# Image of the thin accretion disk around compact objects in the Einstein-
Gauss-Bonnet gravity

Galin Gyulchev1111E-mail<EMAIL_ADDRESS>Petya Nedkova1222E-mail:
<EMAIL_ADDRESS>Tsvetan Vetsov1333E-mail<EMAIL_ADDRESS>sofia.bg,
Stoytcho Yazadjiev1,2444E-mail<EMAIL_ADDRESS>

1 Faculty of Physics, Sofia University,
5 James Bourchier Boulevard, Sofia 1164, Bulgaria
2 Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, Acad. G. Bonchev 8,
Sofia 1113, Bulgaria

###### Abstract

We study the optical appearance of a thin accretion disk around compact
objects within the Einstein-Gauss-Bonnet gravity. Considering static
spherically symmetric black holes and naked singularities we search for
characteristic signatures which can arise in the observable images due to the
modification of general relativity. While the images of the Gauss-Bonnet black
holes closely resemble the Schwarzschild black hole, naked singularities
possess a distinctive feature. A series of bright rings are formed in the
central part of the images with observable radiation $10^{3}$ times larger
than the rest of the flux making them observationally significant. We
elucidate the physical mechanism, which causes the appearance of the central
rings, showing that the image is determined by the light ring structure of the
spacetime. In a certain region of the parametric space the Gauss-Bonnet naked
singularities possess a stable and an unstable light ring. In addition the
gravitational field becomes repulsive in a certain neighbourhood of the
singularity. This combination of features leads to the formation of the
central rings implying that the effect is not specific for the Einstein-Gauss-
Bonnet gravity but would also appear for any other compact object with the
same characteristics of the photon dynamics.

##  1 Introduction

Recently a new static spherically symmetric black hole was proposed which
generalizes the Schwarzschild black hole when quantum corrections are
considered [1]-[3]. It was derived as an exact solution to the semi-classical
Einstein equations with Weyl anomaly of the energy-momentum tensor by
restricting the trace anomaly only to the Gauss-Bonnet invariant. The new
black holes possess a logarithmic term in their Bekenstein-Hawking entropy,
which is consistent with the microscopic statistical interpretation of the
black hole entropy in string theory or quantum gravity. In addition, they
provide a resolution to the singularity problem, since the gravitational force
becomes repulsive at very small distances from the compact object preventing
the particles and light to access the singularity.

The solution was also obtained in a different context considering the
classical four-dimensional (4D) Einstein-Guass-Bonnet equations [4]. It was
suggested that the Einstein-Gauss-Bonnet theory possesses a non-trivial limit
to four spacetime dimensions which circumvents the Lovelock theorem and allows
the contribution of the Gauss-Bonnet term to the local dynamics. While the
proposed regularization procedure is not consistent for general gravitational
fields [5]-[7], it leads to correct predictions in a number of cases with high
symmetries, such as spherically symmetric solutions for example.

Irrespective of the theoretical framework in which the solution is
interpreted, the non-trivial contribution of the Gauss-Bonnet term is expected
to have phenomenological impact. This triggered a range of works investigating
different observational features such as the shadow [8], the gravitational
lensing [9], and the radiation from the accretion disk [10]. Static
relativistic stars in the proposed 4D limit of the Einstein-Gauss-Bonnet
gravity were constructed in [11], while the gravitational collapse of a
spherical cloud of dust was studied in [12].

