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Eur. Phys. J. C (2018) 78:879
https://doi.org/10.1140/epjc/s10052-018-6336-5
Regular Article - Theoretical Physics
Spherical photon orbits in the field of Kerr naked singularities
Daniel Charbuláka, Zdenˇek Stuchlíkb
Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in
Opava, Bezruˇcovo nám. 13, 746 01 Opava, Czech Republic
Received: 23 September 2018 / Accepted: 12 October 2018
© The Author(s) 2018
Abstract For the Kerr naked singularity (KNS) space-
times, we study properties of spherical photon orbits (SPOs)
confined to constant Boyer-Lindquist 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 E>0(E<0), at r=1thereisE=0. All
unstable orbits must have 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 ais proposed, considering the properties
of the SPOs. For selected types of the KNS spacetimes, typ-
ical 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 Keple-
rian 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 infor-
mation 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 influ-
ences 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
ae-mail: daniel.charbulak@fpf.slu.cz
be-mail: zdenek.stuchlik@fpf.slu.cz
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 Sit-
ter) spacetimes has been intensively studied in [63]. How-
ever, 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 Kerr-de Sitter black holes, where the cosmologi-
cal 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 gen-
eralizations of the Einstein theory, e.g., for regular black hole
spacetimes of the Einstein theory combined with non-linear
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 spher-
ical photon orbits, i.e., photons moving along orbits of con-
stant (Boyer–Lindquist) radial coordinate – their motion con-
stants 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 proper-
ties of the spherical photon orbits outside the outer horizon of
Kerr black holes were studied in [58]. Spherical photon orbits
in the field of Kerr-de 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 Hoˇrava and his co-workers, with interior
governed by the String theory and exterior described by the
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879 Page 2 of 25 Eur. Phys. J. C (2018) 78:879
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 ultra-relativistic collisional
processes [53], in the high-frequency quasiperiodic oscilla-
tions in Keplerian disks [27,28,59], or in the Lense-Thirring
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 red-
shifts larger then z=2[49]. The Kerr naked singularity
spacetimes could be applied also in description of the exte-
rior 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 spher-
ical 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 discus-
sion 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, extend-
ing thus an introductory study in more general Kerr-de 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 cir-
cular geodesic motion; we shortly discuss possible observa-
tional effects related to the irradiation of the Keplerian disks
by the photons following the spherical orbits.
2 The Kerr spacetimes
The line element of the Kerr spacetime is in the standard
Boyer-Lindquist spheroidal coordinates t,r,θ,φ, with geo-
metric system of units (c=G=1), described by the well
known formula
ds2=−
dt−asin2θdφ2
+sin2θ
adt−r2+a2dφ2
+
dr2+dθ2,(1)
where
=r2+a2−2Mr,(2)
=r2+a2cos2θ. (3)
Here Mis the gravitational mass parameter of the Kerr space-
time, and ais its angular momentum per unit gravitational
mass. Without any loss of generality we can assume the
parameter ato be positive.
If a<M, the Kerr spacetime describes black holes that
posses two pseudosingularities (horizons) determined by the
condition
=0,(4)
located at radii
r±=M±M2−a2.(5)
If a=M, the horizons coincide at r=Mand the spacetime
describes an extreme Kerr black hole [7]. In the present paper,
we focus ourselves to the case a>Mcorresponding to the
Kerr naked singularity spacetimes where no horizons exist. In
all the Kerr spacetimes, the physical singularity is located at
r=0,θ =π/2. The stationary limits of the Kerr spacetimes,
determined by the condition gtt =0, give the boundary of the
so called ergosphere where extraction of rotational energy is
possible due to the Penrose process [32]. At r<0, allowed in
the Kerr spacetimes, the so called causality-violation region
exists [15] – the Kerr superspinars are constructed in such a
way that both the causality-violation region and the physical
singularity are removed, and it is assumed that they are sub-
stituted by an interior regular solution governed by the String
theory [25,50].
In order to demonstrate clearly behaviour of the Kerr
spacetimes at region close to the physical singularity, it is
convenient to use the (“flat”) Kerr-Schild coordinates that are
connected to the (spheroidal) Boyer-Lindquist coordinate r
by the relations
x2+y2=(r2+a2)sin2θ,z2=r2cos2θ(6)
and enable proper visualization of the Kerr spacetime in the
innermost regions.
3 Carter equations of geodesic motion
The symmetries of the Kerr spacetimes imply existence of the
time Killing vector ξ(t)=∂/∂tand the axial Killing vector
ξ(φ) =∂/∂φ. Then the projections E=−ξ(t)·p(energy)
and =ξ(φ) ·p(angular momentum about the φ-axis) of
a particle four-momentum p=dxμ
dλare constants of the
particle motion. Here, xμare the coordinate components of
the four-momentum and λis an affine parameter. Two another
motion constants are the rest energy mof the particle (m=0
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Eur. Phys. J. C (2018) 78:879 Page 3 of 25 879
for photons) and Q, Carter’s constant connected with total
angular momentum of the particle. The motion of photons
can be described by the Carter equations
dr
dλ2
=R(7)
dθ
dλ2
=W(8)
dφ
dλ=aP
−aE+
sin2θ(9)
dt
dλ=r2+a2
P−a(aEsin2θ−), (10)
where the functions P,R,Ware defined by the relations
P=E(r2+a2)−a(11)
R=P2−[Q+( −aE)2](12)
W=Q−cos2θ2
sin2θ−a2E2.(13)
It is convenient to use following rescaling λ→λ=λEand
put M=1, i.e., express the radius and time (line element) in
units of gravitational mass M.Further, it is usual to introduce
the substitution
m=cos2θ,
dm=2sign(θ −π/2)m(1−m)dθ, (14)
which enables one to replace dealing with the trigonometric
functions by the algebraic ones. The Carter equations then
take the form
( ˙r)2=R(r)
≡(r2−a+a2)2−[(a−)2+q](15)
(1/2˙m)2=M(m)
≡m[q+(a2−2−q)m−a2m2].(16)
˙
φ=2ar +
1−m( −2r)(17)
˙
t=(r2+a2)−2ar[−a(1−m)].(18)
Here the dot indicates differentiation with respect to the
rescaled affine parameter λ. We have introduced new param-
eters (assuming E= 0)
q=Q
E2and =
E,(19)
where is the impact parameter.1Introduction of the Carter
constant Qenables simple classification of the latitudinal
1In case of more complex spacetimes with non-zero cosmological con-
stant and possibly non-zero electric charge of the gravitating source it is
more convenient to introduce the modified impact parameter X=−a,
which simplifies the radial equation of motion 15 and its discussion, as
shown in [17], where the more general case of the Kerr-de Sitter space-
times was considered. However, its introduction is unnecessary in this
paper.
motion in all the Kerr spacetimes, as Q=0 governs the
equatorial motion with θ=π/2, Q>0 governs the so
called orbital motion crossing the equatorial plane2, while
Q<0 governs so called vortical motion when crossing of
the equatorial plane is forbidden [9,19,42]. It is crucial for
the purposes of the present paper that in both Kerr black hole
[58] and Kerr (-de Sitter) naked singularity [17] spacetimes
the spherical orbits with r=const must be of the orbital
type, having thus necessarily Q>0.
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 self-illumination [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
positive-root states where they have positive energy as mea-
sured by local observers, and, equivalently, they evolve to
future ( dt/dλ>0), or in the classically forbidden negative-
root states with negative energy measured by local observers,
evolving into past ( dt/dλ<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 E>0 and are in the
positive-root states. However, the situation is more complex
in the case of Kerr naked singularities.
For distinguishing of the positive-root and negative-root
states in the case of Kerr naked singularities, and under the
inner horizon of Kerr black holes, we can use the time equa-
tion of motion (18); alternatively, the projection of the pho-
ton 4-momentum on the time-like tetrad vector of physical
observers can be used for this purpose [10,17,45]. The time
equation can be written in the form
dt
dλ=E{(r2+a2)−2ar[−a(1−m)]}.(20)
Now it is clear that in the regions of spherical orbits where the
bracket on the r.h.s. is positive, we have positive-root states
for E>0, while at regions where the bracket takes negative
values, the positive-root states must have the energy relative
to infinity E<0. Notice that in the case of E<0 the photons
with <0 have positively-valued impact parameter =
E.
