Photon motion in Kerr–de Sitter spacetimes
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Abstract
We study the general motion of photons in the Kerr–de Sitter blackhole and naked singularity spacetimes. The motion is governed by the impact parameters X, related to the axial symmetry of the spacetime, and q, related to its hidden symmetry. Appropriate ‘effective potentials’ governing the latitudinal and radial motion are introduced and their behavior is examined by the ‘Chinese boxes’ technique giving regions allowed for the motion in terms of the impact parameters. Restrictions on the impact parameters X and q are established in dependence on the spacetime parameters \(M, \Lambda , a\). The motion can be of orbital type (crossing the equatorial plane, \(q>0\)) and vortical type (tied above or below the equatorial plane, \(q<0\)). It is shown that for negative values of q, the reality conditions imposed on the latitudinal motion yield stronger constraints on the parameter X than that following from the reality condition of the radial motion, excluding the existence of vortical motion of constant radius. The properties of the spherical photon orbits of the orbital type are determined and used along with the properties of the effective potentials as criteria of classification of the KdS spacetimes according to the properties of the motion of the photon.
1 Introduction
In the framework of inflationary paradigm [45], recent cosmological observations indicate that a very small relict vacuum energy (equivalently, repulsive cosmological constant \(\Lambda > 0\)), or, generally, a dark energy demonstrating repulsive gravitational effect, has to be introduced to explain dynamics of the recent Universe [3, 7, 16, 40, 41, 49, 92]. These conclusions are supported strongly by the observations of distant Iatype supernova explosions indicating that starting at the cosmological redshift \(z \approx 1\) expansion of the Universe is accelerated [55]. The total energy density of the Universe is very close to the critical energy density \(\rho _{\mathrm{crit}}\) corresponding to almost flat Universe predicted by the inflationary scenario [62], and the dark energy represents about \(70\%\) of the energy content of the observable Universe [15, 63]. These conclusions have been confirmed by recent measurements of cosmic microwave background anisotropies by the space satellite observatory PLANCK [1, 52].
The dark energy equation of state is very close to those corresponding to the vacuum energy [15]. Therefore, it is relevant to study the astrophysical consequences of the effect of the observed cosmological constant implied by the cosmological tests to be \(\Lambda \approx 1.3\times 10^{56}\,\mathrm{cm^{2}}\), and the related vacuum energy \(\rho _{\mathrm{vac}} \sim 10^{29}\,\mathrm{g\,cm^{3}},\) close to the critical density of the Universe. The repulsive cosmological constant changes significantly the asymptotic structure of blackhole, naked singularity, or any compactbody backgrounds as such backgrounds become asymptotically de Sitter spacetimes, and an event horizon (cosmological horizon) always exists, behind which the geometry is dynamic.
A substantial influence of the repulsive cosmological constant has been demonstrated for astrophysical situations related to active galactic nuclei and their central supermassive black holes [69]. The blackhole spacetimes with the \(\Lambda \) term are described in the spherically symmetric case by the vacuum Schwarzschild–de Sitter (SdS) geometry [34, 72], while the internal, uniform density SdS spacetimes are given in [13, 68]. The axially symmetric, rotating black holes are determined by the Kerr–de Sitter (KdS) geometry [17, 25, 44].
In spacetimes with a repulsive cosmological term, the motion of photons was extensively investigated in many papers [8, 38, 39, 43, 47, 57, 58, 71, 73, 91]. The motion of massive test particles was studied in [2, 19, 20, 29, 31, 32, 35, 36, 37, 48, 51, 59, 66, 67, 72, 74, 86]. The KdS geometry can be relevant also for the socalled Kerr superspinars representing an alternative to black holes [14, 26, 27, 83], breaking the blackhole bound on the dimensionless spin and exhibiting a variety of unusual physical phenomena [21, 22, 30, 64, 76, 80, 83, 84]. It is worth to note that the SdS and KdS spacetimes are equivalent to some solutions of the f(R) gravity representing black holes and naked singularities [50, 88].
The role of the cosmological constant can be significant for both the geometrically thin Keplerian accretion discs [46, 61, 69, 72, 86] and the toroidal accretion discs [6, 42, 53, 54, 60, 87, 89] orbiting supermassive black holes (Kerr superspinars) in the central parts of giant galaxies. Both highfrequency quasiperiodic oscillations and jets originating at the accretion discs can be reflected by current carrying string loops in SdS and KdS spacetimes [28, 33, 77, 78, 93]. In the spherically symmetric spacetimes, the Keplerian and toroidal disc structures can be precisely described by a pseudoNewtonian potential of Paczynski type [79, 88], which appears to be useful also in studies of the motion of interacting galaxies [56, 81, 82] demonstrating relation of the gravitationally bound galactic systems to the socalled static radius of the SdS or KdS spacetimes [4, 5, 23, 24, 66, 67]. This idea has been confirmed by the recent study of general relativistic static polytropic spheres in spacetimes with the repulsive cosmological constant [75, 85].
The present paper is devoted to a detailed study of the properties of the motion of the photon in the KdS blackhole and naked singularity spacetimes. We concentrate attention to the behavior of the effective potentials determining the regions allowed for the motion of the photon. Such a study is necessary for a full understanding of the optical phenomena occurring in the blackhole or naked singularity spacetimes with a repulsive cosmological constant. We generalize the previous work concentrated on the properties of the motion of the photon in the equatorial plane [73], discussing the properties of the effective potential of the latitudinal motion in terms of the constant of the motion related to the equatorial plane, and then continuing by study of the effective potential of the radial motion. We concentrate our study on the spherical photon orbits representing a natural generalization of the photon circular geodesics that enables a natural classification of the KdS spacetimes according to the properties of the null geodesics representing the motion of the photon.
2 Kerr–de Sitter spacetime and Carter’s equations of geodesic motion
2.1 Kerr–de Sitter geometry
The physical singularity is located, as in the Kerr geometry, at the ring \(r=0,\) \(\theta =\pi /2.\)
Three event horizons, two black hole \(r_,\) \(r_+,\) and the cosmological horizon \(r_\mathrm{c},\) (\(r_< r_+ < r_\mathrm{c} \)) exist for \(y_{{\text {min}}(h)}(a^2)< y < y_{{\text {max}}(h)}(a^2),\) where the limits \(y_{{\text {min}}/{\text {max}}(h)}(a^2)\) correspond to a local minimum or local maximum of the function \(y_{h}(r;\,a^2),\) respectively, for given rotational parameter a. For \(0<y<y_{{\text {min}}(h)}(a^2)\) or \(y>y_{{\text {max}}(h)}(a^2)\) Kerr–de Sitter naked singularity spacetimes exist. The limit case \(y=y_{{\text {min}}(h)}(a^2)\) corresponds to an extreme blackhole spacetime, when the two blackhole horizons coalesce. If \(y=y_{{\text {max}}(h)}(a^2),\) the outer blackhole and cosmological horizon merge. There exists a critical value of the rotational parameter \(a^2_{\text {crit}}=1.212 02,\) for which the two local extrema of the function \(y_{h}(r;\,a^2)\) coalesce in an inflection point at \(r_{\text {crit}}=1.616 03\) with the critical value \(y_{\text {crit}}=0.0592.\) Thus, for \(a^2>a^2_{\text {crit}}\) only the Kerr–de Sitter naked singularity can exist for any \(y>0.\)
Properties of the event horizons for the more general case of the Kerr–Newman–de Sitter spacetimes can be found in [73].
2.2 Carter’s equations of geodesic motion
A detailed discussion of the equatorial motion of photons in the Kerr–Newman–de Sitter spacetimes has been published in [73]. Circular motion of test particles in the Kerr–de Sitter spacetimes has been presented in [86]. Here we restrict our attention on the general motion of photons in Kerr–de Sitter spacetimes.
3 Latitudinal motion
In the following analysis the relevant range of variable m is, of course, \(0\le m\le 1,\) but somewhere, in order to better understand the behavior of the characteristic functions, we formally permit \(m\in R.\)
The point \(m_\mathrm{d}\) given by the definition (25) determines the loci where the functions \(q^\theta _r(m;\,y,\,a^2),\) \(X_{(\pm )}(m;\,y,\,a)\) and \(X^\theta _(m;\,q,\,y,\,a)\) diverge; it occurs at relevant interval (0; 1) for \(y>1/a^2\) and \(m_\mathrm{d}\rightarrow 1\) for \(a^2y\rightarrow \infty .\) In such case, \(q^\theta _r(m;\,y,\,a^2)\rightarrow +\infty \;(\infty )\) for \(m\rightarrow m_\mathrm{d}\) from the left (right).
 1.Case \(y<1/a^2\)
 \(q<a^2\)

