Spherical photon orbits in the field of Kerr naked singularities
Abstract
For the Kerr naked singularity (KNS) spacetimes, we study properties of spherical photon orbits (SPOs) confined to constant BoyerLindquist radius r. Some new features of the SPOs are found, having no counterparts in the Kerr black hole (KBH) spacetimes, especially stable orbits that could be pure prograde/retrograde, or with turning point in the azimuthal direction. At \(r>1\) (\(r<1\)) the covariant photon energy \(\mathcal{E}> 0\) (\(\mathcal{E}< 0\)), at \(r=1\) there is \(\mathcal{E}= 0\). All unstable orbits must have \(\mathcal{E}> 0\). It is shown that the polar SPOs can exist only in the spacetimes with dimensionless spin \(a < 1.7996\). Existence of closed SPOs with vanishing total change of the azimuth is demonstrated. Classification of the KNS and KBH spacetimes in dependence on their dimensionless spin a is proposed, considering the properties of the SPOs. For selected types of the KNS spacetimes, typical SPOs are constructed, including the closed paths. It is shown that the stable SPOs intersect the equatorial plane in a region of stable circular orbits of test particles, depending on the spin a. Relevance of this intersection for the Keplerian accretion discs is outlined and observational effects are estimated.
1 Introduction
Large effort has been devoted to studies of null geodesics in the gravitational field of compact objects, because the null geodesics govern motion of photons that carry information on the physical processes in strong gravity to distant observers, giving thus direct signatures of the extraordinary character of the spacetime around compact objects that influences both the physical processes and the photon motion. The most detailed studies were devoted to the Kerr geometry that is assumed to describe the spacetime of black holes governed by the Einstein gravity [6, 7, 9, 18, 19, 23, 61]. Extension to the photon motion in the most general black hole asymptotically flat spacetimes, governed by the Kerr–Newman geometry, has been treated, e.g., in [2, 26, 35, 36, 41, 62]. The case of spherical photon orbits in the Reissner–Nordström(de Sitter) spacetimes has been intensively studied in [63]. However, the recent cosmological tests indicate presence of dark energy, probably reflected by a relict cosmological constant, that could play a significant role in astrophysical processes [4, 24, 43, 48, 51, 56]. Therefore, the photon motion in the field of Kerrde Sitter black holes, where the cosmological horizon exists along with the static radius giving the limit on the free circular motion [42, 46], has been studied in [11, 17, 29, 34, 45, 47]; extension to spacetimes containing a charge parameter has been discussed in [12, 22, 44]. There is a large number of studies related to the photon motion in generalizations of the Einstein theory, e.g., for regular black hole spacetimes of the Einstein theory combined with nonlinear electrodynamics [38, 55], and for black holes in alternative approaches to gravity [1, 3, 5, 35, 54, 60].
A crucial role in the photon motion is played by the spherical photon orbits, i.e., photons moving along orbits of constant (Boyer–Lindquist) radial coordinate – their motion constants govern the local escape cones of photons related to any family of observers, and specially the shadow (silhouette) of black holes located in front of a radiating source [7], e.g. an orbiting accretion disk [13, 20, 30, 31, 45, 50]. The properties of the spherical photon orbits outside the outer horizon of Kerr black holes were studied in [58]. Spherical photon orbits in the field of Kerrde Sitter black holes were discussed in [17, 45]  in this case, photons with negative covariant energy could be relevant, in contrast to the case of pure Kerr black holes where only spherical photons with positive covariant energy enter the play [58].
Recently, growing interest in Kerr naked singularity spacetimes is demonstrated, mainly due to the possibility of existence of Kerr superspinars proposed in the framework of String theory by Hořava and his coworkers, with interior governed by the String theory and exterior described by the Kerr naked singularity geometry [25, 50, 52]. The presence of Kerr superspinars in active galactic nuclei or in microquasars could give clear signatures in the ultrarelativistic collisional processes [53], in the highfrequency quasiperiodic oscillations in Keplerian disks [27, 28, 59], or in the LenseThirring precession effects [16]. The instability of test fields in the Kerr naked singularity backgrounds has been studied in [14, 21], possible stabilizing effects were demonstrated for the Kerr superspinars in [33]. The classical instability of Kerr naked singularity (superspinar) spacetimes, converting them to black holes due to standard Keplerian accretion, has been shown to be slow enough in order to enable observation of primordial Kerr superspinars – at least at cosmological redshifts larger then \(z=2\) [49]. The Kerr naked singularity spacetimes could be applied also in description of the exterior of the superspinning quark stars with spin violating the black hole limit \(a=1\) [57]. The optical phenomena related to the Kerr naked singularity (superspinar) spacetimes were treated in [37, 40, 50, 52], and in more general case including the influence of the cosmological constant in [17, 45, 47].
