\partial\psi}\frac{\partial\psi}{\partial r}$ is calculated through the
light bending Eq. (5), and the solid angle equation reads as De Falco  _et
al._ (2016)

$d\Omega=\frac{\frac{\cos i}{D^{2}\ e^{2\Phi(R)}\
\sin^{2}\psi_{E}}\frac{\sin^{2}\alpha_{E}}{\cos\alpha_{E}}}{\int_{R}^{\infty}\frac{dr}{r^{2}}\left\\{\left[1-\frac{b(r)}{r}\right]\left[e^{-2\Phi(r)}-\frac{b_{\rm
ph}^{2}}{r^{2}}\right]\right\\}^{-3/2}}\ dR\ d\varphi.$ (19)

This formula is valid for $0\leq\alpha_{E}\leq\pi/2$. For photons presenting
turning points $\pi/2\leq\alpha_{E}\leq\alpha_{\rm max}$, we follow the same
symmetrization processes already explained for the light bending case, see
Sec. II.2.2.

####  II.2.6 Timelike ISCO and image at infinity

In order to understand the WH spacetime geometry and infer useful information
about it, we calculate the ISCO radius $r_{\rm ISCO}$ for a test particle. We
follow an analogue procedure for deriving the photon sphere radius, see Eq.
(9). We consider the timelike geodesic equation

$\displaystyle
g_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=-1,$ (20)
$\displaystyle\Rightarrow\
\frac{E_{p}^{2}}{e^{2\Phi(r)}}-\frac{\dot{r}^{2}}{1-\frac{b(r)}{r}}-\frac{L_{d,p}^{2}}{r^{2}}=1,$

where $E_{p},L_{d,p}$ are respectively the conserved energy and angular
momentum along the test particle trajectory, and $\tau$ is the affine
parameter along the test particle trajectory. Considering the effective
potential $V_{p}(r)$ for circular orbits $\dot{r}=0$, we obtain

$E_{p}^{2}\equiv
V_{p}(r)=e^{2\Phi(r)}\left(\frac{L_{d,p}^{2}}{r^{2}}+1\right).$ (21)

Imposing $[dV_{p}(r)/dr]_{r=r_{\rm ISCO}}=0$, we obtain

$L^{2}_{d,p}[\Phi^{\prime}(r)r-1]+\Phi^{\prime}(r)r^{3}=0,$ (22)

which solved for the lowest value of $L_{d,p}$, permits to determine the value
of the timelike ISCO radius $r_{\rm ISCO}$.

An emitting source around a WH can be ray-traced from the emission location to
the observer screen using the following coordinates (see Ref. Cunningham and
Bardeen (1973); Luminet (1979), for details)

$\begin{cases}x^{\prime}=-b_{\rm ph}\sin\varphi^{\prime}\equiv-b_{\rm
ph}\frac{\sin\varphi}{\sin\psi_{E}},\\\ \\\ y^{\prime}=-b_{\rm
ph}\cos\varphi^{\prime}\equiv-b_{\rm ph}\frac{\cos
i\cos\varphi}{\sin\psi_{E}}.\end{cases}$ (23)

##  III General relativistic Poynting-Robertson effect in Morris-Thorne-like
wormhole spacetimes

In a previous work De Falco  _et al._ (2020) we derived the equations of
motion of a test particle moving in the gravitational field of a WH and
influenced also by a radiation field, including the general relativistic PR
effect, from a spherical and rigidly rotating emitting source. The metric (1)
can be written in the emission reference frame
$\left\\{\bm{x},\bm{y},\bm{z}\right\\}$, see Sec. II.2.5, by employing
spherical coordinates $\\{r,\theta,\varphi\\}$. In this coordinate system we
have

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}).$ (24)

The radiation field is modelled as a coherent flux of photons, travelling
along null geodesics on the background spacetime (24), and endowed with an
impact parameter $\lambda$, whose expression is

$\lambda=\Omega_{\star}\left[\frac{\mathrm{g_{\varphi\varphi}}}{-\mathrm{g_{tt}}}\right]_{r=R_{\star}},$
(25)

where $R_{\star}$ and $\Omega_{\star}$ are, respectively, the radius and
angular velocity of the emitting surface. The lower label in the right square
bracket indicates that the metric components $g_{\varphi\varphi},g_{tt}$ must
be evaluated at $r=R_{\star}$. It is important to note that $\lambda$ is
different from that of Eq. (4).

The local static observer (SO) frames
$\left\\{\bm{e_{\hat{t}}},\bm{e_{\hat{r}}},\bm{e_{\hat{\varphi}}},\bm{e_{\hat{\psi}}}\right\\}$
are our family of fiducial observers. In the SO frames, we can measure
clockwise from the $\bm{e_{\hat{\varphi}}}$-axis the emission angle $\beta$,
related to the impact parameter $\lambda$, which is

$\cos\beta=\frac{e^{\Phi(r)}}{r}\lambda.$ (26)

We assume that the interaction between the radiation and the test particle is
described by Thomson scattering, which is characterized by a constant
momentum-transfer cross section $\sigma$, which in turn is independent of the
direction and the frequency of the radiation field. The equations of motion
governing the dynamics of a test particle orbiting around a WH in the
equatorial plane and influenced by a radiation field, which also includes the
general relativistic PR effect, are given by De Falco  _et al._ (2020):

$\displaystyle\frac{d\nu}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\sin\alpha}{\gamma}a(n)^{\hat{r}}$
$\displaystyle+\frac{A[1-\nu\cos(\alpha-\beta)][\cos(\alpha-\beta)-\nu]}{e^{2\Phi(r)}\sqrt{r^{2}-e^{2\Phi(r)}\lambda^{2}}},$
$\displaystyle\frac{d\alpha}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\gamma\cos\alpha}{\nu}\left[a(n)^{\hat{r}}+k_{\rm(Lie)}(n)^{\hat{r}}\,\nu^{2}\right]$
$\displaystyle+\frac{A[1-\nu\cos(\alpha-\beta)]\sin(\beta-\alpha)}{e^{2\Phi(r)}\sqrt{r^{2}-e^{2\Phi(r)}\lambda^{2}}\
\nu\cos\alpha},$ $\displaystyle U^{\hat{r}}$ $\displaystyle\equiv$
$\displaystyle\frac{dr}{d\tau}=\frac{\gamma\nu\sin\alpha}{\sqrt{g_{rr}}},$
(29) $\displaystyle U^{\hat{\varphi}}$ $\displaystyle\equiv$
$\displaystyle\frac{d\varphi}{d\tau}=\frac{\gamma\nu\cos\alpha}{\sqrt{g_{\varphi\varphi}}},$
(30) $\displaystyle U^{\hat{t}}$ $\displaystyle\equiv$
$\displaystyle\frac{dt}{d\tau}=\frac{\gamma}{e^{\Phi(r)}},$ (31)

where $\tau$ is the affine parameter (proper time) along the test particle
trajectory, $\bm{U}$ is the test particle four-velocity,
$\gamma(U,n)\equiv\gamma=1/\sqrt{1-||\bm{\nu}(U,n)||^{2}}$ is the Lorentz
factor, $\nu=||\bm{\nu}(U,n)||$ is the magnitude of the test particle spatial
velocity, and $\alpha$ is the azimuthal angle of the vector $\bm{\nu}(U,n)$
measured clockwise from the $\bm{e_{\hat{\varphi}}}$ direction in the SO
frame. Furthermore, $A$ is the luminosity parameter, which can be equivalently
written as $A/M=\mathcal{L}/\mathcal{L}_{\rm EDD}\in[0,1]$ with $\mathcal{L}$
the emitted luminosity at infinity, and $\mathcal{L}_{\rm EDD}$ the Eddington
luminosity. The SO kinematical quantities $\bm{a}(n)$ and $\bm{k_{(\rm
Lie)}}(n)$ denote, respectively, the acceleration and the relative Lie
curvature vector, whose expressions are

$\displaystyle a(n)^{\hat{r}}$ $\displaystyle=$
$\displaystyle\Phi^{\prime}(r)\sqrt{1-b(r)/r},$ (32) $\displaystyle
k_{\rm(Lie)}(n)^{\hat{r}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{1-b(r)/r}}{r}.$ (33)

###  III.1 Critical hypersurfaces

An important implication of the dynamical system (III)–(30) is the _possible_
existence of a critical hypersurface, which is the region where there is a
perfect balance between the radiation and gravitational forces and the test
particle has a perpetual stable motion. The contingency of the critical
hypersurface strictly depends on the considered metric (see discussions in
Sec. II.C.2 in Ref. De Falco  _et al._ (2020), for more details). The equation
defining the critical hypersurface is De Falco  _et al._ (2020)

$a(n)^{\hat{r}}+k_{\rm(Lie)}(n)^{\hat{r}}\cos^{2}\beta-\frac{A\sin^{3}\beta}{r^{2}\
e^{2\Phi(r)}}=0.$ (34)

If a critical hypersurface exists, the test-particle must move on it with
constant velocity $\nu=\cos\beta$. Equation (34) may admit none, one, or
multiple roots, depending on the specific WH solution. However, even in this
general case, the critical hypersurface depends continuously on the luminosity
parameter $A$, because it is written both in terms of the metric components
(which by definition are smooth functions) and of the radiation force
components (which are also smooth functions defined on the whole spacetime).
This is a fundamental propriety which allows the formation of such structures
in regions close to the WH throat, where strong gravitational fields occur.
Therefore, the existence of PR critical hypersurface provides essential
information both on the WH geometry and on those extended theories of gravity
which turn out to be more suited to describe their emission properties in
strong gravitational regimes De Falco  _et al._ (2020).