The purpose of this work is to investigate the optical appearance of four-
dimensional compact objects with a Gauss-Bonnet term. Similar studies were
initiated in order to search for the observational signatures in the
electromagnetic spectrum of various types of compact objects in general
relativity and the modified theories of gravity [13]-[20]. We construct the
observable images of the thin accretion disk around black holes and naked
singularities belonging to the class of solutions obtained in [1]-[4]. While
Gauss-Bonnet black holes appear very similar to the Schwarzschild black hole,
we observe significant distinctions in the case of naked singularities. The
accretion disk possesses multiple images, which form a series of concentric
bright rings in the central region of the primary disk image. Similar effect
was obtained recently for the strongly naked Janis-Newman-Winicour singularity
where its appearance was connected with the absence of a photon sphere [14].
However, in the case of the Gauss-Bonnet naked singularities the multiple
images are related to a different feature of the photon dynamics, and they
appear even when the solution possesses a photon sphere. We elucidate the
physical mechanism which leads to the formation of the image noting that
similar processes may also take place in the case of other compact objects
where multiple ringlike disk images are observed in the presence of a photon
sphere [15]-[16].

The paper is organized as follows. In the next section we describe the exact
solution representing static compact objects within Gauss-Bonnet theory and
some of the characteristics of the geodesic motion in this spacetime, which
are relevant for our studies. In section 3 we present the images of the thin
accretion disk around Gauss-Bonnet black holes and weakly naked singularities
as seen by a distant observer and discuss their properties. In section 4 we
analyze the structure of the images revealing the mechanism of their
formation. In the last section we summarize our results.

##  2 Properties of the exact solution

Static spherically symmetric compact objects within 4D Gauss-Bonnet gravity
can be described by the metric [1]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\phi^{2}),$
(1) $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1+\frac{r^{2}}{2\gamma}\left(1-\sqrt{1+\frac{8\gamma M}{r^{3}}}\right).$

where $\gamma$ is a positive constant and $M$ is the $ADM$ mass. It was
derived originally as an exact solution to the semi-classical Einstein
equations with one loop quantum corrections

$\displaystyle R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta}=8\pi\langle
T_{\alpha\beta}\rangle,$ (2)

when we consider only the Gauss-Bonnet term in the general form of the trace
anomaly of the effective energy-momentum tensor $\langle
T_{\alpha\beta}\rangle$

$\displaystyle
g^{\alpha\beta}T_{\alpha\beta}=-\gamma\left(R^{2}-4R_{\alpha\beta}R^{\alpha\beta}+R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\right),$
(3)

where the coupling constant $\gamma$ depends on the degrees of freedom of the
quantum fields. The metric can be alternatively derived by solving the
classical D-dimensional Einstein-Gauss-Bonnet equations and taking a non-
trivial limit to four-dimensional spacetime [4].

The properties of the solution depend on the value of the dimensionless
coupling constant $\hat{\gamma}=\gamma/M^{2}$. When $\hat{\gamma}$ belongs to
the range $\hat{\gamma}\in[0,1]$ the solution describes a black hole with an
inner and an outer horizon located at $r_{\pm}=M\pm\sqrt{M^{2}-\gamma}$,
respectively. The value $\hat{\gamma}=1$ corresponds to the extremal black
hole, while $\hat{\gamma}=0$ is the Schwarzschild limit. The solutions with
$\hat{\gamma}>1$ are naked singularities. The Kretschmann invariant diverges
at the location of the singularity $r=0$, however in a slower rate than for
the Schwarzschild solution since it behaves as
$R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\sim\frac{1}{r^{3}}$
when approaching the singularity.

In this work we will consider black holes, i.e. solutions with
$\hat{\gamma}\in(0,1]$ and naked singularities with coupling constant taking
the range $1<\hat{\gamma}<3\sqrt{3}/4$. In this region of the parametric space
the solution describes weakly naked singularities, which possess a photon
sphere.555We use the classification of the naked singularities by means of
their lensing properties to weakly and strongly naked, which was introduced by
Virbhadra when studying of the Janis-Newman-Winicour solution [17]. The photon
dynamics is determined additionally by the presence of a stable photon ring
located in the interior of the photon sphere. The location of the photon rings
can be determined by obtaining the stationary points of the effective
potential for the null geodesics

$V^{ph}_{eff}=L^{2}\frac{f(r)}{r^{2}},$ (4)

where $L$ is the specific momentum of the photon. The radial coordinates of
the photon rings correspond to the solutions of the system $V^{ph}_{eff}=0$
and $dV^{ph}_{eff}/dr=0$, which can be reduced to the following algebraic
equation

$\displaystyle r^{3}+8M\gamma-9M^{2}r=0.$ (5)

Figure 1: Effective potential for the null geodesics for the Gauss-Bonnet
black holes with $\hat{\gamma}=1$ and weakly naked singularities with
$\hat{\gamma}=1.15$.

