On the Counter-Rotating Tori and Counter-Rotating Parts of the Kerr Black Hole Shadows

Abstract

We review some aspects of accretion disks physics, spacetime photon shell and photon orbits, related to retrograde (counter-rotating) motion in Kerr black hole (BH) spacetimes. In this brief review, we examine the counter-rotating components of the Kerr BH shadow boundary, under the influence of counter-rotating accretion tori, accreting flows and proto-jets (open critical funnels of matter, associated with the tori) orbiting around the central BH. We also analyze the redshifted emission arising from counter-rotating structures. Regions of the shadows and photon shell are constrained in their dependence of the BH spin and observational angle. The effects of the counter-rotating structures on these are proven to be typical of the fast-spinning BHs, and accordingly can be observed only in the restricted classes of the Kerr BH spacetimes. This review is intended as a concise guide to the main properties of counter-rotating fluxes and counter-rotating disks in relation to the photon shell and the BH shadow boundary. Our findings may serve as the basis for different theoretical frameworks describing counter-rotating accretion flows with observable imprints manifesting at the BH shadow boundary. The results can eventually enable the distinction of counter-rotating fluxes through their observable imprints, contributing to constraints on both the BH spin and the structure of counter-rotating accretion disks. In particular, photon trajectories and their impact parameters can manifest in the morphology of the BH shadow. Such features, when accessible through high-resolution imaging and spectral or polarization measurements, could provide a direct avenue for testing different theoretical models on accretion disk dynamics and their BH attractors.

1. Introduction

The Event Horizon Telescope (EHT) Collaboration in 2019 unveiled an image of M87*, the super-massive black hole (SMBH) at the center of the super-giant elliptical galaxy Messier 87 (M87) [1,2,3,4,5,6]. On the other hand, through VLBI (very-long-baseline interferometry), millimeter-band synchrotron images of the black hole (BH) and accretion disks at the center of SgrA galaxy were also resolved in 2022, providing an image of the emission regions close to SgrA*, the central BH, in the Milky Way [7].

Numerical general relativistic magnetohydrodynamic (GRMHD) torus models were matched to observations in [4,5,6,8,9,10,11,12]. Many analytic disk, accretion flow and jet ejection models have been developed [13,14,15]). Semi-analytical models of the M87* spectra have been also investigated [16]. In addition, semi-analytic GRMHD jet and accretion flow models were developed in [17] to simulate the emission from both M87* and SgrA*. The behavior of high-energy particles within jet–accretion flows, with synchrotron radiation signatures, has been also analyzed [18].

Following these detections, the observational results and theoretical interpretations were further refined and extended. In order to fit astronomical observations, several observables were constructed using special points on the shadow boundary in the celestial coordinates, focusing in particular on the analytical, geometric constraints provided by the gravitational background [11,13,19,20,21,22].

Our analysis fits within this framework, concentrating on the analytical constraints and distinctive features of the shadow boundary arising from flows and disks that counter-rotate relative to the central BH.

In this respect, properties of the BH shadows were investigated since the earlier studies [23,24,25,26,27]. The ring-shaped bright images from EHT analysis are known as the “emission ring” [28]. The image data, obtained for M87* and SgrA*, emerge as a bright, unresolved (thick) ring. Embedded within this, there is a (thin) ring, sometimes called “photon ring”, composed of infinite sub-rings, made of photon orbits around the BH, and directed to the observer at infinity. The set of these orbits approaches the BH shadow boundary, which is fixed by the BH spin and the view-angle only, and it is determined by (photons orbiting) the BH photon shell.

The photons in the region fall in or escape to infinity. The BH shadow corresponds to photon paths falling on its event horizon, and the (dark) region is surrounded by the bright emission ring, due to the accretion flow/disk or lensed photons.

The so-called photon ring corresponds, therefore to photon paths near the (unstable) photon orbits (passing through the emitting plasma, which, if optically thin, will be associated with a bright image. It has been evaluated (from GRMHD simulations and analytic analysis) that the photon ring would correspond to only 10% of the total image flux density [11].

The thick ring diameter is well analyzed, but its compositions, i.e., its internal sub-structure, including the photon orbit’s sub-structure, is not resolved. The photon ring decomposition into sub-rings features a self-similar structure (in the set of orbits) determined by the region orbital instability (see, for more discussion, [11,28,29,30,31,32]).

Each photon ring is indexed with an integer $n\ge 0$, where $n=0$ corresponds to a direct (the primary) image, photon trajectories from the emission point direct to the observer at infinity (if on the line of sight), with no orbit around the central BH. Each $n>0$ corresponds to an exponentially thinner ring-like image (as n increases, the photon orbits become progressively narrower). For $n=1$ (secondary image), photons complete half an orbit around the BH; photons with $n=2$ undergo a full orbit around the BH before reaching the observer. Then, photon trajectories are identified by indexes assigned by the number of half-orbits n completed around the central BH, before they reach the observer1, with each successive image corresponding to increasing n.

The $n=\infty$ photon ring defines the BH shadow boundary. The boundary of the shadow, which the photon rings in the limit $n\to \infty$ approach, is also knoawn as the “critical curve”, and it is independent on the astrophysical context as the emitted plasma and the emission model2. Unlike the emission ring, the critical curve is regulated by the BH spin only and the view angle (intended as the observer inclination relative to the rotational axis).

In our analysis, we investigate the structure of the critical curve, associated with the $n=\infty$ photon ring, in connection with the spacetime photon shell, the accretion disk, and the dynamics of relativistic accretion flows. As photon orbits in the photon shell compose the shadow boundary, the photon shell’s internal structure, defined by $u^{\varphi }$ and ℓ, the photons’ relativistic specific angular momentum (or impact parameter) is reflected in the internal structure of the critical curve. In our analysis, we examine the counter-rotating components, that is, the counter-rotating disks (which always orbit outside the photon shell); the photons orbit with a negative impact parameter and the counter-rotating accretion flow, with $a\ell <0$ and $a{u}^{\varphi }<0$. (Note that generally, the orbiting co-rotating tori can enter the photon shell (or even the ergoregion) directly influencing the photon shell and the BH shadow, while this is not possible for counter-rotating tori. For this reason, we have to consider indirect influence caused by the counter-rotating flow falling into the BH.)