##  IV Radiation flux emitted by a source around a spherically symmetric and
static wormhole

In this geometrical environment, see Sec. II.2, we consider as emission point
a test particle driven by Eqs. (III) – (31), and having coordinates
$P=(r,\varphi)$. If there exists the PR critical hypersurface, which is a
circular orbit in the equatorial plane, then we have that $r=r_{\rm crit}$ is
determined by solving Eq. (34), and $\varphi=\omega_{PR}t$. At $t=0$ the
emission point is closest to the observer direction, and

$\omega_{PR}=\frac{\cos\beta}{r_{\rm crit}},$ (35)

is the PR angular velocity measured by the observer at infinity. The photon
arrival time $T_{\rm obs}$ is the sum of the emission time $T_{\rm
em}=\varphi/\omega_{k}$ and the photon propagation delay $\Delta t(b_{ph})$,
i.e., $T_{\rm obs}=T_{\rm em}+\Delta t(b_{ph})$ (see Eq. (II.2.4)).

The observed flux at the emission frequency $\nu_{\rm em}$ is De Falco  _et
al._ (2016)

$F_{\nu_{\rm
em}}=\int_{\Omega}\frac{\epsilon_{0}\xi^{q}}{4\pi}(1+z)^{-4}d\Omega,$ (36)

where $\epsilon_{0}$ is the vacuum permittivity, $\xi=R/M$ the local surface
emissivity, $q\in\mathbb{R}$ the emission index, and $(1+z)^{-1}=E_{\rm
obs}/E_{\rm em}$ the gravitational redshift, i.e., the ratio between the
observed $E_{\rm obs}$ and emitted $E_{\rm em}$ energies.

###  IV.1 Gravitational redshift

Now, we have all the elements to calculate the gravitational redshift
$(1+z)^{-1}$ associated to the motion of the test particle. When the PR
critical hypersurface exists, the test particle moves on it with velocity

$U^{\alpha}=\frac{1}{|\sin\beta|}\left(\frac{1}{e^{\Phi(r_{\rm
crit})}},0,0,\frac{\cos\beta}{r_{\rm crit}}\right).$ (37)

We transform this velocity from the emission reference frame
$\left\\{\bm{x},\bm{y},\bm{z}\right\\}$ to the observer reference frame
$\left\\{\bm{x^{\prime}},\bm{y^{\prime}},\bm{z^{\prime}}\right\\}$ (see Sec.
II.2.5 and Misner  _et al._ (1973); De Falco  _et al._ (2016)),

$\displaystyle U^{t^{\prime}}$ $\displaystyle=$ $\displaystyle U^{t},\qquad
U^{r^{\prime}}=U^{r},$ (38) $\displaystyle U^{\theta^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{\partial\theta^{\prime}}{\partial\varphi}U^{\varphi}+\frac{\partial\theta^{\prime}}{\partial\theta}U^{\theta},\quad
U^{\varphi^{\prime}}=\frac{\partial\varphi^{\prime}}{\partial\varphi}U^{\varphi}+\frac{\partial\varphi^{\prime}}{\partial\theta}U^{\theta}.$

Since $U^{\theta}=0$ and the transformations between these two reference
frames are De Falco  _et al._ (2016)

$\sin\varphi=\sin\theta^{\prime}\sin\varphi^{\prime},\quad\cos\theta^{\prime}=\sin
i\cos\varphi,$ (39)

then the test particle velocity in the observer frame reads as

$U^{\alpha^{\prime}}=\frac{1}{|\sin\beta|}\left(\frac{1}{e^{\Phi(r_{\rm
crit})}},0,\frac{\cos\beta}{r_{\rm crit}}\frac{\sin
i\sin\varphi}{\sin\psi},U^{\varphi^{\prime}}\right),$ (40)

where the explicit expression of $U^{\varphi}{}^{\prime}$ is not necessary for
the calculation of the gravitational redshift, as it will be clearer soon. We
used also that $\theta^{\prime}\equiv\psi$, corresponding to the light bending
angle, see Sec. II.2.2 and Fig, 1.

The photon velocity is (see Sec. II.2, for details)

$k_{\alpha}=\left(-1,-\sqrt{\frac{e^{-2\Phi(R)}-\frac{\lambda^{2}}{R^{2}}}{1-\frac{b(R)}{R}}},-b_{\rm
ph},0\right).$ (41)

Since the observer is static, its velocity is

$V^{\alpha}=(1,0,0,0).$ (42)

Since $E_{\rm em}=(U^{\alpha}k_{\alpha})_{\rm em}$ and $E_{\rm
obs}=(V^{\alpha}k_{\alpha})_{\rm obs}$, the gravitational redshift is Misner
_et al._ (1973); De Falco  _et al._ (2016)

$\displaystyle(1+z)^{-1}$ $\displaystyle\equiv\frac{E_{\rm obs}}{E_{\rm em}}$
(43) $\displaystyle=\frac{|\sin\beta|r_{\rm crit}e^{\Phi(r_{\rm
crit})}}{r_{\rm crit}+b_{\rm ph}\cos\beta e^{\Phi(r_{\rm crit})}\frac{\sin
i\sin\varphi}{\sin\psi}}.$

####  IV.1.1 Emission from a disk

Let us consider an accretion disk lying in the equatorial plane $\theta=\pi/2$
around a WH and extending from $R_{\rm in}$ to $R_{\rm out}$. We assume that
the matter inside the disk moves on circular orbits with the Keplerian angular
velocity $\Omega_{K}$, which has a proper expression for each WH metric. In
order to estimate the gravitational redshift, we first need to derive the
expression of $\Omega_{K}$.

We consider the timelike geodesic equation (20) for stable circular orbits
(i.e., $\dot{r}=0$ and $dV_{p}/dr=0$). From Eq. (22) we obtain the test
particle’s angular momentum,

$L_{d,p}^{2}=\frac{r^{3}\Phi^{\prime}(r)}{1-r\Phi^{\prime}(r)}.$ (44)

Substituting this expression in Eq. (21), we derive the expression of the test
particle’s energy

$E_{p}^{2}=\frac{e^{2\Phi(r)}}{1-r\Phi^{\prime}(r)}.$ (45)

The two explicit expressions (44) and (45) of the test particle’s constants of
motion permit to obtain the test particle’s velocity components in the
following way:

$\displaystyle U^{\hat{t}}$ $\displaystyle\equiv$ $\displaystyle
g^{tt}U_{t}=\frac{1}{e^{\Phi(r)}\sqrt{1-r\Phi^{\prime}(r)}},$ (46)
$\displaystyle U^{\hat{\varphi}}$ $\displaystyle\equiv$ $\displaystyle
g^{\psi\psi}U_{\psi}=\sqrt{\frac{\Phi^{\prime}(r)}{r(1-r\Phi^{\prime}(r))}}.$
(47)

Therefore, the Keplerian angular velocity $\Omega_{K}$ for the WH metric (1)
is given by

$\Omega_{K}\equiv\frac{U^{\hat{\varphi}}}{U^{\hat{t}}}=e^{\Phi(r)}\sqrt{\frac{\Phi^{\prime}(r)}{r}}.$
(48)

A test particle moving on a circular orbit has the following velocity field
Misner  _et al._ (1973); De Falco  _et al._ (2016)

$\displaystyle U^{\hat{\alpha}}$
$\displaystyle=U^{\hat{t}}(1,0,0,\Omega_{K}\sin\varphi^{\prime})$ (49)
$\displaystyle=U^{\hat{t}}\left(1,0,0,\Omega_{K}\frac{\sin
i\sin\varphi}{\sin\psi}\right),$

where we have used the transformation between the observer frame and the
emission frame De Falco  _et al._ (2016).

The gravitational redshift for the accretion disk is De Falco  _et al._ (2016)

$(1+z)^{-1}=\frac{e^{\Phi(r)}\sqrt{1-r\Phi^{\prime}(r)}}{1+b_{\rm
ph}\Omega_{K}\frac{\sin i\sin\varphi}{\sin\psi}}.$ (50)

##  V Examples of ray-tracing applications to some WH solutions

In this section, we apply the general ray-tracing method of Sec. II to
different WH solutions belonging to distinct extended theories of gravity. We
produce lightcurves and spectra of a PR critical hypersurface. In order to
discuss and compare the results among the different metrics, we pose such ring
at the same distance from the origin of the coordinate system in all WH
spacetimes. The PR critical hypersurfaces are defined in terms of two
parameters $(A,\lambda)$, see Eq. (34). These cannot be chosen arbitrarily, on
the contrary they should range in some astrophysically realistic intervals
(see Table I and discussions in Ref. De Falco  _et al._ (2020)). In addition,
we produce also the image of an accretion disk around a WH, as seen by an
observer at infinity. This step is important to better inquire the WH
spacetime geometries.