the definition range of the potentials is an empty set; the latitudinal motion is not possible;

 \(q=a^2\)
 the potentials \(X^\theta _\pm (m;\,q,\,y,\,a)\) are defined only for \(m=1,\) wherephotons with such values of parameters are the special case of the so called PNC photons ‘radially’ moving along the spin axis [10];$$\begin{aligned} X^\theta _+(1;\,q,\,y,\,a)=X^\theta _(1;\,q,\,y,\,a)=a; \end{aligned}$$

 \(a^2<q<0\) (Fig. 2a, b)
 the two potentials are defined for \(m\in \langle m_l;1\rangle ,\) where the lower limitis the solution of the equation \(q=q^\theta _r(m;\,y,\,a)\) (see Fig. 1b); the limits of the interval are the common points of the potentials, where$$\begin{aligned} m_l = \frac{q(a^2y1)}{a^2(qy+1)}>0 \end{aligned}$$(46)$$\begin{aligned} X^\theta _\pm (m=m_l;\,q,\,y,\,a)=X^\theta _{(\pm )}(m_l)=\frac{a(1+qy)}{a^2y1}<0; \end{aligned}$$(47)


 the latitudinal motion is allowed for values of the parameter X between some local minimum \(X^\theta _{{\text {min}}()}=X^\theta _(m_{{\text {min}}()};\,q,\,y,\,a)\) and maximum \(X^\theta _{{\text {max}}(+)}=X^\theta _+(m_{{\text {max}}(+)};\,q,\,y,\,a),\) for whichthe loci \(m_{{\text {min}}()}\) of minimum \(X^\theta _{{\text {min}}()}\) is given by the Eq. (39), the loci \(m_{{\text {max}}(+)}\) of maximum \(X^\theta _{{\text {max}}(+)}\) is determined by Eq. (37);$$\begin{aligned} X^\theta _{{\text {min}}()}<a<X^\theta _{{\text {max}}(+)}<0; \end{aligned}$$

if X takes one of these extremal values, then the trajectory of such photon lies entirely on cones \(\theta =\arccos \sqrt{m_\mathrm{{ex}}},\) \(\theta =\pi \arccos \sqrt{m_\mathrm{{ex}}},\) where \(m_\mathrm{{ex}}\in \{m_{min()},m_{max(+)}\};\) such photons are called PNC photons [10];
 for \(X^\theta _{{\text {min}}()}<X<X^\theta _{{\text {max}}(+)}\) there are two solutions \(m_1<m_2\) of each of the two equations \(X=X^\theta _\pm (m;\,q,\,y,\,a),\) implying that the photon executes a socalled vortical motion, which is restricted between two pairs of cones, symmetrically placed relative to equatorial plane:and$$\begin{aligned} 0< \arccos \sqrt{ m_2} \le \theta \le \arccos \sqrt{m_1} < \frac{\pi }{2} \end{aligned}$$$$\begin{aligned} \frac{\pi }{2}< \pi  \arccos \sqrt{m_1} \le \theta \le \pi \arccos \sqrt{m_2} < \pi ; \end{aligned}$$

in the special case \(X=a\) one of the turning points is \(m_2=1,\) which represents a transition through the spin axis; such photons therefore oscillate above one of the poles in the cone which is delimited by the angle \(\theta =\arccos \sqrt{m_{1}};\)

from the preceding discussion it follows that we can expect that the case \(X=a\) represents a change in azimuthal direction with respect to some privileged family of observers;
 \(q=0\) (Fig. 2c)
 the expression in the definition (28) can be reduced towhich validity can be enlarged, without any repercussion on the correctness of the analysis, even for \(m=0;\) the definition range of the potentials is thus \(\langle 0;1\rangle ;\)$$\begin{aligned} X^\theta _\pm (m;\,y,\,a)=\frac{a(1\mp \sqrt{(1m)\Delta _m})}{\Delta _ma^2y}, \end{aligned}$$(48)

from the equality \(W(\theta =\pi /2;\,X,\,q,\,y,\,a)=q\) it follows that at least in the equatorial plane the (radial) motion always exists for \(q=0,\) where it can be both stable or unstable (see below); for \(q>0\) the equatorial plane is crossed, for \(q<0\) it cannot be reached;
 there are no extrema of the potentials – \(X^\theta _+(m;\,q,y,\,a)\) is decreasing, \(X^\theta _(m;\,q,\,y,\,a)\) is increasing; the permissible values of X for which \(\,\mathrm{d}\theta /\mathrm{d}\lambda >0\) are still confined to an interval with limits$$\begin{aligned} X^\theta _{{\text {min}}()}= & {} X^\theta _(m=0;\,q=0,\,y,\,a) \nonumber \\= & {} \frac{2a}{(a^2y1)},\end{aligned}$$(49)where \(X^\theta _{{\text {min}}()}<X^\theta _{{\text {max}}(+)};\)$$\begin{aligned} X^\theta _{{\text {max}}(+)}= & {} X^\theta _+(m=0;\,q=0,\,y,\,a)=0, \end{aligned}$$(50)

if \(X\le X^\theta _{{\text {min}}()}\) or \(X\ge X^\theta _{{\text {max}}(+)}\) then the requirement \(W(\theta )\ge 0\) is fulfilled only if \(\theta =\pi /2,\) and in such case \(\,\mathrm{d}\theta /\mathrm{d}\lambda =0,\) thus the motion is stably confined to the equatorial plane;

for \(X^\theta _{{\text {min}}()}<X<X^\theta _{{\text {max}}(+)}\) photon initially released in the direction off the equatorial plane is once reflected at \(\theta =\arccos \sqrt{m_{0}}\) or \(\theta =\pi \arccos \sqrt{m_{0}}\), respectively, where \(m_0\) denotes the only solution of \(X=X^\theta _\pm (m;\,q,\,y,\,a);\) another point where \(\,\mathrm{d}\theta /\mathrm{d}\lambda =0\) is now in the equatorial plane, however, the equality \(\,\mathrm{d}^2\theta /\mathrm{d}\lambda ^2=0\) implies halting in the latitudinal direction; the function \(W(\theta )\) has at \(\theta =\pi /2\) local minimum, which indicates, as follows from perturbation analysis, instability in the equatorial plane;

if specially \(X=a\) then \(m_0=1,\) thus photon initially directed off the equatorial plane crosses the spin axis and finally is captured in the equatorial plane;