Distinguishing counter-rotating from co-rotating BH accretion disks3 based on image morphology, such as brightness asymmetry induced by relativistic Doppler boosting, or from the geometry of photon orbits, remains extremely challenging. This difficulty arises due to strong dependencies on several factors, including the disk modelization (e.g., whether the accretion flow follows a standard and normal evolution (SANE) or a magnetically arrested disk (MAD) configuration), the resolution of photon–ring measurements, and various disk-emission model characteristics. These include the geometry and strength of the magnetic field, jet alignment, and jet energetics, all of which can significantly influence the observable features and complicate the interpretation of VLBI images4.

A discussion on the BH spin constraints from the interferometric measurements of the first photon ring is in5 [29]. More generally, constraints from photon rings are supported by different analyses—see, for example, [28,30,31,33,34].

In the analysis of the BH–accretion disk relative rotation orientation, the system under consideration, as inferred from BH imaging analysis conducted by the EHT, often consists of a central Kerr or Schwarzschild BH, surrounded by a (single) axially symmetric and equatorial accretion disk, accompanied by a jet and centered on the BH. The determination of the disk rotational orientation relative to the central BH, in some systems, often relies on assessing the alignment among the BH, jet, and disk spin axes. This alignment can be differently treated as a variable assumption, forming the basis for numerous best-fit model approaches. The spin of M87* is studied, for example, in [35], favoring a MAD disk for M87*, with respect to a SANE disk, and a co-rotating disk with a BH spin $a\ge 0.4$, or a BH spin $a\ge 0.5$ for counter-rotating disks. In [36], two models with counter-rotating (MAD) disk (with a M87 BH with spin $a\approx 0.5$) and a co-rotating (MAD) accretion disk (with faster spinning M87 BH, with $a\approx 0.95$) were examined.

Hence, current counter-rotating/co-rotating claims concerning, respectively, the rotation orientation of M87* and SgrA* accretion disks, are (strongly) model-dependent. A counter-rotating (MAD) disk for M87* has been seen as favored in some analyses, within the (complex set of) assumptions made for the fitting models. However, other studies also constrain these expectations to different model assumptions, including, for example, the magnetization and the relativistic jet alignment, and features of the ray-tracing and GRMHD simulations6—see [5,32,37,38,39,41]—whereas SgrA* would be described as a spinning BH with a co-rotating accretion flow (and disk)7—see also [37].

However, the study of counter-rotating disks proves particularly relevant for the characterization of the accretion dynamics and their impact on the BH shadow formation, and observational diagnostics in Kerr spacetimes (constraining key parameters such as BH spin and disk rotation orientation). Their dynamics and observational signatures are influenced by several factors, including the effects of the relativistic frame dragging in Kerr spacetime, the stability and efficiency of accretion processes, and the distinct spectral and redshift features they produce.

Furthermore, multiple lines of evidence suggest the presence of counter-rotating disks, for example, from diverse features of the observed emission and luminosity. Notably, in the X-ray binaries, BH systems that lack a detectable ultra-soft component above 1–2 keV in their high-luminosity state may indicate the presence of a rapidly spinning, counter-rotating BH. We refer, for example, to refs. [43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] for an extensive discussion of the distinguishing properties of the accretion discs, particularly in terms of emission and luminosity as shaped by their rotational orientation.

On the other hand, although current EHT observations remain substantially inconclusive regarding whether the accretion discs around SgrA* and M87* are co-rotating or counter-rotating [28,29,30,31,33,34], one could expect that forthcoming observations, combined with more refined analyses, may prove essential for the determination of the rotation orientation.

The constraints presented in this work could ground the construction of libraries’ counter-rotating models, to be systematically matched with the observational data. These constraints might be confronted with astrophysical observations, directly compared to current measurements.

Additionally, although counter-rotating disks themselves do not orbit within the photons’ shell region, the central BH accretes counter-rotating flows. The material they supply can penetrate both the photon shell, as well as the ergoregion, and photons with counter-rotating components in the photon shell. Within the ergoregion, only matter with positive azimuthal velocity, $a{u}^{\varphi }>0$, is allowed. However, accreting material with negative angular momentum $a\ell <0$ can also reside inside the spacetime ergoregion, provided it still satisfies $a{u}^{\varphi }>0$. This indicates that frame-dragging acts on counter-rotating (with $a\ell <0$) matter, as it accretes toward the central BH, before crossing the outer ergosurface, leading to the formation of points where $u^{\varphi }=0$ (inversion surface and the inversion points).

Moreover, in [20,58] the counter-rotating cases were found to be significantly more constrained than the co-rotating case, which further motivates a focus on the counter-rotating components as effective discriminants of the viewing angle, BH spin, and other disk properties. More specifically the analysis in [20,58] is an extensive investigation of the co-rotating and counter-rotating photon trajectories in the BH shadows, which led to a map of the shadow boundaries. Focusing on the properties of the Kerr BH photons’ shell, it delved into the influence of aggregates of co-rotating and counter-rotating toroids orbiting within the photons’ shell, an effect referred to as BH photon shell obscuration. Since counter-rotating toroids were proven not to orbit in the photon shell for any BH spin, the investigation focused on the specific case of a co-rotating accretion disks. Five distinct classes of BHs were identified, each distinguished by unique features of their photon shells and shadow boundaries, in the context of co-rotating structures orbiting within the photon shell.