The WH literature encompasses a plethora of solutions, however we focus our
attention only on those fulfilling the following requirements: geometrical
proprieties of a standard WH solution as described in Sec. II.1.1 (in
particular the asymptotic flatness, i.e., $b(r)/r\to 0$ and $\Phi(r)\to 0$ as
$r\to\infty$444We have highlighted this propriety, since it is not trivial at
all. Indeed, in the literature there are several WH solutions, which are not
explicitly asymptotically flat. However, this property is recovered when the
solution is smoothly matched with the Schwarzschild metric at a certain
radius.), and a non constant-redshift function (i.e., $\Phi^{\prime}(r)\neq 0$
for all $r\in[b_{0},\infty]$). The chosen WH solutions belong to the following
theories: classical GR (see Sec. V.1), metric (see Sec. V.2), metric-affine
(see Sec. V.3), and teleparallel (see Sec. V.4). We underline that this
section simply represents an application of the method outlined in Sec. II and
it has the only aim of showing the potentialities and limits of the PR effect
in distinguishing the WH features among different theories.

Each WH solution generally depends on some free parameters, which naturally
stem out from the underlying theory. In order to fix them (being an essential
step for the development of the forthcoming numerical simulations), we impose
some external constraints. First of all, we require that $r_{\rm ISCO}>r_{\rm
ps}>b_{0}$, where $r_{\rm ISCO}$ and $r_{\rm ps}$ are calculated by employing
Eqs. (9) and (22), respectively. Then, since we are interested in the study of
BH mimickers, we impose that $r_{\rm ISCO}$ and $r_{\rm ps}$ coincide with
those of the Schwarzschild BH metric (being $3M$ and $6M$, respectively, see
Table 1). We also recall that the Schwarzschild BH solution is described by
the following redshift and shape functions Misner  _et al._ (1973):

$\Phi(r)=\frac{1}{2}\log\left(1-\frac{2M}{r}\right),\quad b(r)=2M.$ (51)

###  V.1 The case of General Relativity

Within the GR framework, we examine a WH sourced by Casimir energy density and
pressure and whose metric satisfies the semiclassical Einstein field equations
(see Ref. Garattini (2019), for details). The redshift and shape functions are
given by, respectively, Garattini (2019)

$\displaystyle\Phi(r)$
$\displaystyle=\frac{1}{2}(\omega-1)\log\left(\frac{r\omega}{r\omega+b_{0}}\right),$
(52) $\displaystyle b(r)$
$\displaystyle=\left(1-\frac{1}{\omega}\right)b_{0}+\frac{b_{0}^{2}}{\omega
r},$

where the positive-definite parameter $\omega$ and the WH throat radius
$b_{0}$ are related by the relation

$\omega=\dfrac{90}{\pi^{3}}\left(\dfrac{b_{0}}{\ell_{\rm P}}\right)^{2},$ (53)

$\ell_{\rm P}$ being the Planck length. The fundamental radii $r_{\rm ps}$ and
$r_{\rm ISCO}$ are given by, respectively,

$r_{\rm ps}=\frac{b_{0}}{6}(\omega-3),\quad r_{\rm
ISCO}=\frac{b_{0}}{3}(\omega-3).$ (54)

Therefore, we can obtain positive-definite radii provided that $\omega>3$,
whereas $\omega=3$, which is the condition leading to Casimir stress-energy
tensor, gives vanishing radii. In addition, the requirement $r_{\rm
ISCO}>r_{\rm ps}>b_{0}$ yields $\omega>9$. In order to simplify our
calculations, we have considered the peculiar case $\omega=4$, where $r_{\rm
ps}$ and $r_{\rm ISCO}$ are naturally in order, but located inside the throat.
This implies that in such spacetime both timelike and null circular orbits can
be stable everywhere outside the WH throat. Moreover, Eq. (53) permits to
calculate $b_{0}$, which expressed in gravitational radii reads as

$b_{0}=1.3\times 10^{-38}\left(\frac{M_{\odot}}{M}\right)M.$ (55)

The periastron $p$ can be obtained by looking at the real root of the cubic
algebraic equation

$64p^{3}+48p^{2}+4(3-16b_{\rm ph}^{2})p+1=0,$ (56)

jointly with the condition $p\geq r_{\rm ps}$. The real solution is:

$\displaystyle p$ $\displaystyle=\frac{1}{12}\left[-3b_{0}-\frac{8\
3^{2/3}b_{\rm ph}^{2/3}}{(\sqrt{3}\sqrt{27b_{0}^{2}-64b_{\rm
ph}^{2}}-9b_{0})^{1/3}}\right.$ (57) $\displaystyle\left.+2\sqrt[3]{3}b_{\rm
ph}^{2/3}(\sqrt{3}\sqrt{27b_{0}^{2}-64b_{\rm ph}^{2}}-9b_{0})^{1/3}\right].$

The condition for which the terms underneath the square root sign are non-
negative leads to the inequality $b_{\rm ph}\leq\sqrt{27}b_{0}/8\approx
0.65b_{0}$. Since the critical impact parameter is given by $b_{c}=0.66\
b_{0}$ (i.e., $b_{\rm ph}\geq b_{c}$), we obtain no real solution for the
periastron $p$ and this in turn implies that _no turning points exist in such
WH spacetime._

Since for emissions occurring at distances of order $M$ the throat radius
$b_{0}$ turns out to be extremely small, it is reasonable to consider the
approximation $b_{0}\to 0$. In this case, we can tremendously simplify the
integrals underlying the gravitational effects, being even able to determine
their analytical expressions, which are as follows:

$\displaystyle\psi$ $\displaystyle=$ $\displaystyle\arctan\left(\frac{b_{\rm
ph}}{\sqrt{R^{2}-b_{\rm ph}^{2}}}\right),$ (58) $\displaystyle\Delta t(b_{\rm
ph})$ $\displaystyle=$ $\displaystyle R-\sqrt{R^{2}-b_{\rm ph}^{2}},$ (59)
$\displaystyle\Delta t_{E-p}(b_{\rm ph})$ $\displaystyle=$
$\displaystyle\left|\sqrt{R^{2}-b_{\rm ph}^{2}}-\sqrt{p^{2}-b_{\rm
ph}^{2}}\right|,$ (60) $\displaystyle d\Omega$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{R^{2}-b_{\rm ph}^{2}}}.$ (61)

In this case, the critical impact parameter and the periastron are given by
$b_{c}=0$ and $p=b_{\rm ph}$, respectively. Finally, by performing accurate
numerical simulations we have found that the PR critical hypersurfaces are
allowed in the following parameters range: $A\in(0,0.3],\lambda\in[0,0.8]$. A
concise summary of the obtained results is reported in Table 1.

###  V.2 Metric theories

In the context of metric theories of gravity, let us consider $f(R)$ gravity
which is a generalization/extension of Einstein’s GR, defined by a generic
function, $f$, of the Ricci scalar, $R$ Capozziello  _et al._ (2010). A wide
range of phenomena can be addressed by this theory by adopting different forms
of $f$ Capozziello and De Laurentis (2011, 2012). In particular, we can take
into account models like

$f(R)=R-\mu R_{c}\left(\dfrac{R}{R_{c}}\right)^{l},$ (62)

where $\mu$, $R_{c}$ and $l$ are constants such that $\mu>0$, $R_{c}>0$ and
$0<l<1$ (see Ref. Godani and Samanta (2020) and references therein, for more
details). This kind of models are particularly relevant at infrared scales,
because they potentially address problems like: clustering of structures and
accelerated expansion of the Hubble flow Starobinsky (2007); Capozziello and
De Laurentis (2012). Specifically, instead of searching for new form of exotic
matter as fundamental constituents for dark energy and dark matter, the
approach is devoted to solve the dark side problem through geometry. In other
words, further degrees of freedom related to the gravitational field are
considered instead of choosing a priori the Hilbert-Einstein action.

Besides these fundamental problems, $f(R)$ gravity has several spherically
symmetric solutions which can be interesting both for BH and WH physics. In
particular a WH solution can be Godani and Samanta (2020)

$\Phi(r)=-\frac{\alpha}{r},\quad b(r)=\frac{r}{e^{(r-b_{0})}},$ (63)

where $\alpha>0$. We find that the fundamental radii $r_{\rm ps}$ and $r_{\rm
ISCO}$ are given by, respectively,

$r_{\rm ps}=\alpha,\quad r_{\rm ISCO}=2\alpha.$ (64)

In this case the model sets no constraints on $b_{0}$ and hence we can easily
choose $b_{0}=M$ and $\alpha=3M$, which permits to recover the two fundamental
Schwarzschild radii. We find that $b_{c}=8.15M$ and $p=-3M/W(-3M/b_{\rm ph})$,
where $W(x)$ is the _Lambert $W$ function_ (also known as ProductLog in
computer algebra framework), which is the inverse function of $xe^{x}$, i.e.,
$W(xe^{x})=x$ (see Ref. Fukushima (2013), for further details and its
numerical implementation in numerical codes). Imposing the condition $p\geq
3M$, which implies that the photon can reach the observer at infinity, we
derive that $b_{\rm ph}\geq 3e$ and the function $W(x)$ must be evaluated in
the branch $W_{0}(x)$ (see Ref. Fukushima (2013), for more details). After
having performed some numerical simulations, we have found that the PR
critical hypersurfaces exist for $A\in(0,0.6],\ \lambda\in[0,6]$. In Table 1,
a summary of the obtained results is displayed.