We present the results of the determination of the covari-
ant energy of the SPOs later, and then we use them in the
2The orbital motion is allowed also for q=0andl2<a2when the
motion in the equatorial plane is unstable [9].
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879 Page 4 of 25 Eur. Phys. J. C (2018) 78:879
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
The simultaneous solution of the equations
R(r)=0,dR/dr =0,(21)
where R(r)denotes the r.h.s. of the equation (15), yields the
motion constants of the spherical orbits3
q=qsph(r;a2)≡−r3
a2
r(r−3)2−4a2
(r−1)2,(22)
=sph(r;a2)≡r3−3r2+a2r+a2
a(1−r).(23)
The functions (22), (23) determine the constants of motion
of the SPOs in dependence on their radius, hence, they rep-
resent the basic characteristics of the spherical orbits and
require careful analysis.
4.3 Existence of spherical orbits
First, we shall devote attention to the function in Eq. (22)
giving limits on the parameter q. Discussion of this function
was already performed in [58] for the case of the Kerr black
holes, here we extend the discussion to the case of the Kerr
naked singularities, including also the properties of the SPOs
under the inner horizon of Kerr black holes that were not
discussed in [58]. The function qsph(r;a)is well defined for
any radius r= 1,while it diverges at r=1fora= 1, with
lim
r→1qsph =∓∞ as a≶1.
In the case of extreme Kerr black holes, one can find by
substituting a=1 into the conditions (21) that the function
in (22) should be replaced by
qsph(r;a=1)=(4−r)r3,(24)
which is now continuous even for r=1, c. f. [58]. As we
shall see bellow, only non-negative values of the parameter
qpermit the motion of constant radius.4
The zeros of (22) determining the photon equatorial cir-
cular orbits are given by the condition
a=aco(r)≡1
2r(r−3)2.(25)
For a=0 this function gives the SPO around the
Schwarzschild black hole, located at r=3; if 0 <a<1,
3There exists a second family of solution of these equations, but it is
not physically relevant – see [17,58].
4The same applies to a more general case of the Kerr–de Sitter space-
times, see [17].
it determines one circular orbit under the inner black hole
horizon with radius
rph0=21−cos π
3−1
3arccos(2a2−1),(26)
and two circular orbits located above the outer black hole
horizon, which are the main subject of astrophysical interest.
The inner one being co-rotating, the outer being counter-
rotating, both being unstable with respect to radial pertur-
bations. Their radii rph+,rph−,where rph+<rh−,can be
expressed by the relation [7]
rph±=21+cos 2
3arccos(∓a).(27)
Above the outer horizon of the KBH spacetimes, the SPOs
are thus located at radii
rph+<r<rph−.(28)
In the extreme KBH case a=1, the counter-rotating orbit
is located at r=4, while the co-rotating orbit shares the
same (Boyer-Lindquist) radius r=1 with the black hole
horizon, although they are in fact separated by a non-zero
proper radial distance [7].
In the Kerr naked singularity spacetimes, only the counter-
rotating equatorial photon orbit exists. For a>1, we express
the photon orbit radii rph−by the expression [40]
rph−=21+cosh 1
3arg cosh(2a2−1).(29)
The co-rotating circular photon orbit should be (formally)
located at r=0,θ =π/2, representing thus the limit of
unstable circular equatorial orbits of test particles; however,
such an orbit has no physical meaning as it coincides with
the physical singularity of the Kerr spacetimes. Therefore, in
the KNS spacetimes, the SPOs can be located at all radii
0<rsph ≤rph−.(30)
4.4 Stability of spherical photon orbits
Stability of the spherical null geodesics against radial pertur-
bations is governed by the condition d2R/dr2<0 consid-
ered in the loci of the spherical orbits, i.e., by the extrema
of the function qsph(r;a). The function qsph(r;a)has one
local extreme qex =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., ˙
φ(r=3,θ =π/2)=0(c.f.[58]).
For the stability criterion, another extreme of the function
qsph(r;a)is relevant that is determined by the condition
a=astab(r)≡(r−1)3+1,(31)
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Eur. Phys. J. C (2018) 78:879 Page 5 of 25 879
0 1 2 3 4 5 6
0.0
0.5
1.0
1.5
r
a
apol r
astab r
aco r
ahr
Fig. 1 Characteristic functions a(r)determining the behaviour of the
function qsph. They govern existence and stability of the SPOs and
existence of polar spherical orbits. The event horizons for the black
hole spacetimes are given by the function ah(r), the function aco(r)
determines the equatorial circular photon orbits, the function apol(r)
determines loci of the polar SPOs with =0 crossing the rotary axis.
The function astab(r)governs the stability of the SPOs – the orbits
above/bellow the curve astab(r)are stable/unstable
which is implied also directly by the condition d2R/dr2=
0 determining the marginal stability of the spherical orbits.
The stability condition d2R/dr2<0 can be thus written
in the form a>astab(r). The function astab(r)governing
the marginally stable SPOs is increasing with increasing r,
having an inflexion point at the special radius r=1; its
behaviour is demonstrated in Fig. 1. It is immediately clear
that the marginally stable spherical orbits are located at r>1
in KNS spacetimes, while in the KBH spacetimes it must be
located under the inner horizon r<r−<1. The radius of
the marginally stable SPO, rms±, can be directly expressed
by the simple formulas
rms−=1−(1−a2)1/3for a2≤1(32)
or
rms+=1+(a2−1)1/3for a2≥1.(33)
The stable (unstable) spherical orbits are located for given
parameter aat r<rms±(r>rms±).
4.5 Polar spherical photon orbits
Now we consider behaviour of the function sph(r;a).In
the case of non-extreme Kerr BH spacetimes, this function
is monotonically decreasing in the stationary region above
the outer horizon and its point of discontinuity at r=1is
hidden between the black hole horizons. In the extreme KBH
case a=1, the equation (21) reduces to the form having no
discontinuity
sph =r(2−r)+1.(34)
For the KNS spacetimes, a>1, there is
lim
r→1∓sph =±∞
and the discontinuity occurs at r=1.
In order to find the special case of spherical orbits covering
whole the range of the latitudinal coordinate, reaching thus
the symmetry axis at θ=0, we have to find when the function
sph(r;a)takes the significant value of sph(r;a)=0 cor-
responding to photons with zero angular momentum, since
only such photons can reach the symmetry axis. We adhere
notation introduced in [17] and denote the radii of the polar
SPOs crossing the symmetry axis by rpol. These radii can be
found by solving the equation
a=apol(r)≡(3−r)r2
r+1.(35)
The function apol(r)has zeros at r=0 and at r=3, and a
local maximum at rpol =√3for
a=apol(max)≡6√3−9=1.17996.(36)
The solution of Eq. (35) can be written in whole the relevant
region of the rotation spacetime parameter a∈(0,apol(max))
in the form
rpol±=1+21−a2
3cos[π
3±1
3arccos a2−1
(1−a2
3)2
3],
(37)
where rpol+<rpol−.
In the case of KBH spacetimes, 0 <a<1, only the
formula for rpol−is relevant and it gives the only polar
SPO in the stationary region, where rpol−<3[58]. In
the KNS spacetimes with 1 <a<apol(max)=1.17996,
two polar SPOs exist at the radii rpol±given by Eq. (37).
For a=apol(max)these radii coalesce at rpol =√3. For
a>apol(max)no polar SPOs exist. Combining Eq. (22) and
Eq. (35) one can find the values of parameter qof the polar
orbits to be given by
q=qpol(r)≡r2(r+3)
r−1.(38)
The graph of the function (38) is depicted in Fig. 3.Itsmin-
imum value is qpol(min)=19.3923 for r=rpol and in the
limit points of its definition range it is
qpol(r)→∞ as r→1+
and
qpol(r=3)=27.
We shall still mention the polar SPOs in section devoted to
latitudinal motion.