Therefore, in the present analysis, we concentrate on the counter-rotating disk, specifically analyzing the photon shell counter-rotating components (counter-rotating photon spherical orbits) and the effects on the shadow boundary of the counter-rotating matter orbiting around the central Kerr BH. We start by reviewing the main constraints on the counter-rotating flows, and we will further detail the properties of counter-rotating tori. We delve deeper into the analysis of shadow boundary variation under the influence of photons from counter-rotating structures outside the photon shell. Finally, we investigate the emission from counter-rotating structures, analyzing the redshift properties of the radiation from counter-rotating particles. The photon shell, as the region populated by unstable spherical photon orbits, plays a decisive role in shaping the shadow boundary and in mediating the interaction between counter-rotating accreting structures and observable emission. Within this framework, counter-rotating fluxes and disks represent a fundamental class of configurations whose properties are tightly constrained by the geometry of the Kerr spacetime (its dimensionless spin also through the effects of frame dragging). The investigation aims to clarify the conditions under which we could provide potential observational discriminants for identifying counter-rotating contributions in accretion flows. We believe this brief review can serve as a concise guide to several key effects that warrant closer analysis in light of the comparison with observational data.

Six different systems are under investigation: 1. counter-rotating accretion disks and structures; 2. the counter-rotating flux of accreting materials in photon shell; 3. the counter-rotating spherical photon orbits; 4. the bound unstable spherical photon orbits in the ergoregion and on the outer ergosurface; and 5. the photon spherical orbits with null impact parameter $\ell =0$ and 6. with $u^{\varphi }=0$.

The article is divided into two parts. In the first part, from Section 2 to Section 4, we introduce the model and investigate the constraints of the counter-rotating orbits on the BH shadow boundary. In detail, in Section 2 we introduce the spacetime metric. Constants of motion and geodesic equations are summarized in Section 2.1. In Section 3 we analyze the properties of the Kerr BH photon shell. In Section 3.1 we focus on the counter-rotating tori orbiting the central BH. In Section 3.2 we define the inversion surfaces, whereas the BH shadow boundary is the focus of Section 4.

The second part of this analysis is in Section 5, where we discuss the redshift emission from counter-rotating structures.

Final remarks follow in Section 6.

2. The Spacetime Metric

Adopting Boyer–Lindquist (BL) coordinates $\{t,r,\theta ,\varphi \}$, we can write the Kerr spacetime line element as follows:

$$\begin{array}{c}d{s}^2=-\left(1-{\displaystyle \frac{2Mr}{\Sigma }}\right)d{t}^2+{\displaystyle \frac{\Sigma }{\Delta }}d{r}^2+\Sigma d{\theta }^2+\hfill \\ \left[(r^2+a^2)+{\displaystyle \frac{2Mr{a}^2}{\Sigma }}{sin}^2\theta \right]{sin}^2\theta d{\varphi }^2-{\displaystyle \frac{4rMa}{\Sigma }}{sin}^2\theta dtd\varphi \hfill \end{array}$$

(1)

where

$$\Delta \equiv a^2+r^2-2rM\mathrm{and}\Sigma \equiv a^2(1-{sin}^2\theta )+r^2$$

(We use geometrical units, where $c=1=G$.) Parameter $a\in [0,M]$ is the metric spin, while M is the (gravitational) mass parameter. Below, we will use dimensionless units with $M=1$ (where $r/M\to r$ and $a/M\to a$). According to the values of the spin parameter, metric (1) reduces to the Schwarzschild (non-rotating) BH solution by $a=0$ or to the Kerr BH by the condition $a\in ]0,1]$. This includes the extreme Kerr BH solution, when $a=1$.

The Kerr BH outer horizon and outer ergosurface are

$$r_{+}\equiv 1\pm \sqrt{1-a^2},\mathrm{and}{r}_{ϵ}^{+}\equiv 1+\sqrt{1-a^2(1-\sigma )}\mathrm{with}\sigma \equiv {sin}^2\theta \in [0,1],$$

(2)

respectively. The region $]r_{+},r_{ϵ}^{+}]$ is known as the outer ergoregion.

2.1. Geodesics Equations and Constants of Motion

In this section we introduce the Kerr spacetime constants of motion and geodesic equations. Let us start by defining the quantities $d{x}^a/d\tau \equiv u^a\equiv \{\dot{t},\dot{r},\dot{\theta },\dot{\varphi }\}$, the geodesic four-vector tangent. (Here, $\tau$ is the affine parameter.) $u^a{u}_a=-1$, for test particles, or otherwise, $u^a{u}_a=0$ for a null geodesic.

Below, we also consider the constants of motion $\{\mathcal{E},\mathcal{L},\mathcal{Q}\}$, where $\mathcal{Q}$ is the Carter constant of motion and

$$\mathcal{E}=-(g_{t\varphi }\dot{\varphi }+g_{tt}\dot{t}),\mathcal{L}=g_{\varphi \varphi }\dot{\varphi }+g_{t\varphi }\dot{t}.$$

(3)

From $(\mathcal{L},\mathcal{E})$ in Equation (3), there is the constant of motion

$$\ell \equiv {\displaystyle \frac{\mathcal{L}}{\mathcal{E}}}=-{\displaystyle \frac{g_{\varphi \varphi }{u}^{\varphi }+g_{\varphi t}{u}^t}{g_{tt}{u}^t+g_{\varphi t}{u}^{\varphi }}},$$

(4)

known as the specific angular momentum (also called the impact parameter).

The introduction of the constant ℓ enables a precise definition of the rotational orientation of particles, fluxes and disks, with respect to the central BH, in terms of counter-rotating or co-rotating matter, according to $\ell a<0$ or $\ell a>0$, respectively.