 \(q>0\) (Fig. 2d)

the potentials are defined for \(m\in (0,1\rangle ;\) they are monotonous in the same manner as in the case \(q=0,\) but \(X^\theta _+(m;\,q,\,a,\,y)\rightarrow +\infty \) and \(X^\theta _(m;\,q,\,y,\,a)\rightarrow \infty \) as \(m\rightarrow 0;\)

from the behavior of the potentials it follows that for \(X\ne a\) a photon is forced to oscillate in the \(\theta \) direction through the equatorial plane between two cones governed by \(\arccos \sqrt{m_{0}}\le \theta \le \pi \arccos \sqrt{m_{0}},\) with \(m_0\) of the same meaning as above;

case \(X=a\) represents the motion above both poles;

the foregoing conclusion is a reason to have a suspicion that cases \(X<a\) and \(X>a\) differ in the azimuthal direction relative to some family of stationary observers, it corresponds to \(\ell >0\) and \(\ell <0\).

 2.Case \(y=1/a^2\)
 the potentials simplify into the form$$\begin{aligned} X^\theta _\pm (m;\;q,\;a)=\frac{a\pm \sqrt{(1m^2)(q+a^2)}}{m}; \end{aligned}$$(51)
 \(q<a^2\)

the potentials are not defined, thus the latitudinal motion is not allowed;


 \(q=a^2\)
 the curves \(X=X^\theta _\pm (m;\;q=a^2,\,y=1/a^2,\,a)\) coalesce, since$$\begin{aligned}&X^\theta _+(m;q=a^2,y=1/a^2,a) \nonumber \\&\quad = X^\theta _(m;q=a^2,y=1/a^2,a)= X^\theta _{(\pm )}(m;\;a)\nonumber \\&\quad \equiv \frac{a}{m}; \end{aligned}$$(52)


for \(X\le a\) there is one solution of the equation \(X=X^\theta _{(\pm )}(m;\;a),\) which gives \(m=m_{(\pm )}\equiv a/X;\) this corresponds to PNC photons moving along cones \(\theta =\arccos \sqrt{m_{(\pm )}},\) \(\theta =\pi \arccos \sqrt{m_{(\pm )}};\)

for \(X\rightarrow \infty \) the cones approach the equatorial plane;

if specially \(X=a\) the cones degenerate to spin axis, therefore, such PNC photons move along the spin axis;

for \(X>a\) there is no motion allowed;
 \(a^2<q<0\) (Fig. 2e, f)

the potentials are both defined for \(m\in (0;1\rangle \); there is one local maximum \(X^\theta _{{\text {max}}(+)}\) given by (37) of the function \(X^\theta _+(m;\,q,\,y,\,a)\) and no extremum of \(X^\theta _(m;\,q,\,y,\,a);\) we have \(X^\theta _(m;\,q,\,y,\,a)<X^\theta _+(m;\,q,\,y,\,a)<0\) and \(X^\theta _(m;\,q,\,y,\,a),X^\theta _+(m;\,q,y,\,a)\rightarrow \infty \) as \(m \rightarrow 0\) from the right;

if \(X<a\) or \(a<X<X^\theta _{{\text {max}}(+)},\) the vortical motion exists;

for \(X=a\) the ‘inner’ cones coalesce with the spin axis, thus the vortical motion involves crossing the poles;

for \(X=X^\theta _{{\text {max}}(+)}\) both the ‘inner’ and ‘outer’ cones coalesce, giving thus rise to PNC photons;

if \(X>X^\theta _{{\text {max}}(+)},\) no motion is allowed;

 \(q=0\) (Fig. 2g)

the same discussion holds as in the case \(y<1/a^2,\) except that the motion exists for X arbitrarily small;

 3.Case \(y>1/a^2\)
 \(q<a^2\) (Fig. 2i)

the definition range of both potentials is an interval \((0;m_u\rangle \) (see the purple curve in Fig. 1d), where the upper limit \(m_u<1\) is given as \(m_l\) in the previous case by (46);

there is \(X^\theta _+(m;\,q,\,y,\,a)\rightarrow \infty \) and \(X^\theta _(m;\,q,y,\,a)\rightarrow +\infty \) as \(m\rightarrow 0,\) moreover, \(X^\theta _(m;\,q,y,\,a)\) now diverge at \(m=m_\mathrm{d},\) which is the solution of (33), and \(X^\theta _(m;\,q,\,y,\,a)\rightarrow +\infty \;(\infty )\) as \(m\rightarrow m_\mathrm{d}\) from the left (right);

there are thus two regions of permissible values X in the (m, X)plane for which the motion can exist; the lower one bounded by the graph of \(X^\theta _+\) and the lower branch of \(X^\theta _,\) which at \(m=m_u\) join into continuous curve, and the upper region given by the upper branch of \(X^\theta _;\) the motion is therefore allowed for \(X\le X^\theta _{{\text {max}}(+)}<a\) or \(X\ge X^\theta _{{\text {min}}()}>0,\) where the loci of local extrema \(X^\theta _{{\text {max}}(+)},\) \(X^\theta _{{\text {min}}()}\) are given by (39) (see the blue curve in Fig. 1d);

if \(X<X^\theta _{{\text {max}}(+)}\) or \(X>X^\theta _{{\text {min}}()}\) photon executes vortical motion, the cases \(X=X^\theta _{{\text {max}}(+)}\), \(X=X^\theta _{{\text {min}}()}\) correspond to PNC photons;

for \(X=X^\theta _(m_u)=X^\theta _+(m_u)=a(1+qy)/(a^2y1),\) the inner cones delimiting the vortical motion are the narrowest;
 for \(X\rightarrow \infty \) or \(X\rightarrow +\infty \) the outer cones given by anglesapproach the equatorial plane since \(m_1\rightarrow 0;\) for the inner cones$$\begin{aligned}\theta =\arccos \sqrt{m_1},\quad \theta =\pi \arccos \sqrt{m_1}\end{aligned}$$we have$$\begin{aligned}\theta =\arccos \sqrt{m_2},\quad \theta =\pi \arccos \sqrt{m_2},\end{aligned}$$$$\begin{aligned}m_2\rightarrow m_\mathrm{d}=11/a^2y;\end{aligned}$$