On the other hand, the Gauss-Bonnet black holes possess only a photon sphere
similar to the Schwarzschild black hole. In fig. 1 we illustrate the behavior
of the effective potential for the two types of compact objects as the photon
sphere corresponds to the maximum of the potential, while the stable light
ring is located at the minimum.

The structure of the timelike circular orbits for the Gauss-Bonnet black holes
resembles the Schwarzschild black hole. The stable circular orbits extend from
a certain marginally stable orbit to infinity. For weakly naked singularities
the stable circular orbits are located in two disconnected region. There
exists an inner annular region located between the inner limit of existence of
timelike circular orbits $r_{in}$ and the stable photon ring, and an outer
region extending from the marginally stable timelike orbit $r_{ms}$ to
infinity. In the gap between the two regions we have the following behavior.
Timelike circular orbits are not allowed between the stable and unstable
photon rings, while after passing the photon sphere they exist but remain
unstable until reaching $r_{ms}$. The inner radius of existence of timelike
circular orbits is given by the solution of the equation

$\displaystyle r^{3}+2M\gamma-r^{3}\sqrt{1+\frac{8M\gamma}{r^{3}}}=0,$ (6)

while the location of the marginally stable orbit corresponds to the
inflection point of the effective potential $V_{eff}$ for the timelike
geodesics satisfying $V_{eff}=0$, $dV_{eff}/dr=0$ and $d^{2}V_{eff}/dr^{2}=0$,
where

$V_{eff}=f(r)\left(1+\frac{L^{2}}{r^{2}}\right).$ (7)

In the following analysis the position of the $r_{ms}$ is obtained
numerically.

For our purposes the described characteristic features of the circular
geodesic orbits are sufficient to determine the structure of the thin
accretion disk for coupling constant in the range
$0<\hat{\gamma}<3\sqrt{3}/4$. A detailed qualitative analysis of the geodesic
motion will be presented in a further work.

##  3 Image of the thin accretion disk around Einstein-Gauss-Bonnet black
holes and naked singularities

We consider the image a thin accretion disk around Einstein-Gauss-Bonnet black
holes and weakly naked singularities with $1<\hat{\gamma}<3\sqrt{3}/4$. The
disk is described by the Novikov-Thorne model [24], [25], and consists of
radiating particles moving on stable circular geodesics in the compact
object’s spacetime. We construct its image as seen by a distant observer by
obtaining the optical appearance of the timelike circular orbits at spacetime
infinity and evaluating the observable radiation flux.

The apparent shape of the timelike circular orbits is obtained numerically
using a ray-tracing procedure. We apply additionally a semi-analytical scheme
requiring only numerical integration, which is more convenient for the
interpretation of the images. Both methods are described in detail in our
previous works [13]-[14]. In essence, the trajectories of the photons emitted
at a certain point on the circular orbits are obtained numerically up to some
radial distance corresponding effectively to the spacetime infinity.
Afterwards, their projections on the observer’s sky are visualized by means of
two celestial angles $\alpha$ and $\beta$, which are related to the photon’s
4-momentum as

$\displaystyle
p_{\theta}=\sqrt{g_{\theta\theta}}\sin\alpha,\qquad\qquad\,\,\,p_{\phi}=\sqrt{g_{\phi\phi}}\sin\beta\,\cos\alpha,$
$\displaystyle p_{r}=\sqrt{g_{rr}}\cos\beta\,\cos\alpha.$ (8)

Practically it is more convenient to use $\alpha$ and $\beta$ as initial data
and integrate the photon trajectory backwards to its emission point. We
consider the full range of the celestial angles $\alpha\in[0,\pi]$ and
$\beta\in[-\pi/2,\pi/2]$, and select those values which correspond to null
geodesics originating from the accretion disk, i.e. solutions to the geodesic
equation passing through the equatorial plane at a radial coordinate with
range of stability of the timelike circular orbits. In this way we obtain the
set of the celestial angles, which represent the image of the accretion disk
on the observer’s sky.

