Shadows and Accretion Disk Images of Compact Objects

Abstract

The black hole shadow was predicted theoretically already in the 1960s as a strong gravitational lensing phenomenon. Recently, its observation became feasible and opened a major experimental channel for probing the gravitational interaction in the strong field regime. Although considered historically as a property of the black hole spacetimes, the shadow is not exclusively a black hole effect and does not require the presence of an event horizon. Other compact objects can also cast a shadow if their gravitational field is sufficiently strong bringing up the issue of how we can differentiate between self-gravitating systems based on their shadow images. In this chapter we discuss the analytical or semi-analytical methods for obtaining the observable images produced by compact objects in some basic physical settings such as a uniform spherical distribution of distant light sources or the presence of a thin accretion disk. We review the calculation of the shadow boundary for the Kerr black hole and focus on recent research on the images created by wormholes and naked singularities. These compact objects can look qualitatively very similar to black holes in some cases, but they can also possess clear-cut observational signatures.

1 Introduction

General relativity has been tested in the weak field regime with a remarkable precision [93, 94], but its validity in strong gravitational fields is still hypothetical. A central question in this respect is whether the Kerr solution indeed represents the unique stationary and axisymmetric black hole in vacuum. Testing this statement known as the Kerr hypothesis is equivalent to confirming whether general relativity describes the black holes in our Universe or another more refined gravitational theory takes its place in the extreme regime.

In the last decade we witnessed a major experimental breakthrough in this direction with the detection of gravitational waves which opened a new window for exploring gravity [1, 2]. On the other hand, the Event Horizon Telescope (EHT) collaboration provided the first direct images of the supermassive compact objects in the nearby galactic targets M87 and Sgr A with a horizon-scale resolution [7, 9, 10]. These developments launched a new era in fundamental physics allowing to combine information from different observational channels in order to test its predictions. Thus, the investigation of compact objects turned for the first time from a predominantly theoretical field into an active area in observational astrophysics.

Black hole imaging experiments such as the Event Horizon Telescope have their roots in the foundations of general relativity governing the propagation of particles and light in curved spacetime. These phenomena are described by the theory of gravitational lensing which explains the observational effects caused by the interplay of light with extreme curvature. In particular, the strong gravitational field in the vicinity of black holes gives rise to a specific phenomenon known as the black hole shadow. This effect arises when the black hole is surrounded by a distribution of light sources and we observe a dark spot on the luminous sky. It can be easily predicted by considering the definition of black hole. Assuming a bundle of photon trajectories with a broad range of initial condition, there will be always such ones that will enter the black hole region and consequently never come back to us. In this way they leave dark directions in the observer’s sky, which form the black hole shadow.

The shadow provides a means to map the black hole vicinity into an optical image which encodes essential information about the properties of the underlying spacetime. The main observational characteristic is the shadow boundary and its size and shape determine the black hole spin and other relevant charges [6, 44, 52, 58, 86]. Ideally, if the boundary curve is measured with a very high precision, it enables us to specify the black hole giving rise to it. This correspondence can be further elaborated and extended into a procedure for testing the Kerr hypothesis and imposing constraints on the modified theories of gravity [17, 54, 60, 75].

The study of the black hole shadow has a long history dating back to the classical works of general relativity. The phenomenon was described theoretically already in the 1960s in the early works [82, 100], which obtained the viewing angle of the shadow for the Schwarzschild black hole. In the next decade Bardeen investigated the shadow of the Kerr black hole and taking advantage of the separability of the null geodesic equations developed a general formalism for obtaining the shadow boundary [16]. Although of fundamental importance, these developments considered the black hole shadow as a purely theoretical phenomenon which is unlikely to be experimentally detected. The idea that observing the shadow of the black hole at the center of our galaxy may be feasible was first suggested in [38], where the necessary experimental conditions were also discussed. This seminal paper put the foundations for the development of the global interferometer Event Horizon Telescope which recently produced the first black hole images.

The analytical construction of the black hole shadow boundary has some fundamental consequences. Revealing the explicit mechanism for the shadow formation it became evident that the phenomenon is not limited exclusively to black holes but a much broader class of compact objects will lead to a similar image [48, 64, 77, 80]. The reason is that the mathematical structure which determines the shadow is not the event horizon but another fundamental surface called a photon sphere. The photon sphere represents a separatrix between two families of infalling null geodesics. The first class consists of trajectories which are reflected by the gravitational field and manage to reach a distant observer. The second class of trajectories get trapped in the gravitational potential and fail to scatter away to infinity, thus forming the shadow. For black holes these are the geodesics which enter the event horizon. In other spacetimes such behavior can develop for different physical reasons. For example, wormhole geometries suggest that part of the geodesics will pass through the wormhole throat and continue to propagate in another universe. In naked singularity spacetimes shadows are formed due to the geodesics which end at the singularity.

Most of the classical results in the compact objects astrophysics were developed considering the Kerr black hole. However, ideas from fundamental physics suggest that general relativity may not be the final theory of gravity. It will probably need to be modified in order to explain mysterious phenomena such as the dark energy and dark matter and describe the gravitational interaction at the Planck scale. In order to address these issues various alternative theories of gravity were proposed motivated as a low-energy limit of unification theories or effective theories of quantum gravity. Modifying the properties of the gravitational interaction they allow for a much greater variety of compact objects than general relativity including black holes with different kinds of hair and wormholes which do not violate the energy conditions. Other solitonic self-gravitating configurations are also studied such as regular black holes, boson stars and gravastars. These exotic compact objects from the perspective of general relativity are no longer purely theoretical ideas. Their experimental detection has become a solid part of the agenda of the major astrophysical missions both in the gravitational wave and the electromagnetic spectrum inspiring a range of works investigating and predicting their observational signatures.

The aim of this review is to describe the theoretical foundations of the black hole shadow and to demonstrate how the effect can be extended to other compact objects such as wormholes and naked singularities. We will consider spacetimes with high degree of symmetries which allow for the integrability of the geodesic equations and analytical construction of the shadow boundary. The classical calculation of the black hole shadow assumes a uniform distribution of the light sources around the compact object. This set-up is a simplified toy-model but it allows to extract the most essential information about the phenomenon without additional complications from more restricting initial conditions. In realistic astrophysical scenarios the compact objects are surrounded by accretion disks which represent the main source of the electromagnetic emission. Therefore, as a next step we will demonstrate how the observational picture is modified if we adopt a more realistic light sources distribution. We will consider the basic model of a geometrically thin and optically thick disk around a spherically symmetric compact object and construct the observable image for the Schwarzschild black hole and certain types of naked singularities. Although simplified, such models provide valuable intuition how the observational signatures arise and how the properties of the compact object influence the images which can be useful for interpreting the results of more complicated simulations.

2 Black Hole Shadow in Static Spherically Symmetric Spacetime

In order to describe the effect of black hole shadow we will discuss initially static spherically symmetric spacetime in vacuum. This simple geometrical setting allows to illustrate clearly the theoretical ideas which we will generalize subsequently in more complicated scenarios. On the relevant scales for gravitational lensing light propagation is described by the geometric optics approximation. Light follows null geodesics and the geodesic equations can be derived from the least action principle by introducing the Lagrangian

$$\begin{aligned} \mathcal{L} = \frac{1}{2}g_{\mu \nu }\dot{x^\mu }\dot{x^\nu }, \end{aligned}$$

(1)

where $g_{\mu \nu }$ is the metric, $x^{\mu }$ are the spacetime coordinates, and the dot denotes differentiation with respect to the affine parameter along the geodesics.¹ In this way we obtain a system of four ordinary differential equations and we have in addition the constraint

$$\begin{aligned} g_{\mu \nu }\dot{x^\mu }\dot{x^\nu } = 0, \end{aligned}$$

(2)

which ensures that the type of geodesic is preserved for any value of the affine parameter. The geodesic equations can be simplified if the spacetime possesses higher degree of symmetries. Stationary and axisymmetric spacetimes are particularly important since they describe the quasi-equilibrium configurations of the astrophysical objects such as black holes surrounded by an accretion disk or galaxies with a supermassive central compact object. These symmetries are manifested by the presence of a Killing vector associated with time translations $\partial /\partial t$ and a Killing vector associated with rotations with respect to the symmetry axis $\partial /\partial \varphi$ which induce conservation laws on the geodesics. In particular they lead to the conservation of the photon’s specific energy E and specific angular momentum L.