The constant $\mathcal{Q}$ regulates the motion in the $\theta$ direction and appears in the geodesic equations:

$$\begin{array}{c}\dot{t}={\displaystyle \frac{1}{\Sigma }}\left[{\displaystyle \frac{P\left(a^2+r^2\right)}{\Delta }}-a\left[a\mathcal{E}\sigma -\mathcal{L}\right]\right],\dot{r}=\pm {\displaystyle \frac{\sqrt{R}}{\Sigma }};\hfill \\ \dot{\theta }=\pm {\displaystyle \frac{\sqrt{T}}{\Sigma }},\dot{\varphi }={\displaystyle \frac{1}{\Sigma }}\left[{\displaystyle \frac{aP}{\Delta }}-\left[a\mathcal{E}-{\displaystyle \frac{\mathcal{L}}{\sigma }}\right]\right];\hfill \end{array}$$

(5)

ref. [59], where

$$\begin{array}{c}\mathcal{Q}\equiv T+{(cos\theta )}^2\left[a^2\left({\mu }^2-{\mathcal{E}}^2\right)+\left({\displaystyle \frac{{\mathcal{L}}^2}{\sigma }}\right)\right],\mathrm{and}\hfill \\ R\equiv P^2-\Delta \left[{(\mathcal{L}-a\mathcal{E})}^2+{\mu }^2{r}^2+\mathcal{Q}\right],\mathrm{with}P\equiv \mathcal{E}\left(a^2+r^2\right)-a\mathcal{L},\hfill \end{array}$$

(6)

and $\mu =1$ for test particles, while there is $\mu =0$ for photons.

From $(\mathcal{E},\mathcal{Q})$ in Equations (3) and (6), there is the constant of motion

$$q\equiv {\displaystyle \frac{\mathcal{Q}}{{\mathcal{E}}^2}}.$$

(7)

3. Shadows

In this section we introduce the concept of the boundary of the BH shadow. This object is the set of the (unstable) photon orbits with

$$R={\partial }_{r}R=0,{\partial }_r^{2}R>0$$

(8)

(see Equation (5)).

The concept of the BH shadow boundary is strongly related to the region of the spacetime known as the photon shell. Solutions of Equation (8) are contained in the photon shell. This region is divided into an outer part, with orbits $\ell <0$, and an inner region, which includes the solutions $\ell >0$. The inner region of the photon shell is close to the central BH and, depending on the spin of the BH, partially contained in the outer ergoregion—Figure 1 and Figure 2.

Solutions of Equation (8) can be found in terms of the impact parameter ℓ and the constant q, obtaining the two functions

$${\ell }_{\Theta }\equiv {\displaystyle \frac{a^2(r+1)+(r-3)r^2}{a(1-r)}}\mathrm{and}{q}_{\Theta }\equiv -{\displaystyle \frac{r^3\left[{(r-3)}^{2}r-4a^2\right]}{a^2{(r-1)}^2}}.$$

(9)

Solutions can also be found in terms of the radius r, the orbit of the photons in the photon shell. By solving the equations ${\ell }_{\Theta }(r,a)=\ell$, we obtain:

$$\begin{array}{c}r=r_{\lambda }\equiv {\displaystyle \frac{1}{6}}\left(6-{\displaystyle \frac{2\sqrt[3]{2}{V}_{\circ }}{\sqrt[3]{V_{★}}}}+2^{2/3}\sqrt[3]{V_{★}}\right),\mathrm{with}\hfill \\ V_{\circ }\equiv 3(a^2+a\ell -3),\mathrm{and}\hfill \\ V_{★}\equiv \sqrt{{\left[54(1-a^2)\right]}^2+4V_{\circ }^3}+54(1-a^2).\hfill \end{array}$$

(10)

From $T=0$, in Equations (5) and (6), and using $q_{\Theta }$ and ${\ell }_{\Theta }$ from Equation (9), we find the angles

$${\sigma }_{\pm }\equiv {\displaystyle \frac{r^2\left(a^2+r^2-3\right)+a^2}{a^2{(r-1)}^2}}-2\sqrt{{\displaystyle \frac{r^2\Delta \left[a^2+r^2(2r-3)\right]}{a^4{(r-1)}^4}}},$$

(11)

defining the photon shell’s inner and outer boundaries.

Hence, the photon shell is defined in the polar angular range $\sigma \in [{\sigma }_{-},{\sigma }_{+}]$ (with $\varphi \in [0,2\pi [$)—Figure 3.

Furthermore, ${\sigma }_{\pm }\left(r_{lim}\right)=0$ and ${\ell }_{\Theta }\left(r_{lim}\right)=0$ on the orbit of radius $r_{lim}\left(a\right)$, defined as follows:

$$\begin{array}{c}{r}_{lim}\equiv -{\displaystyle \frac{a^2-3}{\sqrt[3]{3}\sqrt[3]{S_{\circ }}}}+{\displaystyle \frac{\sqrt[3]{S_{\circ }}}{3^{2/3}}}+1,\mathrm{with}\hfill \\ S_{\circ }\equiv 9(1-a^2)+\sqrt{3}\sqrt{a^2\left(a^4+18a^2-27\right)}.\hfill \end{array}$$

(12)

Radius $r_{lim}$ is therefore a limiting radius separating the counter-rotating orbits, in the outer region $r>r_{lim}$, from co-rotating photon orbits, in the inner region $r<r_{lim}$—Figure 1, Figure 2 and Figure 3.

3.1. Counter-Rotating Tori Orbiting the Central BH

Let us consider the radii $r_{mbo}^{+}$ and $r_{mso}^{+}$. On the equatorial plane, these radii coincide with the counter-rotating test particles’ marginally stable circular orbit and the marginally bounded circular orbit, respectively.

There is

$$r_{\gamma }^{-}<r_{\lambda }\left({\ell }_{mso}^{+}\right)<r_{\lambda }\left({\ell }_{mbo}^{+}\right)<r_{\gamma }^{+}<r_{mbo}^{+}<r_{mso}^{+},$$

(13)

and $r_{\gamma }^{\pm }$ values are, on the equatorial plane, the counter-rotating and co-rotating circular photon orbit, respectively. (Note that only orbits in the photon shell can be observed on the shadow boundary (see for further details [20,58]).)

The photon shell is included in the region $[r_{\gamma }^{-},r_{\gamma }^{+}]$. Hence, from Equation (13), it follows that there are no solutions on the spherical surfaces defined by constant radii8 $r_{mbo}^{+}$ and $r_{mso}^{+}$. However, radii $\{r_{\lambda }({\ell }_{mso}^{+},r_{\lambda }\left({\ell }_{mbo}^{+}\right)\}$ can be inside the photon shell.