We associate further with each point of the image an observable flux emitted
by the accreting particles. The local flux radiated by a portion of the disk
delimited by the radial coordinates $r_{0}$ and $r$ is evaluated according to
the Novikov-Thorne model by the integral [24], [25]

$F(r)=-\frac{\dot{M}_{0}}{4\pi\sqrt{-g^{(3)}}}\frac{\Omega_{,r}}{(E-\Omega
L)^{2}}\int_{r_{0}}^{r}(E-\Omega L)L_{,r}dr,$ (9)

where $r_{0}$ is corresponds to the edge of the disk, $\dot{M}_{0}$ is the
accretion rate, and $g^{(3)}$ is the determinant of the induced metric in the
equatorial plane. We denote by $\Omega$, $E$ and $L$ the angular velocity, the
energy and the angular momentum of the particles on the circular orbits. For a
general static spherically symmetric metric in the form

$ds^{2}=g_{tt}\,dt^{2}+g_{\phi\phi}\,d\phi^{2}+g_{rr}\,dr^{2}+g_{\theta\theta}\,d\theta^{2}\,,$
(10)

they are given by the expressions

$\displaystyle E$ $\displaystyle=$
$\displaystyle-\frac{g_{tt}}{\sqrt{-g_{tt}-g_{\phi\phi}\Omega^{2}}},$ (11)
$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{g_{\phi\phi}\Omega}{\sqrt{-g_{tt}-g_{\phi\phi}\Omega^{2}}},$
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{d\phi}{dt}=\sqrt{-\frac{g_{tt,r}}{g_{\phi\phi,r}}}.$

In order to obtain the observable flux $F_{obs}$ at a given point of the
celestial sphere we should take into account the gravitational redshift $z$.
Thus, we obtain the relation

$F_{obs}=\frac{F}{(1+z)^{4}},$ (12)

where the gravitational redshift for a general spherically symmetric spacetime
can be expressed by means of the metric functions and the impact parameter
$b=L/E$ as [21]

$1+z=\frac{1+\Omega b}{\sqrt{-g_{tt}-\Omega^{2}g_{\phi\phi}}}.$ (13)

We illustrate the qualitative behavior of the images for the Einstein-Gauss-
Bonnet black holes and the weakly naked singularities by choosing a
representative value of the coupling constant $\hat{\gamma}$ for each class of
solutions. Their optical appearance is given in Figs. 2 and 3 representing the
distribution of the apparent flux intensity as seen by a distant observer,
where $\hat{\gamma}=1$ for the black hole, and $\hat{\gamma}=1.15$ for the
naked singularity. The image of the Schwarzschild black is also given for
comparison. The observable flux is normalized by its maximal value, and its
range $F_{obs}/F^{max}_{obs}\in[0,1]$ is mapped continuously to the color
spectrum from red to blue, as the highest values are depicted in blue. The
effective infinity is assumed to correspond to the radial coordinate
$r=5000M$, and we choose an inclination angle $i=80^{\circ}$, since for larger
inclination angles the relativistic effects are more pronounced.

The images of the Einstein-Gauss-Bonnet black holes resemble closely the
Schwarz-
schild black hole. Although we choose a Guass-Bonnet solution with a value of
the coupling constant maximally deviating from the Schwarzschild black hole,
only slight quantitative differences are present. The disk size is smaller in
the Einstein-Gauss-Bonnet case and the peak of the observable radiation is
slighly lower compared to the Schwarzschild black hole. Still, the observable
flux distribution follows the same pattern for the two solutions with a
similar characteristic location of the maximum of the radiation in the
vicinity of the ISCO on the left-hand side of the image.