Spherical symmetry increases the number of constants of motion since it generates two additional Killing vectors which lead to conservation of the plane of motion. Thus, geodesics propagate in a single plane $\theta = \text {const.}$ which we assume for convenience to correspond to the equatorial plane. In this way the geodesic equations become completely integrable and reduce to a one-dimensional problem. The photon trajectories can be obtained by either integrating the radial geodesic equation, or more conveniently considering the constraint given by Eq. (2).

Let us introduce a static spherically symmetric metric in the general form

$$\begin{aligned} {} & {} ds^2= -A(r)dt^2 + A^{-1}(r)dr^2 + r^2 D(r)(d\theta ^2 + \sin ^2\theta d\varphi ^2), \end{aligned}$$

(3)

where the metric functions A(r) and D(r) depend only on the radial coordinate. Then, the constants of motion and the constraint equation take the form

$$\begin{aligned} {} & {} E = A(r)\dot{t}, \quad ~~~ L = r^2 D(r)\dot{\varphi }, \nonumber \\ {} & {} \dot{r}^2 + V_{\text {eff}} = E^2, \quad ~~~V_{\text {eff}} = L^2\frac{A(r)}{r^2 D(r)}. \end{aligned}$$

(4)

We see that the geodesic equations reduce effectively to a familiar problem representing the motion in the field of a spherically symmetric potential $V_{\text {eff}}$. Although this analogy is only formal, we can analyse the qualitative behaviour of the photon trajectories by taking advantage of our intuition from classical mechanics. In general we can rescale the affine parameter by the photon’s specific energy E. Thus, we see than there is a single dynamically important parameter defined by the radio of the specific energy and angular momentum $b= L/E$. It is called an impact parameter and the type of the photon trajectories is determined by its value and the particular form of the effective potential.

Let’s examine this problem for the Schwarzschild black hole. The effective potential is given explicitly by

$$\begin{aligned} V_{\text {eff}} = \frac{b^2}{r^2}\left( 1-\frac{R_s}{r}\right) , \end{aligned}$$

(5)

where $R_s = 2M$ is the Schwarzschild radius and M is the mass of the black hole. The effective potential tends to zero at the spacetime infinity $r\rightarrow \infty$ and behaves as $V_{\text {eff}}\rightarrow -\infty$ at the curvature singularity $r=0$ (see Fig. 1). It possesses a single maximum in between which determines an unstable circular photon orbit by the conditions

$$\begin{aligned} V_{\text {eff}}= 1, \quad ~~~ V^{'}_{\text {eff}} =0, \quad ~~~ V^{''}_{\text {eff}} < 0, \end{aligned}$$

(6)

where the prime denotes derivative with respect to the radial coordinate. Solving these equations we obtain that the circular orbit is located at $r=3M$ and corresponds to the value of the impact parameter $b_{\text {crit}}= 3\sqrt{3}M$. Let us consider photon trajectories approaching the black hole from infinity. They will separate in two qualitatively different classes. Photon trajectories with impact parameters larger than that of the circular orbit $b > b_{\text {crit}}$ will be reflected by the effective potential and scatter away to infinity, while those with impact parameters $b<b_{\text {crit}}$ will manage to cross the potential barrier and end up plunging into the black hole. Thus, the unstable circular orbit serves as a critical curve which separates gravitational scattering from gravitational capture. Its impact parameter represents the limiting value for the impact parameters of infalling trajectories which are capable of reaching back to a distant observer. Hence, it determines the boundary of the black hole shadow.

Fig. 1

Reprinted from [23]. with the permission of AIP Publishing

Effective potential for the null geodesics for the Schwarzschild black hole. Infalling photon trajectories from large distances scatter away from the potential if their impact parameter satisfies $b>b_{\text {crit}}$, and plunge into the black hole if $b<b_{\text {crit}}$.

Let’s imagine the collection of all the unstable circular orbits in all the possible planes of motion for the null geodesics. They will build up a sphere with a radius $r=3M$, which is called a photon sphere. The shadow of the Schwarzschild black hole represents a lensed image of the photon sphere projected on the observer’s sky. By the described argument we see that any static spherically symmetric compact object which allows for both families of scattering and plunging photon trajectories will possess a photon sphere and therefore cast a shadow. Thus, the shadow is not exclusively a black hole phenomenon but an observational characteristic of compact objects possessing a photon sphere. These are also called ultra-compact objects and include soliton-like self-gravitating configurations of various physical nature like wormholes, naked singularities, boson stars, gravastars as particular examples.

The photon sphere is a fundamental surface which is important not only for its association with observational signatures. Compact objects spacetimes can be classified using this structure similar to the black hole classification based on the properties of the event horizon. In this way it was demonstrated that the Schwarzschild spacetime is the unique static and asymptotically flat solution to the Einstein equations in vacuum possessing a photon sphere [21]. This result was extended to electro-vacuum spacetimes [22, 97] and to the Einstein-scalar field theory where it was proven that any static and asymptotically flat solution which possesses a photon sphere is spherically symmetric and isometric to the Janis–Newman–Winicour weakly naked singularity [96]. Further generalizations were developed considering the Einstein–Maxwell-dilaton theory [98] and multiple scalar fields [99]. These theorems provide a much broader classification than the black hole uniqueness theorems assuming an analytical event horizon since they include also horizonless compact objects.

Let us get more intuition about the properties of the photon sphere by considering some explicit solutions of the geodesic equations for the Schwarzschild black hole. When our aim is to obtain particular solutions for the trajectories it is convenient to express the constraint given by Eq. (2) as a differential equation for the variation of the radial coordinate with respect to the azimuthal angle. Thus, using the constants of motion we obtain

$$\begin{aligned} \left( \frac{dr}{d\varphi }\right) ^2 = r^2 D(r)\,A(r)\left( \frac{r^2 D(r)}{b^2A(r)}-1\right) = f(r), \end{aligned}$$

(7)

for a general spherically symmetric spacetime. For the Schwarzschild black hole it is convenient to make the substitution $u=R_s/r$. Plugging in the particular form of the metric functions, we see that the right-hand side of the trajectory equation reduces to a polynomial in the new variable

$$\begin{aligned} \left( \frac{du}{d\varphi }\right) ^2 = u^3 -u^2 +\frac{R_s^2}{b^2} = f(u). \end{aligned}$$

(8)

This form of the equation allows for straightforward qualitative analysis of the photon motion without introducing an effective potential. The possible types of photon trajectories are determined by the root structure of the function f(u) at the right-hand side of the equation since its roots correspond to the possible turning points of the trajectories. We can demonstrate that the function always possesses a negative root which is irrelevant for the geodesic motion. In addition it possesses two distinct positive real roots for large values of the impact parameter b, which approach each other when b declines, merge into a double real root for a certain critical value $b_{\text {crit}}$, and turn into a pair of complex conjugate roots for lower impact parameters $b<b_{\text {crit}}$. The first type of root structure containing two distinct real roots implies that infalling null geodesics from infinity will always possess a turning point at the smaller root of f(u) and therefore will scatter away from the black hole. On the contrary, complex roots correspond to the absence of turning points so geodesics with such impact parameters will always plunge into the black hole.

Let’s examine the marginal case between these two types of behaviour. Having a double real root implies that the equations

$$\begin{aligned} f(u)=0, \quad ~~~ f^{'}(u)=0, \end{aligned}$$

(9)

are satisfied. Consequently, this case represents a circular orbit. We can further demonstrate that $f^{''}(u)>0$ is satisfied at this point, i.e. the circular orbit is unstable, and solving Eq. (9) we obtain its position $r=3M$ and the corresponding impact parameter $b_{\text {crit}} = 3\sqrt{3}M$. In this way we have obtained the photon sphere from a different perspective.

The photon sphere consists of unstable circular orbits so photons propagating on such orbits with either scatter away or fall into the black hole by the slightest perturbation. Thus, the photon sphere should be rather interpreted as a limiting surface for two families of infalling and outgoing geodesics with the impact parameter $b_{\text {ph}} = 3\sqrt{3}M$. We can derive the explicit form of these trajectories by solving Eq. (8) for the value of the impact parameter $b = b_{\text {ph}}$, however requiring that $\frac{du}{d\varphi }\ne 0$. Assuming that we have infalling geodesics with initial conditions $r>r_{\text {ph}}$ we obtain [24] (see also [23])

$$\begin{aligned} u= -\frac{1}{3} + \tanh ^2{\frac{1}{2}(\varphi -\varphi _0)}, \end{aligned}$$

(10)

where $\varphi _0$ is an integration constant, while the solution

$$\begin{aligned} u= -\frac{1}{3} + \coth ^2{\frac{1}{2}(\varphi -\varphi _0)}, \end{aligned}$$

(11)

corresponds to outgoing trajectories which originate at radial distances between the horizon and the photon sphere ($R_s<r<r_\text {ph}$). Both solutions represent geodesics which approach the photon sphere either from its exterior or its interior and spiral infinitely around its surface. We see that when $r\rightarrow 3M$, i.e. $u\rightarrow {2}/{3}$, the azimuthal angle tends to infinity, i.e. the trajectory performs an infinite number of turns around the photon sphere.