###  V.3 Metric-affine theories

Assuming the metricity in a theory of gravity means accepting the validity of
the Equivalence Principle and then the coincidence of the causal and geodesic
structures of the spacetime Capozziello  _et al._ (2012a). On the other hand,
more general theories can be formulated relaxing this strong hypothesis. In
this perspective, one can consider the so called metric-affine theories of
gravity (i.e., the so called Palatini formulation Olmo (2011)), where metric
and affine connections are not related through the Levi-Civita connection.

Within metric-affine theories, particularly relevant is the hybrid metric-
Palatini gravity Tamanini and Boehmer (2013); Capozziello  _et al._ (2015),
where a $f(R)$ term is constructed _à la_ Palatini and it is added to the
usual (metric) Einstein-Hilbert Lagrangian, linear in the Ricci scalar $R$.
Also in these models, it is possible to find out WH solutions, as shown in
Capozziello  _et al._ (2012b). In this case, we have the following functions:

$\Phi(r)=\Phi_{0}\left(\frac{b_{0}}{r}\right)^{\gamma},\quad
b(r)=b_{0}\left(\frac{b_{0}}{r}\right)^{\beta},$ (65)

where $\gamma>0$, $\beta>-1$, and $\Phi_{0}<0$. In this general framework, we
obtain the photon sphere radius,

$r_{\rm ps}=b_{0}\left(-\Phi_{0}\gamma\right)^{1/\gamma},$ (66)

whereas the ISCO radius can be obtained by solving the following algebraic
equation:

$L_{d,p}^{2}r^{\gamma}+\left(\gamma
b_{0}^{\gamma}\Phi_{0}\right)r^{2}+L_{d,p}^{2}\gamma
b_{0}^{\gamma}\Phi_{0}=0,$ (67)

and then minimizing the real solution in terms of $L_{d,p}$, see Sec. II.2.6.
The degree of Eq. (67) is determined once the value of $\gamma$ is known. To
differentiate this metric from the previous examples, we choose $\gamma=3$,
obtaining thus for the photon sphere radius

$r_{\rm ps}=b_{0}\left(-3\Phi_{0}\right)^{1/3},$ (68)

while for $r_{\rm ISCO}$ we should solve the following equation in terms of
the test particle angular momentum $L_{d,p}$:

$r^{3}+3b_{0}^{3}\Phi_{0}+\frac{3r^{2}b_{0}^{3}\Phi_{0}}{L_{d,p}^{2}}=0.$ (69)

For simplicity, we chose $\beta=1$, $b_{0}=M$, and $\Phi_{0}=-1$. The explicit
expression(s) of the real solution(s) of a cubic algebraic equation can be
obtained by exploiting the Cardano formula Abramowitz and Stegun (1965). In
our case, Eq. (69) admits one real solution which can written in terms of the
delta function $\sqrt{3L^{6}_{d,p}+4M^{6}}$. Imposing that the quantity under
the square root be non-negative, we obtain

$L^{6}_{d,p}=-\frac{4M^{6}}{3}.$ (70)

This entails that $L_{d,p}$ should assume imaginary values, representing a
non-physical possibility. Therefore, we conclude that there are no stable
circular orbits in such WH spacetime. The critical impact parameter and the
periastron are respectively $b_{c}=2.01M$ and
$p=-\sqrt[3]{3}M/\sqrt[3]{W\left[-3(M/b_{\rm ph})^{3}\right]}$, which implies
$b_{\rm ph}\geq\sqrt[3]{3e}$ in order to have meaningful results (see
discussions of Sec. V.2). After having performed some numerical simulations,
we have found that the PR critical hypersurfaces exist for $A\in(0,0.56],\
\lambda\in[0,2]$. In Table 1, we report a short summary of the obtained
results.

We note that _the PR effect configures as one of the viable mechanisms
allowing for the occurrence of timelike stable circular orbits in such WH
spacetime_.

###  V.4 Teleparallel theories

As a final example, let us consider teleparallel equivalent gravity and its
possible generalizations Cai  _et al._ (2016). Here the Lagrangian of the
gravitational field is a function of the torsion scalar $T$ and instead of
using the Levi-Civita connection, the Weitzenböck connection is adopted. It is
important to stress that the teleparallel formulation of gravity is equivalent
to GR, because the field equations written in terms of $T$ can be reconciled
to those written in terms of $R$. Nevertheless it is important to note that
two main differences exists between $f(T)$ and $f(R)$ gravity theories: (1) in
the metric formulation of $f(R)$, the field equations are of fourth-order,
while in $f(T)$, they remain of second-order as it also occurs in GR; (2) in
teleparallel gravity the dynamical variables are the tetrad fields, while in
metric formulation, the dynamics is related to tensor metric $g_{\mu\nu}$.

In particular, we can consider a (modified) teleparallel model as
$f(T,T_{\mathcal{G}})$ gravity, where $T$ is the torsion scalar and
$T_{\mathcal{G}}$ is the teleparallel equivalent of the Gauss-Bonnet term (see
Ref. Kofinas and Saridakis (2014); Bahamonde  _et al._ (2016); Capozziello
_et al._ (2016), for more details).

Also in this case, it is possible to build up WH solutions. Considering the
model

$f(T,T_{\mathcal{G}})=a_{0}\sqrt{a_{1}\,T_{\mathcal{G}}+T^{2}}-T,$ (71)

where $a_{0}$ and $a_{1}$ are dimensionless coupling constants Kofinas  _et
al._ (2014), a WH solution is Sharif and Nazir (2018)

$\Phi(r)=-\frac{\xi}{r},\quad b(r)=\frac{b_{0}^{\zeta+1}}{r^{\zeta}},$ (72)

where $\xi>0$ and $\zeta>-1$. The fundamental radii assume the same form as
those of the metric theory (cf. Eqs. (63) and (64)), i.e.,

$r_{\rm ps}=\xi,\quad r_{\rm ISCO}=2\xi,$ (73)

and hence we choose $b_{0}=M$ and $\xi=3M$, and to ease the calculations
$\zeta=2$. The critical impact parameter is $b_{c}=8.15M$, whereas the
periastron is $p=-3M/W(-3M/b_{\rm ph})$, with $b_{\rm ph}\geq 3e$ and $W(x)$
evaluated in the branch $W_{0}(x)$ (see Sec. V.2). After some numerical
simulations we have obtained that the PR critical hypersurfaces exist for
$A\in(0,0.6],\ \lambda\in[0,6]$. In Table 1, we report a brief summary of the
obtained results.

Table 1:  Summary of important information related to the different WH
solutions presented in this paper. The photon sphere radius $r_{\rm ps}$ and
ISCO radius $r_{\rm ISCO}$ have been calculated through Eqs. (9) and (22),
respectively. The lightgray cells refer to the Schwarzschild BH solution,
while the others to the WH solutions.

---
Theory |  WH Eq. |  Critical Hypersurface |  WH throat |  Photon sphere |  ISCO |  Ref.
|  BH Eq. |  |  BH horizon |  |  |
Schwarzschild | (51) | $A\in(0,0.6],\ \ \lambda\in[0,6]$ | $2M$ | $3M$ | $6M$ | Misner  _et al._ (1973)
GR | (52) | $A\in(0,0.3],\ \ \lambda\in[0,0.8]$ |  $b_{0}$555For the WH in GR theory it is possible to determine the WH throat within the theory itself as $b_{0}=1.3\times 10^{-38}\left(\frac{M_{\odot}}{M}\right)M$, see Eq. (55). | $\frac{b_{0}}{6}$ | $\frac{b_{0}}{3}$ | Garattini (2019)
Metric | (63) | $A\in(0,0.6],\ \ \lambda\in[0,6]$ | $M$ | $3M$ | $6M$ | Godani and Samanta (2020)
Affine-Metric | (65) | $A\in(0,0.56],\ \ \lambda\in[0,2]$ | $M$ | $\sqrt[3]{3}M$ |  NOT EXIST | Capozziello  _et al._ (2012b)
Teleparallel | (72) | $A\in(0,0.6],\ \ \lambda\in[0,6]$ | $M$ | $3M$ | $6M$ | Sharif and Nazir (2018)

Figure 2: Shapes of the WH solutions on which have been applied the related
redshift functions. They are all plotted in $M$ units. The inset figure in GR
theory represents the WH shape plotted in $b_{0}$ unity with the same legend
bar of the main figure.