 \(q=a^2\) (Fig. 2j)

there is no local extremum of the function \(X^\theta _+(m;\,q,\,y,\,a),\) which is now increasing; we have \(m_u=1,\) \(X^\theta _(m_u)=X^\theta _+(m_u)=X^\theta _{+({\text {max}})}=a,\) hence for \(X=a\) both the inner and outer cones coalesce with the spin axis, which again corresponds to ‘axial’ PNC photon;

other PNC photons exist for \(X=X^\theta _{{\text {min}}()}>0;\)

there are no other qualitative differences from the case \(q<a^2;\)

 \(a^2<q<0\) (Fig. 2k)

the definition range is an interval \((0;1\rangle \) and the divergencies of the potentials are the same as above;

the function \(X^\theta _+(m;\,q,\,y,\,a)\) has now local maximum \(X^\theta _{{\text {max}}(+)},\) \(a<X^\theta _{{\text {max}}(+)}<0,\) \(X^\theta _{{\text {max}}(+)}\rightarrow 0\) for \(q\rightarrow 0,\) determined by Eq. (37);

case \(X=a\) now corresponds to vortical motion above the poles – the inner cones have coalesced with the spin axis, the outer ones stay open;

the vortical motion exists as in the previous cases and above that for \(a<X<X^\theta _{{\text {max}}(+)};\)

 \(q=0\) (Fig. 2l)

the definition (48) holds, the functions \(X^\theta _{+()}(m;q,\,y,\,a)\) are defined at \(\langle 0,1\rangle \) (\( \langle 0,1\rangle \setminus \{m_\mathrm{d}\});\) the values for \(m=0\) are given by (50), but now \(X^\theta _{{\text {max}}(+)}<X^\theta _{{\text {min}}()};\)

the potential \(X^\theta _+(m;\,y,\,a)\) is decreasing in its whole definition range, \(X^\theta _(m;\,y,\,a)\) is piecewise increasing because of the divergent point \(m_\mathrm{d};\)

if \(X\le X^\theta _{{\text {max}}(+)}=0\) or \(X\ge X^\theta _{{\text {min}}()}\) then the same conclusions can be made as in the case \(y<1/a^2\) for \(X^\theta _{{\text {min}}()}\le X\le X^\theta _{{\text {max}}(+)};\)

for \(X^\theta _{{\text {max}}(+)}< X< X^\theta _{{\text {min}}()}\) we have \(W(\theta =\pi /2;\,X,\,q=0,\,y,\,a) =0\) again, otherwise \(W(\theta ;\,X,\,q=0,\,y,\,a)<0,\) therefore photons can radially move in the equatorial plane;

 \(q>0\) (Fig. 2m, n)

the function \(X^\theta _+(m;\,q,\,y,\,a)\) is defined at \(\langle m_l,1\rangle ,\) the function \(X^\theta _(m;\,q,\,y,\,a)\) at \(\langle m_l,1\rangle \setminus \{m_\mathrm{d}\},\) where \(m_l\) is given by (46) with the difference that now \(X^\theta _{(\pm )}(m_l)>0\); the graphs of both functions now form a single open curve, which intersects a line \(X={\text {const.}}\) at a single point;

in the interval \(m \in \langle 0; m_l \rangle \) the latitudinal motion is allowed for arbitrarily large or small value of the motion constant X;

for arbitrary \(X\ne a\) there exists oscillatory motion through the equatorial plane as described in the case \(y<1/a^2, q>0;\)
 if \(X=X_{(\pm )}(m_l)\) the boundary cones are closest to equatorial plane, they are given by angles$$\begin{aligned} \theta =\arccos {\sqrt{m_l}},\quad \theta =\pi \arccos {\sqrt{m_l}}; \end{aligned}$$

the case \(X=a\) corresponds to orbits above both poles crossing also the equatorial plane;

there is no vortical motion or PNC photons.


 Case \(y<1/a^2\) (Fig. 3a)$$\begin{aligned} q_{\text {min}}(X,\,y,\,a) \equiv \left\{ \begin{array}{ll} 0, &{} \text {for}\quad X< \frac{2 a}{a^2 y  1}\quad \text {or}\quad X > 0;\\ q_2(X;\,y,\,a), &{} \text {for}\quad \frac{2 a}{a^2y1} \le X < a;\\ q_1(X), &{} \text {for}\quad a \le X \le 0;\\ \end{array}\right. \end{aligned}$$(56)
 Case \(y=1/a^2\) (Fig. 3b)$$\begin{aligned} q_{\text {min}}(X,\,y,\,a) \equiv \left\{ \begin{array}{ll} a^2, &{} \text{ for }\quad X \le a;\\ \\ q_1(X),&{} \text{ for }\quad a \le X \le 0;\\ 0,&{}\text {for}\quad X\ge 0; \end{array}\right. \end{aligned}$$(57)
 Case \(y>1/a^2\) (Fig. 3c)$$\begin{aligned} q_{\text {min}}(X,\,y,\,a) \equiv \left\{ \quad \begin{array}{ll} q_2(X;\,y,\,a),&{}\text {for}\quad X< a \quad \text {or} \quad \frac{2 a}{a^2 y  1}< X;\\ q_1(X),&{}\text {for}\quad a \le X \le 0;\\ 0,&{}\text {when}\quad 0 < X \le \frac{2 a}{a^2 y  1}; \end{array}\right. \end{aligned}$$(58)
4 Radial motion
The function \(a^2_{\mathrm{{ex}}(z(\mathrm{{ex}})+)}(r)\) should be excluded from further analysis since for \(r>0\) we have \(a^2_{\mathrm{{ex}}(z(\mathrm{{ex}})+)}(r)<0.\)
The functions \(y_{\mathrm{{ex}}(\mathrm{{ex}})\pm }(r;\,a^2)\) have the divergency point at \(r=0\) and \(y_{\mathrm{{ex}}(\mathrm{{ex}})\pm }(r;\,a^2)\rightarrow \pm \infty \) for \(r\rightarrow 0.\) For \(r\rightarrow \infty \) we find that \(y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,a^2)\rightarrow \infty \) and \(y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2)\rightarrow 0\) from above.
The zero point of \(y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2)\) is at \(r=3\) and the function is increasing for all \(r>0.\)
If we compare the asymptotic behavior of all characteristic functions \(y(r;\,a^2),\) we find that the following inequality is satisfied: \(1/a^2>y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2)>y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2)>y_h(r;\,a^2)>y_\mathrm{d}(r;\,a^2)>y_{\mathrm{d}(\mathrm{{ex}})}(r;\,a^2)\) \(>0>y_{z(\mathrm{{ex}})}(r;\,a^2)>1/a^2\) as \(r\rightarrow \infty .\)

\(a^2_{z(h)}(r)\);

\(a^2_{\mathrm{{ex}}(h)+}(r)\)=\(a^2_{\mathrm{{ex}}(z(\mathrm{{ex}}))+}(r)\)=\(a^2_{{\text {inf}}(\mathrm{{ex}}(\mathrm{{ex}})\pm )+}(r)\);

\(a^2_{{\text {max}}(\mathrm{d})}(r)\);

\(a^2_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(r)\);

\(a^2_{z(z(\mathrm{{ex}})+)}(r)\);

\(a^2_{r(\mathrm{{ex}}(\mathrm{{ex}})\pm )+}(r)\);