Here we focus our attention to the properties of the spherical photon orbits in the Kerr naked singularity spacetimes, generalizing thus the study of spherical photon orbits in the Kerr black hole spacetimes [58]. For the spherical photon orbits we give the motion constants in dependence on their radius and dimensionless spin, and present detailed discussion of their latitudinal and azimuthal motion. We introduce detailed classification of the Kerr spacetimes according to the properties of the spherical photon orbits, including the stability of the spherical orbits and the role of the spherical photon orbits with negative energy relative to infinity, extending thus an introductory study in more general Kerrde Sitter spacetimes [17]. We also discuss astrophysically important interplay of the spherical photon orbits and the Keplerian accretion disks, with matter basically governed by the circular geodesic motion; we shortly discuss possible observational effects related to the irradiation of the Keplerian disks by the photons following the spherical orbits.
2 The Kerr spacetimes
3 Carter equations of geodesic motion
4 Spherical photon orbits
The SPOs have a crucial role in characterization of the Kerr spacetimes as they govern the shadows of Kerr black holes or Kerr superspinars, and could have influence on the behaviour of the Keplerian disks or more complex accretion structures due to effect of selfillumination [6, 41, 50].
4.1 Covariant energy of photons following spherical orbits
The photons (test particles) whose motion is governed by the Carter equations of motion can be in the classically allowed positiveroot states where they have positive energy as measured by local observers, and, equivalently, they evolve to future (\(\,\mathrm {d}t/\,\mathrm {d}\lambda > 0\)), or in the classically forbidden negativeroot states with negative energy measured by local observers, evolving into past (\(\,\mathrm {d}t/\,\mathrm {d}\lambda < 0\)) [32]. Above the outer horizon of a Kerr black hole, the situation is simple and all photons on the SPOs have positive covariant energy \(\mathcal{E}>0\) and are in the positiveroot states. However, the situation is more complex in the case of Kerr naked singularities.
We present the results of the determination of the covariant energy of the SPOs later, and then we use them in the classification of the Kerr spacetimes where this property is considered as one of the criteria of the classification.
4.2 Motion constants of spherical photon orbits
4.3 Existence of spherical orbits
4.4 Stability of spherical photon orbits
Stability of the spherical null geodesics against radial perturbations is governed by the condition \(d^2R/dr^2<0\) considered in the loci of the spherical orbits, i.e., by the extrema of the function \(q_{sph}(r;a)\). The function \(q_{sph}(r;a)\) has one local extreme \(q_{ex}=27\) located at \(r=3\), independently of the rotational parameter a. This extreme is a local maximum for \(0<a<3,\) while for \(a>3\) it becomes a minimum. The significance of this extreme, as follows from (17), is that the photon orbit at \(r=3\) intersects the equatorial plane perpendicularly, i.e., \(\dot{\phi }(r=3,\theta =\pi /2)=0\) (c.f. [58]).
4.5 Polar spherical photon orbits
4.6 Spherical photon orbits with negative energy
4.7 Spherical photon orbits with \(\mathcal{E}= 0\)
5 Trajectories of photons on the spherical null geodesics
In order to construct trajectories of the photons following the spherical null geodesics, we have to discuss in detail the latitudinal and azimuthal motion at the \(r=const\) surfaces. In the context of the spherical motion of photons the natural question arises, what is the range of the latitudinal coordinate in dependence on the allowed motion constants and the dimensionless spin of the Kerr spacetime. Simultaneously, the important question is on the possible existence and number of the turning points of the azimuthal motion.
5.1 Latitudinal motion
The latitudinal motion can be of the so called orbital type, where the photons oscillate between two latitudes \(\theta _0,\) \(\pi \theta _0\), crossing repeatedly the equatorial plane or even being confined to the equatorial plane^{6}, or of the so called vortical type, where the photons oscillate ’above’ or ’bellow’ the equatorial plane between two pairs of cones coaxial with the symmetry axis of the spacetime, with latitudes \(\theta _1, \theta _2\), \(\theta _1< \theta _2\) and \(\pi \theta _1,\pi  \theta _2\). The special case is the vortical motion along the symmetry axis \(\theta =0\), or the motion at any constant latitude – such photons are called PNC photons and have a generic role in the Kerr spacetimes [9]. Now one can ask, which of these types is possible for the spherical photon motion.