###  V.5 Numerical simulations and discussions

Figure 3: The PR critical hypersurface, extending from $R=1.2M$ to $R=1.3M$
(except in the GR theory that extends from $R=1.2\ b_{0}$ to $R=1.3\ b_{0}$)
for different WH solutions belonging to the four distinct extended theories of
gravity are plotted, considering an observer inclined by an angle
$i=80^{\circ}$. _First row:_ The PR critical hypersurface (yellow surface),
the WH throat (dark gray surface), and the photon sphere (dashed red line) and
ISCO (violet dashed line) radii are plotted. _Second row:_ The lightcurve of
the PR critical hypersurface is plotted. _Third row:_ The spectrum of the PR
critical hypersurface is plotted.

In order to investigate the geometry of the different WH solutions we plot in
Fig. 2 their shapes on which we highlight also the behaviour of the redshift
function (see Ref. Morris and Thorne (1988), for the spatial geometrical
visualization of a WH embedded in the 3D Euclidean space). The WH solution
obtained in GR theory using the Casimir effect as a stress-energy tensor gives
rise to a _micro-WH_ of few Planckian size (i.e, $b_{0}\approx 0.37\ell_{\rm
P}$). Therefore, it is very challenging to detect it with the actual
technologies, because the metric outside the WH throat at distances of order
of $M$ is generally flat. However, they are very intriguing because they may
hide critical information both on the nature of the spacetime geometry and on
gravity at microscopic levels. The other WH solutions clearly show that
$\Phi(r),b(r)$ are monotone increasing functions, and the geometry of the WH
neck changes in each WH solution. This plot permits to have a fast overview of
the WH solutions to quickly learn their proprieties for further theoretical
and observational speculations.

Figure 4: Image of an accretion disk seen by a static, and no-rotating
observer at infinity inclined by an angle $i=80^{\circ}$. The disk extends:
from $R_{\rm in}=6M$ to $R_{\rm out}=100M$ for WH solutions in the metric and
teleparallel theories; from $R_{\rm in}=6\ b_{0}$ to $R_{\rm out}=100\ b_{0}$
for the WH solution in the GR theory; from $R_{\rm in}=6M$ to $R_{\rm
out}=20M$ for the WH solution in the metric-affine theory, where the disk is
composed by PR critical hypersurfaces in which $\lambda=0.3$ at all radii. The
lateral legend bar indicates the level of gravitational redshift in each point
of the disk, being different for each WH solution.

In Fig. 3 we consider the PR critical hypersurface extending from $R_{i}=1.2M$
to $R_{f}=1.3M$, except in GR theory where it spans from $R_{i}=1.2b_{0}$ to
$R_{f}=1.3b_{0}$. In all cases (except in GR theory), the PR critical
hypersurface is located under the photon sphere, where possible metric-changes
could occur De Falco  _et al._ (2020). We consider that the matter on the PR
crtitical hypersurface moves with constant angular velocity. For this reason,
we fix $\lambda_{0}=0.5$ at $R_{i}=1.2M$ (at $R_{i}=1.2b_{0}$ in GR theory)
and then we let $\lambda(r)$ vary with $r\in[R_{i},R_{f}]$ according to this
law

$\lambda(r)=\frac{e^{\Phi(R_{i})}}{e^{\Phi(r)}}\left(\frac{r}{R_{i}}\right)^{2}\lambda_{0}.$
(74)

In this way we can calculate the intervals over which the PR photon impact
parameter $\lambda$ and the luminosity parameter $A/M$ range. We obtain the
following results:

  1. 1.

GR theory: $\lambda\in[0.50,0.54]$, $A/b_{0}\in[0.10,0.12]$;

  2. 2.

Metric theory: $\lambda\in[0.50,0.52]$, $A/M\in[0.01,0.02]$;

  3. 3.

Metric-affine theory: $\lambda\in[0.50,0.54]$, $A/M\in[0.37,0.46]$;

  4. 4.

Teleparallel theory: $\lambda\in[0.50,0.52]$, $A/M\in[0.01,0.02]$.

All WH solutions admit physically reasonable luminosity parameter ranges,
except maybe for the WH solution in metric-affine theory which is relatively
high.

The lightcurves of the PR critical hypersurfaces are plotted for just one
period, time that the test particle takes to complete one loop from the
starting position. In these plots, it is possible to examine the dynamical
evolution of the emitted fluxes. We note that all plots show almost the same
trend, similar to a peaked Gaussian at half period, corresponding to the
material on the approaching side toward the observer which is greatly enhanced
by the gravitational field. In all cases (except in GR theory), the PR
critical hypersurface is located under the photon sphere radius, therefore
only the part of the PR critical hypersurface which is located inside the cone
of avoidance reaches the observer at infinity. The WH solution in the metric-
affine theory shows some visible differences at the begin and end of its
periodic motion, while the WH solutions in metric and teleparallel cases
exhibit only very tiny differences, which are almost indistinguishable. The
explanation of such behaviours relies on the evidence that the redshift
functions $\Phi(r)$ of these two theories are equal and dominate over their
shape functions, albeit they have different functional forms. Instead in the
metric-affine case, it is worth to note that the shape function is always
dominated in the blueshifted region, but it contributes at the begin and end
of the period by producing small enhancements in the flux.

Another fundamental enquiry relies on the spectral analysis, which
investigates how the flux changes in terms of the energy. We note that the
spectral profiles of the WH solutions in metric and teleparallel theories are
still very similar, and are also defined in the same energy interval, i.e.
$E_{\rm obs}/E_{\rm em}\in[0.010,0.104]$. The explanation of such analogy
resumes the same argument exposed in the lightcurve case. In the other cases
alternately, the profiles are well distinguishable and therefore it is very
easy to recognise them. From such example we learn that for some WH solutions
it is difficult to reconstruct $\Phi(r)$ and $b(r)$ functions only looking at
their PR critical hypersurfaces. Other alternative ways to better disentangle
two (or more) similar WH solutions belonging to different or same extended
theories of gravity could be: (i) looking for other emitting surfaces or (ii)
searching for other strategies which can be combined with that developed here
to crosscheck the results and extract new information.

Finally, in Fig. 4 we consider an accretion disk forming around a WH, where we
numerically simulate its appearance on the screen of a static and no-rotating
observer at infinity. For the WH solution in GR theory we consider a small
accretion disk extending from $R_{\rm in}=6\ b_{0}$ to $R_{\rm out}=100\
b_{0}$, otherwise if it is too far from the WH throat the results become
trivial, because the disk would lie in a quasi flat spacetime. For the WH
solutions in metric and teleparallel theories, a real disk extending from
$R_{\rm in}=6M$ to $R_{\rm out}=100M$ is considered. Instead, for the WH
solution in metric-affine theory it is not possible to have an accretion disk
supported uniquely by the background gravitational field, because stable
circular orbits do not exist (see Sec. V.3). However, we consider a disk made
by PR critical hypersurfaces, extending from $R_{\rm in}=6M$ to $R_{\rm
out}=20M$, and endowed with PR photon impact parameter $\lambda=0.3$ in all
points of the disk. This case is not fictitious at all, because it
realistically reproduces the physics occurring in the inner regions of an
astrophysical disk, which are radiation-pressure dominated (see Refs. Shakura
and Sunyaev (1973); Lančová  _et al._ (2017), for further details). With our
parameter choices, we have that the luminosity parameter ranges in
$A/M\in[0.003,0.0067]$. This result gives rise to a counterintuitive issue,
because we expect that the more far we move from an emitting source (located
close to the WH throat), the stronger the luminosity should be. This
controversy can be better understood if we think that the geometrical
proprieties of the background spacetime are different from those of classical
GR, where the PR effect is normally framed. In the other cases, we consider
that the matter moves inside the disk with Keplerian angular velocity (see
Table 2, for a summary of the equations employed for obtaining the WH image).
In all cases, we plot next to each image a legend bar showing the
gravitational redshift in each point of the accretion disk. This permits to
understand not only the WH geometrical structure, but also how the matter
interacts and behaves with the gravity in each WH spacetime. We immediately
note that also in this case, the WH solutions in metric and teleparallel
theories are still similar, being hard to spot the differences. Therefore,
although the ray-tracing and imaging procedures are very powerful and
successful diagnostic tools in several cases, sometimes they should be
complemented by alternative strategies in order to achieve more solid and
robust results.

Table 2:  Summary of the accretion disk information around the WH solutions.
Critical impact parameter, Keplerian velocity, periastron, and gravitational
redshift are calculated respectively through Eqs. (10), (48), (13), and (50).