On the other hand for weakly naked singularities we observe clear qualitative
distinctions. In the central region of the image a series of bright rings are
formed which are absent in the case of the Schwarzschild or the Gauss-Bonnet
black holes. For weakly naked singularities the stable circular orbits are
distributed into two disconnected regions which form an inner and an outer
accretion disk. The outer disk produces two types of images - a hat-like image
similar to the Schwarzschild black hole, and a sequence of rings located in
the interior part of the image of the marginally stable orbit. On the other
hand the inner disk is observed only in the form of central ring images.

The ring images resulting from the two disconnected parts of the accretion
disk radiate with different intensity. The observable flux of the rings
resulting from the outer disk can be evaluated to be less than $30\%$ of the
apparent flux maximum from the outer disk. The maximum of the observable
radiation from the outer disk $F^{out}_{max}$ is located on the hat-like image
similar to the Schwarzschild and Gauss-Bonnet black holes. In contrast the
observable flux from the inner disk rings is with three orders of magnitude
larger as its maximum $F^{in}_{max}$ is related to the radiation from the
outer disk as $F^{in}_{max}/F^{out}_{max}\sim 10^{3}$. Thus, it represents the
most pronounced intensity in the radiation from the whole accretion disk
image.

In order to facilitate the interpretation of the image we present the optical
appearance of the outer and the inner disks separately in the left and right
panels of Fig. 3. The complete observable image of the accretion disk is
composed by the superposition of the two images. For the chosen value of the
coupling constant $\hat{\gamma}=1.15$ the outer accretion disk extends from
the marginally stable orbit $r_{ms}=5.08M$ to infinity. The inner disk is
located between the inner limit of existence of the timelike circular orbits
$r_{in}=1.05M$ and the stable photon ring $r_{ph}=1.23M$.

A characteristic feature of the inner disk is that the emitted flux $F(r)$
decreases when the radius of the particle orbit $r$ moves towards its inner
edge. This is contrary to the behavior of the flux for the outer disk, or the
accretion disk for the Schwarzschild or Gauss-Bonnet black holes, which has a
maximum in the vicinity of the ISCO. The reason is the opposite gradient of
the angular velocity on the circular orbits in this region, which decreases in
direction to the naked singularity becoming zero at the disk inner edge. Such
anomaly can be associated with the repulsive action of the gravitational field
in the vicinity of the singularity, which prevents the particles from falling
in, thus screening the singularity.

Figure 2: Apparent radiation flux for the Gauss-Bonnet black hole with
$\hat{\gamma}=1$ (left), and for the Schwarzschild black hole (right). The
observer is located at $r_{obs}=5000M$, and at the inclination angle
$i=80^{\circ}$.  Figure 3: Apparent radiation flux for the Gauss-Bonnet weakly
naked singularity with $\hat{\gamma}=1.15$. The radiation from the outer disk
(left) and the inner disk (right) are represented separately in order to make
the image more transparent. The maximum of the observable radiation from the
outer disk is $F^{out}_{max}=6.07\times 10^{-5}M\dot{M}$, while for the inner
disk the flux reaches the value $F^{in}_{max}=4.94\times 10^{-2}M\dot{M}$. The
observer is located at $r_{obs}=5000M$, and at the inclination angle
$i=80^{\circ}$.