From the behaviour of the function \(q_{m}(m;a,\ell )\), which is shown in Fig. 4, it can be seen that the vortical motion exists for negative values of the parameter q. According to the relation (48), the negative values of the parameter q allowing the latitudinal motion exist for \(\ell \le a\), where the lowest value is \(q=a^2\) and occurs for \(\ell =0\). However, as can be verified by calculation, at radii where \(l_{sph}\le a\), there is \(q_{sph}>0\), which confirms that the vortical motion of constant radius is impossible for the spherical orbits.
For the KNS spacetimes with \(a<a_{pol(max)}\) (Fig. 5c), two maxima exist at \(m_{\theta (max)}=1\) that are given by the function \(a_{pol}(r)\), and one local minimum located at \(r=r_{ms+}\). If \(a=a_{pol(max)}\), the three extrema coalesce into maximum \(m_{\theta (max)}=1\) (Fig. 5d). In these KNS spacetimes thus polar SPOs can exist for properly chosen motion constants at the properly chosen radii.
For the KNS spacetimes with \(a>a_{pol(max)}\) (Fig. 5e, f), the function \(m_{\theta }(r;a)\) has a local maximum at \(m_{\theta (max)}<1\) at \(r=r_{ms+}\). In such KNS spacetimes the polar SPOs cannot exist.
5.2 Azimuthal motion
5.3 Shift of nodes
5.4 Periodic orbits
The nodal shift function becomes to be continuous for the KNS spacetimes with \(a>a_{pol(max)}\). Therefore, in KNS spacetimes with \(a>a_{pol(max)} \sim 1.18\), there exist “oscillatory” trajectories with \(\Delta \phi =0\), i. e., the photons in such trajectories are following a closed path with finite extent in azimuth, ending at the starting point. They are of octallike shape (see, e. g., Fig. 9 case \(k=0\)). However, a detailed calculation using numerical procedure reveals that the radius \(r_{\Delta \phi =0}\) of the zero nodal shift \(\Delta \phi =0\) first occurs for \(a=a_{\Delta \phi =0(min)}\equiv 1.179857\) at \(r=r_{\Delta \phi =0(max)}=1.71473\) (see Fig. 6d). In Fig. 7 we show that the function \(r_{\Delta \phi =0}(a)\) is descending, hence the subscript ’\(\Delta \phi =0(max)\)’. Notice that the value of \(a_{\Delta \phi =0(min)}\) is only very slightly lower than \(a_{pol(max)}\). Since the point \(r_{\Delta \phi =0(max)}\) and corresponding point of the \(4\pi \)discontinuity \(r_{pol+}(a_{\Delta \phi =0(min)})\) are infinitesimally separated, we can claim that the SPO at \(r_{pol+}(a_{\Delta \phi =0(min)})\) is a special case of polar oscillatory orbit. We illustrate its trajectory explicitly in Fig. 10 and for comparison we give illustration of the orbit at \(r=r_{pol}\) for \(a=a_{pol(max)}\).
In the Fig. 8, the nodal shift is shown for the polar spherical orbits \(r_{pol\pm }\), i. e., for the isolated points in Fig. 6, in dependence on the spin parameter a. As follows from the above, varying a, the first occurrence of the zero nodal shift appears when there is \(\Delta \phi [r_{pol+}(a)]=2\pi \). According to Figs. 6, 7, 8, we can summarize that for the very tiny interval \(a_{\Delta \phi =0(min)}\le a \le a_{pol(max)}\) there exist orbits with \(\Delta \phi <0,\) i. e., globally retrograde, with radii \(r_{\Delta \phi =0(max)}<r<r_{pol+}(a)\), followed by orbits with \(\Delta \phi >0\), i. e., globally prograde, at \(r_{pol+}(a)<r<r_{pol}(a)\), and again orbits globally retrograde with \(r>r_{pol}\). Otherwise, the SPOs of this class of the KNS spacetimes possesses no new essential features in comparison with the other cases, therefore, its character is explained sufficiently (Figs. 1, 10).
Further, we consider a possibility of orbits having a plurality of revolutions about the spacetime symmetry axis per one latitudinal oscillation (\(k>1\)), which are of a helixlike shape, and that of greater number of latitudinal oscillations per one revolution about the axis (\(k<1\)), both with, or without a turning point in the \(\phi \)direction. Note that the cases \(k\le 1\) cannot be realized, since from Eq. (55) it follows that the nodal shift \(\Delta \phi >2\pi \). This is demonstrated in Fig. 12 on the left, which depicts the shift of nodes at \(r\rightarrow r_{ph}\).