The local extrema of the function sph(r;a)are deter-
mined by the relation (31), i.e., they are located at the same
radii as the local extrema of the function qsph(r;a).The
functions (25), (31), (35) are, together with the function
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879 Page 6 of 25 Eur. Phys. J. C (2018) 78:879
Fig. 2 Depiction of the
functions qsph(r;a)(left) and
sph(r;a)(right) for
appropriately chosen values of
the dimensionless spin a giving
characteristic types of their
behaviour
ah(r)≡2r−r2,(39)
determining the loci of the event horizons, illustrated in Fig.
1. Behaviour of the functions qsph(r;a)and sph(r;a)is
demonstrated for typical values of the dimensionless spin
parameter ain Fig. 2.
4.6 Spherical photon orbits with negative energy
For further discussion it is now necessary to determine the
loci of the SPOs with negative energy E<0 (Fig. 3). Sub-
stituting =sph into the time equation (20), we arrive to
equation
dt
dλ=E·T(r;m,a), (40)
where
T(r;m,a)≡r2(r+3)+a2m(r−1)
r−1.(41)
Clearly, the equation (40) describes the motion of photons
with positive energy E>0 directed to the future (dt/dλ>
0) on spherical orbits at radii r>1.5On the contrary, for
0<r<1 the condition dt/dλ>0 demands E<0 and
m<mzE(r;a), or E>0 and m>mzE(r;a), where
mzE(r;a)≡2r(3−r2)
a2(1−r).(42)
In the region of interest, 0 <r<1, the function mzE(r;a)
is increasing, and mzE(0;a)=0 and mzE(r;a)→∞as
r→1. As we shall see bellow, at the interval 0 <r<
1, the inequality m>mzE(r;a)is inconsistent with the
reality condition for the latitudinal motion, hence the latter
alternative is irrelevant and the range 0 <r<1 corresponds
to region of the SPOs with E<0. Of course, this region must
be hidden under the inner black hole horizon in case of the
Kerr black holes. From discussion of the stability of the SPOs
it follows that all the SPOs with E<0 must be stable against
5Here we restrict our discussion on the stationary region >0.
radial perturbations. Of course, photons from the stable SPOs
cannot escape to infinity due to a perturbation.
4.7 Spherical photon orbits with E=0
As we have shown above, the radii 0 <r<1 correspond
to range of SPOs with negative energy, which suggests that
there are SPOs with zero energy at radii r=1. It is supported
by the fact that the functions (22,23,41) diverge at this radii.
For this reason, let us consider the Carter equations of motion
for the photons (7–9) with explicitly expressed energy Eand
revise the discussion assuming E=0. The conditions (21)
then imply r=1 as expected, and the ratio of the motion
constants of the SPO with E=0 in the KNS spacetimes
reads
2/Q=a2−1.(43)
Notice that the impact parameters and qare not defined in
this special case. In the extreme KBH spacetimes, =0is
required for arbitrary Q>0 for the spherical orbits at r=1.
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 coordi-
nate 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 num-
ber 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 θ0,π−θ0,
crossing repeatedly the equatorial plane or even being con-
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Eur. Phys. J. C (2018) 78:879 Page 7 of 25 879
Fig. 3 Dependence of the
parameter qpol of the polar
spherical orbits on its radii rpol
depicted with corresponding
dimensionless spin apol (left)
and its projection qpol(rpol )
onto the (r−q)plane (right)
132 3
r
19.4
27
qpol
fined to the equatorial plane6, or of the so called vortical type,
where the photons oscillate ’above’ or ’bellow’ the equato-
rial plane between two pairs of cones coaxial with the sym-
metry axis of the spacetime, with latitudes θ1,θ
2,θ1<θ
2
and π−θ1,π −θ2. The special case is the vortical motion
along the symmetry axis θ=0, or the motion at any con-
stant 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.
First, we demonstrate that for the spherical orbits there
is q≥0 necessarily, i.e., the motion is of the orbital type.
This can be shown easily using the Carter equation of the
latitudinal motion 16), from which it can be immediately
seen that for q<0 the condition M(m)≥0 ensuring the
existence of the latitudinal motion is fulfilled only if
a2−q−l2>0.(44)
On the other hand, we can show that a2−qsph(r;a)+
l2
sph(r;a)<0 so that the possibility q<0 must be rejected
[17,58].
In fact, there is even more restrictive condition for the
motion constants than that given by (44). To show this, let
us express the reality condition M(m)≥0 using linearity in
parameter qby
q≥qm(m;a,) ≡m2
1−m−a2.(45)
The equality gives the turning points in the latitudinal coor-
dinate. Notice that using the coordinate m=cos2θ,wehave
to restrict the range of the solutions to m∈0;1. Of course,
there is obvious zero of (16) given by m=0, emerging due
to the used substitution, which indicates just a transit through
the equatorial plane. Behaviour of the function qm(m;a,)
in the limit points of the interval 0;1is as follows:
qm(0;a,) =0,
6This 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.
1
m
a2
qm
aa0a
0
Fig. 4 Typical behaviour of the function qm(m;a,)for appropriately
chosen values of the parameter
lim
m→1qm(m;a,) =∞.
Another zero of qm(m;a,) is at
m=a2−2
a2
for ||≤a.The local extrema can be expressed from the
condition dq/dm=0 in an implicit form
2=a2(1−m)2,(46)
or, equivalently
=min±≡±a(1−m). (47)
Clearly, the extrema exist for ||≤a.The subscript ’min’ in
(47) indicates that these extrema must be minima, as follows
from the inequality d2qm/dm2=2a2/(1−m)>0. The
values of these minima read qmin =−(a−l)2for 0 ≤≤a,
or qmin =−(a+l)2for −a≤≤0.The above results can
be written in a compact form
q≥qm(min)≡⎧
⎨
⎩
−(|l|−a)2,for |l|<a;
0,for |l|≥a.
(48)
This is the relation giving stronger limitation on the impact
parameter in case q<0 than that given by (44).
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879 Page 8 of 25 Eur. Phys. J. C (2018) 78:879
From the behaviour of the function qm(m;a,), 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 rela-
tion (48), the negative values of the parameter qallowing the
latitudinal motion exist for ||≤a, where the lowest value is
q=−a2and occurs for =0. However, as can be verified
by calculation, at radii where |lsph|≤a, there is qsph >0,
which confirms that the vortical motion of constant radius is
impossible for the spherical orbits.
The latitudinal turning points of the SPOs are given by
zeros of the r. h. s. of Eq. (16) with substitution of =
sph. The maximum latitude reached by a photon following
a spherical null geodesic at a particular radius can be inferred
from the relation
m=mθ(r;a),
where the latitudinal turning function (relevant in the range
m∈(0,1)) is defined by
mθ(r;a)≡r2
a2
4a2−9r+6r2−r3
r3−3r+2a2+2(2r3−3r2+a2)
.
(49)
The function mθ(r;a)is real everywhere in the stationary
region. Its zeros are determined by the function aco(r).For
KBHs (0 <a<1), the function mθ(r;a)has two local
maxima (Fig. 5a); one under the inner black hole horizon,
located at radius determined by the function astab(r), i.e., at
r=rms−. The second maximum of mθ(max)=1 is deter-
mined by the function apol(r). In extreme KBHs case where
a=1 (Fig. 5b), the function mθ(r;a)has one local maxi-
mum mθ(max)=1 given by apol(r). For fixed rotation param-
eter a, the radii rpol±are given implicitly by relation (35) and
explicitly by Eq. (37). The corresponding impact parameter
qpol =qsph(rpol(a), a)is represented in Fig. 3; the other
motion constant, lpol =0 by definition.
For the KNS spacetimes with a<apol(max)(Fig. 5c), two
maxima exist at mθ(max)=1 that are given by the function
apol(r), and one local minimum located at r=rms+.If
a=apol(max), the three extrema coalesce into maximum
mθ(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>apol(max)(Fig. 5e, f),
the function mθ(r;a)has a local maximum at mθ(max)<1
at r=rms+. In such KNS spacetimes the polar SPOs cannot
exist.