---
Theory |  $\bm{b_{c}}$ | $\bm{p}$ | $\bm{\Omega_{K}}$ | $\bm{(1+z)^{-1}}$
GR | $0.66\ b_{0}$ | NOT EXIST |  $4\sqrt{6}\sqrt{\frac{b_{0}}{R^{2}(3R+b_{0})}}\left(\frac{R}{4R+b_{0}}\right)^{3/2}$ |  $\frac{8\left(\frac{R}{4R+b_{0}}\right)^{3/2}\sqrt{1-\frac{3b_{0}}{2(3R+b_{0})}}}{1+b_{\rm ph}\Omega_{K}\frac{\sin i\sin\varphi}{\sin\psi}}$
Metric |  $8.15\ M$ |  $-\frac{3M}{W(-3M/b_{\rm ph})}$666We remind that $W$ is the Lambert (or ProductLog) function Fukushima (2013). |  $\sqrt{3}e^{-3M/r}\sqrt{\frac{M}{r^{3}}}$ |  $\frac{e^{-3M/r}\sqrt{1-3M/r}}{1+b_{\rm ph}\Omega_{K}\frac{\sin i\sin\varphi}{\sin\psi}}$
Affine-Metric | $2.01M$ | $-\frac{\sqrt[3]{3}M}{\sqrt[3]{W\left[-3\left(\frac{M}{b_{\rm ph}}\right)^{3}\right]}}$ |  $\sqrt{3}e^{-\frac{M^{3}}{r^{3}}}\sqrt{\frac{M^{5}}{r^{5}}}$777It is important to note that in such WH spacetime there not exist stable circular orbits, see Sec. V.3. Therefore, the Keplerian angular velocity and the gravitational redshift (for a disk) are not globally defined. However, we are able to derive their analytical expressions, as those reported in the table, but they can be interpreted only as expressions valid locally. | $\frac{e^{-\frac{M^{3}}{r^{3}}}\sqrt{1-\frac{3M^{3}}{r^{3}}}}{1+b_{\rm ph}\Omega_{K}\frac{\sin i\sin\varphi}{\sin\psi}}$
Teleparallel | $8.15\ M$ | $-\frac{3M}{W(-3M/b_{\rm ph})}$ | $\sqrt{3}e^{-3M/r}\sqrt{\frac{M}{r^{3}}}$ | $\frac{e^{-3M/r}\sqrt{1-3M/r}}{1+b_{\rm ph}\Omega_{K}\frac{\sin i\sin\varphi}{\sin\psi}}$

##  VI Conclusions

This paper configures as a follow up analysis to our previous work De Falco
_et al._ (2020). This paper configures as a follow up analysis to our former
work De Falco  _et al._ (2020). Indeed, in our previous publication we
employed the flux emitted by the PR critical hypersurfaces, modeled in the BH
Schwarzschild metric, through the ray-tracing equations in GR as a probe to
detect possible metric changes occurring in the transition surface layer,
located between the BH event horizon and the photon sphere. Since this
scenario entails the possible existence of a WH, we are naturally led to look
for a procedure to distinguish among the different WH solutions which are the
most adapt to fit and interpret the observational data.

To solve this issue, we have proposed a strategy for testing static and
spherically symmetric WH solutions described by the general Morris-Thorne-like
metric (1) against the observational data. This method is based on the
following three fundamental steps:

  1. 1.

development of general ray-tracing equations in metric (1), which are: light
bending (5), travel time delay (II.2.4), and solid angle (19), see Sec. II.2;

  2. 2.

employing the general relativistic PR effect model framed in the general
metric (1) De Falco  _et al._ (2020) to determine the existence of critical
hypersurfaces (34) in strong field regime regions, see Sec. III;

  3. 3.

calculation of the flux (36) emitted by the PR critical hypersurface (or also
by an accretion disk in the equatorial plane around a WH and located only in
one universe) toward a distant observer. This model is so general and flexible
that it is capable to perfectly model all different spherically symmetric and
static WH solutions, without using any approximation. This permits to fit all
observational data presenting metric-changes, and to consequently determine
the profile of the unknown functions $\Phi(r),b(r)$ relative to the region
where the emitting surface is placed. The observational data can thus impose
tight constraints not only on the WH solutions, but also on the theories of
gravity.

We would like to underline that this work and the previous publication De
Falco  _et al._ (2020) are conceptually distinct. Moreover, this paper
configures as a logical and direct consequence of the previous article. The
fundamental differences between the two papers can be summarized as follows:

  1. 1.

_practical intents_ , because in the first paper we looked for observational
WH existence, while in this manuscript we have constrained the WH solutions
within general gravity frameworks through the fit of the data;

  2. 2.

_geometrical approach_ , because in the first work we only employed the BH
Schwarzschild metric, while in this manuscript we have dealt with generic
static and spherically symmetric WH metrics;

  3. 3.

_philosophy and methodology of investigation_ , because in the former
publication we have only considered a single specific metric and we based our
conclusions on the dihcotomic answer “yes” or “no” to the question of the WH
existence; on the other hand, in this paper we have handled a two-parameters
class of general metrics and we have ontologically dealt with several WH
solutions, which can be reduced and constrained only through the data,
producing as a final result a “procedure”, rather than a “specific answer”.

The fundamental element of this work is represented by metric (1), which has a
general character and valid both in GR and in extended/alternative theories of
gravity. The ray-tracing technique combined with the imaging procedure permit
to infer fundamental information on the WH solutions through the fit of the
observational data. In some cases, we are not able to identify just one single
WH solution, but we can reconstruct a class of possible WH solutions. To
properly determine the right WH solution, it is necessary to exploit
alternative procedures for extracting the missing information, see Sec. V.5.

This procedure allows also to provide new tests of gravity in strong field
regimes, understanding thus whether GR or extended/alternative theories of
gravity are needed. Another advantage of this work is represented by the
general set of ray-tracing equations, which can be employed to calculate and
model the flux from any x-ray source, regardless of its geometry. The same
argument holds also for any configuration of the observer’s geometrical screen
structure and dynamical setup.

To show the potentiality of our approach, we have applied the general ray-
tracing equations, see Sec. II, to four different WH solutions belonging to
distinct extended theories of gravity, which are: GR (see Sec. V.1), metric
(see Sec. V.2), metric affine (see Sec. V.3), and teleparallel (see Sec. V.4).
For each WH solution we have analysed the geometry close to the WH neck (see
Fig. 2), the lightcurve and spectrum of the PR critical hypersurface around a
WH toward a distant observer (see Fig. 3), and the image of an accretion disk
seen by an observer at infinity (see Fig. 4). We have conducted an accurate
study on these WH solutions, showing how to extract important information on
the WH geometrical proprieties (see Tables 1 and 2, for more details). In
particular the calculus of the photon sphere and ISCO radii are important
a-priori checks for having a quick overview of a WH spacetime’s geodesic
structure.

A limiting case of this model is represented by the fact that all the ray-
tracing equations are _elliptic integrals_ , which do not admit a solution in
terms of elementary functions De Falco  _et al._ (2016). However, the
calculation of such integrals strongly depends also on the functional form of
$\Phi(r),b(r)$, but generally, they give rise to complicate functions, which
are very time-consuming. However, following the same ideas adopted in the
Schwarzschild case, it would be helpful to develop a set of high-accurate
approximate equations (see Refs. De Falco  _et al._ (2016); Semerák (2015); La
Placa  _et al._ (2019); Poutanen (2019), for more details), which drastically
reduce the computational times. Our ray-tracing approach could be further
complemented by the model-independent framework developed by Rezzolla and
collaborators Rezzolla and Zhidenko (2014), where through a finite set of
coefficients it would be possible to measure possible metric-deviations from
GR for all static and spherically symmetric geometries mimicking that of a BH.

As future perspectives, we aim at extending this treatment also to axially
symmetric spacetimes in the equatorial plane and then in the 3D space.

## Acknowledgements

V.D.F. thanks Gruppo Nazionale di Fisica Matematica of Istituto Nazionale di
Alta Matematica (INDAM) for support. S.C. acknowledges the support of Istituto
Nazionale di Fisica Nucleare (INFN) sez. di Napoli, Iniziative Specifiche
MOONLIGHT2 and QGSKY. M.D.L. acknowledges INFN sez. di Napoli, Iniziative
Specifiche QGSKY and TEONGRAV.