##  4 Image formation for the weakly naked singularities

Using a semi-analytical scheme for constructing the optical appearance of the
circular orbits we can study the process of the image formation for the weakly
naked singularities. This procedure is described in detail in [13]-[14],
therefore here we online only the basic steps. The central quantity in the
scheme is the evaluation of the variation of the azimuthal angle $\phi$ along
the photon trajectory. For a general spherically symmetric metric in the form
(1) it is given by the integral

$\phi(b)=\int^{r_{obs}}_{r_{source}}{\frac{dr}{r^{2}\sqrt{\frac{1}{b^{2}}-\frac{f(r)}{r^{2}}}}},$
(14)

where $b=L/E$ is the impact parameter on the geodesic and the integration is
performed between the photon’s emission point $r_{source}$ and the location of
the observer $r_{obs}$. Taking advantage of the spherical symmetry, the
azimuthal angle can be expressed by means of the inclination angle $i$ and a
properly defined celestial coordinate $\eta$ as [21], [22]

$\phi=-\arccos{\frac{\sin\eta\tan i}{\sqrt{\sin^{2}\eta\tan^{2}i+1}}}.$ (15)

The equality of the two expressions $(\ref{phi_b})$ and $(\ref{phi_eta})$
determines the impact parameter of the photon trajectories which can be
observed at a given inclination angle and a given celestial coordinate666For
an observer located at the asymptotic infinity the celestial coordinate $\eta$
is related to the impact parameters $\alpha$ and $\beta$, which we used in the
visualization of the disk images as $\eta=\arctan{\frac{\beta}{\alpha}}$..
Considering the full range of the celestial coordinate $\eta\in[0,\pi]$ we get
the impact parameters of all the photon trajectories emitted by a particular
circular orbit with a radial coordinate $r=r_{source}$, which can reach an
observer located at $r=r_{obs}$ and inclination angle $i$. So far we
considered only trajectories which reach the observer directly without
revolving around the compact object. In general we can observe also
trajectories performing an arbitrary large number of turns around the origin
of the coordinate system. Such trajectories are taken into account by
including in the equation an integer number $k$, parameterizing the number of
half-loops around the coordinate origin. Direct trajectories with $k=0$ lead
to the primary observable image, but we can observe also secondary images of
higher order $k$. Thus, we obtain the basic equation

$\int^{r_{obs}}_{r_{source}}{\frac{dr}{r^{2}\sqrt{\frac{1}{b^{2}}-\frac{f(r)}{r^{2}}}}}=k\pi-\arccos{\frac{\sin\eta\tan
i}{\sqrt{\sin^{2}\eta\tan^{2}i+1}}}.$ (16)

We should further consider that some trajectories possess a radial turning
point. Then, the integral $(\ref{phi_b})$ is represented as a sum of two
integrals including the turning point in the integration limits. The turning
point $r_{0}$ is a function of the impact parameter $b$ determined by the
largest root of the equation

$\displaystyle b=\frac{r_{0}}{\sqrt{f(r)}}.$ (17)

We can illustrate graphically the solutions of eq. (16) by using the following
argument. For every order $k$ the right-hand side of the equation determines a
minimal and a maximal observable azimutal angle, which we denote by
$\phi^{k}_{min}$ and $\phi^{k}_{max}$, respectively. They correspond to the
minimal and maximal values of the celestial coordinate $\eta$. Then, for every
order $k$ we define an observational window
$\Delta\phi^{k}=\phi^{k}_{max}-\phi^{k}_{min}$. By definition it gives the
possible variation of the azimuthal angle on the photon trajectories starting
at a given radial coordinate $r_{source}$, which can be observed at a given
inclination angle $i$ and observer position $r_{obs}$ after making $k/2$ loops
around the coordinate origin. Then, the solutions of eq. (16) for a given
boundary data $\\{r_{source},r_{obs},i\\}$ will correspond to all the possible
intersections of the curve $\phi(b)$ with the different observational windows
$\Delta\phi^{k}$. From this graphical representation we can deduce information
about the existence of the images of a given order $k$ and some of their
qualitative features like the formation of multiple images for example [14].
Multiple images of order $k$ arise if the intersection of the curve $\phi(b)$
with the observation window $\Delta\phi^{k}$ consists of multiple disconnected
portions. Each disconnected piece of the curve $\phi(b)$ leads to a separate
image in the observer’s sky.