5.2 Azimuthal motion
Finally, let us consider the relations of the function mθ(r;a)
with a function mφ(r;a)determining the latitude at which a
turning point of the azimuthal motion occurs. The equation
(17), after performing substitution =sph, can be written
in the form
dφ
dλ=E·Φ(r;a,m), (50)
where
Φ(r;a,m)≡r2(r−3)+a2m(r+1)
a(1−m)(1−r).(51)
The condition dφ/ dλ≥0 implies
m≤mφ(r;a),
where the azimuthal turning function is defined as
mφ(r;a)≡(3−r)r2
a2(r+1).(52)
The change of sign in denominator of (51), while crossing the
divergence point r=1, now plays no role, since, as follows
from the preceding discussion, in order to have dt/dλ>
0, the relation E/(1−r)<0 holds at any radius rsince
E<0(E>0)at r<1(r>1). Therefore, the motion in
the φ-direction is fully governed by the function (52) and the
inequalities presented above. If we compare the expressions
in (35) and (52), we see that
apol(r)=amφ(r;a). (53)
The turning point of the azimuthal motion thus exist for all
KBHs and KNSs. The functions mθ(r;a)and mφ(r;a)have
common points at r=1, where they have value mθ(1;a)=
mφ(1;a)=1/a2, and at the local extrema determined by
the curve apol(r). As is evident from Eq. (53), the function
mφ(r;a)has a local maximum at r=√3=rpol with
mφ(max)=(6√3−9)/a2=a2
pol(max)/a2.
The position of the SPOs with turning point of the
azimuthal motion is represented in Fig. 14. The latitudinal
angle where the azimuthal turning point occurs is given by
the relation
θturn(φ)(r;a)=±arccos ⎛
⎝(3−r)r2
a2(r+1)⎞
⎠.(54)
The existence of the azimuthal turning points enables exis-
tence of “oscillatory” orbits with change of the azimuthal
angle for half period in the latitudinal motion φ(a)=0.
5.3 Shift of nodes
Now we examine the dragging of the nodes, i.e., we thus
determine the azimuthal angle between the two points, where
123
Eur. Phys. J. C (2018) 78:879 Page 9 of 25 879
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 0.9
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 1
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 1.1
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 1.18
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 2
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
r
m
a 4
(a) (b)
(c)
(f)(e)
(d)
Fig. 5 The graphs of the characteristic functions mθ(r;a)(full black
curve), mφ(r;a)(dashed black curve) and mzE(r;a)(grey curve) are
given for appropriately chosen values of the spin parameter a, demon-
strating all qualitatively different cases of their behaviour. Grey ver-
tical line is an asymptote of mzE(r;a). Dark shading demarcates the
dynamic region <0, light shading corresponds to negative energy
orbits. Region of the prograde orbits with dφ/dλ>0 is located under
the curve mφ(r;a). Therefore, all the SPOs with negative energy are
prograde
the ascending, i.e., the ’northwards’ directed parts of the pho-
ton track intersects the equatorial plane. For this purpose we
need to evaluate the change in the azimuthal coordinate φ
by integrating dφ/dm expressed from the Carter‘s equations
16,17 with =sph(r;a),q=qsph(r;a)inserted.7The
exact expression for the nodal shift φ was found in [58]–
with our notification it reads
7The more general case of the function M(m)with non-zero cosmo-
logical parameter was studied in detail in [17].
φ =4
√m+−m−2r−asph
Km+
m+−m−
+
a(1+m+)m+
m+−1,m+
m+−m−.(55)
Here m±are the positive and negative roots of the function
M(m), while
K(s)=
π/2
0
(1−ssin2θ)−1/2dθ(56)
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879 Page 10 of 25 Eur. Phys. J. C (2018) 78:879
(a) (b) (c)
(f)(e)(d)
Fig. 6 Dependence of the node shift φ on the radius of the spheri-
cal orbit. A discontinuity occurs at r=rpol±the jump change is 2π
between the single point and each of the end points of the curve, c. f.
[58]. For a=apol(max)=1.18 the points of the discontinuity coalesce
at r=rpol =√3.For a>apol(max)the function is continuous. The
zero of φ first occurs for parameter a=aφ=0(min)=1.179857
slightly lower than apol(max);it indicates an existence of an oscillating
orbit, which forms a boundary between prograde (φ > 0) and retro-
grade orbits (φ < 0.)Fora<apol(max)the boundary is at r=rpol−.
and
(v, s)=
π/2
0
(1−vsin2θ)−1(1−ssin2θ)−1/2dθ(57)
are the complete elliptic integrals of the first and third kind,
respectively. The dependence of the nodal shift φ on the
radial coordinate qualitatively differs in dependence on the
presence of the polar orbits. If they are present, discontinu-
ities occur at the radii rpol±– see Fig. 6a–d. The jump drop is
always 4π, and the single points, which are inherent values in
the discontinuity points rpol±, are halfway between the end-
points of the interrupted curve. This is a simple consequence
of the singular behaviour of the Boyer-Lindquist coordinates
at the poles, as explained in [58]. For apol(max)these dis-
continuities coalesce at r=rpol , for which the nodal shift
function φ < 0 (Fig. 6e).
5.4 Periodic orbits
The nodal shift function becomes to be continuous for the
KNS spacetimes with a>apol(max). Therefore, in KNS
spacetimes with a>apol(max)∼1.18, there exist “oscil-
latory” trajectories with φ =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 octal-like
shape (see, e. g., Fig. 9case k=0). However, a detailed cal-
a0min apol max 1.18
1.7
1.7147
3
1.18 2 3 4
1.66
1.7147
3
1.75
a
r
rms
r0
Fig. 7 Radii rφ =0of orbits with zero nodal shift φ in dependence
on the spin parameter aand comparison with the loci of marginally
stable orbits rms.For given dimensionless spin athe radii of the stable
orbits are r<rms(a). Hence all such orbits are stable. Note that the
detail is approx. 104-times horizontally stretched
culation using numerical procedure reveals that the radius
rφ=0of the zero nodal shift φ =0 first occurs for
a=aφ=0(min)≡1.179857 at r=rφ =0(max)=1.71473
(see Fig. 6d). In Fig. 7we show that the function rφ=0(a)is
descending, hence the subscript ’φ =0(max)’. Notice
that the value of aφ=0(min)is only very slightly lower
than apol(max). Since the point rφ=0(max)and correspond-
ing point of the 4π-discontinuity rpol+(aφ=0(min))are
infinitesimally separated, we can claim that the SPO at
rpol+(aφ=0(min))is a special case of polar oscillatory orbit.
We illustrate its trajectory explicitly in Fig. 10 and for com-
parison we give illustration of the orbit at r=rpol for
a=apol(max).
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Eur. Phys. J. C (2018) 78:879 Page 11 of 25 879
1.1798 a0min 1.17986 apol max 1.17996
1
0.982741
0 1 apol max
a
1
2
3
4
5
6
2
rpol
rpol
π
Fig. 8 Shift of nodes for the polar spherical orbits rpol±represented
by the isolated points in Fig. 6in independence on the dimensionless
spin a.
In the Fig. 8, the nodal shift is shown for the polar spher-
ical orbits rpol±, 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 φ[rpol+(a)]=2π. According to
Figs. 6,7,8, we can summarize that for the very tiny inter-
val aφ=0(mi n)≤a≤apol(max)there exist orbits with
φ < 0,i. e., globally retrograde, with radii rφ=0(max)<
r<rpol+(a), followed by orbits with φ > 0, i. e., glob-
ally prograde, at rpol+(a)<r<rpol−(a), and again orbits
globally retrograde with r>rpol−. Otherwise, the SPOs of
this class of the KNS spacetimes possesses no new essen-
tial features in comparison with the other cases, therefore, its
character is explained sufficiently (Figs. 1,10).
The radius of the oscillatory orbits rφ=0is the only
point dividing the trajectories which are globally prograde
(φ > 0), and those that are globally retrograde (φ <
0) in case a>apol(max). For the KNS spacetimes with
aφ=0(min)<a<apol(max )the role of such dividing points
play the radii rφ=0and the discontinuity points rpol±.For
a<aφ=0(min)only the radius rpol−is the divider, how-
ever, the oscillatory orbits are not possible (c.f. orbits in the
BH background with a=0.9atrrpol−,r=rpol−and
rrpol−in Fig. 11). The function rφ=0(a)is compared in
Fig. 21 with another relevant functions.