## References

  * Winkler  _et al._ (2003) C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, A&A 411, L1 (2003).
  * Burrows  _et al._ (2005) D. N. Burrows, J. E. Hill, J. A. Nousek, J. A. Kennea, A. Wells, J. P. Osborne, A. F. Abbey, A. Beardmore, K. Mukerjee, A. D. T. Short, G. Chincarini, S. Campana, O. Citterio, A. Moretti, C. Pagani, G. Tagliaferri, P. Giommi, M. Capalbi, F. Tamburelli, L. Angelini, G. Cusumano, H. W. Bräuninger, W. Burkert, and G. D. Hartner, SSR 120, 165 (2005), astro-ph/0508071 .
  * Beckwith and Done (2004) K. Beckwith and C. Done, MNRAS 352, 353 (2004), arXiv:astro-ph/0402199 [astro-ph] .
  * Tomsick  _et al._ (2014) J. A. Tomsick, M. A. Nowak, M. Parker, J. M. Miller, A. C. Fabian, F. A. Harrison, M. Bachetti, D. Barret, S. E. Boggs, F. E. Christensen, W. W. Craig, K. Forster, F. Fürst, B. W. Grefenstette, C. J. Hailey, A. L. King, K. K. Madsen, L. Natalucci, K. Pottschmidt, R. R. Ross, D. Stern, D. J. Walton, J. Wilms, and W. W. Zhang, ApJ 780, 78 (2014), arXiv:1310.3830 [astro-ph.HE] .
  * Chael  _et al._ (2016) A. A. Chael, M. D. Johnson, R. Narayan, S. S. Doeleman, J. F. C. Wardle, and K. L. Bouman, ApJ 829, 11 (2016), arXiv:1605.06156 [astro-ph.IM] .
  * Akiyama  _et al._ (2019) K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA] .
  * Barcons  _et al._ (2017) X. Barcons, D. Barret, A. Decourchelle, J. W. den Herder, A. C. Fabian, H. Matsumoto, D. Lumb, K. Nandra, L. Piro, R. K. Smith, and R. Willingale, Astronomische Nachrichten 338, 153 (2017).
  * (8) S. Zhang, M. Feroci, A. Santangelo, Y. Dong, H. Feng, _et al._
  * Soffitta  _et al._ (2013) P. Soffitta, X. Barcons, R. Bellazzini, J. Braga, E. Costa, G. W. Fraser, S. Gburek, J. Huovelin, G. Matt, M. Pearce, J. Poutanen, V. Reglero, A. Santangelo, R. A. Sunyaev, G. Tagliaferri, M. Weisskopf, R. Aloisio, E. Amato, P. Attiná, M. Axelsson, L. Baldini, S. Basso, S. Bianchi, P. Blasi, J. Bregeon, A. Brez, N. Bucciantini, L. Burderi, V. Burwitz, P. Casella, E. Churazov, M. Civitani, S. Covino, R. M. Curado da Silva, G. Cusumano, M. Dadina, F. D’Amico, A. De Rosa, S. Di Cosimo, G. Di Persio, T. Di Salvo, M. Dovciak, R. Elsner, C. J. Eyles, A. C. Fabian, S. Fabiani, H. Feng, S. Giarrusso, R. W. Goosmann, P. Grandi, N. Grosso, G. Israel, M. Jackson, P. Kaaret, V. Karas, M. Kuss, D. Lai, G. La Rosa, J. Larsson, S. Larsson, L. Latronico, A. Maggio, J. Maia, F. Marin, M. M. Massai, T. Mineo, M. Minuti, E. Moretti, F. Muleri, S. L. O’Dell, G. Pareschi, G. Peres, M. Pesce, P.-O. Petrucci, M. Pinchera, D. Porquet, B. Ramsey, N. Rea, F. Reale, J. M. Rodrigo, A. Różańska, A. Rubini, P. Rudawy, F. Ryde, M. Salvati, V. A. r. de Santiago, S. Sazonov, C. Sgró, E. Silver, G. Spandre, D. Spiga, L. Stella, T. Tamagawa, F. Tamborra, F. Tavecchio, T. Teixeira Dias, M. van Adelsberg, K. Wu, and S. Zane, Experimental Astronomy 36, 523 (2013), arXiv:1309.6995 [astro-ph.HE] .
  * Gourgoulhon  _et al._ (2019) E. Gourgoulhon, A. Le Tiec, F. H. Vincent, and N. Warburton, A&A 627, A92 (2019), arXiv:1903.02049 [gr-qc] .
  * Abuter  _et al._ (2020) R. Abuter _et al._ (GRAVITY), (2020), arXiv:2004.07187 [astro-ph.GA] .
  * Visser (1995) M. Visser,  _Lorentzian wormholes: From Einstein to Hawking_ (1995).
  * Capozziello and De Laurentis (2011) S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167 (2011), arXiv:1108.6266 [gr-qc] .
  * Damour and Solodukhin (2007) T. Damour and S. N. Solodukhin, PRD 76, 024016 (2007), arXiv:0704.2667 [gr-qc] .
  * Bambi (2012) C. Bambi, ApJ 761, 174 (2012), arXiv:1210.5679 [gr-qc] .
  * Bambi (2013) C. Bambi, Phys. Rev. D 87, 107501 (2013), arXiv:1304.5691 [gr-qc] .
  * Cardoso  _et al._ (2016) V. Cardoso, E. Franzin, and P. Pani, PRL 117, 089902 (2016).
  * Konoplya and Zhidenko (2016) R. A. Konoplya and A. Zhidenko, JCAP 2016, 043 (2016), arXiv:1606.00517 [gr-qc] .
  * Paul  _et al._ (2019) S. Paul, R. Shaikh, P. Banerjee, and T. Sarkar, arXiv e-prints , arXiv:1911.05525 (2019), arXiv:1911.05525 [gr-qc] .
  * Dai and Stojkovic (2019) D.-C. Dai and D. Stojkovic, PRD 100, 083513 (2019), arXiv:1910.00429 [gr-qc] .
  * Banerjee  _et al._ (2019) P. Banerjee, S. Paul, R. Shaikh, and T. Sarkar, arXiv e-prints , arXiv:1912.01184 (2019), arXiv:1912.01184 [astro-ph.HE] .
  * Hashimoto and Tanahashi (2017) K. Hashimoto and N. Tanahashi, PRD 95, 024007 (2017), arXiv:1610.06070 [hep-th] .
  * Dalui  _et al._ (2019) S. Dalui, B. R. Majhi, and P. Mishra, Physics Letters B 788, 486 (2019), arXiv:1803.06527 [gr-qc] .
  * De Falco  _et al._ (2020) V. De Falco, E. Battista, S. Capozziello, and M. De Laurentis, Phys. Rev. D 101, 104037 (2020), arXiv:2004.14849 [gr-qc] .
  * Ballantyne and Strohmayer (2004) D. R. Ballantyne and T. E. Strohmayer, The Astrophysical Journal Letters 602, L105 (2004), astro-ph/0401154 .
  * Ballantyne and Everett (2005) D. R. Ballantyne and J. E. Everett, The Astrophysical Journal 626, 364 (2005), astro-ph/0502463 .
  * Worpel  _et al._ (2013) H. Worpel, D. K. Galloway, and D. J. Price, The Astrophysical Journal 772, 94 (2013), arXiv:1303.4824 [astro-ph.HE] .
  * Ji  _et al._ (2014) L. Ji, S. Zhang, Y. Chen, S.-N. Zhang, D. F. Torres, P. Kretschmar, and J. Li, The Astrophysical Journal 782, 40 (2014), arXiv:1312.4166 [astro-ph.HE] .
  * Keek  _et al._ (2014) L. Keek, D. R. Ballantyne, E. Kuulkers, and T. E. Strohmayer, The Astrophysical Journal 789, 121 (2014), arXiv:1403.4940 [astro-ph.HE] .
  * Worpel  _et al._ (2015) H. Worpel, D. K. Galloway, and D. J. Price, The Astrophysical Journal 801, 60 (2015), arXiv:1501.02070 [astro-ph.HE] .
  * Poynting (1903) J. H. Poynting, Monthly Notices of the Royal Astronomical Society 64, 1 (1903).
  * Robertson (1937) H. P. Robertson, Monthly Notices of the Royal Astronomical Society 97, 423 (1937).
  * Bini  _et al._ (2009) D. Bini, R. T. Jantzen, and L. Stella, Classical and Quantum Gravity 26, 055009 (2009), arXiv:0808.1083 [gr-qc] .
  * Bini  _et al._ (2011) D. Bini, A. Geralico, R. T. Jantzen, O. Semerák, and L. Stella, Classical and Quantum Gravity 28, 035008 (2011), arXiv:1408.4945 [gr-qc] .
  * De Falco  _et al._ (2019) V. De Falco, P. Bakala, E. Battista, D. Lančová, M. Falanga, and L. Stella, PRD 99, 023014 (2019).
  * Bakala  _et al._ (2019) P. Bakala, V. De Falco, E. Battista, K. Goluchová, D. Lančová, M. Falanga, and L. Stella, PRD 100, 104053 (2019).
  * Wielgus (2019) M. Wielgus, MNRAS 488, 4937 (2019), arXiv:1907.11268 [astro-ph.HE] .
  * De Falco  _et al._ (2020) V. De Falco, P. Bakala, and M. Falanga, Phys. Rev. D 101, 124031 (2020).
  * De Falco  _et al._ (2018) V. De Falco, E. Battista, and M. Falanga, Phys. Rev. D D97, 084048 (2018), arXiv:1804.00519 [gr-qc] .
  * De Falco and Battista (2019) V. De Falco and E. Battista, EPL 127, 30006 (2019), arXiv:1907.13354 [gr-qc] .
  * De Falco and Battista (2020) V. De Falco and E. Battista, Phys. Rev. D 101, 064040 (2020), arXiv:2003.04416 [gr-qc] .