\(a^2_{r(\mathrm{{ex}}(\mathrm{{ex}})\pm )}(r)\);

\(a^2_{z(\mathrm{{ex}}(\mathrm{{ex}})\pm )}(r).\)

\(y_{h}(r;\,a^2)\);

\(y_\mathrm{d}(r;\,a^2)\);

\(y_{\mathrm{{ex}}(r)}(r;\,a^2)=y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2)\);

\(y_{\mathrm{d}(\mathrm{{ex}})}(r;\,a^2)\);

\(y_{z(\mathrm{{ex}})+}(r;\,a^2)\);

\(y_{\mathrm{{ex}}(\mathrm{{ex}})\pm }(r;\,a^2).\)

\(a^2=1\)– the common local maximum of the functions \(a^2_{z(h)}(r)\) and \(a^2_{z(z(\mathrm{{ex}}))}(r)\) at \(r=1,\) which coincides with the inflection point of the function \(a^2_{z(\mathrm{{ex}}(\mathrm{{ex}})\pm )}(r;\,a^2)\) and with the intersection with the curve \(a^2_{\mathrm{{ex}}(h)+}(r)\);

\(a^2=a^2_{\text {crit}}=1.21202\)– the local maximum \(a^2_{\mathrm{{ex}}(h)+}(r)\) which is the intersection of the curves \(a^2_{r(\mathrm{{ex}}(\mathrm{{ex}})\pm )+}(r)\), \(a^2_{r(\mathrm{{ex}}(\mathrm{{ex}})\pm )}(r)\) and \(a^2_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(r).\)

\(y_{{\text {max}}(h)}(a^2)=y_{{\text {max}}(z(\mathrm{{ex}})+)}(a^2)=y_{{\text {inf}}(\mathrm{{ex}}(\mathrm{{ex}}))}(a^2)= y_{\mathrm{d}(\mathrm{{ex}})\text{ }h\text{ }(z(\mathrm{{ex}})+)\text{ }\mathrm{{ex}}(\mathrm{{ex}})}(a^2)\);

\(y_{{\text {min}}(h)}(a^2)=y_{{\text {min}}(z(\mathrm{{ex}})+)}(a^2)=y_{{\text {inf}}(\mathrm{{ex}}(\mathrm{{ex}})+)}(a^2)= y_{\mathrm{d}(\mathrm{{ex}})\text{ }h\text{ }(z(\mathrm{{ex}})+)\text{ }\mathrm{{ex}}(\mathrm{{ex}})+}(a^2)\);

\(y_{{\text {max}}(\mathrm{d})}(a^2)=y_{\mathrm{d}\text{ }\mathrm{d}(\mathrm{{ex}})\text{ }(z(\mathrm{{ex}})+)\text{ }\mathrm{{ex}}(\mathrm{{ex}})}(a^2)\);

\(y_{\mathrm{d}\text{ }(z(\mathrm{{ex}})+)}(a^2)\);

\(y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)=y_{\mathrm{d}(\mathrm{{ex}})\text{ }\mathrm{{ex}}(\mathrm{{ex}})}(a^2)\);

\(y_{\mathrm{{ex}}(\mathrm{{ex}})\text{ }(\mathrm{{ex}}(\mathrm{{ex}})+)}(a^2)\).

the functions \(y_{{\text {max}}(h)}(a^2)\) and \(y_{{\text {min}}(h)}(a^2)\) are both determined by \(a^2_{\mathrm{{ex}}(h)+}(r)\) and \(y_h(r;\,a^2=a^2_{\mathrm{{ex}}(h)+}(r));\)

\(y_{{\text {max}}(\mathrm{d})}(a^2)\) we obtain from \(a^2_{{\text {max}}(\mathrm{d})}(r)\) with \(y_\mathrm{d}(r;\,a^2=a^2_{{\text {max}}(\mathrm{d})}(r));\)