Of course, another closed paths corresponding to the gen-
eral periodic orbits occur whenever
nφ =2mπ. (58)
In Fig. 9, we present basic types of the closed SPOs for
several ratios k=m/n, giving the cases of prograde (k>0),
oscillate (k=0), and retrograde (k<0) orbits. Of course, of
special astrophysical relevance could be the oscillate orbits
with φ =0, as they immediately return to the starting point
at the azimuthal coordinate related to distant static observers.
Further, we consider a possibility of orbits having a plu-
rality of revolutions about the spacetime symmetry axis per
one latitudinal oscillation (|k|>1), which are of a helix-like
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 φ-direction. Note that the cases
k≤−1 cannot be realized, since from Eq. (55) it follows
that the nodal shift φ > −2π. This is demonstrated in Fig.
12 on the left, which depicts the shift of nodes at r→rph−.
5.5 Spherical photon orbits with zero energy and their
nodal shift
For the SPOs with zero energy, located at radii r=1, the
constants of motion have to fulfil the relation 2/Q=a2−1.
The latitudinal equation (8) with the substitution m=cos2θ
then reads
1/2dm
dλ2
=M(m;a,Q)≡Qm(1−a2m), (59)
and the azimuthal equation (9) reads
dφ
dλ=Φ(m;a,) ≡1−a2m
(a2−1)(m−1).(60)
Therefore, the turning points of the latitudinal and the
azimuthal motion coalesce at m=1/a2, in concordance
with the result for intersection of the functions mθ(r;a),
mφ(r;a), as we have found earlier. In this case, the point
with dφ/dλ=0 is not a real azimuthal turning point, it
is only the point of vanishing of the azimuthal velocity, as
shown in Fig. 13.
The complete change of the azimuthal coordinate per one
latitudinal oscillation, denoted φ, is now given by the rela-
tion
φ =2πa
√a2−1−1,(61)
which corresponds to values of the local maxima of the func-
tion (55). Behaviour of this function is shown in Fig. 12.
5.6 Summary of properties of spherical photon orbits
We can summarize properties of the SPOs in the following
way. The covariant energy E>0(E<0) have the SPOs
at r>1(r<1) for all KBHs and KNSs; there is SPO
with E=0atr=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<rms−, the unstable ones at
rms−<r<rph0, all being co-rotating (prograde), none of
these orbits can be polar. Above the outer event horizon only
123
879 Page 12 of 25 Eur. Phys. J. C (2018) 78:879
k=6 k=2 k=1,E>0
1
2
3
k=1,E<0k=2/3k=1/2,E<0
1
2
3
k=0 k=−1/2k=−2/3
Fig. 9 Various types of the periodic SPOs characterized by k-the
number of revolutions about the φ- axis per one latitudinal oscillation.
Here and hereafter we shall depict in green the sphere of the photon orbit
with a negative energy, and in red/blue that parts of the spheres, where
the photon moves in the positive/negative azimuthal direction, having
a positive energy. Figures include cases of the prograde orbits (k>0),
both with and without the turning point of the azimuthal motion, the
oscillating orbit (k=0), and the retrograde orbits (k<0), with and
without turning point in the azimuthal motion. Cases |k|>> 1cor-
respond to orbits with multiple revolution about the spin axis per one
latitudinal oscillation, tending to have a helix-like shape. However, the
case k≤−1 can not occur, hence such orbits are solely prograde.
Details, including the minimum attained latitude θmin, and latitude θφ
at which the turning point in the φ-direction occurs, sign of the pho-
ton’s energy Eand the time period of the orbit (to be discussed bellow),
characterizing the orbits are presented in the attached Table 1
unstable SPOs spread in the interval rph+≤r≤rph−.They
are purely co-rotating for rph+<r<rpol−; with turning
point in the azimuthal direction but globally counter-rotating
(retrograde) for rpol−<r<3; and purely counter-rotating
at 3 <r<rph−. One polar orbit exist above the outer
horizon of the KBH spacetimes.
In the KNS spacetimes the SPOs exist at 0 <r<rph−,
being stable/unstable for r≶rms+. For the KNS spacetimes
with 1 <a<aφ=0(min)the SPOs at r<1 are purely
co-rotating; for 1 <r<rpol+they have turning point in
the azimuthal direction but are co-rotating globally; between
the inner stable and the outer unstable polar orbit, i. e., at
rpol+<r<rpol−, they are purely co-rotating; at rpol−<
r<3 they have turning point in the azimuthal direction and
are counter-rotating globally; for 3 <r<rph−they are
purely counter-rotating. Two polar orbits can exist in such
KNS spacetimes.
There exists extremely small interval of the spin parameter
aφ=0(min)<a<apol(max ), for which the KNS spacetimes
possess two polar orbits and one orbit of zero nodal shift
rφ=0(a)–for1 <r<rφ =0(a)there are orbits with
turning point in the azimuthal motion, being co-rotating in
global; such orbits occur also for rφ =0<r<rpol+, being
globally counter-rotating; at the radii rpol+<r,thesame
behaviour occurs as in the previous case.
For the KNS spacetimes with a>apol(max), the SPOs at
r<1 are purely corotating; at 1 <r<3 there are orbits with
turning points of the azimuthal motion, which can be both
co-rotating or counter-rotating globally, in dependence on the
spin a; spherical orbits at r>3 are purely counter-rotating.
For the KNS spacetimes with a>3 there is 3 <rms+,
hence stable purely counter-rotating spherical orbits exist at
3<r<rms+. All the above results are clearly illustrated in
Fig. 14.
123
Eur. Phys. J. C (2018) 78:879 Page 13 of 25 879
Tabl e 1 Characteristics of the
periodic spherical photon orbits
in Fig. 9
karqθmin θφsign Et
61.01 1.11 1.58 13.15 22.9◦–+174.76
21.11.72 0.62 16.42 8.5◦–+130.13
11.11.22 −0.16 30.32 1.6◦6.7◦+132.98
11.10.77 3.26.85 49.8◦–−114.08
2/31.18 1.24 −1.47 41.44 12.5◦21.2◦+126.36
1/21.10.53 2.25 0.92 64.6◦–−15.69
0√31.69 −3.63 30.80 32.3◦47.1◦+119.36
−1/21.12.52 −0.72 24.38 8.2◦32.2◦+129.45
−2/3√34.34 −7.31 8.70 67.5◦–+134.30
(a) (b)
Fig. 10 Illustration of the special cases of the polar SPOs: oscillatory
orbit with a=1.17986 =aφ=0(min)at r=1.7147 =rφ =0(max)a
and the SPO with a=1.17996 =apol(max)at the coalescing radius
r=rpol+=rpol−=√3=rpol b. Note that the latter orbit is globally
retrograde (c. f. detail in Fig. 6e)
r=2.5 Δφ= 518.4◦r=2.56 = rpol−Δφ=151.7◦r=2.6 Δφ=−212.5◦
Fig. 11 Demonstration of the 4πjump drop in value of the φ func-
tion when overpassing the discontinuity point at r=rpol−in case of
the Kerr BH with the spin parameter a=0.9 (see Fig. 6a). In general,
the SPOs are globally prograde at r<rpol−and globally retrograde
at r>rpol−, however, there is not zero nodal shift at r=rpol−.The
jump of the shift is a simple consequence of a singular behaviour of the
φ- coordinate at the poles θ=0,π
123
879 Page 14 of 25 Eur. Phys. J. C (2018) 78:879
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
1.5
2.0
a
2
π
Fig. 12 Behaviour of the function (61) in dependence on the rotational
parameter a,depicted as the full curve, which represents the shift of
nodes φ of orbits with zero energy.The dot-dashed curve is a depiction
of the lowest negative shift of nodes of a retrogressive spherical orbits
approaching the position of the equatorial circular counter-rotating orbit
at r=rph−.It can be seen that the increase of the rotational parameter
aweakens the negative shift due to strong dragging of the spacetime.