---
$a)$ $b)$
Figure 4: Image formation diagram for a circular orbit from the outer disk a),
and the corresponding observable image b). The orbit is located at the radial
coordinate $r=10M$, while the observer’s position is at
$\\{r_{obs}=5000M,i=80^{\circ}\\}$. In a) we have denoted the value of the
impact parameter of the photon sphere with a vertical dashed line.

In fig. 4 we illustrate our argument by presenting the solutions of eq.
$(\ref{orbit_im})$ for the boundary data
$\\{r_{source}=10M,r_{obs}=5000M,i=80^{\circ}\\}$. This diagram determines the
image of a circular orbit located at $r=10M$ observed at the inclination angle
$i=80^{\circ}$. The red curve represents the solutions of the integral
$(\ref{phi_b})$ when the photon trajectories possess no radial turning point,
while the green curve corresponds to the solutions when a turning point is
present. Although we consider a particular orbit, the properties of its image
are representative for any circular orbit from the outer disk.

We see that the graph of the function $\phi(b)$ consists of two disconnected
curves, which diverge when $b\rightarrow b_{ph}^{+}$ and $b\rightarrow
b_{ph}^{-}$, respectively, where $b_{ph}$ is the value of the impact parameter
corresponding to the photon sphere. The curve approaching the photon sphere
with values of the impact parameter larger than $b_{ph}$ possesses similar
behavior to the function $\phi(b)$ for the Schwarzschild black hole (see
[14]). For the Schwarzschild black hole eq. (16) has a single solution for
every $k$, which leads to the formation of an infinite sequence of images
approaching the image of the photon sphere when $k\rightarrow\infty$. This
reflects the behavior of the trajectories with impact parameters close to
$b_{ph}$, since they can perform arbitrary large number of turns abound the
photon sphere before reaching the distant observer.

The distinctive feature of the weakly naked Gauss-Bonnet singularity is the
existence of a second solution of equation (16) for every $k\geq 1$, forming a
second curve $\phi(b)$, which tends to infinity when approaching the photon
sphere, however with values of the impact parameter lower that $b_{ph}$. Thus,
we get a second infinite sequence of images resulting from trajectories which
revolve a certain number of times around the photon sphere before scattering
to infinity.

Figure 5: Dependence of the impact parameter on the photon trajectory’s
turning point. Figure 6: Images of the circular orbit located at $r=10M$ as
seen by an observer at $\\{r_{obs}=5000M,i=80^{\circ}\\}$. We illustrate the
secondary images up to the order $k=9$.

This behavior can be explained by looking at the effective potential for the
null geodesics or equivalently at the dependence of the impact parameter on
the trajectory’s turning point $b(r_{0})$ presented in fig. 5. The minimum of
the curve with coordinates $\\{r_{ph},b_{ph}\\}$ corresponds to the photon
sphere, while the maximum is located at the stable light ring. We see that
there are two classes of photon trajectories which can scatter away to
infinity after being emitted at a radial distance larger than the location of
the photon sphere. The first type has impact parameters $b>b_{ph}$, and radial
turning points $r>r_{ph}$. This is the only possible case of scattering null
geodesics for the Schwarzschild and Gauss-Bonnet black holes, as well as for
some other weakly naked singularities like the Janis-Newman-Winicour solution.
The second type has impact parameters lower than $b_{ph}$ and turning points
approaching the singularity. Such geodesics are possible if the gravitational
field becomes repulsive in some characteristic neighbourhood of the compact
object and prevents particles and light from reaching it. As a result, all the
geodesics emitted in the exterior of the photon sphere will scatter away to
infinity either from the first or the second type of potential barriers except
for the limit case of trajectories revolving eternally around the photon
sphere.