5.5 Spherical photon orbits with zero energy and their nodal shift
5.6 Summary of properties of spherical photon orbits
We can summarize properties of the SPOs in the following way. The covariant energy \(\mathcal{E}>0\) (\(\mathcal{E}<0\)) have the SPOs at \(r>1\) (\(r<1\)) for all KBHs and KNSs; there is SPO with \(\mathcal{E}=0\) at \(r=1\). The other properties depend on the dimensionless spin a.
In the KBH spacetimes, the stable SPOs are located under the inner event horizon at \(0<r<r_{ms}\), the unstable ones at \(r_{ms}<r<r_{ph0}\), all being corotating (prograde), none of these orbits can be polar. Above the outer event horizon only unstable SPOs spread in the interval \(r_{ph+}\le r\le r_{ph}\). They are purely corotating for \(r_{ph+}<r<r_{pol}\); with turning point in the azimuthal direction but globally counterrotating (retrograde) for \(r_{pol}<r<3\); and purely counterrotating at \(3<r<r_{ph}\). One polar orbit exist above the outer horizon of the KBH spacetimes.
Characteristics of the periodic spherical photon orbits in Fig. 9
k  a  r  \(\ell \)  q  \(\theta _\text {min}\)  \(\theta _{\phi }\)  \(\,\mathrm {sign\,}\mathcal{E}\)  \(\Delta t\) 

6  1.01  1.11  1.58  13.15  \(22.9^\circ \)  –  \(+1\)  74.76 
2  1.1  1.72  0.62  16.42  \(8.5^\circ \)  –  \(+1\)  30.13 
1  1.1  1.22  \(0.16\)  30.32  \(1.6^\circ \)  \(6.7^\circ \)  \(+1\)  32.98 
1  1.1  0.77  3.2  6.85  \(49.8^\circ \)  –  \(1\)  14.08 
2 / 3  1.18  1.24  \(1.47\)  41.44  \(12.5^\circ \)  \(21.2^\circ \)  \(+1\)  26.36 
1 / 2  1.1  0.53  2.25  0.92  \(64.6^\circ \)  –  \(1\)  5.69 
0  \(\sqrt{3}\)  1.69  \(3.63\)  30.80  \(32.3^\circ \)  \(47.1^\circ \)  \(+1\)  19.36 
\(1/2\)  1.1  2.52  \(0.72\)  24.38  \(8.2^\circ \)  \(32.2^\circ \)  \(+1\)  29.45 
\(2/3\)  \(\sqrt{3}\)  4.34  \(7.31\)  8.70  \(67.5^\circ \)  –  \(+1\)  34.30 
There exists extremely small interval of the spin parameter \(a_{\Delta \phi =0(min)}<a<a_{pol(max)}\), for which the KNS spacetimes possess two polar orbits and one orbit of zero nodal shift \(r_{\Delta \phi =0}(a)\) – for \(1<r<r_{\Delta \phi =0}(a)\) there are orbits with turning point in the azimuthal motion, being corotating in global; such orbits occur also for \(r_{\Delta \phi =0}<r<r_{pol+}\), being globally counterrotating; at the radii \(r_{pol+}<r\), the same behaviour occurs as in the previous case.
6 Classification of the Kerr spacetimes due to properties of the spherical photon orbits
 Class I: Kerr black holes \(0<a<1\) (see Fig. 15) endowed with the SPOs of two families. First family is limited by radii \(0<r<r_{ph0}<r_{},\) where$$\begin{aligned} r_{ph0}\equiv 4\sin ^2 \left[ \frac{1}{6}\arccos (12a^2)\right] \end{aligned}$$(63)
is the radius of corotating equatorial circular photon orbit located under the inner black hole horizon \(r_{}\) (green dot). These are the orbits with negative energy \(\mathcal{E}<0\); at \(0<r<r_{ms}\) they are stable (rich green area), for \(r_{ms}<r<r_{}\) they are unstable with respect to radial perturbations (light green area). The radius \(r_{ms}\), given by (32), denotes marginally stable spherical orbit with negative energy. All the first family orbits are prograde and span small extent in latitude in vicinity of the equatorial plane – its maximum at \(r=r_{ms}\) is approaching the value \(\theta _{min(z\mathcal{E})}(a=1)=\arccos \sqrt{2\sqrt{3}3}=47.1^\circ \) as \(r\rightarrow 1\) when \(a\rightarrow 1\). Here and in the following, we restrict our discussion on the ’northern’ hemisphere (\(0\le \theta \le \pi /2\)), the situation in the ’southern’ hemisphere is symmetric with respect to the equatorial plane. Second family of the SPOs spreads between the inner corotating (red dot) and outer counterrotating (blue dot) equatorial circular orbits with radii given by (27). There is no turning point of the azimuthal motion for orbits with radii \(r_{ph+}<r<r_{pol}\), where \(r_{pol}\) is the radius of the polar spherical orbit (purple ellipse) given by (37), and all such photons are prograde. At radii \(r_{pol}<r<3\) (black dotted incomplete ellipse), there exist orbits with one turning point of the azimuthal motion in each hemisphere, such that the photons become retrograde as they approach the symmetry axis. At radii \(3<r<r_{ph}\), all spherical orbits are occupied by retrograde photons with no turning point of the azimuthal motion.