The shift is always φ > −2π
6 Classification of the Kerr spacetimes due to
properties of the spherical photon orbits
Using the knowledge of the behaviour of the above described
functions, we are now able to summarize the properties of the
SPOs in dependence on the value of the Kerr spacetime spin
parameter a, giving thus the corresponding classification of
these spacetimes. The following description of the individual
classes is supplemented by an explicit illustrations of the
spatial structures of the SPOs, which can regarded as a spatial
representation of the Fig. 14. In the figures we use the so
called Kerr-Schild coordinates x,y,zthat are connected to
the Boyer-Lindquist coordinates r,θ by the relations
x2+y2=(r2+a2)sin2θ, z2=r2cos2θ. (62)
We present a meridional sections of the SPOs with the y-
coordinate being suppressed, hence, the surfaces of constant
Boyer-Lindquist radius are depicted as an oblate ellipses.
Class I: Kerr black holes 0 <a<1(seeFig.15)
endowed with the SPOs of two families. First family is
limited by radii 0 <r<rph0<r−,where
rph0≡4sin
21
6arccos(1−2a2)(63)
is the radius of co-rotating equatorial circular photon
orbit located under the inner black hole horizon r−(green
dot). These are the orbits with negative energy E<0;
at 0 <r<rms−they are stable (rich green area), for
rms−<r<r−they are unstable with respect to radial
perturbations (light green area). The radius rms−,given
by (33), denotes marginally stable spherical orbit with
negative energy. All the first family orbits are prograde
Tabl e 2 Characteristics of the spherical photon orbits with zero energy
in Fig. 13
a2/Qφ/2πθ
min
1.0001 0.0002 69.70.8◦
1.001 0.002 21.42.6◦
1.02 0.042 4 11.5◦
1.06 0.125 2 19.5◦
1.10.21 1.424.6◦
1.15 0.33 1 30.0◦
1.34 0.80.541.8◦
380.06 70.5◦
and span small extent in latitude in vicinity of the equa-
torial plane – its maximum at r=rms−is approaching
the value θmin(zE)(a=1)=arccos 2√3−3=47.1◦
as r→1 when a→1. Here and in the following,
we restrict our discussion on the ’northern’ hemisphere
(0 ≤θ≤π/2), the situation in the ’southern’ hemi-
sphere is symmetric with respect to the equatorial plane.
Second family of the SPOs spreads between the inner co-
rotating (red dot) and outer counter-rotating (blue dot)
equatorial circular orbits with radii given by (27). There
is no turning point of the azimuthal motion for orbits with
radii rph+<r<rpol−, where rpol−is the radius of the
polar spherical orbit (purple ellipse) given by (37), and
all such photons are prograde. At radii rpol−<r<3
(black dotted incomplete ellipse), there exist orbits with
one turning point of the azimuthal motion in each hemi-
sphere, such that the photons become retrograde as they
approach the symmetry axis. At radii 3 <r<rph−,all
spherical orbits are occupied by retrograde photons with
no turning point of the azimuthal motion.
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,8where
also non-spherical bound photon orbits with two turn-
ing points of the radial motion exist. Such orbits are not
present in any kind of the most general case of the Kerr-
Newmann-(anti) de Sitter black hole spacetimes. At radii
1<r<rph−=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 appar-
ently corresponds to the SPO with zero energy, which is
marginally stable and prograde. This zero energy orbit
has minimum latitude θmin(zE)(a=1)=47.1◦; note
that for a>1, there is θmin(zE)(a)=arccos (1/a).The
8From the perspective of the locally non-rotating observer such pho-
tons appear to be retrograde, see [17].
123
Eur. Phys. J. C (2018) 78:879 Page 15 of 25 879
Fig. 13 The SPOs with zero
energy, located solely at radius
r=1, are presented for KNS
spacetimes with increasing
value of the rotational parameter
a. The turning points of the
latitudinal and the azimuthal
motion coalesce in these cases,
hence the turning loops have
shrinked into peaks. In the
attached Table 2we demonstrate
the descending total change in
azimuth per one latitudinal
oscillation, and the descending
range in the latitudinal motion,
with increasing spin parameter
a. In some cases closed
spherical orbits emerge due to
properly chosen parameters of
the SPOs
a=1.0001 a=1.001
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879 Page 16 of 25 Eur. Phys. J. C (2018) 78:879
Fig. 14 Characteristic functions a(r)with supplementary informa-
tions about stability and latitudinal and azimuthal motion for wider
interval of the parameter a(above) and in greater detail (bellow). Sig-
nificance of the characteristic functions is given in Fig. 1. Gray shad-
ing demarcates the dynamic region, colouring highlights region of the
SPOs. Orbits with E<0,all being purely prograde, i. e., without turning
points in φ-direction, are highlighted in green; purely prograde orbits
with E>0 in red; orbits with turning points in φ-direction and with
E>0 in purple and purely retrograde with E>0 in blue. Full/faint hues
correspond to stable/unstable orbits. Thin black curve is the zero nodal
shift function azs(r)dividing regions of globally prograde/retrograde
orbits marked by +/−sign
SPOs at radii 0 <r≤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<aφ=0(min)=
1.17986 (see Fig. 17). Two polar SPOs appear at radii
rpol+,rpol−(inner and outer purple ellipse, respectively)
given by Eq. (37). Photons at 1 <r<rpol+are sta-
ble, they have 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 lat-
itude reads θmin =arccos √mθ(r;a). At the region
θmin ≤θ<θ
φ, the photon motion is in negative φ-
direction (deep blue area), while for θφ≤θ≤π/2itis
in positive φ-direction (deep red area). The break point
dividing the globally prograde orbits from the globally
retrograde ones is at r=rpol+. The motion constants
sph →−∞,qsph →∞,asr→1 from the right,
and sph =0, qsph =27 at r=rpol+. The orbits
at the radii rpol+<r<rpol−are prograde, with no
change in the azimuthal direction, for rpol+<r<rms+
being stable (deep red area), for rms+≤r≤rpol−being
unstable (light red area). The function mθ(r;a)has a
local minimum at r=rms+, hence, the marginally sta-
ble 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 rms−at the KBH spacetimes,
where they have the widest extent (see detail in Fig. 15).
The motion constant qsph of photons on this orbit cor-
responds to the local minimum 0 <qsph(min)<27
of the function defined in (22), and the local maximum
0<
sph(max)of the function defined by (23). The turn-
ing point of the azimuthal motion appears for the SPOs
at the radii rpol−<r<3, and such SPOs appear to be
retrograde as whole. The motion constants for r=rpol−
are qsph(min)<qsph <27, and sph =0. At r=3,
there is qsph =27, corresponding to the local maximum
of (22), i.e., to the photons crossing the equatorial plane
with zero velocity component in the φ-direction ([58]),
and sph <0. The SPOs in the region 3 <r<rph−
are purely retrograde with sph <0 and qsph →0as
r→rph−.
Class IV: KNS spacetimes with aφ=0(mi n)≤a<
apol(max)=1.17996. In the limit case a=aφ=0(mi n),
there exist radius r=rφ=0(max )=1.7147 of spherical
orbit with zero nodal shift φ =0, infinitesimally dis-
tant from the 4π-discontinuity point rpol+(see Fig. 6d).
The radius rpol+therefore corresponds to special case of
polar oscillatory orbit. For aφ=0(min)<a<apol(max),
the radius which separates the globally prograde SPOs
from the globally retrograde ones is at r=rφ =0
rφ=0(max)<rpol+(see the detail of Fig. 14). For
123
Eur. Phys. J. C (2018) 78:879 Page 17 of 25 879
Fig. 15 Structure of the SPOs in the KBH spacetime with a=0.8
corresponding to the Class I. Colouring is as in Fig. 14 - in green it is
depicted the region of the spherical orbits with negative energy E,in
red the region of photon prograde motion in the azimuthal direction,
and in blue the region of the retrograde azimuthal motion. Note that
not all the orbits are purely prograde, nor purely retrograde, but can
have a turning point of the azimuthal motion, hence the colouring of a
particular curve representing some fixed Boyer-Lindquist radius rcan
be change. Full/light hues correspond to stable/unstable orbits. Grey
shading demarcates the area where gtt ≥0 (ergosphere), grey curve
is its boundary (ergosurface). Purple curve is the polar orbit. Loci of
turning points of the azimuthal motion are designated by red curves,
loci of turning points of the latitudinal motion are distinguished by blue
curves. In all figures some outstanding radii are highlighted, namely
r=1 (full black ellipse), r=3 (black dotted incomplete ellipse).