In lensing problems we frequently consider the deflection angle defined as the variation of the azimuthal angle along the photon trajectory. For static spherically symmetric spacetimes it is given by

$$\begin{aligned} \varphi = \int _{r_{source}}^{r_{obs}} \frac{dr}{r \sqrt{D(r)\,A(r)}\sqrt{\left( \frac{r^2 D(r)}{b^2A(r)}-1\right) }}, \end{aligned}$$

(12)

where we integrate between the radial positions of the light source $r_\text {source}$ and the observer $r_\text {obs}$. For the Schwarzschild black hole the deflection angle can be expressed explicitly in terms of elliptic functions for a general value of the impact parameter (see e.g. [24]). However, the behaviour in the vicinity of the photon sphere can be represented by the approximate solution [43, 59]

$$\begin{aligned} {} & {} \varphi = \log {\frac{C_{\pm }}{|b-b_{ph}|}}, \quad b\rightarrow b^{\pm }_{ph}, \\ {} & {} C_{\pm } = const. \nonumber \end{aligned}$$

(13)

which is valid for impact parameters approaching the photon sphere with lower values $b\rightarrow b^{-}_\text {ph}$ or higher ones $b\rightarrow b^{+}_\text {ph}$. Both solutions for the deflection angle are illustrated in Fig. 2. As we already know, the deflection angle diverges logarithmically at the photon sphere. In addition, we can see that a hierarchy of infalling trajectories is formed in the neighbourhood of the photon sphere which circle around it an arbitrary large but finite number of turns n as the winding number n grows when the trajectory’s impact parameter approaches $b_\text {ph}$. These trajectories ultimately scatter away to infinity and carry important information for the observations. Even for comparatively low values of the winding number $n\sim 1.25$ they probe sufficiently well the photon sphere and provide a reasonable estimate for its image [43].

Fig. 2

Reprinted figure with permission from [43]. Copyright 2019 by the American Physical Society

Deflection angle for the null geodesics for the Schwarzschild black hole as a function of the impact parameter. The dashed line represents the exact solution of the integral, while the solid line corresponds to the approximation given by Eq. (13). The region denoted in red already provides a good approximation for the photon sphere.

Section summary: We introduced the notion of photon sphere in the simplest geometrical set-up of static spherically symmetric spacetimes using the Schwarzschild black hole as a particular example. We illustrated its role for the light propagation from different perspectives discussing the effective potential for the null geodesics, particular explicit solutions for the photon trajectories and the properties of the deflection angle. These concepts can be generalized and provide intuition in technically more involved geometries such as stationary axisymmetric spacetimes and light sources distributions such as accretion disks. We will consider such generalizations in the next sections.

3 Shadow of Stationary and Axisymmetric Compact Objects

In this section we will reduce the symmetry assumptions and consider the shadow cast by stationary and axisymmetric compact objects when illuminated by a uniform distribution of distant light sources. In this setting we have two Killing vectors associated with the conservation of the photon’s specific energy E and angular momentum L on the geodesics and the constraint given by Eq. (2) which ensures that the geodesic’s type is preserved. These integrals of motion are not sufficient for the complete integrability of the geodesic equations and in general solutions are obtained numerically. However, there exist particular spacetimes possessing additional symmetries called hidden symmetries. They are associated with the existence of irreducible Killing and Killing–Yano tensors which do not generate isometries but lead to additional integrals of motion along the geodesics. This property allows to separate the variables in the geodesic equations and reduce formally the problem of light propagation to a one-dimensional motion in the field of an effective potential. Then, we can obtain the shadow boundary analytically by analysing the radial motion similar to the spherically symmetric case.

Black hole solutions in general relativity belong to this type of integrable spacetimes since it was proven that the Kerr–Newman-NUT-(A)dS family possesses a Killing tensor. Its explicit form was derived initially by Carter [20] and the corresponding integral of motion was called a Carter constant. Penrose and Walker obtained this result simultaneously in an independent way relating the presence of a Killing tensor to the algebraic type of the spacetime [91].

Let us demonstrate how we can obtain the shadow boundary in stationary axisymmetric spacetimes possessing hidden symmetries. In this case it is convenient to consider the Hamilton–Jacobi formulation of the geodesic problem. Due to the presence of a Killing tensor the Hamilton–Jacobi equation

$$\begin{aligned} \frac{\partial S}{\partial \lambda }=-\frac{1}{2}g^{\mu \nu }\frac{\partial S}{\partial x^{\mu }}\frac{\partial S}{\partial x^{\nu }}, \end{aligned}$$

(14)

is separable, and introducing a spheroidal coordinate system $x^\mu = \{t, r, \theta , \varphi \}$ there exists a solution in the form

$$\begin{aligned} S=\frac{1}{2}\mu ^2 \lambda - E t + L \varphi + S_{r}(r) + S_{\theta }(\theta ). \end{aligned}$$

(15)

In this expression we denote by $\mu$ the mass of the test particle, $\lambda$ is the affine parameter on the geodesics, while the functions $S_r(r)$ and $S_\theta (\theta )$ depend only on the specified coordinates.The geodesic equations are obtained by the standard procedure using the fact that the partial derivatives of the Jacobi action S with respect to the constants of motion should vanish. Setting $\mu =0$ for null geodesics, we obtain two decoupled equations for the radial and polar motion in the form

$$\begin{aligned} {} & {} \frac{dr}{d\lambda } = R(r, E,L, Q), \end{aligned}$$

(16)

$$\begin{aligned} {} & {} \frac{d\theta }{d\lambda } = T(\theta , E,L, Q), \end{aligned}$$

(17)

where Q is the integral of motion associated with the hidden symmetries. These equations combined with the conservation laws associated with the Killing vectors determine the photon propagation.

In Eqs. (16)–(17) we see that the photon trajectories depend on the conserved charges $\{E, L, Q\}$. However, the number of independent parameters can be reduced by recognizing that the specific energy E is merely a scale parameter, and can be eliminated by rescaling the affine parameter as $\lambda \rightarrow E\lambda$. Then, the geodesic motion will depend only on the ratios $\xi =L/E$ and $\eta =Q/E^2$, which play the role of impact parameters.

We can obtain the shadow boundary by analysing the radial equation similar to the spherically symmetric case. In analogy we can transform Eq. (16) into an energy-like equation for a certain effective potential $V(r,\eta , \xi )$

$$\begin{aligned} \left( \frac{dr}{d\lambda }\right) ^2 + V_{eff} = 1. \end{aligned}$$

(18)

Performing the qualitative analysis which we described in Sect. 2 we can determine the trajectories which scatter away from the gravitational potential and those which overcome the potential barrier and plunge into the black hole. The critical orbits which separate gravitational scattering from gravitational capture correspond to the highest maximum of $V_\text {eff}$. Thus, they satisfy the conditions

$$\begin{aligned} V_{eff}=1, \quad ~~~V^{'}_{eff}=0, \quad ~~~V^{''}_{eff} < 0. \end{aligned}$$

(19)

An important distinction from the spherically symmetric case is that the critical orbits are in general not circular. They represent unstable spherical orbits, i.e. they lie on a sphere with a certain radius, but the $\theta$-motion can be very complicated.² Due to this distinction we cannot define a photon sphere as in the spherically symmetric case. However, we can introduce an appropriate generalization. The collection of all the unstable spherical orbits is now called a photon region and the shadow boundary arises as its lensed image on the observer’s sky.

We can outline the following general procedure for constructing the shadow boundary for stationary axisymmetric spacetimes with integrable geodesic equations.

• From the definition of the unstable spherical orbits given by Eq. (19) we obtain two algebraic equations for the impact parameters $\xi$ and $\eta$. They determine a curve in the impact parameter space $\eta =\eta (\xi )$, which can be also represented in a parametric form using the radial coordinate, i.e. $\xi =\xi (r)$, $\eta =\eta (r)$. This curve corresponds to the shadow boundary in the impact parameter space.