Characteristics of the spherical photon orbits with zero energy in Fig. 13
a  \(\Phi ^2/Q\)  \(\Delta \phi /2\pi \)  \(\theta _\text {min}\) 

1.0001  0.0002  69.7  \(0.8^\circ \) 
1.001  0.002  21.4  \(2.6^\circ \) 
1.02  0.042  4  \(11.5^\circ \) 
1.06  0.125  2  \(19.5^\circ \) 
1.1  0.21  1.4  \(24.6^\circ \) 
1.15  0.33  1  \(30.0^\circ \) 
1.34  0.8  0.5  \(41.8^\circ \) 
3  8  0.06  \(70.5^\circ \) 

Class II: Extreme KBH with \(a=1\) (Fig. 16). A family of stable prograde spherical orbits with negative energy occurs at radii \(0<r<1\) near equatorial plane,^{8} where also nonspherical bound photon orbits with two turning points of the radial motion exist. Such orbits are not present in any kind of the most general case of the KerrNewmann(anti) de Sitter black hole spacetimes. At radii \(1< r < r_{ph}=4\), there are orbits with positive energy and properties similar to the second family orbits in the previous KBH case. The special case of \(r=1\) apparently corresponds to the SPO with zero energy, which is marginally stable and prograde. This zero energy orbit has minimum latitude \(\theta _{min(z\mathcal{E})}(a=1)=47.1^\circ \); note that for \(a>1\), there is \(\theta _{min(z\mathcal{E})}(a)=\arccos {(1/a)}\). The SPOs at radii \(0<r\le 1\) have the same properties as those that occur in all KNS spacetimes, and we shall not repeat them in the following cases. Similarly, for all KNS spacetimes the orbits at \(r>1\) have positive energy.

Class III: KNS spacetimes with \(1<a<a_{\Delta \phi =0(min)}=1.17986\) (see Fig. 17). Two polar SPOs appear at radii \(r_{pol+}, r_{pol}\) (inner and outer purple ellipse, respectively) given by Eq. (37). Photons at \(1<r<r_{pol+}\) are stable, they have \(\mathcal{E}>0\) and one turning point of the azimuthal motion in each hemisphere. The latitudinal coordinate where the SPO has the azimuthal turning point is given by Eq. (54), the minimum allowed latitude reads \(\theta _\text {min}=\arccos \sqrt{m_{\theta }(r;a)}\). At the region \(\theta _\text {min}\le \theta <\theta _{\phi }\), the photon motion is in negative \(\phi \)direction (deep blue area), while for \(\theta _{\phi }\le \theta \le \pi /2\) it is in positive \(\phi \)direction (deep red area). The break point dividing the globally prograde orbits from the globally retrograde ones is at \(r=r_{pol+}\). The motion constants \(\ell _{sph}\rightarrow \infty \), \(q_{sph} \rightarrow \infty \), as \(r\rightarrow 1\) from the right, and \(\ell _{sph}=0\), \(q_{sph}=27\) at \(r=r_{pol+}\). The orbits at the radii \(r_{pol+}<r<r_{pol}\) are prograde, with no change in the azimuthal direction, for \(r_{pol+}<r<r_{ms+}\) being stable (deep red area), for \(r_{ms+}\le r\le r_{pol}\) being unstable (light red area). The function \(m_{\theta }(r;a)\) has a local minimum at \(r=r_{ms+}\), hence, the marginally stable SPO is of the least extent in the latitude (black dashed curve), contrary to the case of the marginally stable SPOs with negative energy at \(r_{ms}\) at the KBH spacetimes, where they have the widest extent (see detail in Fig. 15). The motion constant \(q_{sph}\) of photons on this orbit corresponds to the local minimum \(0<q_{sph(min)}<27\) of the function defined in (22), and the local maximum \(0<\ell _{sph(max)}\) of the function defined by (23). The turning point of the azimuthal motion appears for the SPOs at the radii \(r_{pol}<r<3\), and such SPOs appear to be retrograde as whole. The motion constants for \(r=r_{pol}\) are \(q_{sph(min)}<q_{sph}<27\), and \(\ell _{sph}=0\). At \(r=3\), there is \(q_{sph}=27\), corresponding to the local maximum of (22), i.e., to the photons crossing the equatorial plane with zero velocity component in the \(\phi \)direction ([58]), and \(\ell _{sph}<0\). The SPOs in the region \(3<r<r_{ph}\) are purely retrograde with \(\ell _{sph}<0\) and \(q_{sph}\rightarrow 0\) as \(r \rightarrow r_{ph}\).