The counter-rotating disk is therefore separated from the outer boundary of the photon shell (on the equatorial plane, this is $r=r_{\gamma }^{+}$) by the radial interval $]r_{\gamma }^{+},r_{mbo}^{+}]$.

In Figure 3, the solutions9 ${\ell }_{\Theta }(r,a)={\ell }_{mso}^{+}\left(a\right)$ and ${\ell }_{\Theta }(r,a)={\ell }_{mbo}^{+}\left(a\right)$ are shown. A part of these curves is thus in the region of the photon shell. Therefore, orbits with these impact parameter values can appear on the BH shadow boundary.

On the other hand, the inner edge $r_{inner}^{+}$ (and inner part) of a counter-rotating quiescent torus with a strong angular momentum magnitude10 can approach the photon shell on the equatorial plane, i.e., $r_{inner}^{+}⪆r_{\gamma }^{+}$—Figure 2.

In these structures, there could be inversion surfaces where, for particles or photons, $\dot{\varphi }=0$ for time-like and light-like particles alike, with $\ell <0$.

Only a part of the photon inversion surfaces correspond to spherical, bound, and unstable photon orbits, as found in solutions of Equation (8); consequently, only some may manifest on the BH shadow boundary [70].

In fact, the rotating BH always accretes matter with $\dot{\varphi }>0$, and this could have $\ell <0$. On the other hand, the photon shell is included in the ergoregion only for the faster-spinning BHs—Figure 3 left panel; for larger $\sigma$—Figure 4 [58].

3.2. The Inversion Surfaces

An inversion surface encloses the central Kerr BH and is situated outside the BH outer ergosurface. On the inversion surface, there are inversion points, defined by the condition $\dot{\varphi }=0$, for particles and photon trajectories—Figure 1 [70].

Defined only by the parameter $\ell <0$ and the spin a, inversion points approach the outer ergosurface, for ℓ values that are large in magnitude (or, in general, small $\sigma$ values)—see Figure 4.

Hence, within the condition $\dot{\varphi }=0$, $\ell ={\ell }_{\mathbf{T}}\equiv -g_{t\varphi }/g_{tt}$ with radius $r_{\mathbf{T}}(\ell ,\sigma ,a):\ell ={\ell }_{\mathbf{T}}(a,r.\sigma )$, for timelike or light-like particles—see [58].

On the equatorial plane, there is always $r_{\mathbf{T}}<r_j^{+}<r_{\times }^{+}$ (where $r_j^{+}<r_{\times }^{+}$ are the counter-rotating proto-jet and tori cusps respectively; see [58,70]) for the parameter ${\ell }^{+}\le {\ell }_{mso}^{+}$, and therefore, it can be included in the photon shell if $r_{\mathbf{T}}\le r_{\lambda }$—Figure 3 and Figure 4.

We examine solutions $r=r_{\mathbf{T}}$ for general timelike and lightlike geodesics, and in particular orbits $r=r_{(\mathbf{T},\Theta )}$, constituting a small set of all $r_{\mathbf{T}}$ orbits, made up by photon spherical orbits with $\dot{\varphi }=0$.

Hence, the condition ${\ell }_{\mathbf{T}}(a,\sigma ,r)={\ell }_{\Theta }(a,r)$ leads to the radius

$$\begin{array}{c}{r}_{(\mathbf{T},\Theta )}\equiv -{\displaystyle \frac{\sqrt[3]{2}{W}_{•}}{3\sqrt[3]{W_{\circ }}}}+{\displaystyle \frac{\sqrt[3]{W_{\circ }}}{3\sqrt[3]{2}}}+1,\mathrm{with}\hfill \\ W_{\circ }\equiv W_{★}+\sqrt{4W_{•}^3+W_{★}^2},W_{★}\equiv 54[1+a^2(\sigma -1)],W_{•}\equiv -3[3+a^2(\sigma -1)],\hfill \end{array}$$

(14)

where there is

$$r_{(\mathbf{T},\Theta )}\in [r_{lim},r_{\gamma }^{+}]\subset ]\sqrt{2}+1,3],$$

(15)

(for $(a\in ]0,1\left]\right)$).

Therefore, radius $r_{(\mathbf{T},\Theta )}$ approaches $r_{lim}$ for slowly spinning BHs or for decreasing angle $\sigma$ (Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6).

Note, in a fixed spacetime, for each value of $\ell <0$, there are infinite inversion surfaces $r_{\mathbf{T}}(a,\sigma ,\ell )$; on the other hand, there is only one inversion surface with $r_{(\mathbf{T},\Theta )}(a,\sigma )$ per plane $\sigma$.

In Figure 4, radii $r_{\mathbf{T}}$ are shown for different values of the impact parameter $\ell <0$, with the photon shell boundary, the radius $r_{lim}$, the ergosurface, and $r_{(\mathbf{T},\Theta )}$ for two observational angle values and for all values of the spin.

As clear from Figure 5 and Figure 6, $r_{\mathbf{T}}\left({\ell }^{+}\right)>r_{\gamma }^{-}$ is true on the equatorial plane only for sufficiently large spin. In Figure 5, we show the radius r solution of ${\sigma }_{\pm }={\sigma }_{\mathbf{T}}$, where ${\sigma }_{\mathbf{T}}\left(a,{\ell }_{*}\right):r_{(\mathbf{T},\Theta )}\left(a,{\sigma }_{\mathbf{T}},{\ell }_{*}\right)=r$, for ${\ell }_{*}\in \{{\ell }_{mso}^{+},{\ell }_{mbo}^{+},{\ell }_{\gamma }^{+}\}$. As made clear in Figure 6, the radius sets constraints on the intersection between the (inner) boundary of the photon shell and the inversion radius $r_{\mathbf{T}}$.