The ring singularity is depicted by the horizontal abscissa, the spin
axis by the vertical dashed line. The bold dots represent the photon
equatorial circular orbits, namely, the co-rotating orbits with negative
energy E<0 (green), the co-rotating orbits with E>0atr=rph+
(red), and the counter-rotating orbits at r=rph −(blue). In addition, we
included images of a possible Keplerian accretion discs, represented by
the first family of the equatorial circular orbits of the test particles [40]
(bold black horizontal abscissas). Their inner edge is the marginally
stable circular orbit
rφ=0<r<rpol+there appear globally retrograde
orbits. The other properties remain the same as in pre-
vious case and the SPOs structure is represented by the
Fig. 17.
Class V: KNS spacetimes with 1.17996 =apol(max)≤
a<3.For a=apol(max), the two polar SPOs coalesce
at r=rpol =√3(seeFig.18 above).
For KNS spacetimes with a>apol(max), there are no
spherical polar orbits. As a consequence, no purely prograde
spherical orbits, neither stable nor unstable, are possible. The
radius which separates the globally prograde spherical orbits
Fig. 16 Structure of the SPOs in the extreme Kerr BH a=1
Fig. 17 Structure of the SPOs in the KNS spacetime with a=1.1
corresponding to the Class III
from the globally retrograde ones is at r<rφ=0(max)(black
dot-dashed curve in Fig. 18 below) and it slowly decreases
as a→∞(see Fig. 7). In addition to this, the discussion is
qualitatively same as in the previous case.
Class VI: Kerr naked singularity spacetimes with a≥3.
For a=3 the local extrema of the function qsph(r;a)
coalesce at the inflex point at r=3=rms+with qinf =
123
879 Page 18 of 25 Eur. Phys. J. C (2018) 78:879
Fig. 18 Structure of the SPOs in the KNS spacetime with a=
apol(max)(above) and with a=1.7 (below) corresponding to the Class
V
27, where it becomes local minimum qsph,min =27 for
a>3. The local maximum is then at rms+>3(see
Fig. 2), which is also locus of the local maximum of
latitudinal turning function mθ(r;a)(Fig. 5f) at the radius
of the marginally stable spherical orbit. The orbits at 3 <
r<rms+are stable and purely retrograde; in the range
rms+<r<rph−, there are unstable retrograde orbits
(see Fig. 19).
We present a systematic construction of the spherical pho-
ton trajectories for the outstanding radii and other appropri-
Fig. 19 Enlargement of part of structure of the SPOs in the KNS space-
time with a=4 corresponding to the Class VI. The spin axis is outside
of the drawing. Note the reversed order of orbits r=3 (black dotted)
and r=rms+(black dashed)
ately chosen representative radii for the KNS spacetime of
the Class III with dimensionless spin parameter a=1.1in
Fig. 20. Basic characteristics of these orbits, i. e., their radii
r, impact parameters , q, total nodal shift φ, minimum
attained latitude θmin, latitude of change in the φ-direction
θφ, sign of energy Eand type, are presented in Table 3.
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 vicin-
ity 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 trap-
ping process was for the Kerr superspinars (naked singularity
spacetimes) studied in detail [50] where also possible self-
irradiation of accreting matter was briefly discussed.9Note
9Recall that even general self-irradiation (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].
123
879 Page 20 of 25 Eur. Phys. J. C (2018) 78:879
Tabl e 3 Characteristics of the
spherical photon orbits in Fig.
20
rqφθmin θφsign EType
0.95.452.5 469◦36.7◦–−1Stable
1.15 −0.950.5 426◦7.4◦14.1◦+1Stable
rpol+=1.24 0 27 728◦0◦0◦+1 Stable,in.polar
1.40.517.8 912◦6.9◦–+1Stable
rms =1.54 0.66 =max 16 =qmin 781◦9.1◦–+1 Marg.stable
20.33 18.8 630◦4.3◦–+1 Unstable
rpol−=2.2 0 21 227◦0◦0◦+1 Unst.,out.polar
2.6−0.925−189◦10.4◦38.0◦+1 Unstable
3−2.2=−2a27 =qmax −223◦22.6◦90◦+1 Unstable
3.4−3.824−246◦37.0◦–+1 Unstable
that the self-irradiation 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 pho-
ton orbits that return to a fixed azimuthal position, and the
self-irradiation thus occurs repeatedly at the fixed position
relative to distant observers.
7.1 Connection of the spherical photon orbits and the stable
circular geodesics
The existence of the trapped photon orbits motivates us to
examine the spread of the SPOs in relation to a possible dis-
tribution of radiating matter related to accretion disks. Of
special interest are the stable equatorial circular orbits of test
particles, which are assumed to be governing the structure
of the thin accretion (Keplerian) disks. Particularly impor-
tant are the marginally stable circular orbits, representing the
inner boundary of the Keplerian disks – if a mass element
of a Keplerian disk reaches the marginally stable orbit after
loss of the energy and angular momentum due to viscous fric-
tion, any additional loss of its energy causes its direct free
fall onto the naked singularity (black hole). The thorough
discussion of the equatorial circular orbits in the field of the
Kerr naked singularities was done in [40]. Since the SPOs
can exist only at radii r<rph−, we can restrict our attention
to the equatorial circular orbits of the so called first family,
which are co-rotating at large distance from the KNS, but
they could become counter-rotating at close vicinity of the
ring singularity; the unstable orbits of the first family could
extend down to the ring singularity [40]. On the contrary, the
circular orbits of the second family, which are all counter-
rotating, are located at r>rph−and hence are irrelevant for
the study of the interaction of the SPOs with the Keplerian
disks. The marginally stable circular orbits of the first family
are determined by [8,40]
rmsc =3+Z2−(3−Z1)(3+Z1+2Z2), (64)
where
Z1=1+3
1−a2(3
√1+a+3
√1−a),
Z2=3a2+Z2
1.
We introduce the subscript ‘msc’ in order to distinguish
the marginally stable SPOs. In the case of thick accretion
disks governed by axi-symmetric toroidal structures of per-
fect fluid, the radiation matter could reach the radius of the
marginally bound orbit at [8,40]
rmb =2+a+2√1+a.(65)
In the present paper, we focus our attention on the Keplerian
disks, because the case of toroidal accretion configurations
requires more detailed study of the relation of the SPOs and
the orbiting matter, as in some Kerr spacetimes the existence
of the SPOs could be excluded by the presence of the accre-
tion torus.
The stable circular orbits of the first family are located at
r>rmsc. It is relevant from the astrophysical point of view
that these stable circular orbits interfere with the area of the
stable SPOs at r<rms for KNS spacetimes with rotation
parameter 1 ≤a≤ai=4.468, since just in these space-
times there is rms >rmsc. The limits a=1,aicorrespond to
equality rms =rmsc =1fora=1, and rms =rmsc =3.667
for a=ai. Moreover, due to the results of [40], it follows
that even the circular orbits with negative energy with respect
to infinity (E<0) lie at the region of stable spherical orbits.
Their radii satisfy inequality rzE1<r<rzE2, where
rzE1=8
3cos2π
3+1
3arccos 27
32a,(66)
rzE2=8
3cos2π
3−1
3arccos 27
32a(67)
are radii of the zero energy orbits.
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Eur. Phys. J. C (2018) 78:879 Page 21 of 25 879
1.001.02 1.041.061.08
0.6
0.7
0.8
0.9
1.0
1 2.478 4.468
1
2
3
a
r
rph
rms
rmsc
r0
rpol
rpol
rzE1
rzE2
r
Fig. 21 Relative position of the equatorial circular orbits of the test
particles and the spherical photon orbits. The overlap region of the
stable circular orbits of test particles and stable photon spherical orbits
is highlighted in grey, the light grey shading correspond to radii of the
circular orbits with negative energy. The function rφ=0(bold dashed
curve) corresponds to radii of the closed spherical photon orbits with
zero total change in azimuth φ =0.It interferes with the area of
possible occurrence of the thin Keplerian accretion disc, hence, some
self-illumination effects manifesting itself as periodic light echo are
plausible here. The remaining functions are defined in the text
We can expect observationally significant optical and
astrophysical effects, particularly in the region of negative
energy orbits connected with an interplay between the possi-
ble extraction of the rotational energy of the KNS (Kerr super-
spinar) and a subsequent trapping of the radiated energy, both
for the radiated electromagnetic and gravitational waves.