• The range of the radial coordinate on the curve $\{\xi (r), \eta (r) \}$ is constrained by the geodesic equation for the polar motion. The impact parameters $\xi =\xi (r)$ and $\eta =\eta (r)$ should be such that the function $T(r, \xi , \eta )$ on the right-hand side of Eq. (17) is well-defined. This ensures the existence of the unstable spherical orbits determining the photon region.

• Similar to the spherically symmetric case, for every radial coordinate $r=r_0$ belonging to the photon region there exist ingoing geodesics with the same impact parameters $\xi (r_0)$ and $\eta (r_0)$ which spiral towards the corresponding spherical orbit as a limit curve. Then, the observable image of the photon region is determined by the projection of these geodesics on the observer’s sky.

The last step of the procedure is constructing explicitly the projected image. The projection on the observer’s sky is not defined uniquely and there are different types of celestial coordinates proposed in the literature. In our approach we will follow the Bardeen procedure which was applied initially to obtain the Kerr black hole shadow [16] (see also [28]). For the purpose we should first define the observer’s frame by introducing a local orthonormal frame (tetrad) at the observer’s position. Considering the general form of a stationary axisymmetric metric in the spheroidal coordinates $x^\mu = \{t, r, \theta , \varphi \}$ a natural choice of the orthonormal tetrad is given by

$$\begin{aligned} {} & {} e_{(r)} = \frac{1}{\sqrt{g_{rr}}}\partial _r, \,\,\,\,\, e_{(\theta )} = \frac{1}{\sqrt{g_{\theta \theta }}} \partial _{\theta }, \,\,\,\,\, e_{(\phi )} = \frac{1}{\sqrt{g_{\varphi \varphi }}}\partial _{\varphi },\\ \nonumber {} & {} e_{(t)} = \zeta \,\partial _t + \gamma \,\partial _{\varphi }, \end{aligned}$$

(20)

where the quantities $\zeta$ and $\gamma$ are defined as

$$\begin{aligned} \zeta = \sqrt{\frac{g_{\varphi \varphi }}{g^2_{t\varphi } - g_{tt}g_{\varphi \varphi }}}, \quad ~~~\gamma = -\frac{g_{t\varphi }}{g_{\varphi \varphi }}\zeta . \end{aligned}$$

(21)

Then, we can obtain the locally measured 4-momentum $p^{(\mu )}$ by projecting the tangent vector to the geodesics, i.e. the photon’s 4-momentum $p_\mu$, in the observer’s frame

$$\begin{aligned} {} & {} p^{(r)}=\frac{p_r}{\sqrt{g_{rr}}}, \quad ~~~p^{(\theta )}=\frac{p_\theta }{\sqrt{g_{\theta \theta }}}, \quad ~~~p^{(\phi )}=\frac{L}{\sqrt{g_{\phi \phi }}}, \\ \nonumber {} & {} p^{(t)}=\zeta E-\gamma L. \end{aligned}$$

(22)

The projection is determined by two celestial angles $\alpha$ and $\beta$ which serve as coordinates on the observer’s sky. They are expressed explicitly by the components of the local momentum as

$$\begin{aligned} \alpha = \arcsin {\frac{p^{(\theta )}}{p^{(t)}}}, \quad ~~~\beta = \arctan {\frac{p^{(\phi )}}{p^{(r)}}}. \end{aligned}$$

(23)

We can further set $p^{(t)} =1$ by rescaling the affine parameter along the geodesics.

In order to obtain the image of the shadow boundary on the observer’s sky we should select the celestial angles $\alpha$ and $\beta$, which correspond to the photon region. Since the 4-momentum $p_\mu$ is determined from the geodesic equations, we should simply substitute in these expressions the relevant values of the impact parameters $\xi$ and $\eta$. Then, using Eq. (23) we can obtain the celestial angles describing the shadow boundary

$$\begin{aligned} \alpha =\alpha (\xi , \eta ,r_\text {obs}, \theta _\text {obs}), \quad ~~~ \beta =\beta (\xi , \eta ,r_\text {obs}, \theta _\text {obs}), \end{aligned}$$

(24)

for a given position of the observer $(r_\text {obs}, \theta _\text {obs})$. Since $\alpha$ and $\beta$ decrease with the radial distance of the observer, i.e. $\alpha \sim \frac{1}{r}$, $\beta \sim \frac{1}{r}$, it is convenient to rescale them as $\alpha \rightarrow \alpha r_\text {obs}$, $\beta \rightarrow \beta r_\text {obs}$ introducing celestial coordinates with dimension of mass.

By the described procedure we can obtain the image of the shadow boundary for any radial position of the observer. However, if we aim at describing the observational data from the Event Horizon Telescope, we should assume that the observer is located at spacetime infinity. In the limit $r_\text {obs}\rightarrow \infty$ we further obtain simpler expressions for the celestial coordinates.

3.1 Example: Kerr Black Hole

As an example we will discuss the shadow of the Kerr black hole. The Kerr solution is characterized by two parameters representing its ADM mass M and angular momentum J, which is commonly substituted by the spin parameter $a= J/M$. Following the outlined procedure we can derive the impact parameters which define the photon region [16]

$$\begin{aligned} {} & {} \xi (r)=-\frac{r^3-3Mr^2+a^2(M+r) }{a(r-M)}, \\ {} & {} \eta (r)=\frac{r^{3}[4a^{2}M-r(r-3M)^{2}]}{a^2(r-M)^2}, \nonumber \end{aligned}$$

(25)

where the radial coordinate satisfies the inequality

$$\begin{aligned} \eta + \cos ^2\theta \left( a^2 -\frac{\xi ^2}{\sin ^2\theta }\right) \ge 0. \end{aligned}$$

(26)

In this way we obtain the shadow boundary in the impact parameter space. Assuming an asymptotic observer, i.e. $r_\text {obs}\rightarrow \infty$, these expressions lead to the following celestial coordinates

$$\begin{aligned} \alpha = & {} -\frac{\xi }{\sin \theta _\text {obs}}, \nonumber \\ \beta = & {} \left[ \eta + a^2\cos ^{2}\theta _\text {obs}-\xi ^{2}\cot ^{2}\theta _\text {obs}\right] ^{1/2}, \end{aligned}$$

(27)

which determine the observable shadow image. We see that the shadow depends on the spin of the black hole and the angular position of the observer called also an inclination angle. In Fig. 3 we illustrate the shadow boundary for the Kerr black hole for different spin parameters and inclination angles. For static black holes the shadow is circular as its size depends on the black hole mass. Introducing rotation leads to characteristic deformation of the boundary curve. The particular shape of the shadow boundary contains enough information to estimate the black hole spin if it is restored from the observational data with sufficient resolution [52, 58, 86]. It can further determine whether the spacetime is characterized by higher multipole moments, in this way serving as a means for detecting black hole hair or existence of more exotic compact objects.

Fig. 3

Reprinted from [102]. ©  AAS. Reproduced with permission

Shadow of the Kerr black hole for different inclinations angles i and spin parameters a. In each panel the spin parameter grows continuously from $a=0$ (black curve) to the near-extremal value $a=0.999$ (red curve). Increasing the spin and the inclination angle leads to stronger deformation of the shadow boundary.

Section summary: We demonstrated how the notion of photon sphere can be generalized for a stationary and axisymmetric spacetime possessing an irreducible Killing tensor. Due to the ’hidden’ symmetries the geodesic motion becomes completely integrable. We showed how we can obtain the shadow boundary analytically using the Hamilton–Jacobi formulation of the geodesic problem. We considered the Kerr black hole as a particular example.

4 Shadow of Traversable Wormholes

One of the most exciting theoretical predictions is that our universe may contain traversable wormholes which we have not detected so far because they mimic the observational features of black holes. As we already explained the shadow is such a strong gravity phenomenon which is common for a range of compact objects of different physical nature. In this section we will construct explicitly the shadow of a certain class of traversable wormholes and discuss how closely its boundary curve resembles the Kerr black hole.

For the purpose we consider the general form of the metric describing stationary axisymmetric wormholes suggested by Teo [83] as a rotating generalization of the Morris–Thorne wormhole

$$\begin{aligned} ds^2=-N^2\textrm{d}t^2+\left( 1-{\frac{b}{r}}\right) ^{-1}\textrm{d}r^2+r^2K^2 \left[ \textrm{d}\theta ^2+\sin ^2\theta (\textrm{d}\varphi -\omega \textrm{d}t)^2\right] . \end{aligned}$$

(28)

The metric functions depend only on the spherical coordinates r and $\theta$ and under some mild conditions define a completely regular geometry which represents a tunnel connecting two distant regions in spacetime. The wormhole throat corresponds to the minimal surface located at the constant radius $b=r$, where the metric function $g_{rr}$ becomes divergent. This behavior reduces to an apparent singularity if $\partial _\theta b(r, \theta )=0$ at the throat. The spacetime does not contain any curvature singularities or event horizons, if we require further that the redshift function N is finite and non-zero in all the coordinate range. In addition, satisfying $\frac{db}{dr} < 1$ at the throat provides the characteristic flaring out shape of the embedding of the constant t and $\theta$ cross-sections of the wormhole spacetime into three-dimensional Euclidean space.