Class IV: KNS spacetimes with \(a_{\Delta \phi =0(min)}\le a<a_{pol(max)}=1.17996\). In the limit case \(a=a_{\Delta \phi =0(min)}\), there exist radius \(r=r_{\Delta \phi =0(max)}=1.7147\) of spherical orbit with zero nodal shift \(\Delta \phi =0\), infinitesimally distant from the \(4\pi \)discontinuity point \(r_{pol+}\) (see Fig. 6d). The radius \(r_{pol+}\) therefore corresponds to special case of polar oscillatory orbit. For \(a_{\Delta \phi =0(min)}<a<a_{pol(max)}\), the radius which separates the globally prograde SPOs from the globally retrograde ones is at \(r=r_{\Delta \phi =0}\lessapprox r_{\Delta \phi =0(max)}<r_{pol+}\) (see the detail of Fig. 14). For \(r_{\Delta \phi =0}<r<r_{pol+}\) there appear globally retrograde orbits. The other properties remain the same as in previous case and the SPOs structure is represented by the Fig. 17.

Class V: KNS spacetimes with \(1.17996=a_{pol(max)}\le a<3.\) For \(a=a_{pol(max)}\), the two polar SPOs coalesce at \(r=r_{pol}=\sqrt{3}\) (see Fig. 18 above).

Class VI: Kerr naked singularity spacetimes with \(a\ge 3.\) For \(a=3\) the local extrema of the function \(q_{sph}(r;a)\) coalesce at the inflex point at \(r=3=r_{ms+}\) with \(q_{inf}=27\), where it becomes local minimum \(q_{sph,min}=27\) for \(a>3\). The local maximum is then at \(r_{ms+}>3\) (see Fig. 2), which is also locus of the local maximum of latitudinal turning function \(m_{\theta }(r;a)\) (Fig. 5f) at the radius of the marginally stable spherical orbit. The orbits at \(3<r<r_{ms+}\) are stable and purely retrograde; in the range \(r_{ms+}<r<r_{ph}\), there are unstable retrograde orbits (see Fig. 19).
Characteristics of the spherical photon orbits in Fig. 20
r  \(\ell \)  q  \(\Delta _{\phi }\)  \(\theta _\text {min}\)  \(\theta _{\phi }\)  \(\,\mathrm {sign\,}\mathcal{E}\)  Type 

0.9  5.4  52.5  \(469^\circ \)  \(36.7^\circ \)  –  −1  Stable 
1.15  \(0.9\)  50.5  \(426^\circ \)  \(7.4^\circ \)  \(14.1^\circ \)  \(+\)1  Stable 
\(r_{pol+}=1.24\)  0  27  \(728^\circ \)  \(0^\circ \)  \(0^\circ \)  \(+\)1  Stable,in.polar 
1.4  0.5  17.8  \(912^\circ \)  \(6.9^\circ \)  –  +1  Stable 
\(r_{ms}=1.54\)  \(0.66=\ell _\text {max}\)  \(16=q_\text {min}\)  \(781^\circ \)  \(9.1^\circ \)  –  \(+\)1  Marg.stable 
2  0.33  18.8  \(630^\circ \)  \(4.3^\circ \)  –  \(+\)1  Unstable 
\(r_{pol}=2.2\)  0  21  \(227^\circ \)  \(0^\circ \)  \(0^\circ \)  \(+1\)  Unst.,out.polar 
2.6  \(0.9\)  25  \(189^\circ \)  \(10.4^\circ \)  \(38.0^\circ \)  \(+1\)  Unstable 
3  \(2.2=2a\)  \(27=q_\text {max}\)  \(223^\circ \)  \(22.6^\circ \)  \(90^\circ \)  \(+1\)  Unstable 
3.4  \(3.8\)  24  \(246^\circ \)  \(37.0^\circ \)  –  \(+1\)  Unstable 
7 Spherical photon orbits related to the Keplerian disks and possible observational consequences
It is well known that in the KBH spacetimes the SPOs define the light escape cones in the position of an emission, i.e., they represent a boundary between the photons captured by the black hole and the photons escaping to infinity. In case of the KNS spacetimes, there is in addition a possibility of existence of trapped photons, which remain imprisoned in the vicinity of the ring singularity [39, 50]. Such a “trapping” region spreads in the neighbourhood of the stable SPOs, where small radial perturbations cause that the photons oscillate in radial direction between some pericentre and apocentre, but remain trapped in the gravitational field – efficiency of the trapping process was for the Kerr superspinars (naked singularity spacetimes) studied in detail [50] where also possible selfirradiation of accreting matter was briefly discussed.^{9} Note that the selfirradiation of the disk is possible also due to the unstable SPOs, but they could be send away from the sphere (to infinity) due to any small perturbative influence – for this reason we focus our attention on the influence of the stable spherical photons on the Keplerian disk in a special and observationally interesting case of the oscillatory photon orbits that return to a fixed azimuthal position, and the selfirradiation thus occurs repeatedly at the fixed position relative to distant observers.