Both forms could substantially violate the structure and sta-
bility 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 mat-
ter and the stable SPOs, where repeated re-absorption of the
radiated heat, and self-illumination and self-reflection phe-
nomena 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 ais given in Fig. 21.
Now we concentrate attention on the special case of the
repeated irradiation of a Keplerian disk at an azimuthal posi-
tion fixed to the distant observers that could be observa-
tionally relevant, enabling even an efficient determination of
spacetime parameters of the KNS (Kerr superspinar) space-
times allowing existence of this effect.
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.
The time interval that elapses during one latitudinal oscil-
lation can be computed by combining equations (16) and
(18). The searched period of the latitudinal motion at a fixed
radius rcan be expressed by the integral
t=4
m+
0
(r2+a2)−2ar[sph −a(1−m)]
2M(m,a,
sph,qsph),(68)
where sph and qsph are taken at the fixed radius r. Of course,
the nodal period, i.e. the period between two subsequent
crossing of the equatorial plane, is half of the time interval
givenbyEq.(68). The nodal period can be expressed, using
the standard procedures (see Gradshtein, S., Ryzhik,M.,
Tables of integrals, series and products), in terms of the ellip-
tic integrals by the formula
tnod =2r2(r2+a2)−2ar(sph−a)
a√m+−m−Km+
m+−m−
+4am−
√m+−m−Km+
m+−m−
+√m+−m−E(m+
m+−m−).(69)
Here m±are the positive/negative roots of M(m),K(s)
denotes the complete elliptic integral of the first kind given
by (56) and
E(s)=
π/2
01−ssin2θdθ(70)
is the complete elliptic integral of the second kind.
Using the general formula for the time period of the nodal
motion, we are able to calculate the time period of the spe-
cial nodal motion of closed oscillatory orbits with vanishing
change of the azimuthal coordinate for one node. We give
the time periods tnod=0in dependence on the dimension-
less spin parameter aof the spacetime in Fig. 22.
Notice that there is relatively strong decline of the time
period tnod=0with increase of the spin parameter a,
although the dependence of the radius of these orbits, rφ=0,
on the spin ademonstrates only very slow descent, as shown
in Figs. 7,21. Numerical computation reveals that
lim
a→∞rφ=0(a)=5/3.(71)
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879 Page 22 of 25 Eur. Phys. J. C (2018) 78:879
2 4 6 8 10
a
5
10
15
20
25
30
35
t
0
r1.73
r5
3
Fig. 22 Dependence of the period of the photon circulation over a
closed spherical orbit with zero nodal shift on the spin parameter a.
The graph is defined for a>1.18.Such orbits are of octal shape (see
case k=0inFig.9)
We have found that the condition rφ=0>rmsc is satis-
fied for the KNS spacetimes with dimensionless spin in the
interval a∈(apol(max)=1.7996,2.47812), therefore, just
for this interval of the KNS spacetimes we can consider the
effect of the irradiation at a fixed azimuth. Observation of
such an effect enables to estimate the spin parameter a,if
the mass parameter of the spacetime can be fixed by an inde-
pendent method.10 In this case we have considered only the
spherical orbits with positive covariant energy E>0, as the
oscillatory orbits with azimuthal turning point are necessarily
located at r>1.
Now we have to distinguish the case of the SPOs demon-
strating φ =0 with the turning point of the azimuthal
motion, and the analogous cases of the periodic SPOs where
φ =k2π, with kbeing an integer, giving also the return to
the fixed azimuthal coordinate in the equatorial plane where
the Keplerian disk is located.
The case of the SPOs with tnod=2πis given in Fig. 23.
We found by numerical methods that the SPOs with E>0
and φ =2πare limited by the radii r=1.22854 for
a=1.09533 and, by solving Eq. 61, by the radii r=1
for a=√4/3=1.1547.A representative of such orbits is
depicted in Fig. 9(see case k=1).
The case of the orbits with E>0 demonstrating tnod=4π
is illustrated in Fig. 24. These orbits exist for the spin param-
eter 0 <a≤aφ=0(min)=1.17986 with r→2asa→0,
and r=rφ=0(max)=1.7147 as a=aφ =0(min).This
limit case is depicted in Fig. 10a, for the values from the
inside of the interval the Fig. 9, case k=2, is representative.
In the figures we depict together with the graphs the radii
corresponding to the limits of their definition range.
10 We could slightly extend the range of the considered spin of the
KNS spacetime, down to the value of aφ=0(min)=1.79857 when
the SPOs with φ =0 start to appear; however, in the case of KNS
spacetimes with a∈(aφ=0(min),apol(max ))there is an extended region
of spherical orbits allowing for φ ∼2πthat invalidates applicability
of the orbits with φ =0.
1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15
a
5
10
15
20
25
30
35
t
2
π
r 1.2286
r1
Fig. 23 Dependence of time periods of the closed SPOs with φ =2π
on the spin parameter adefined for a∈(1.095,1.155).
0.0 0.2 0.4 0.6 0.8 1.0 1.2
a
20
40
60
80
100
t
4
π
rrpol max
r 2
Fig. 24 Case φ =4π. The shape of such orbits represents second
picture in Fig. 9
Now we can consider also the SPOs at r<1 (with E<0),
in the KNS spacetimes with spin 1 <a<apol(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 aindependently of their
mass parameter. For comparison, we also compute the time
intervals for the closed spherical orbits in Fig. 9in 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→
0 the formula (69)forr=3 gives the result tSchwarz =
32.6484, in accordance with the exact solution tSchwarz =
6√3πobtained by integration from the Schwarzschild line
element after inserting ds=dr=dθ=0 and θ=π/2.
The nodal time delay is governed by half of the quantity:
tnod(Schwarz)=16.3242.
In order to have dimensional estimates governing the time
delays, we have to express the resulting time delays in the
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Eur. Phys. J. C (2018) 78:879 Page 23 of 25 879
dimensional form that takes in the standard units the form
tnod(dim)=GM
c3tnod.(72)
Therefore, in the systems with stellar mass compact object
(KNS) we can estimate the time delay of the level of
tnod(dim)∼0.01s, but in the case of the supermassive
object in the Galaxy centre we can estimate tnod(dim)∼
103s∼1/4hour, while in the case of the galaxy M87 the
central object can demonstrate time delay by three orders
higher ∼106s∼12 days. Clearly, from the point of view of
the observational abilities of the recent observational tech-
niques, the most convenient candidate for testing the time
delay effect seems to be the Galaxy centre SgrA* object.
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 rela-
tive to the radial perturbations has been demonstrated. The
photons at spherical orbits located at r>1 are of standard
kind, having covariant energy E>0, as outside the black
hole horizon, but at r<1 they must have E<0; for the
special position at r=1, the photons at spherical orbits have
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 co-rotating
or counter-rotating relative to distant observers, also orbits
changing orientation of the azimuthal motion occur in the
field of Kerr naked singularities. Existence of polar spher-
ical orbits reaching the symmetry axis of the Kerr space-
time is limited to the naked singularity spacetimes with spin
a<apol(max)=1.17996. On the other hand, the KNS space-
times with a>apol(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 dimension-
less spin of the KNS spacetimes allowing their existence,
if their mass parameter is known due to other phenomena.
11 The photons at the spherical orbits located under the inner black hole
horizon have also E<0.
This is the case of the KNS spacetimes with the spin param-
eter a∈(apol(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
φ =2kπwith 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.
Acknowledgements Daniel Charbulák acknowledge the Czech Sci-
ence Foundation Grant No. 16-03564Y.
Open Access This article is distributed under the terms of the Creative
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