Since the spherical coordinates break down at the wormhole throat, we can represent only one of the asymptotic regions in this coordinate system. However, we can introduce a global radial coordinate l defined as

$$\begin{aligned} dl = \pm \frac{dr}{\sqrt{1-\frac{b}{r}}}, \end{aligned}$$

(29)

which extends the wormhole metric across the throat. It takes the range $-\infty < l < +\infty$ and describes smoothly the transition between two asymptotic regions for positive and negative values of l connected by the wormhole throat at $l=0$.

For our purposes we will impose restrictions on the general wormhole geometry (28) by requiring that the metric functions N, b, $\omega$ and K depend only on the radial coordinate r. In this way we obtain a family of traversable wormholes with integrable geodesic equations and we can calculate the shadow boundary analytically as described in Sect. 3. Separating the variable in the Hamilton–Jacobi equation we obtain for the null geodesics [64]

$$\begin{aligned} {} & {} N\left( 1-\frac{b}{r}\right) ^{-1/2}\frac{dr}{d\lambda } =\left[ \left( 1-\omega \xi \right) ^2 - \eta \frac{N^2}{r^2K^2}\right] ^{1/2} = \sqrt{R(r)}, \\ {} & {} r^2K^2\frac{d\theta }{d\lambda } = \left[ \eta - \frac{\xi ^2}{\sin ^2\theta }\right] ^{1/2}, \quad ~~~ \frac{dt}{d\lambda } = \frac{1- \omega \xi }{N^2}, \nonumber \\ {} & {} \frac{d\varphi }{d\lambda } = \frac{\omega (1 - \omega \xi )}{N^2} + \frac{\xi }{r^2\sin ^2\theta K^2}, \nonumber \end{aligned}$$

(30)

where $\xi$ and $\eta$ are the impact parameters determined by the integrals of motion. The radial equation can be transformed into an energy-like equation

$$\begin{aligned} \left( \frac{dr}{d\lambda }\right) ^2 + V_{eff} = 1, \end{aligned}$$

by introducing the effective potential

$$\begin{aligned} \quad V_{eff} = 1 - \frac{1}{N^2}\left( 1-\frac{b}{r}\right) R(r). \end{aligned}$$

(31)

Analysing its behavior we obtain two families of unstable spherical orbits [48, 78]. The first family corresponds to maxima of the effective potential lying outside the wormhole throat and obeys the conditions

$$\begin{aligned} R(r) = 0,\quad ~~~ \frac{dR}{dr}=0,\quad ~~~\frac{d^2R}{dr^2}>0. \end{aligned}$$

(32)

Solving these equations for the impact parameters $\xi$ and $\eta$ we obtain the shadow boundary in the impact parameter space [64]

$$\begin{aligned} {} & {} \eta = \frac{r^2K^2}{N^2}(1-\omega \xi )^2, \nonumber \\ {} & {} \xi = \frac{\varSigma }{\varSigma \omega -\omega '}, \quad ~~~ \varSigma = \frac{1}{2}\frac{d}{dr}\ln \left( \frac{N^2}{r^2K^2}\right) . \end{aligned}$$

(33)

The corresponding observable image at the asymptotic infinity is determined by the celestial coordinates

$$\begin{aligned} \alpha = -\frac{\xi }{\sin i},\quad ~~~ \beta = \left( \eta - \frac{\xi ^2}{\sin ^2 i}\right) ^{1/2}, \end{aligned}$$

(34)

where i is the inclination angle of the observer.

In addition, there exists a second type of unstable spherical orbits located at the wormhole throat. They are described by the conditions

$$\begin{aligned} 1-\frac{b(r)}{r}=0, \quad ~~~ R(r)=0, \quad ~~~ \frac{dR}{dr}> 0. \end{aligned}$$

(35)

Using the expressions for the celestial coordinates these conditions reduce to the implicit relation [78]

$$\begin{aligned} (\omega ^2r^2_0 K^2\sin ^2i - N^2)\alpha ^2 + 2\omega r^2_0 K^2 \sin i\alpha + r^2_0 K^2 - N^2\beta ^2\mid _{r_0} =0, \end{aligned}$$

(36)

where $r_0$ is the location of the wormhole throat. The observable shadow boundary at large distances and inclination angle i is constructed as the collection of images determined by the relations given by Eqs. (33) and (36). Thus, both families of spherical orbits correspond only to certain portions of the boundary curve.

In the following we will illustrate explicitly the shadow boundary for some particular wormhole geometries. We consider the metric functions

$$\begin{aligned} b = r_0, \quad ~~~ K =1,\quad ~~~ \omega = \frac{2J}{r^3}, \end{aligned}$$

(37)

where $r_0$ is the mass of the wormhole, J is its angular momentum, and two different choices for the redshift function

$$\begin{aligned} N^{(1)} = \exp {\left( -\frac{r_0}{r}\right) }, \quad ~~~ N^{(2)} = \exp {\left( -\frac{r_0}{r}-\frac{r^2_0}{r^2}\right) }. \end{aligned}$$

(38)

The shadow for the first redshift function $N^{(1)}$ is presented in Fig. 4 in comparison with the Kerr black hole shadow for the same spin parameter. The shadow boundary determined by the spherical orbits outside the throat corresponds to positive values of the celestial coordinate $\alpha$ (orange curve), while the blue curve for negative $\alpha$ represents the spherical orbits at the throat. Both portions of the boundary curve join smoothly at $\alpha =0$.

Fig. 4

Adapted from [48]

Wormhole shadow for the redshift function $N^{(1)} = \exp {\left( -\frac{r_0}{r}\right) }$ and different values of the spin parameter. The shadow of the Kerr black hole is presented with a dashed line for comparison. As the spin increases the shadow of the wormholes grows and approaches the Kerr black hole.

In Fig. 5 we illustrate the shadow for the second type of redshift function $N^{(2)}$ using the same conventions. A major distinction from the previous case is that the two portions of the shadow boundary defined by the two families of spherical orbits do not merge smoothly but form a cusp at the intersection. Such features were also discovered in the shadow of hairy black holes [31, 92] and in general they can serve as observational signatures for refuting the Kerr hypothesis if sufficiently high resolution is reached.

Fig. 5

Adapted from [48]

Wormhole shadow for the redshift function $N^{(2)} = \exp {\left( -\frac{r_0}{r}-\frac{r^2_0}{r^2}\right) }$ and different values of the spin parameter. The shadow of the Kerr black hole is presented with a dashed line for comparison. This class of wormholes resembles the shadow of the Kerr black hole more closely.

Comparing the wormhole shadow images with the Kerr black hole, we see that for the first redshift function the shadow is significantly smaller than for the Kerr black hole. For the second redshift function the differences decrease and when the spin parameter grows the shadows of the two types of compact objects approach a similar size. However, the deformation of the shadow boundary due to rotation is more pronounced for the wormholes and may be observationally significant.

Despite these deviations the wormholes generally produce qualitatively similar shadows as the Kerr black hole. Keeping in mind that we examined two particular examples we can conjecture that the quantitative discrepancies can be minimized at least for some spin parameters by fine-tuning the wormhole metric. Thus, some of the wormhole solutions will closely mimic the Kerr black hole. For example, in [35] we studied a static wormhole geometry leading to a shadow which cannot be distinguished from the Schwarzschild black hole at the current resolution of EHT.

Section summary: We demonstrated that horizonless spacetimes can cast a shadow by considering a particular class of traversable wormholes. The geodesic equations in these geometries are completely integrable and we calculated explicitly the boundary of the wormhole shadow by using the Hamilton–Jacobi approach. The wormhole shadow is qualitatively similar to the Kerr black hole and by fine-tuning the wormhole geometry we can obtain images which resemble so closely black holes that the two types of compact objects cannot be distinguished at the current resolution of EHT.