7.1 Connection of the spherical photon orbits and the stable circular geodesics
We can expect observationally significant optical and astrophysical effects, particularly in the region of negative energy orbits connected with an interplay between the possible extraction of the rotational energy of the KNS (Kerr superspinar) and a subsequent trapping of the radiated energy, both for the radiated electromagnetic and gravitational waves. Both forms could substantially violate the structure and stability of the accretion disk, generating thus an observationally relevant feedback, which we presume to be the subject of our further study. Of course, similar effects can be expected in the whole overlap region of the stable circular orbits of matter and the stable SPOs, where repeated reabsorption of the radiated heat, and selfillumination and selfreflection phenomena take place, as indicated in the preliminary study of the Kerr superspinars in [50]. The illustration of the overlap region of the stable circular orbits with the stable SPOs, and the other relevant orbits in dependence on the Kerr spacetime rotation parameter a is given in Fig. 21.
7.2 Time periods of the latitudinal oscillations and time sequences related to the oscillatory orbits
In order to have a clear observational signature of the effects of the SPOs, we have to calculate dependence of the time period of the nodal motion, i.e., the time interval measured by a distant static observer between the two subsequent crossing of the equatorial plane by the photons following the spherical orbits. We first give a general formula and then we discuss the case of the closed spherical orbits finishing their azimuthal motion at the starting point in the equatorial plane. Such orbits could be observationally relevant, as the time interval of the nodal motion could give exact information on the Kerr naked singularity (black hole) spin, if its mass is determined by an independent method.
Now we have to distinguish the case of the SPOs demonstrating \(\Delta \phi = 0\) with the turning point of the azimuthal motion, and the analogous cases of the periodic SPOs where \(\Delta \phi = k2\pi \), with k being an integer, giving also the return to the fixed azimuthal coordinate in the equatorial plane where the Keplerian disk is located.
Now we can consider also the SPOs at \(r<1\) (with \(\mathcal{E}<0\)), in the KNS spacetimes with spin \(1<a<a_{pol(max)}\), since we are not limited by the necessity of having the azimuthal turning point. Of course, for the KNS spacetimes allowing for existence of closed spherical orbits of all three types (or two of them), we can use the time sequences (the ratio of the time delays) corresponding to the relevant types of the closed orbits to obtain relevant restrictions on the allowed values of the dimensionless spin a independently of their mass parameter. For comparison, we also compute the time intervals for the closed spherical orbits in Fig. 9 in associated Table 1.
To estimate the astrophysical relevance of the time delay effect, let us present the time interval for the circular photon orbit in case of the Schwarzschild black hole: in the limit \(a\rightarrow 0\) the formula (69) for \(r=3\) gives the result \(\Delta t_{Schwarz}=32.6484\), in accordance with the exact solution \(\Delta t_{Schwarz}=6\sqrt{3}\pi \) obtained by integration from the Schwarzschild line element after inserting \(\,\mathrm {d}s=\,\mathrm {d}r=\,\mathrm {d}\theta =0\) and \(\theta =\pi /2\). The nodal time delay is governed by half of the quantity: \(\Delta t_{nod(Schwarz)}=16.3242\).