In fig. 4 b) and fig. 6 we present the images of the circular orbit at $r=10M$
up to the order $k=9$ as seen by an observer located at
$\\{r_{obs}=5000M,i=80^{\circ}\\}$. The direct image ($k=0$) leads to the
characteristic hat-like shape of the accretion disk primary image observed
also for the Schwarzschild and the Gauss-Bonnet black holes at large
inclination angles. This is the only order which produces a single image. For
all the higher orders $k\geq 1$ a pair of secondary images is formed, as the
two images come closer to each other when $k$ increases both approaching the
image of the photon sphere in the limit $k\rightarrow\infty$. All the images
of the circular orbit up to the order $k=9$ are superposed in fig. 4 b)
showing how the image of the accretion disk is built up.

---
$a)$ $b)$
Figure 7: Image formation diagram for a circular orbit from the inner disk a),
and the corresponding observable image b). The orbit is located at the radial
coordinate $r=1.227M$, while the observer’s position is at
$\\{r_{obs}=5000M,i=80^{\circ}\\}$. In a) we have denoted the value of the
impact parameter of the photon sphere with a vertical dashed line. Figure 8:
Images of the circular orbit located at $r=1.227M$ as seen by an observer at
$\\{r_{obs}=5000M,i=80^{\circ}\\}$. We illustrate the primary image of order
$k=0$ and the secondary images up to the order $k=5$.

We can use a similar argument to describe the image of the inner disk. The
image formation diagram in this case is presented in fig. 7 for a
representative orbit located at $r_{source}=1.227M$, and an observer at
$\\{r_{obs}=5000M,i=80^{\circ}\\}$. The images of the orbit up to the order
$k=5$ and their superposition are presented in figs. 7 b) and 8. We observe
again a single direct image with $k=0$, while we have a pair of secondary
images for all the higher orders $k\geq 1$. When $k\rightarrow\infty$ the
couple of images approach the image of the photon sphere both with impact
parameters $b<b_{ph}$. In contrast to the case of the outer disk, one of the
images for each $k$ arises from photon trajectories without turning point,
while for the second one we have a turning point. The pattern for the
formation of the images can be understood qualitatively by looking at the
effective potential for the null geodesics in fig. 1. The photons emitted from
the inner disk, i.e. at radial distances smaller than the location of the
stable light ring, can reach the spacetime infinity only if their impact
parameter is lower than that of the photon sphere. Their trajectories can be
either direct, i.e. without a radial turning point, or they can scatter away
from the repulsive gravitational field in the vicinity of the singularity. In
both cases for suitable impact parameters they can perform arbitrary number of
turns around the photon sphere before reaching the observer, thus creating an
infinite sequence of higher order images.

##  5 Conclusion

We study the images of the thin accretion disk around static spherically
symmetric black holes and naked singularities in the 4D Einstein-Gauss-Bonnet
gravity. On the one hand our goal is to get some insights about the
observational signatures of the Gauss-Bonnet gravity compared to the general
relativity. In this respect we observe no qualitative distinction in the
appearance of the thin accretion disk around the Gauss-Bonnet black holes and
the Schwarzschild black hole. Only small quantitative differences are present
in the disk size and the maximum of the radiation flux.

On the other hand we aim at investigating the observable features of the naked
singularities, which are not present for black holes and could serve as an
experimental test for distinguishing the two types of compact objects. For
certain values of the coupling constant Gauss-Bonnet gravity allows for an
interesting case of naked singularities with respect to their lensing
properties. In addition to a photon sphere they possess a stable light ring
and the gravitational field becomes repulsive in a certain region around the
singularity. Due to these properties broader classes of photon trajectories
scatter away from the compact object and reach a distant observer leading to
the formation of multiple observable images of the accretion disk. These
images appear as a series of bright rings in the center of the primary disk
image, and some of them emit radiation $10^{3}$ times larger that the flux
from the primary disk image. Thus, they represent a significant observational
feature. We describe in detail the physical mechanism for the formation of the
accretion disk image showing that the phenomenon is expected also for other
spacetimes possessing the same properties of the photon dynamics.

## Acknowledgments

We gratefully acknowledge support by the Bulgarian NSF Grant KP-06-H38/2.
Networking support by the COST Actions CA16104 and CA16214 is also gratefully
acknowledged.

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