the curve \(y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)\) is given by functions \(a^2_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(r)\) and \(y_{\mathrm{d}(\mathrm{{\mathrm{{ex}}}})}(r;\,a^2=a^2_{{\text {max}}(\mathrm{d}(ex))}(r));\)
 \(y_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(a^2)\) is determined byand \(y_\mathrm{d}(r;\,a^2=a^2_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(r)),\) where the function \(a^2_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(r)\) is a solution of \( y_\mathrm{d}(r;\,a^2)=y_{z(\mathrm{{ex}})+}(r;\,a^2) \) with respect to parameter \(a^2;\) all such functions are obtained by analogous manner;$$\begin{aligned} a^2_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(r)\equiv \frac{r}{8}(14r+\sqrt{40r+1}) \end{aligned}$$
 \(y_{\mathrm{{ex}}(\mathrm{{ex}})\text{ }\mathrm{{ex}}(\mathrm{{ex}})+}(a^2)\) are constructed from$$\begin{aligned} a^2_{\mathrm{{ex}}(\mathrm{{ex}})\text{ } \mathrm{{ex}}(\mathrm{{ex}})\pm }(r) \end{aligned}$$and \( y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,a^2=a^2_{\mathrm{{ex}}(\mathrm{{ex}})\text{ } \mathrm{{ex}}(\mathrm{{ex}})+}(r)).\)$$\begin{aligned} \equiv \frac{r}{2}(4r^2{}12r+3\pm \sqrt{16r^496r^3{+}156r^236r+9}) \end{aligned}$$
 I:
\(y \le y_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(a^2)\) for \(a^2\le 0.5\);
 II:
\(y_{\mathrm{d}\text{ }z(\mathrm{{ex}})+}(a^2)\le y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(a^2\le 0.5,\) or \(y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(0.5 \le a^2\le 1,\) or \(y_{{\text {min}}(h)}(a^2)\le y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(1\le a^2\le 1.08316;\)
 III:
\(y_{{\text {max}}(\mathrm{d})}(a^2)\le y \le y_{{\text {max}}(h)}(a^2)\) for \(a^2\le 1.08316,\) or \(y_{{\text {min}}(h)}(a^2)\le y \le y_{{\text {max}}(h)}(a^2)\) for \(1.08316\le a^2\le 1.21202=a^2_{\text {crit}};\)
 IVa:
\(y\le y_{{\text {min}}(h)}(a^2)\) for \(1\le a^2\le 1.08316,\) or \(y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(1.08316\le a^2\le 1.28282,\) or \(y\le y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) for \(1.28282\le a^2\le 6\sqrt{3}9=1.3923;\)
 IVb:
\(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\le y\le y_{{\text {max}}(d)}(a^2)\) for \(1.28282\le a^2\le 1.3923,\) or \(y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(1.3923\le a^2\le 9,\) or \(y_{\mathrm{{ex}}(\mathrm{{ex}})\text{ }\mathrm{{ex}}(\mathrm{{ex}})+}(a^2)\le y\le y_{{\text {max}}(\mathrm{d})}(a^2)\) for \(a^2\ge 9;\)
 V:
\(y\le y_{\mathrm{{ex}}(\mathrm{{ex}})\text{ }\mathrm{{ex}}(\mathrm{{ex}})+}(a^2)\) for \(a^2\ge 9;\)
 VIa:
\(y_{{\text {max}}(\mathrm{d})}(a^2)\le y\le y_{{\text {min}}(h)}(a^2)\) for \(1.08316\le a^2\le 1.21202,\) or \(y_{{\text {max}}(\mathrm{d})}(a^2)\le y\le y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) for \(1.21202\le a^2\le 1.28282;\)
 VIb:
\(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\le y\le y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)\) for \(1.21202\le a^2\le 1.28282,\) or \(y_{{\text {max}}(\mathrm{d})}(a^2)\le y\le y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)\) for \(a^2\ge 1.28282;\)
 VII:
\(y_{{\text {max}}(h)}(a^2)\le y\le 1/a^2\) for \(a^2\le 1.21202,\) or \(y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)\le y\le 1/a^2\) for \(a^2\ge 1.21202;\)
 VIII:
\(y\ge 1/a^2.\)
In regions I and II, which describe blackhole spacetimes with divergent repulsive barrier, we have to compare the two local maxima \(q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) located under the inner horizon and \(q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2)=q_{{\text {min}}(r)}(y,\,a^2)\) between the outer and cosmological horizons, respectively (see Fig. 8a, b). The function \(q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) is given parametrically by the functions \(y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,y,\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,y,\,a^2),\,a^2)\) with r being the parameter; similarly \(q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2)\) is given by \(y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,y,\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,y,\,a^2),\,a^2).\)
The extrema function \(q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2)\) diverges at the curve \(y_{{\text {max}}(\mathrm{d})}(a^2)\), which forms the boundary between regions II–III and IV–VI, i.e. \(q_{{\text {max}}(ex)}(y=y_{{\text {max}}(\mathrm{d})}(a^2),\,a^2) \rightarrow +\infty \) (cf. Fig. 8b–f). In region III, corresponding to blackhole spacetimes with the restricted repulsive barrier, only local maximum \(q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) located under the inner horizon remains.
The function \(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) corresponds to the local minima of the potential \(X_{}(r;\,q,\,y,\,a)\) reaching the value \(X=a,\) i.e., \(\ell =0.\) The photons corresponding to these minima, and having appropriate constant of the motion q, persist on ’spherical’ orbits with \(r={\text {const}}\), which are crossing the spacetime rotation axis alternately above both poles. Below we shall call them ’polar’ spherical orbits – in the following section we shall see that such polar spherical orbits form a border surface between prograde and retrograde spherical photon orbits, as related to the locally nonrotating observers. The function \(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) has therefore an important meaning, since it represents a boundary between regions of qualitatively different KdS spacetimes in the \(a^2\)–y plane. From this point of view, the function \(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) creates another qualitative shift in the parameter plane \((a^2y)\) with regard to the character of the motion of the photon, however, no qualitative shift in the mathematical properties of the characteristic functions \(q_{r}(r;\,y,\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y,\,a^2)\) in their relevant values \(q\ge 0.\) The parts of the \(a^2\)–y plane corresponding to different behavior of the characteristic functions are in Fig. 7 distinguished by Roman numerals, the curve \(y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) then induces an additional division a / b.
Further we have to relate the minima \(q_{{\text {min}}(\mathrm{{ex}}\pm )}(y,\,a^2)\) with the maxima \(q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2)=q_{{\text {min}}(r)}(y,\,a^2).\) In the region V, the minima of \(q_{\mathrm{{ex}}}(r;\,y,\,a^2)\) coalesce with the minima of \(q_{r}(r;\,y,\,a^2)\) (Fig. 8e). We therefore have to compare the minima function \(q_{{\text {min}}(\mathrm{{ex}})}(y,\,a^2)= q_{{\text {min}}(r)}(y,\,a^2)\) determined by \(y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,y,\,a^2)\) and, e.g., \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})}(r;\,y,\,a^2),\,a^2),\) with the maxima function \(q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) parametrized by \(y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,y,\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,y,\,a^2),\,a^2).\) The boundary curve \(y_{\mathrm{{ex}}(\mathrm{{ex}})\text{ }\mathrm{{ex}}(\mathrm{{ex}})+}(a^2)\) then represents such combinations of parameters \(a^2,y\) for which the local extrema of \(q_{\mathrm{{ex}}}(r;\,y,\,a^2)\) have coalesced into an inflection point. For parameters from region VI, corresponding to naked singularity spacetimes with restricted repulsive barrier (as well as from the remaining regions VII, VIII), the function \(q_{\mathrm{{ex}}}(r;\,y,\,a^2)\) has one local minimum (Fig. 8f), and we therefore construct a function \(q_{{\text {min}}(\mathrm{{ex}}\pm )}(y,\,a^2)\) determined by functions \(y_{\mathrm{{ex}}(\mathrm{{ex}})\pm }(r;\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})\pm }(r;\,a^2),\,a^2),\) where the minus sign has to be chosen for \(1.17007 \le a^2 \le a^2_{\text {crit}}\) and \(y_{{\text {max}}(\mathrm{d})}(a^2)\le y \le y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))1}(a^2),\) or \(a^2_{\text {crit}} \le a^2 \le 1.2828\) and \(y_{{\text {max}}(\mathrm{d})}(a^2) \le y \le y_{(\mathrm{{ex}}(\mathrm{{ex}})+)\text{ }(\mathrm{{ex}}(\mathrm{{ex}}))2}(a^2)\) (see Fig. 7). For \(y\rightarrow y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2)\) and \(a^2\ge a^2_{\text {crit}}\) we have \(q_{{\text {min}}(\mathrm{{ex}}+)}(y,\,a^2)\rightarrow +\infty ,\), and for \(y> y_{{\text {max}}(\mathrm{d}(\mathrm{{ex}}))}(a^2),\) i.e. in region VII, it converts into the local maximum (cf. Fig. 8f, g). The transition into region VII from region III can be inferred from a comparison of Fig. 8c with g. Therefore, in region VII we have to follow up the values of the function \(q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) determined by the functions \(y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,a^2)\) and \(q_{\mathrm{{ex}}}(r;\,y=y_{\mathrm{{ex}}(\mathrm{{ex}})+}(r;\,a^2),\,a^2).\) The functions \(q(y,\,a^2)\) are demonstrated in Fig. 9.