  * De Falco (2020) V. De Falco, Emerging Science Journal 4, 3 (2020).
  * Walker and Meszaros (1989) M. A. Walker and P. Meszaros, The Astrophysical Journal 346, 844 (1989).
  * Walker (1992) M. A. Walker, The Astrophysical Journal 385, 642 (1992).
  * Lančová  _et al._ (2017) D. Lančová, P. Bakala, K. Goluchová, M. Falanga, V. De Falco, and L. Stella, in  _Proceedings of RAGtime 17-19: Workshops on black holes and neutron stars, 17-19/23-26 Oct., 1-5 Nov. 2015/2016/2017, Opava, Czech Republic, Z. Stuchlík, G. Török and V. Karas editors, Silesian University in Opava, 2017, ISBN 978-80-7510-257-7, ISSN 2336-5676, p. 127-136_ (2017) pp. 127–136.
  * Fragile  _et al._ (2020) P. C. Fragile, D. R. Ballantyne, and A. Blankenship, Nature Astronomy , 6 (2020).
  * Visser (1989) M. Visser, Nuclear Physics B 328, 203 (1989), arXiv:0809.0927 [gr-qc] .
  * Barcelo and Visser (1999) C. Barcelo and M. Visser, Physics Letters B 466, 127 (1999), arXiv:gr-qc/9908029 [gr-qc] .
  * Böhmer  _et al._ (2012) C. G. Böhmer, T. Harko, and F. S. N. Lobo, Phys. Rev. D 85, 044033 (2012), arXiv:1110.5756 [gr-qc] .
  * Anchordoqui  _et al._ (2000) L. A. Anchordoqui, S. Capozziello, G. Lambiase, and D. F. Torres, Mod. Phys. Lett. A15, 2219 (2000), arXiv:gr-qc/0011097 [gr-qc] .
  * Bahamonde  _et al._ (2016) S. Bahamonde, U. Camci, S. Capozziello, and M. Jamil, Phys. Rev. D94, 084042 (2016), arXiv:1608.03918 [gr-qc] .
  * Capozziello  _et al._ (2020) S. Capozziello, R. D’Agostino, and D. Gregoris, Phys. Dark Univ. 28, 100513 (2020), arXiv:2002.04875 [gr-qc] .
  * Morris and Thorne (1988) M. S. Morris and K. S. Thorne, American Journal of Physics 56, 395 (1988).
  * Hochberg and Visser (1998a) D. Hochberg and M. Visser, Phys. Rev. D 58, 044021 (1998a), arXiv:gr-qc/9802046 [gr-qc] .
  * Hochberg and Visser (1998b) D. Hochberg and M. Visser, Phys. Rev. Letters 81, 746 (1998b), arXiv:gr-qc/9802048 [gr-qc] .
  * Capozziello  _et al._ (2015) S. Capozziello, F. S. N. Lobo, and J. P. Mimoso, Phys. Rev. D91, 124019 (2015), arXiv:1407.7293 [gr-qc] .
  * Capozziello  _et al._ (2014) S. Capozziello, F. S. N. Lobo, and J. P. Mimoso, Phys. Lett. B730, 280 (2014), arXiv:1312.0784 [gr-qc] .
  * Hochberg  _et al._ (1997) D. Hochberg, A. Popov, and S. V. Sushkov, Phys. Rev. L. 78, 2050 (1997), arXiv:gr-qc/9701064 [gr-qc] .
  * Bronnikov  _et al._ (2013) K. A. Bronnikov, L. N. Lipatova, I. D. Novikov, and A. A. Shatskiy, Gravitation and Cosmology 19, 269 (2013), arXiv:1312.6929 [gr-qc] .
  * Garattini (2019) R. Garattini, European Physical Journal C 79, 951 (2019), arXiv:1907.03623 [gr-qc] .
  * Lobo and Oliveira (2009) F. S. N. Lobo and M. A. Oliveira, PRD 80, 104012 (2009), arXiv:0909.5539 [gr-qc] .
  * Harko  _et al._ (2013) T. Harko, F. S. N. Lobo, M. K. Mak, and S. V. Sushkov, Phys. Rev. D 87, 067504 (2013), arXiv:1301.6878 [gr-qc] .
  * Di Grezia  _et al._ (2017) E. Di Grezia, E. Battista, M. Manfredonia, and G. Miele, European Physical Journal Plus 132, 537 (2017), arXiv:1707.01508 [gr-qc] .
  * Misner  _et al._ (1973) C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _San Francisco: W.H. Freeman and Co., 1973_ (1973).
  * Beloborodov (2002) A. M. Beloborodov, ApJL 566, L85 (2002), astro-ph/0201117 .
  * De Falco  _et al._ (2016) V. De Falco, M. Falanga, and L. Stella, A&A 595, A38 (2016), arXiv:1608.04574 [astro-ph.HE] .
  * Press  _et al._ (2002) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,  _Numerical recipes in C++ : the art of scientific computing by William H. Press. xxviii, 1,002 p. : ill. ; 26 cm. Includes bibliographical references and index. ISBN : 0521750334_ (2002).
  * Chandrasekhar (1983) S. Chandrasekhar,  _Research supported by NSF. Oxford/New York, Clarendon Press/Oxford University Press (International Series of Monographs on Physics. Volume 69), 1983, 663 p._ (1983).
  * Pechenick  _et al._ (1983) K. R. Pechenick, C. Ftaclas, and J. M. Cohen, ApJ 274, 846 (1983).
  * Poutanen and Beloborodov (2006) J. Poutanen and A. M. Beloborodov, MNRAS 373, 836 (2006), astro-ph/0608663 .
  * Cunningham and Bardeen (1973) C. T. Cunningham and J. M. Bardeen, ApJ 183, 237 (1973).
  * Luminet (1979) J. P. Luminet, A&A 75, 228 (1979).
  * Capozziello  _et al._ (2010) S. Capozziello, M. De Laurentis, and V. Faraoni, Open Astron. J. 3, 49 (2010), arXiv:0909.4672 [gr-qc] .
  * Capozziello and De Laurentis (2012) S. Capozziello and M. De Laurentis, Annalen Phys. 524, 545 (2012).
  * Godani and Samanta (2020) N. Godani and G. C. Samanta, New Astron. 80, 101399 (2020), arXiv:2004.14209 [gr-qc] .
  * Starobinsky (2007) A. A. Starobinsky, JETP Lett. 86, 157 (2007), arXiv:0706.2041 [astro-ph] .
  * Fukushima (2013) T. Fukushima, Journal of Computational and Applied Mathematics 244, 77 (2013).
  * Capozziello  _et al._ (2012a) S. Capozziello, M. De Laurentis, L. Fatibene, and M. Francaviglia, Int. J. Geom. Meth. Mod. Phys. 9, 1250072 (2012a), arXiv:1202.5699 [gr-qc] .
  * Olmo (2011) G. J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011), arXiv:1101.3864 [gr-qc] .
  * Tamanini and Boehmer (2013) N. Tamanini and C. G. Boehmer, Phys. Rev. D 87, 084031 (2013), arXiv:1302.2355 [gr-qc] .
  * Capozziello  _et al._ (2015) S. Capozziello, T. Harko, T. Koivisto, F. Lobo, and G. Olmo, Universe 1, 199 (2015), arXiv:1508.04641 [gr-qc] .
  * Capozziello  _et al._ (2012b) S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo, and G. J. Olmo, Phys. Rev. D86, 127504 (2012b), arXiv:1209.5862 [gr-qc] .
  * Abramowitz and Stegun (1965) M. Abramowitz and I. A. Stegun,  _Dover Books on Advanced Mathematics, New York: Dover, —c1965, Corrected edition, edited by Abramowitz, Milton; Stegun, Irene A._ (1965).
  * Cai  _et al._ (2016) Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Rept. Prog. Phys. 79, 106901 (2016), arXiv:1511.07586 [gr-qc] .
  * Kofinas and Saridakis (2014) G. Kofinas and E. N. Saridakis, Phys. Rev. D 90, 084044 (2014).
  * Capozziello  _et al._ (2016) S. Capozziello, M. De Laurentis, and K. F. Dialektopoulos, Eur. Phys. J. C76, 629 (2016), arXiv:1609.09289 [gr-qc] .
  * Kofinas  _et al._ (2014) G. Kofinas, G. Leon, and E. N. Saridakis, Classical and Quantum Gravity 31, 175011 (2014).
  * Sharif and Nazir (2018) M. Sharif and K. Nazir, Annals of Physics 393, 145 (2018).
  * Shakura and Sunyaev (1973) N. I. Shakura and R. A. Sunyaev, A&A 500, 33 (1973).
  * Semerák (2015) O. Semerák, ApJ 800, 77 (2015), arXiv:1412.5650 [astro-ph.HE] .
  * La Placa  _et al._ (2019) R. La Placa, P. Bakala, L. Stella, and M. Falanga, Research Notes of the American Astronomical Society 3, 99 (2019), arXiv:1907.11786 [gr-qc] .
  * Poutanen (2019) J. Poutanen, arXiv e-prints , arXiv:1909.05732 (2019), arXiv:1909.05732 [astro-ph.HE] .
  * Rezzolla and Zhidenko (2014) L. Rezzolla and A. Zhidenko, PRD 90, 084009 (2014), arXiv:1407.3086 [gr-qc] .