5 Image of the Thin Accretion Disk Around Compact Objects

In the previous sections we discussed the images of compact objects when they are illuminated by a uniform distribution of distant light sources. This physical setting is useful for obtaining the photon region and its projection on the observer’s sky since it is independent of the light sources location as long as they reside outside the last spherical orbit. However, if we are interested in constructing a realistic image of the compact object’s vicinity we should take into account the accretion disk surrounding them. The accretion disk is the major source of electromagnetic radiation in most of the astrophysical scenarios. It produces a bright image on the observer’s sky with a characteristic shape and intensity which contain information about the compact object’s spacetime and the properties of the accreting plasma.

In this section we will consider one of the simple analytical models of accretion representing geometrically thin and optically thick disk [66, 68]. It assumes stationary and axisymmetric fluid distribution such that the disk’s height is negligible compared to its radial dimension and the fluid motion is Keplerian. Then, the disk can be approximated by a collection of particles moving on stable circular geodesics in the equatorial plane and the radiation flux can be calculated by means of their kinematic quantities as

$$\begin{aligned} F(r)=-\frac{\dot{M}_{0}}{4\pi \sqrt{-g^{(3)}}}\frac{\varOmega _{,r}}{(E-\varOmega L)^{2}}\int _{r_{ISCO}}^{r}(E-\varOmega L)L_{,r}dr, \end{aligned}$$

(39)

where the integration starts from the innermost stable circular orbit (ISCO). In this expression we denote the accretion rate by $\dot{M}_0$, $g^{(3)}$ is the determinant of the induced metric in the equatorial plane, while E, L and $\varOmega$ are the specific energy, angular momentum and angular velocity on the circular orbits. Considering a general stationary and axisymmetric metric we can express the kinematic quantities on the circular orbits in terms of the metric functions as

$$\begin{aligned} {} & {} E=-\frac{g_{tt} + g_{t\phi }\varOmega }{\sqrt{-g_{tt}- 2g_{t\phi }\varOmega - g_{\phi \phi }\varOmega ^2}}, \quad ~~~ L=\frac{g_{t\phi }+ g_{\phi \phi }\varOmega }{\sqrt{-g_{tt}- 2g_{t\phi }\varOmega - g_{\phi \phi }\varOmega ^2}}, \nonumber \\ {} & {} \varOmega =\frac{d\phi }{dt}=\frac{ -g_{t\phi ,r} + \sqrt{g^2_{t\phi ,r}- g_{tt,r}g_{\phi \phi ,r}}}{g_{\phi \phi ,r}}. \end{aligned}$$

(40)

where the terms $(...)_{,r}$ denote differentiation with respect to the radial coordinate.

The observable image of the thin disk is constructed by obtaining the projection of the photon trajectories originating from the disk on the observer’s sky. This is achieved by solving numerically the null geodesic equations with the appropriate boundary conditions and associating with each geodesic the appropriate celestial angles $\alpha$ and $\beta$ according to the procedure described in Sect. 3 (see Eqs. (20)–(23)). Practically, it is more convenient to scan all the celestial angles in the range $\alpha \in [0,\pi ]$ and $\beta \in [-\frac{\pi }{2}, \frac{\pi }{2}]$ and integrate the corresponding photon trajectories backwards towards their emission point. Then, we select these trajectories which intersect the disk, i.e. pass through the equatorial plane at a radial distance within the range of stability of the timelike circular geodesics. The corresponding set of celestial angles $\alpha$ and $\beta$ build up the image of the accretion disk on the observer’s sky.

The observable radiation intensity at each point of the image is calculated by modifying the emitted flux given by Eq. (39) by the gravitational redshift z

$$\begin{aligned} F_\text {obs} = \frac{F}{(1+z)^4}, \end{aligned}$$

(41)

where the gravitational redshift for a trajectory with an impact parameter $b=L/E$ can be expressed as

$$\begin{aligned} 1+z=\frac{1+\varOmega b}{\sqrt{ -g_{tt} - 2g_{t\phi }\varOmega -\varOmega ^2 g_{\phi \phi }}}, \end{aligned}$$

(42)

for a general stationary and axisymmetric spacetime.

5.1 Thin Accretion Disk Images in Static Spherically Symmetric Spacetime

In static spherically symmetric spacetime the construction of the accretion disk image simplifies due to the integrability of the null geodesic equations. Similar to the calculation of the shadow boundary it reduces to the integration of the radial geodesic equation or equivalently to the estimation of the deflection angle for the photon trajectories which originate from the disk and reach the observer [59, 62]. Thus, for the general static spherically symmetric metric

$$\begin{aligned} ds^2= -A(r)dt^2 + A^{-1}(r)dr^2 + r^2 D(r)(d\theta ^2 + \sin ^2\theta d\varphi ^2), \end{aligned}$$

(43)

we should calculate the integral

$$\begin{aligned} \varphi = \int _{r_{source}}^{r_{obs}} \frac{dr}{r \sqrt{D(r)\,A(r)}\sqrt{\left( \frac{r^2 D(r)}{b^2A(r)}-1\right) }}, \end{aligned}$$

(44)

where $b = L/E$ is the impact parameter on the geodesic, $r_\text {source}$ denotes the radial coordinate of the photon’s emission point, while $r_\text {obs}$ is the position of the observer. This reduces the construction of the lensed image of the thin accretion disk to a semi-analytic procedure and allows to make predictions about the morphology of the image based on the qualitative behavior of the deflection angle [50].

As we already discussed, if the spacetime possesses a photon sphere, the null geodesics can perform an arbitrary large number of turns around the compact object before reaching the observer resulting in a diverging behaviour of the deflection angle in the vicinity of the photon sphere. Therefore, it is convenient to classify the trajectories according to the number of half-loops k which they perform. Direct trajectories are characterized with $k=0$ and deflection angles in the range $\phi \in [0, \pi )$, while trajectories of higher order k lead to deflection angles $\phi \in [k\pi , (k+1)\pi )$. The images on the observer’s sky are also classified into direct ($k=0$) and indirect ($k\ge 1$) according to the order of the null geodesics which give rise to them.

In Fig. 6 we illustrate the image of the thin accretion disk around the Schwarzschild black hole for a distant observer located at $r = 5000M$ and inclination angle $i=80^\circ$. On the left-hand side we present the apparent shape of the disk by constructing its projection on the observer’s sky without considering its radiation. On this diagram we can differentiate between the photon trajectories of different order k which build up the image. Direct trajectories with $k=0$ give rise to the main hat-like image in orange as the images of some particular circular geodesics with radius in the range $r\in [r_\text {ISCO}, 30M]$ are highlighted with solid lines. The image of order $k=1$ is depicted in blue, while the higher order images with $k\ge 2$ correspond to the central black circle. These images are located in a very close neighbourhood of the image of the photon sphere approaching it asymptotically when $k\rightarrow \infty$.

Fig. 6

Adapted from [49]

Image of the thin accretion disk around the Schwarzschild black hole as seen by a distant observer at the inclination angle $i=80^\circ$. The left panel illustrates the disk morphology by highlighting the images of the particular circular orbits. The right panel represents the observable radiation from the disk. The most intensive flux is presented in blue, while the dimmest regions are in red.

In the right panel of Fig. 6 we present the observable radiation associated with each point of the image. We depict the observable flux calculated by Eq. (41) normalized by the maximal value $F^\text {max}_\text {obs}$ reached in the image. We further map continuously this quantity $F_\text {obs}/F^\text {max}_\text {obs}\in [0, 1]$ to the color spectrum from red to blue, as the highest values correspond to dark blue.

Section summary: We discussed the Novikov-Thorne model of thin accretion disk. We demonstrated how we can obtain its observable image giving as a particular example the Schwarzschild black hole.

6 Images of Thin Accretion Disks Around Naked Singularities

In this section we will discuss three types of static spherically symmetric naked singularities which possess qualitatively different behavior of the null geodesics. The first type is characterized by a single maximum of the effective potential for the null geodesics which corresponds to a photon sphere similar to the Schwarzschild black hole. The second spacetime contains no photon sphere but the gravitational field becomes repulsive in the close vicinity of the singularity which results into reflective behavior of the effective potential in this region. The third example combines the presence of a photon sphere with a reflective potential for the null geodesics near the singularity. These features lead to a different appearance of the thin disk around the corresponding compact objects. While the first type of naked singularities mimic closely black holes, the other two cases show distinctive phenomenology.