5 Spherical photon orbits and classification of the Kerr–de Sitter spacetimes due to properties of the motion of the photon
We shall demonstrate by using the behavior of the effective potentials \(X_{\pm }(r;\,q,\,a,\,y)\) that the null geodesics create qualitatively different structures in the various cases of Kerr–de Sitter spacetimes with the spacetime parameters chosen from different parts of the \(a^2\)–y plane labeled by numerals I–VIII. Hence the regions of the spacetime parameter space of these labels can be considered as representatives of the classification of the Kerr–de Sitter spacetimes due to the motion of the photon (null geodesics). Similarly to [73], there are three (four) criteria used – the main criterion for the classification is the existence (number) of the event horizons. The other differentiating factors follow from the nature of the motion of the photon. First, there is some kind of repulsive barrier preventing light from reaching the ring singularity, which is always created in its vicinity for photons with \(q>0\). However, a similar barrier can emerge between the outer blackhole horizon and the cosmological horizon in blackhole and naked singularity spacetimes, repelling photons towards one of these horizons. In the naked singularity spacetimes, the occurrence of an additional barrier, which reflects photons towards the ring singularity, leads to the occurrence of the phenomenon of bound photon orbits. Such bound photon orbits are not present in the case of the blackhole spacetimes. The presence and character of this barrier we take as another criterion in the following classification. The other aspect that authorizes us to make such a distinction between the KdS spacetimes will be the existence and character of the spherical photon orbits. In the KdS naked singularity spacetimes the bound orbits are concentrated around the stable spherical photon orbits.
5.1 Spherical photon orbits
5.2 Classification
 Class I:
Blackhole spacetimes with the divergent repulsive barrier of the radial motion of the photon, having one equatorial counterrotating circular unstable orbit with negative energy located under the inner blackhole horizon (\(0<r<r_{}\)), which is limiting the range of the spherical photon orbits with negative energy. There exist stable orbits, corresponding to local minima \(X_{{\text {min}}(+)}\) of the effective potential \(X_{+}\) at \(0<r<r_{{\text {max}}(\mathrm{{ex}})1},\) and unstable orbits, corresponding to local maxima \(X_{{\text {max}}(+)}\) of \(X_{+}\) at \(r_{{\text {max}}(\mathrm{{ex}})1}<r<r_{z(\mathrm{{ex}})1}\) for \(0<q<q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) (Fig. 12a). We denote by \(r_{{\text {min}}/{\text {max}}(\mathrm{{ex}})}\) the local extrema, and by \(r_{z(\mathrm{{ex}})}\) the zero point, of the function \(q_{ex}(r;\,y,\,a^2)\) hereafter. Such a structure is present under the inner horizon of any KdS blackhole spacetime. Outside the ergosphere, one unstable corotating equatorial circular orbit, located at \(r=r_{{\text {ph}}+}=r_{z(\mathrm{{ex}})2}\), and a polar spherical orbit with \(r=r_{\text {pol}},\) \(r_{{\text {ph}}+}<r_{\text {pol}},\) limit the range of unstable prograde spherical orbits given by the local minima \(X_{{\text {min}}()}\) of the effective potential \(X_{},\) for which \(X_{{\text {min}}()}>a\). The radius of the polar spherical orbit is found by solving \(X_{}(r_{\text {pol}};q_{\mathrm{{ex}}}(r_{\text {pol}}))=a.\) The counterrotating equatorial circular orbit at \(r=r_{\mathrm{{ph}}}=r_{z(\mathrm{{ex}})3}\) gives the limit of the region of unstable retrograde spherical orbits, given by the local minima \(X_{{\text {min}}()}<a\), and maxima \(X_{{\text {max}}(+)}<a\) for \(0<q<q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2),\) such that \(r_{\text {pol}}<r_{{\text {ph}}}.\)
 Class II:
Blackhole spacetimes with the same features as in the class I, but now the ergosphere enters the region of the spherical photon orbits (Fig. 12b). No spherical orbit is fully immersed in the ergosphere and photons at all the spherical orbits have positive energy. The presence of the ergosphere in region of the spherical photon orbits influences character of the light escape cones [18].
 Class III:
Blackhole spacetimes with the restricted repulsive barrier of the radial motion of the photon. The ergosphere spreads over all radii. The prograde spherical orbits are given by the local minima \(X_{{\text {min}}()}>a\) at \(r_{{\text {ph}}+}<r<r_{\text {pol}},\) while the retrograde spherical orbits with \(E>0\) are given by the minima \(X_{{\text {min}}()}<a\) at \(r_{\text {pol}}<r<r_{\mathrm{d}(\mathrm{{ex}})}.\) The spherical orbits given by the local maxima \(X_{{\text {max}}(+)}\) (see Fig. 10k–n) at \(r_{\mathrm{d}(\mathrm{{ex}})}<r<r_{{\text {ph}}},\) where \(r_{\mathrm{d}(\mathrm{{ex}})}\) denotes the divergence point of \(q_{\mathrm{{ex}}}(r;\,y,\,a^2)\) (see Fig. 8c), are fully immersed in the ergosphere. Such areas are drawn in Fig. 12 in green and the spheres with \(r=r_{\mathrm{d}(\mathrm{{ex}})}\) as full/dashed green ellipses. Photons in such regions have \(E<0\).
 Class IVa:
Naked singularity spacetimes with divergent repulsive barrier of the motion of the photon. At radii \(0<r<r_{\mathrm{d}(\mathrm{{ex}})}\) (Fig. 8d can be used for illustration), there are local minima of the potential \(X_{+}\) (for illustration use Fig. 10p–r) corresponding to the stable retrograde spherical orbits with negative energy (\(E<0\)) (Fig. 12d). The stable retrograde orbits with positive energy corresponding to the local maxima of \(X_{}\) are at \(r_{\mathrm{d}(\mathrm{{ex}})}<r<r_{{\text {pol}}1}\). These maxima exceed the value \(X_{{\text {max}}()}=a\) at \(r_{{\text {pol}}1}<r<r_{{\text {min}}(\mathrm{{ex}})},\) where they yield stable prograde spherical orbits. At radii \(r_{{\text {min}}(\mathrm{{ex}})}<r<r_{pol2}\), we have the local minima of \(X_{}\) with values \(X_{{\text {min}}()}>a,\) – these radii are thus occupied by the unstable prograde orbits. The local minima of \(X_{}\) and the local maxima of \(X_{+}\) at \(r>r_{{\text {pol}}2}\) correspond to the unstable retrograde orbits. There are thus two polar spherical orbits enclosing the region of prograde orbits – the inner at the radius \(r=r_{{\text {pol}}1}\) being stable, the outer at the radius \(r=r_{{\text {pol}}2}\) being unstable.
 Class IVb:
Naked singularity spacetimes with the same features as in the class IVa, but the two polar orbits have coalesced, therefore, there are no prograde spherical orbits (Fig. 12e).
 Class V:
Naked singularity spacetimes having the structure of the spherical orbits corresponding to the previous case (Fig. 12f), but with is a small region of bound orbits for photons with constants of the motion \(q_{{\text {min}}(\mathrm{{ex}})}(y,\,a^2)<q<q_{{\text {max}}(\mathrm{{ex}}+)}(y,\,a^2)\) and X between the appropriate local extrema of \(X_{+}\) (cf. Fig. 10q, u), which is not contained in the other cases.