The following limiting spin values are identified: $a_{(\gamma ,\mathbf{T},s)}\equiv 0.550895:r_{\gamma }^{-}=r_{\mathbf{T}}\left({\ell }_{mso}^{+}\right)$, $a_{(\gamma ,\mathbf{T},b)}\equiv 0.5625:r_{\gamma }^{-}=r_{\mathbf{T}}\left({\ell }_{mbo}^{+}\right)$, $a_{(\gamma ,\mathbf{T},\gamma )}\equiv 0.599433:r_{\gamma }^{-}=r_{\mathbf{T}}\left({\ell }_{\gamma }^{+}\right)$.

4. Shadow Boundary

In this section we relate points and regions discussed in Section 3 to the BH shadow boundary. The location of these points on the shadow boundary and the shadow boundary morphology vary with the BH spin and the view angle $\sigma$. Results of this analysis, discussed below, are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13.

At fixed spin a and angle $\sigma$, the BH shadow boundary, generated by the spherical orbits of the photon shell at a fixed viewing angle $\sigma$, appears as a closed curve in the $(\alpha ,\beta )$ plane, where the celestial coordinates for the null geodesics are defined as

$$\alpha =-{\displaystyle \frac{\ell }{\sqrt{\sigma }}},\beta =\pm \sqrt{q-{\displaystyle \frac{(1-\sigma )\left({\ell }^2-a^2\sigma \right)}{\sigma }}},$$

(16)

(see [23]). Each point of the shadow boundary corresponds to solutions of Equation (8), giving a spherical orbit with constant radius r, constant q and constant parameter ℓ.

In Figure 7, white points correspond to $\beta =0$, which is the intersection of the axis defined by the viewing angle $\sigma =$ constant with the boundary ${\sigma }_{\pm }$ of the photon shell. Hence, at $\sigma =1$, these correspond to the circular orbits $r_{\gamma }^{\pm }$. In general, these points fix the maximum extension (diameter) of the BH shadow boundary on the $\alpha$ axis.

Furthermore, the section of the outer ergoregion (the ergosurface corresponds to the orange points) is included in the photon shell, along with the angles where, instead, the photon shell is out of the ergoregion. The counter-rotating region on the shadow boundaries is clearly shown and bounded by the limiting radius $r_{lim}\left(a\right)$ (gray points) where $\alpha =0$.

Radius $r_{(\mathbf{T},\Theta )}$ (blue points) is obviously in the range of counter-rotating photon orbits $r>r_{lim}$. As expected, in this range, the orbits with impact parameters ${\ell }_{mso}^{+}$ (red points) and ${\ell }_{mbo}^{+}$ (purple points) are also located. The white right point corresponds to ${\ell }_{\gamma }^{+}$. The points in the external region of the photon shell correspond to larger values in magnitude of the impact parameter, as confirmed in Figure 8—left panel—where points on the shadow for impact parameter $\{{\ell }_{mso}^{+},{\ell }_{mbo}^{+},{\ell }_{\mathbf{T}}\}$ are shown for all values of the observational angle $\sigma \in [0,1]$ and for fixed BH spin a. (Each curve in the panel is for a fixed spin, and each point of a curve is for a different value of $\sigma$. Arrows indicate the increasing values of $\sigma$ along each curve.)

In Figure 8, the right panel shows the results for $r_{(\mathbf{T},\Theta )}$, in agreement with the results for ${\ell }_{\mathbf{T}}$ in Figure 8—left panel. The curve corresponding to ${\ell }_{mbo}^{+}$ is more external than the curve for ${\ell }_{mso}^{+}$. The static case ($a=0$)—black curve in the panel—is more internal to the $(\alpha ,\beta )$ plane. The greater the $\sigma$ is and the greater $\beta$ is, the smaller $\alpha$ is in magnitude. The situation is the opposite for the curves corresponding to the inversion impact parameter ${\ell }_{\mathbf{T}}$ located around $\alpha =0$ (corresponding to $r=r_{lim}$). There is no solution for the static case (at $a=0$). On the other hand, coordinate $\left|\alpha \right|$ decreases and $\left|\beta \right|$ increases with decreasing BH spin. When $\sigma$ increases, the coordinates $\left|\alpha \right|$ and $\left|\beta \right|$ increase.

Counter-rotating cusped tori have momentum $\ell \in [{\ell }_{mbo}^{+},{\ell }_{mso}^{+}]$ and an inner edge in $r\in [r_{mbo}^{+},r_{mso}^{+}]$. Within these ranges, the differences between the curves remain minimal, even at high spin values, and this behavior persists in general across all viewing angles $\sigma$.

These results are also confirmed in Figure 9, where there are limiting (bottom values of the) spins and view angles for the existence of the spherical orbits with impact parameter ${\ell }_{mbo}^{+}$ and ${\ell }_{mso}^{+}$—[20]. Solutions do not exist for all combinations of spin parameters and viewing angles. For small $\sigma$ and small a, there are no curves for $\ell ={\ell }_{mbo}^{+}$. In general, the separation between two curves corresponding to different ℓ values increases as the viewing angle decreases ($\beta$ increases with spin and angle).

The separation between the two curves at different angular momentum is reduced for viewing angles close to the equatorial plane, while it becomes more pronounced for angles with $\sigma >0.5$. The difference between the curves at different spins is larger for angles close to the equatorial plane; this provides different constraints for spin and viewing angle. It is also evident from Figure 9 (right panel) that there is a clear distinction between the ranges of values below and above $a=0.48$.

Figure 10 shows similar analysis for the outer ergosurface $r_{ϵ}^{+}$ (orange points of Figure 7). The curves of the outer ergosurface exist only for $a>0.71$ (where $r_{\gamma }^{-}=r_{ϵ}^{+}$). Increasing the spin increases $\left|\beta \right|$. The greater $\sigma$ is, the greater $\beta$ is in magnitude. Clearly, the portion of the photon shell (and the shadow boundary) included in the ergoregion increases with the spin (and angle)—Figure 5.

In Figure 11, we show the coordinates $\beta$ for the orbits with radius $r_{lim}$ and $r_{(\mathbf{T},\Theta )}$, as a function of the BH spin and viewing angles.