6.1 Weakly Naked Janis–Newman–Winicour Singularities

Spherically symmetric naked singularities can be classified into weakly and strongly naked according to their lensing properties. We can consider as an example the Janis–Newman–Winicour naked singularity which arises as a solution to the Einstein-scalar field equations [39, 53, 90, 95]. In this case spherically symmetric black holes do not exist and it was proven that the Janis–Newman–Winicour naked singularity represents the unique static spherically symmetric solution. It is given by the metric

$$\begin{aligned} ds^2 = {} & {} -\left( 1-\frac{2M}{\gamma r}\right) ^{\gamma } dt^2 +\left( 1-\frac{2M}{\gamma r}\right) ^{-\gamma } dr^2 \nonumber \\ {} & {} + \left( 1-\frac{2M}{\gamma r}\right) ^{1-\gamma } r^2\left( d\theta ^2 + \sin ^2\theta d\phi ^2\right) , \end{aligned}$$

(45)

and the scalar field takes the form

$$\begin{aligned} \varphi = \frac{q\gamma }{2M} \ln \left( 1-\frac{2M}{\gamma r}\right) , \end{aligned}$$

(46)

where the parameter $\gamma$ is determined by the conserved charges of the solution, i.e. its ADM mass M and scalar charge q. It takes the range $\gamma \in [0,1]$ as $\gamma =1$ corresponds to vanishing scalar charge and the solution reduces to the Schwarzschild black hole.

The Janis–Newman–Winicour class of solutions describes both weakly and strongly naked singularities depending on the value of the scalar field parameter $\gamma$. The weakly naked regime is realized in the range $\gamma \in (0.5,1)$. Then, the spacetime contains a photon sphere with radial position determined by the expression

$$\begin{aligned} r_\text {ph} = (2\gamma + 1)M/\gamma , \end{aligned}$$

(47)

while the innermost stable circular orbit for the particle motion corresponds to the solutions of the equation [49]

$$\begin{aligned} r^2\gamma ^2 - 2r\gamma (3\gamma + 1) + 2(2\gamma ^2 + 3\gamma + 1) =0. \end{aligned}$$

(48)

The effective potential for the null geodesics possesses only a single maximum determining the photon sphere and resembles qualitatively that for the Schwarzschild black hole. Under these conditions the shadow and the accretion disk images are expected to mimic the Schwarzschild black hole with only quantitative deviations. The shadow was calculated in [80] and it was demonstrated in [55] that its apparent size can become by approximately $20\%$ smaller than for the Schwarzschild black hole. Still, for most values of the scalar field parameter, i.e. in the range $\gamma \in [0.53, 1)$, its dimensions are compatible with the EHT observations of the supermassive compact object at the center of the galaxy M87.

Fig. 7

Adapted from [49]

Image of the thin accretion disk around the weakly naked Janis–Newman–Winicour singularity (right) compared to the Schwarzschild black hole (left) for the scalar field parameter $\gamma =0.51$. The disk image for the naked singularity closely resembles the black hole. Direct images are presented in orange, indirect images of order $k=1$ are denoted in blue, while the higher order images are in black. The observer is located at $r=5000M$, while the inclination angle is $i=80^\circ$.

Fig. 8

Adapted from [49]

Observable radiation from the thin accretion disk around the weakly naked Janis–Newman–Winicour singularity for the scalar field parameter $\gamma =0.51$ (right) compared to the Schwarzschild black hole (left). The flux is normalized to its maximum value for each solution. The observer is located at $r=5000M$, while the inclination angle is $i=80^\circ$.

The image of the thin accretion disk was obtained in [49] leading to similar conclusions. We observe a qualitatively similar apparent shape as for the Schwarzschild black hole with quantitative deviations in the observable size of the circular orbits since the naked singularity causes a stronger focussing effect (see Fig. 7). The deviation depends on the radius of the circular orbit as it is stronger in the inner part of the disk reaching ${\sim }18\%$ for the ISCO for $\gamma =0.51$. It decreases in the regions with weaker gravitational field as for more distant orbits located in outskirts of the disk at $r=30M$ it becomes ${\sim }3\%$ for the same value of the scalar field parameter.³

The observable radiation from the disk also resembles qualitatively the distribution for the Schwarzschild black hole with a similar position of its maximum (see Fig. 8). Quantitatively the maximum value of the observable flux for the naked singularity is higher than for the Schwarzschild black hole reaching approximately two times difference for the scalar field parameter $\gamma = 0.51$. Since we plot the normalized flux $F_\text {obs}/F^\text {max}_\text {obs}$ this leads to a dimmer appearance of the outskirts of the naked singularity thin disk.

We discussed the shadow and accretion disk images for a particular type of naked singularities represented by the Janis–Newman–Winicour solution. However, qualitatively similar phenomenology is expected for any solution with the same behaviour of the effective potential for the null geodesics, i.e. type of extrema and asymptotics, and the same structure of the circular geodesics for the massive particles. This includes a broad class of compact objects such as black holes in the modified theories of gravity [51], wormholes [70], and regular back holes [15].

6.2 Strongly Naked Janis–Newman–Winicour Singularities

In the range of the scalar field parameter $\gamma \in (0, 0.5)$ the Janis–Newman–Winicour singularity classifies as strongly naked. The spacetime contains no photon sphere and the effective potential for the null geodesics possesses no further extrema (see Fig. 9). Instead, it diverges in the vicinity of the singularity preventing the null geodesics from reaching it. In this way, if the naked singularity is illuminated by a uniform distribution of light sources all the photon trajectories will scatter away to infinity and no shadow will be observed.

Fig. 9

Adapted from [50]

Behavior of the effective potential for the null geodesics for the strongly naked Janis–Newman–Winicour singularity. The effective potential for the Schwarzschild black hole and the weakly naked Janis–Newman–Winicour singularity are presented for comparison. We specify the values of the scalar field parameter $\gamma$ for each solution and the location of the singularity $r_{cs}$ or the event horizon $r_h$ in the case of black holes.

Another important feature of this geometry is that the stable circular geodesics for the massive particles are located into two disconnected regions with a gap in between. Thus, we have an inner disk delimited by two marginally stable orbits and an outer disk spanning from another marginally stable orbit to infinity. The emission from both regions gives contribution to the image of the thin accretion disk. The location of the marginally stable orbits is determined by the roots of Eq. (48). For $\gamma \le 1/\sqrt{5}$ it has no real solutions and the stable circular orbits extend up to the singularity.

In Fig. 10 we present the optical appearance of the thin disk for $\gamma = 0.48$ without taking into account its radiation. For clarity we provide separate images for the outer and the inner disk as the complete observable image is constructed as their superposition. As in the previous sections, we denote the parts of the image created by photon trajectories of different order k by distinct colors. The direct image of the outer disk only partially resembles the case of the Schwarzschild black hole. On the one hand, it produces the characteristic hat-like image similar to the Schwarzschild black hole. However, it contains also a second disconnected part which represents a bright ring at the center of the disk. This phenomenon of creating double images of the disk distinguishes observationally the strongly naked singularities. In addition, the inner disk gives rise to another central bright ring. This image is further absent for the Schwarzschild black hole since it does not possess a similar disconnected region of stable circular orbits.

Fig. 10

Adapted from [50]

Image of the outer disk (left) and inner disk (right) for the strongly naked Janis–Newman–Winicour singularity with scalar field parameter $\gamma =0.48$. Direct images are presented in orange, indirect images of order $k=1$ are denoted in blue, while the higher order images are in black. The observer is located at $r=5000M$, and the inclination angle is $i=80^\circ$.

The properties of the accretion disk image can be predicted by examining the behavior of the deflection angle on the photon trajectories [50]. In Fig. 11 we consider the photon trajectories emitted by the ISCO for the outer disk for naked singularities with $\gamma = 0.48$. We assume that the observer is located at the radial coordinate $r=5000 M$ and the inclination angle is $i=80^\circ$. Then, we plot the deflection angle of the trajectories which reach the observer’s position as a function of their impact parameter b as the solutions for the different order k are outlined in distinct color strips. The intersection of the curve $\phi (b)$ with each color strip corresponds to an image of the ISCO of the same order. If there exist two disconnected intersections of the deflection angle with a certain color strip they will produce a couple of disconnected images of the corresponding order on the observer’s sky.

Fig. 11

Adapted from [50]

Behavior of the deflection angle for photon trajectories originating from the ISCO and reaching an asymptotic observer at the inclination angle $i=80^\circ$ (left panel). The color strips correspond to images of different order k. Disconnected intersections of the deflection angle with a certain color strip give rise to a couple of disconnected images of the corresponding order. The ISCO possesses a double direct image with $k=0$ and an indirect image of order $k=1$ presented in the right panel in orange and blue respectively. We consider the scalar field parameter $\gamma = 0.48$.