 Class VIa:
Naked singularity spacetimes with the restricted repulsive barrier of the radial motion of the photon. For \(0<r<r_{\mathrm{d}(\mathrm{{ex}})1}\) (the function \(q_\mathrm{{ex}}(r;\,y,\,a^2)\) has two divergence points \(r_{\mathrm{d}(\mathrm{{ex}})1}, r_{\mathrm{d}(\mathrm{{ex}})2}\) – see Fig. 8f) the minima of \(X_{+}\) correspond to stable retrograde orbits with \(E<0\); for \(r_{\mathrm{d}(\mathrm{{ex}})1}<r<r_{{\text {pol}}1}\), there are the local maxima of \(X_{}\) with values \(X_{{\text {max}}()}<a\) giving retrograde orbits with \(E>0.\) The local minima of \(X_{}\) at \(r_{{\text {pol}}1}<r<r_{{\text {min}}(\mathrm{{ex}})}\) give the stable prograde orbits. At radius \(r=r_{{\text {pol}}1}\) the stable polar orbit is located. For \(r_{{\text {min}}(\mathrm{{ex}})<r<r_{{\text {pol}}2}}\), the function \(X_{}\) has minima with values \(X_{{\text {min}}()}>a\), giving the unstable prograde orbits. For \(r_{{\text {pol}}2}<r<r_{\mathrm{d}(\mathrm{{ex}})2}\), they correspond to the unstable retrograde orbits. At radius \(r=r_{{\text {pol}}2}\), the unstable polar orbit exists. The local maxima of the function \(X_{+}\) at \(r_{\mathrm{d}(\mathrm{{ex}})2<r<r_{{\text {ph}}}}\) correspond to the retrograde unstable spherical orbits with \(E<0.\)
 Class VIb:
The structure of the spherical orbits corresponds to the class VIa with the exception that the local extrema of the potential \(X_{}\) have values \(X<a\), implying that there are no polar spherical orbits, nor the prograde spherical orbits (Fig. 12h).
 Class VII:
Naked singularity spacetimes with the restricted repulsive barrier of the radial motion of the photon having stable retrograde spherical orbits at \(0<r_{{\text {max}}(\mathrm{{ex}})}\) corresponding to local minima of \(X_{+}\) (Fig. 10 \(\alpha \)), and unstable retrograde spherical orbits at \(r_{{\text {max}}(\mathrm{{ex}})}<r<r_{{\text {ph}}}\) corresponding to local maxima of \(X_{+}.\) All these spherical orbits, including the counterrotating equatorial circular orbit at \(r=r_{{\text {ph}}}\), correspond to photons with \(E<0.\)
 Class VIII:
Special class of the naked singularity spacetimes demonstrating the same features of the radial motion of photons with \(q\ge 0\) as the class VII, but differing from all previous cases by the existence of null geodesics for arbitrary \(q<0.\) The allowed values of the impact parameter X are for \(q<0\) confined to the intervals \(X<X^\theta _{{\text {max}}(+)}<0\) or \(X>X^\theta _{{\text {min}}()}>0\) (see Sect. 3). The potentials governing the radial motion of the photon are fully immersed in the forbidden region (Fig. 10 (\(\gamma \)) and (\(\delta \))), thus in the radial direction the photons with such parameters move freely in the whole range between the ring singularity and the cosmological horizon.
6 Conclusions
 1.
In any kind of the blackhole spacetimes, there are no radially bound null geodesics in the stationary region, i.e., the trajectory of a photon has at most one turning point in radial direction between the outer and cosmological horizon, or the photons can move freely between the outer blackhole and the cosmological horizons. However, such bounded photon orbits exist in each naked singularity spacetime for photons with parameters \(q>0\) and X chosen appropriately.
 2.
No photons with \(q>0\) can reach the ring singularity at \(r=0\) in any of the Kerr–de Sitter spacetimes.
 3.
In the Kerr–de Sitter spacetimes of classes I–VII, i.e., with the spacetime parameters satisfying the condition \(y<1/a^2,\) there is a lower limit \(q=a^2\) of the parameter \(q<0,\) for which the motion of the photon is allowed. The range of the allowed values of the impact parameter X is then an interval given by Eqs. (53)–(56). Photons with such tuned parameters have no turning point in radial direction, since the effective potential lies entirely in the forbidden region (Fig. 10e–g). Furthermore, by the results of Sect. 3, only such photons execute the vortical motion, or their trajectory lies completely on the cones of \(\theta = {\text {constant}}\). We can therefore reject the possibility of the existence of a vortical motion of the photon of constant radius, or offequatorial circular photon orbits.
 4.
In the Kerr–de Sitter spacetimes of class VIII (\(y>1/a^2\)), the motion of the photon is allowed for any \(q<0.\) The permissible values of the parameter X are then two disjunct unlimited intervals determined by Eq. (58). In the extreme case \(y=1/a^2,\) we must have \(q\ge a^2\) again and for negative q, the parameter X can take a smaller value than a certain negative value, given by (57). The consequences for the motion of the photon are then the same as in previous note (Fig. 10 (\(\gamma \)) and (\(\delta \))).
 5.
In the Kerr–de Sitter spacetimes with the divergent repulsive barrier of the radial motion of the photon, there exists a critical value \(q_{{\text {max}}(\mathrm{{ex}})}(y,\,a^2),\) for which this barrier becomes impermeable between the outer blackhole horizon and cosmological horizon, or, in naked singularity spacetimes, between the ring singularity and cosmological horizon, for photons with any impact parameter X. In spacetimes with the restricted repulsive barrier of the radial motion of the photon, the height of this barrier slowly grows with increasing parameter q, but it stays finite for any \(q>0\) (Fig. 10n, y).
 6.
In the Kerr–de Sitter spacetimes of classes I–III, IVa and VIa, there exist spherical photon orbits, which can be both prograde or retrograde as seen by the family of locally nonrotating observers. Additionally, each of the two types can be stable or unstable with respect to radial perturbations. The regions of spherical orbits of different orientations are separated by the socalled polar spherical orbit, at which photons cross the spacetime rotation axis alternately above both poles. In the naked singularity spacetimes of class IVa, VIa, there are two polar spherical orbits, the inner one being stable, the outer one being unstable.
 7.
In the Kerr–de Sitter spacetimes of classes IVb, V, VIb, VII, VIII there are no prograde or polar spherical orbits.
 8.
In each class of the Kerr–de Sitter spacetimes, there exists a region where the effective potential \(X_{+}\) have positive values. Photons with impact parameter X exceeding these values appear to move in retrograde direction as seen in LNRFs. This region must be located inside the ergosphere, and photons with such impact parameters must have negative energy, \(E<0\). In the blackhole spacetimes with the divergent barrier of the radial motion of the photon, the ergosphere has two parts above the blackhole outer event horizon – the inner one, which is limited to the outer vicinity of the outer event horizon, and the outer one, limited to the inner vicinity of the cosmological horizon. In the blackhole spacetimes with restricted repulsive barrier of the radial motion of the photon, the two regions of the ergosphere merge in the equatorial plane, and they spread at any radii except for a certain region in the vicinity of the rotation axis.
 9.
In the LNRFs, trajectories of photons moving along any spherical orbit have no turning point of the azimuthal motion.
Footnotes
 1.
However, we have to note that, similarly to the case of Kerr black holes [90], the sign of the variation of the azimuthal coordinate can be changed at some latitude, if related to distant observers.
 2.
Keeping \(E>0\) means \(k^{(t)}<0,\) i.e., a photon in negativeroot state with time evolution directed to the past – for details see [11].
Notes
Acknowledgements
Z.S. acknowledges the Albert Einstein Centre for Gravitation and Astrophysics supported by the Czech Science Foundation Grant no. 1437086G. D.Ch. acknowledges the Silesian University in Opava Grant no. SGS/14/2016.
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