The celestial coordinate (on the shadow boundary) $\beta$ evaluated at $r_{(\mathbf{T},\Theta )}$ is consistently smaller than the corresponding value at $r_{lim}$ (but for $a=0$). For $r=r_{(\mathbf{T},\Theta )}$ and $r=r_{lim}$, coordinate $\beta$ decreases with the BH spin. The curves, for fixed spin, approach one other for small values $\sigma$ (near the rotational axis). Coordinate $\beta$, for $r=r_{(\mathbf{T},\Theta )}$, increases increasing $\sigma$. For all values of the spin parameter, the curves at $r=r_{(\mathbf{T},\Theta )}$ converge toward one another on the equatorial plane. Whereas $\beta$ for $r=r_{lim}$ decreases increasing $\sigma$. Consequently, the largest separation between the curves at different radii and for fixed spin, occurs on the equatorial plane and becomes most pronounced at high spin values.

In Figure 12 and Figure 13 there is the analysis of the solutions $\beta =0$, correspondent to the intersection points of the surfaces ${\sigma }_{\pm }$ with the axis defined by the viewing angle $\sigma$. This analysis constrains the diameter of the BH shadow boundary in the celestial plane. This quantity increases with spin and $\sigma$. The situation differs for $\alpha$ that is positive or negative. For $\alpha <0$ (where $\ell >0$), the trend is consistent for all values of the BH spin and view angle. For $\alpha >0$ (there is $\ell <0$), the trend is different for viewing angles $\sigma <0.1$. The corresponding parameter $\ell <0$ increases in magnitude with $\sigma$ and spin a, for counter-rotating photons, and decreases with the spin for co-rotating photons—Figure 13, left panel. Meanwhile, there are generally two (orbital) radii r for fixed $\sigma$ and spin, as shown in Figure 13, right panel. In general, the distance between the two orbits increases with the spin at large $\sigma$ (close to the equatorial plane). The situation is instead more complex for angles close to the axis ($\sigma \approx 0$).

5. Redshift Imaging

The frame-dragging effects also modulate the radiation emitted by the counter-rotating material in the vicinity of the Kerr BH. In this section, we focus on the redshift of signals produced by particles from counter-rotating accretion disks orbiting the central Kerr BH.

We introduce the redshift function $g_{red}$, defined as the energy (frequency) ratio, between the energy as measured by the observer at infinity (with $u^a=(1,0,0,0)$), and the emitter particle in a circular orbit, with $u^a=(u^t,0,0,u^{\varphi })$. The emitted signal is blueshifted if $g_{red}>1$, redshifted when $g_{red}<1$, and conserved for $g_{red}=1$.

The quantity $g_{red}$ can be expressed in terms of the relativistic angular velocity $\Omega$ of the emitter, which is defined as

$$\Omega ={\displaystyle \frac{u^{\varphi }}{u^t}}.$$

(17)

On the equatorial plane, for a counter-rotating particle with specific angular momentum $\ell ={\ell }^{+}$, the relativistic angular velocity of Equation (4) reads

$$\Omega ={\displaystyle \frac{1}{a-\frac{1}{{\left(\frac{1}{r}\right)}^{3/2}}}},$$

(18)

as shown in Figure 14.

As expected, $\Omega$ (on the equatorial plane) decreases in magnitude with the spin a and the distance from the central attractor (there is, at a fixed radius r, $\Omega \left({\ell }^{+}\right)\ne 0$ and ${\ell }_{\mathbf{T}}\left(r\right)\ne {\ell }^{+}\left(r\right)$).

The constant $\mathcal{L}$ of Equation (3), for counter-rotating particles, becomes

$${\mathcal{L}}^{+}\equiv -{\displaystyle \frac{a^2+2a\sqrt{r}+r^2}{r^{3/4}\sqrt{-2a+r^{3/2}-3\sqrt{r}}}},$$

(19)

as shown in Figure 15.

In this analysis, we also introduced the radius

$$r_{\chi }^{+}\equiv {\displaystyle \frac{r_{mbo}^{+}-r_{\gamma }^{+}}{2}}+r_{\gamma }^{+}$$

(20)

located in the orbital range $]r_{\gamma }^{+},r_{mbo}^{+}[$ (associated with quiescent counter-rotating disks’ inner edges and proto-jet cusps).11

The angular momentum ${\mathcal{L}}^{+}$ is negative and increases in magnitude with the spin (at the radii indicated in Figure 15). The momentum ${\mathcal{L}}^{+}$ increases in magnitude at fixed spin and as the radius r approaches the attractor.

The constant $\mathcal{E}$ of Equation (3), for the counter-rotating particles, is

$${\mathcal{E}}^{+}={\displaystyle \frac{-a+r^{3/2}-2\sqrt{r}}{r^{3/4}\sqrt{-2a+r^{3/2}-3\sqrt{r}}}},$$

(21)

and it is shown in Figure 16 (according to Equation (4), ${\ell }^{+}\equiv {\mathcal{L}}^{+}/{\mathcal{E}}^{+}$).

The particle orbital energy generally decreases as the distance from the attractor increases, and ${\mathcal{E}}^{+}=1$ at $r=r_{mbo}^{+}$. When evaluated across the counter-rotating orbital domain (defined by the radii $r_{mso}^{+}$ and $r_{mbo}^{+}$), the energy range of values12, associated with counter-rotating cusped disks, decreases as the BH spin increases (the energy at the marginally stable orbit grows with spin)13.