Since the strongly naked singularities do not possess a photon sphere, the deflection angle cannot grow arbitrary large. Instead, it is a bounded function in contrast to the Schwarzschild black hole. We see that it reaches a maximum which corresponds to photon trajectories of order $k=1$ for the ISCO. As a result of this behavior we obtain two branches of solutions for the observable direct trajectories with $k=0$ which correspond to two disconnected intervals for the impact parameter b. Since the impact parameter is directly related to the celestial coordinates, these solutions give rise to two disconnected images in the observer’s sky. The solution for larger b results in the hat-like disk image which is present also in the Schwarzschild case, while the smaller impact parameters produce the central ring.

Although in Fig. 11 we considered a particular example, the described behavior of the deflection angle is representative for any circular orbit from the accretion disk. The deflection angle always possesses a maximum which can be located at most in the range of photon trajectories of order $k=2$. Thus, we obtain an upper limit for the number of half-turns which the null geodesics originating from the disk can perform before reaching infinity. In this case, in addition to the double direct image of the disk we observe also double image of order $k=1$. It is produced by the two disconnected solutions for the trajectories of order $k=1$ which now become possible.

Fig. 12

Adapted from [50]

Observable radiation from the thin accretion disk around the strongly naked Janis–Newman–Winicour singularity with scalar field parameter $\gamma =0.48$ (left panel). The Schwarzschild black hole is presented for comparison in the right panel. The observable flux is normalized to its maximum value for each solution. The observer is located at $r=5000M$, while the inclination angle is $i=80^\circ$.

In Fig. 12 we present the observable radiation from the thin disk around the strongly naked singularity. The central rings are observationally significant within this model of accretion since the maximum of the radiation flux is reached in the image of the inner disk. On the other hand, the central rings which result from the double image of the outer disk emit with around $30\%$ of the maximum radiation.

6.3 Einstein–Gauss–Bonnet Naked Singularities

The four-dimensional Einstein–Gauss–Bonnet gravity admits a static spherically symmetric solution in the form [18, 25, 42, 85]

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

where M is the ADM mass of the solution, while $\hat{\gamma }$ is the Gauss–Bonnet coupling constant. In the range $\hat{\gamma }\in [0, 1]$ the solution describes black holes while for $\gamma >1$ it represents naked singularities. The naked singularities with coupling constant $1<\hat{\gamma }<3\sqrt{3}/4$ lead to particularly interesting lensing properties [51]. The effective potential for the null geodesics possesses a stable and unstable photon ring and diverges in the vicinity of the singularity (see Fig. 13). Thus, it reflects infalling photon trajectories preventing them from reaching the singularity. Such spacetimes cannot cast a shadow although they possess a photon sphere. The photon trajectories which pass through the photon sphere will still be reflected back to infinity due to the potential barrier in the vicinity of the singularity leaving no dark directions on the observer’s sky.

Fig. 13

Adapted from [51]

Behavior of the effective potential for the null geodesics for the Gauss–Bonnet naked singularities with coupling constant in the range $1<\hat{\gamma }<3\sqrt{3}/4$. We plot the effective potential for the representative value $\hat{\gamma }=1.15$ for several values of the specific angular momentum (red line). The effective potential for the Schwarzschild black hole is presented for comparison (black line).

The stable circular timelike orbits are located in two disconnected regions consisting of an inner annulus and an outer disk. The limits of these regions are determined by the marginally stable orbits which are solutions to the equation

$$\begin{aligned} r^3 -9M^2 r + 8 M^3\hat{\gamma }=0. \end{aligned}$$

(49)

Thus, the thin accretion disk consists of two disconnected portions, i.e. inner and outer disk, similar to the strongly naked Janis–Newman–Winicour singularity.

These properties determine the structure of the disk images presented in Fig. 14. The images of the inner and the outer disks are constructed separately as the observable image represents their superposition. The outer disk possesses a double image of any order $k\ge 1$. One of the images produces an observable structure similar to the thin disk for the Schwarzschild black hole. The second image represents a nested multi-ring structure at small inclination angles which is absent for black holes. The inner disk also leads to double images of any order, however both of them correspond to central bright rings. The central rings represent the brightest part of the image radiating with intensity ${\sim } 10^3$ higher than the hat-like disk image.

Fig. 14

Adapted from [51]

Image of the outer disk (left) and inner disk (right) for the Gauss–Bonnet naked singularity with coupling constant $\hat{\gamma }=1.15$. The observable flux is normalized to its maximum value for each image. The observer is located at $r=5000M$, while the inclination angle is $i=80^\circ$.

Examining the behavior of the deflection angle we can predict the formation of the qualitatively different morphology of the disk images [51]. In Fig. 15 we present the deflection angle for photon trajectories originating at the ISCO of the outer disk and reaching an observer located at $r=5000M$ and inclination angle $i=80^\circ$. We see that the deflection angle diverges at the location of the photon sphere. However, in contrast to the Schwarzschild black hole, there exist two families of null geodesics scattering away to infinity which approach the photon sphere as a limit surface with higher and lower values of the impact parameters, respectively. The first family gets reflected from the maximum of the effective potential and gives rise to a similar image as for the Schwarzschild black hole. The second family gets reflected from the potential barrier in the vicinity of the singularity and produces the multi-ring structure at small celestial angles. As a result the ISCO possesses double images of any order $k\ge 1$. The images of lower order are distinguishable but when k grows they converge towards the images of the photon sphere. This behavior is representative for any orbit from the outer disk, thus demonstrating how the disk image is created.

Fig. 15

Adapted from [51]

Behavior of the deflection angle for photon trajectories originating from the ISCO and reaching an asymptotic observer at inclination angle $i=80^\circ$ (left panel). The color strips correspond to images of different order k. Disconnected intersections of the deflection angle with a certain color strip give rise to a couple of disconnected images of the corresponding order. In the right panel we present the corresponding images of the ISCO up to order $k=6$. We consider the coupling constant $\hat{\gamma }= 1.15$.

The multi-ring structure which we described in the thin disk images for the strongly naked JNW singularities and the Gauss–Bonnet singularities appears in much more general cases than these particular spacetimes. As we demonstrated it is governed by the behavior of the deflection angle for the scattering photon trajectories, which on the other hand depends on the form of the effective potential for the null geodesics. This implies that any spacetime with a qualitatively similar behavior of the effective potential will give rise to a qualitatively similar morphology of the disk images encompassing a large class of compact objects of diverse physical nature. According to a general theorem [29] any regular compact object with trivial topology which possesses a photon sphere should further possess a stable light ring.⁴ The thin disk images of these spacetimes would be characterised by the same morphology as for the Gauss–Bonnet naked singularity although they can describe very different self-gravitating objects like regular black holes or exotic stars. Some examples of such compact objects were presented in [19, 36, 37, 47].

Section summary: We discussed the thin accretion disk images for three types of spherically symmetric naked singularities. The first spacetime produces a very similar image to the Schwarzschild black hole, while the other two give rise to a characteristic multi-ring structure at the center of the image. The key concept which determines the morphology of the image is the behavior of the deflection angle on the scattering photon trajectories which reach the observer. Since the deflection angle depends directly on the properties of the effective potential for the null geodesics, compact objects with similar effective potentials will lead to qualitatively similar accretion disk images despite their different physical nature.

Further Reading

• On the calculation of the black hole shadow:

  The analytical calculation of the shadow boundary for the Kerr black hole is described in detail in the books [24, 40], as well as in the recent reviews [32, 74]. The procedure is generalized in the presence of a cosmological constant [73], NUT charge [5, 45], and plasma environment [71, 72]. Shadows of black holes interacting with an external gravitational source are obtained numerically in [3, 4, 46, 65, 81].

• Black hole shadow in the modified theories of gravity:

  These exist a large number of works studying the black hole shadow in various modifications of general relativity. Analytical results include [12,13,14, 56, 61, 69], while [26, 27, 30, 87] investigate the problem numerically.

• On the calculation of the thin disk image:

  The construction of observable image of the thin accretion disk for the Schwarzschild black hole is developed in the classical works [33, 34, 41, 59, 62, 63]. The procedure can be generalized straightforwardly for a general static spherically symmetric compact objects as demonstrated in [49,50,51].

• Accretion disk images of exotic compact objects:

  Exotic compact objects such as wormholes, naked singularities, boson stars and regular black holes are considered as viable alternatives of the Kerr black hole and simulations of their images are used in the interpretation of the EHT results [8, 11, 55]. Accretion disk images of exotic compact objects considering different models of accretion include [57, 67, 76, 79, 88, 89, 101].