8 Concluding remarks
We have shown that in the field of the KNS spacetimes there exist a variety of the SPOs, which are not present in the KBH spacetimes. Existence of spherical photon orbits stable relative to the radial perturbations has been demonstrated. The photons at spherical orbits located at \(r>1\) are of standard kind, having covariant energy \(\mathcal{E}>0\), as outside the black hole horizon, but at \(r<1\) they must have \(\mathcal{E}<0\); for the special position at \(r=1\), the photons at spherical orbits have \(\mathcal{E}=0\).^{11}
The character of the spherical motion in the field of Kerr naked singularities is more complex in comparison with those of spherical orbits above the outer horizon of Kerr black holes – along with the orbits purely corotating or counterrotating relative to distant observers, also orbits changing orientation of the azimuthal motion occur in the field of Kerr naked singularities. Existence of polar spherical orbits reaching the symmetry axis of the Kerr spacetime is limited to the naked singularity spacetimes with spin \(a<a_{pol(max)}=1.17996\). On the other hand, the KNS spacetimes with \(a>a_{pol(max)}=1.17996\) allow for existence of oscillatory orbits with azimuthal turning point that return in the equatorial plane to the fixed original azimuthal coordinate as related to distant observers.
From the astrophysical point of view it seems that the most interesting and relevant is the existence of the closed spherical photon orbits with zero total change in azimuth, which intersect themselves in the equatorial plane, where the stable circular orbits of massive particles can take place. This effect is potentially of high astrophysical relevance as it enables a relatively precise estimation of the dimensionless spin of the KNS spacetimes allowing their existence, if their mass parameter is known due to other phenomena. This is the case of the KNS spacetimes with the spin parameter \(a\in (a_{pol(max)}=1.7996, 2.47812)\) (see Fig. 21). These phenomena could be extended for the case of the closed, periodic orbits demonstrating the azimuthal angle changes \(\Delta \phi =2k\pi \) with integer k, when the orbits are again closed at fixed azimuth as related to distant observers.
In the case of the effects related to a fixed azimuth as related to distant observers it is in principle possible to admit some event (e.g. a collision) in the Keplerian accretion disc causing a release of energy in the form of electromagnetic radiation that would be partly and repeatedly returned back to the same radius as the original event, and at the same azimuth as observed by a distant observer, initializing repetition of the effects under slightly modified internal conditions. It can be expected that such periodic light echo could be characteristic for particular KNS spacetime. Detailed study of the related phenomena will be the task of our future work.
Footnotes
 1.
In case of more complex spacetimes with nonzero cosmological constant and possibly nonzero electric charge of the gravitating source it is more convenient to introduce the modified impact parameter \(X=\ell a,\) which simplifies the radial equation of motion 15 and its discussion, as shown in [17], where the more general case of the Kerrde Sitter spacetimes was considered. However, its introduction is unnecessary in this paper.
 2.
The orbital motion is allowed also for \(q=0\) and \(l^2<a^2\) when the motion in the equatorial plane is unstable [9].
 3.
 4.
The same applies to a more general case of the Kerr–de Sitter spacetimes, see [17].
 5.
Here we restrict our discussion on the stationary region \(\Delta >0.\)
 6.
This is the case of the equatorial circular orbits characterized by the value \(q=0\) that can be regarded as a special case of the spherical orbits.
 7.
The more general case of the function M(m) with nonzero cosmological parameter was studied in detail in [17].
 8.
From the perspective of the locally nonrotating observer such photons appear to be retrograde, see [17].
 9.
Recall that even general selfirradiation (occulation) of an accretion disk orbiting a black hole could have significant influence on the optical phenomena related to the accretion disks, as demonstrated for the first time in [6].
 10.
We could slightly extend the range of the considered spin of the KNS spacetime, down to the value of \(a_{\Delta \phi =0(min)}=1.79857\) when the SPOs with \(\Delta \phi = 0\) start to appear; however, in the case of KNS spacetimes with \(a\in (a_{\Delta \phi =0(min)},a_{pol(max)})\) there is an extended region of spherical orbits allowing for \(\Delta \phi \sim 2\pi \) that invalidates applicability of the orbits with \(\Delta \phi = 0\).
 11.
The photons at the spherical orbits located under the inner black hole horizon have also \(\mathcal{E}<0\).
Notes
Acknowledgements
Daniel Charbulák acknowledge the Czech Science Foundation Grant No. 1603564Y.
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