On the equatorial plane, therefore, the redshift function reads

$$g_{red}={\displaystyle \frac{\sqrt{\frac{r\left(-2a\sqrt{\frac{1}{r}}+r-3\right)}{{\left(r-a\sqrt{\frac{1}{r}}\right)}^2}}\left(r-a\sqrt{\frac{1}{r}}\right)}{\sqrt{\frac{1}{r}}(l_p-a)+r}},$$

(22)

where $l_p\equiv -k_{\varphi }/k_t$ is the photon impact parameter, and the four-momentum of emitted photons is $k^a=(k^t,k^r,k^{\theta },k^{\varphi })$. Function $g_{red}$ is shown in Figure 17. In general, the emission is redshifted for positive $l_p$. The signal emitted by counter-rotating particles, on the equatorial plane, is blueshifted for a small range of negative values of14 $l_p$. The redshift function generally decreases with the distance from the central attractor, and for positive $l_p$15. Our analysis also reveals a maximum of the redshift $g_{red}$, within the blueshifted region, as a function of the radius r (The redshift $g_{red}$ is generally larger for a small spin).

Redshift function $g_{red}$ in Equation (22) is well defined for impact parameter $l_p>l_m$, where

$$l_m\equiv a-\sqrt{r^3},$$

(23)

shown in Figure 18 and Figure 19. The photon impact parameter $l_m$ is negative, and it decreases in magnitude, increasing spin and distance from the attractor.

Hence, we also introduce the impact parameter $l_I$, as follows:

$$l_I\equiv -{\displaystyle \frac{\left(a\sqrt{\frac{1}{r}}-r\right)\left(\sqrt{\frac{r\left(-2a\sqrt{\frac{1}{r}}+r-3\right)}{{\left(r-a\sqrt{\frac{1}{r}}\right)}^2}}-1\right)}{\sqrt{\frac{1}{r}}}},\mathrm{where}{l}_I=l_p:g_{red}=1,$$

(24)

defining conserved energy emission for $r>r_{\gamma }^{+}$, shown in Figure 19. For impact parameters greater than $l_I$, the signal from counteracting matter is blueshifted. This analysis provides the conditions where the energy emitted from the counter-rotating structures is conserved (as measured by the observer at infinity). The $l_I$ curves increase in magnitude as the BH spin increases. The curves of impact parameter $l_I$, associated with the different radii in Figure 19 are very close, although the impact parameter is larger in magnitude for radii farther from the central attractor.

6. Final Remarks

We reviewed properties of the counter-rotating accretion disks and photons motion around a central Kerr BH, investigating their possible effects on the BH shadow profiles, relating these structures to the (outermost region, with respect to the central BH) photon shell, depending on the BH’s dimensionless spin, the (constant) specific angular momentum of the disk, the state of the disk, whether quiescent or cusped, and the observer inclination viewing angle.

For fixed BH spins and observational angles, constraints are given on regions of observations of the shadow boundary, distinguished by regions of counter-rotating matter and photons.

We focused in this analysis on the outer ergosurface and the counter-rotating photons and counter-rotating accretion, in particular focusing on the tri cusps $r_{\times }$ and counter-rotating proto-jet cusps. In the photon shell region we examined the imprint of the radii $r_{lim}$, where there is ${\ell }_{\Theta }=0$ and the inversion surface radius on the photon orbits $r_{(\mathbf{T},\Theta )}$, where $\dot{\varphi }=0$ on the shadow boundary.

Observable regions of the boundary can be identified, at any viewing angle and for any spin value, as an imprint of the BH domains examined in this analysis.

Therefore, the analysis connects segments of the shadow boundary to bound, unstable spherical photon orbits, which are subject to specific constraints on the orbital radius r, or on the impact parameter ℓ. The existence and precise location of these points on the BH shadow boundary vary with the spin parameter a and the viewing angle, thereby serving potentially as a tracer of the BH properties.

Therefore, the setup provides an immediate astrophysical context, constraining the counter-rotating orbiting toroids. There are no counter-rotating tori or proto-jets in the photon shell (hence, no photon shell “obscuration” by counter-rotating toroids). However, the counter-rotating accreting flows, freely falling from the cusps, have been considered, leaving the marginally stable circular orbit with $\ell ={\ell }_{mso}^{+}$, and, more in general, leaving the toroid accreting cusp $r_{\times }^{+}\in ]r_{mbo}^{+},r_{mso}^{+}]$, or $r_j^{+}\in ]r_{\gamma }^{+},r_{mbo}^{+}]$ for the proto-jets. Materials accreting onto the central BH have an inversion surface ($r_{\mathbf{T}}$), which is constrained in this analysis in relation to the BH photon shell. We proved that only a restricted class, $r_{(\mathbf{T},\Theta )}$, of $r_{\mathbf{T}}$ orbits are in the photon shell, and therefore can be observed, for the faster spinning BHs and the observational angles identified here. It is worth noting that the inversion surfaces lie in close proximity even for different values of the impact parameter—see Figure 1, Figure 2 and Figure 4. It should also be noted, however, that we expect such observational properties to depend strongly on the characteristic timescales of the underlying processes, specifically the time required for the flow to reach the inversion points. This fact has relevant consequences. In fact, this region may be filled with remarkably active components of the accreting flux of matter and photons (particularly near the BH poles and along the equatorial plane), as the region is ultimately characterized by enhanced luminosity and elevated flow temperatures. In Section 4, we discussed the conditions under which constraints on the spin, viewing angle, and rotation direction of cusped disks can be extracted from the shadow boundary regions. Moreover, in Section 5, we showed that the emergence of blue-shifted zones introduces additional restrictions, refining the characterization of cusped disk dynamics in the vicinity of the BH. In Section 5, we also studied the redshift properties of the radiation generated by particles from counter-rotating tori. We provided a detailed examination of how the emission is modified by the spacetime geometry (and the particle motion, by relativistic and gravitational Doppler effects). This highlights the redshift signatures arising from counter-rotating flows. The counter-rotating photon components at the shadow boundary reveal distinctive signatures that might be exploited to test theoretical models against observational data. Within the broader context of photon shell structure and shadow boundary formation, the redshift imaging can enable the identification of counter-rotating contributions within accretion flows, providing constraints on the spin parameter and the morphological characteristics of the counter-rotating disk. The results presented here connect the physics of photon shells, accretion dynamics, and shadow imaging, possibly establishing a framework for the modeling of the counter-rotating accreting matter and emission in the vicinity of Kerr BHs.