Completing characterization of photon orbits in Kerr and Kerr-Newman metrics
CCompleting characterization of photon orbits in Kerr andKerr-Newman metrics
D. V. Gal’tsov ∗ Faculty of Physics, Moscow State University, 119899, Moscow, Russia,Kazan Federal University, 420008 Kazan, Russia
K.V. Kobialko † Faculty of Physics, Moscow State University, 119899, Moscow, Russia
Abstract
Recently, several new characteristics have been introduced to describe null geodesic structure of sta-tionary spacetimes, such as photon regions (PR) and transversely trapping surfaces (TTS). The formerare three-dimensional domains confining the spherical photon orbits, while the latter are closed two-surface of spherical topology which fill other regions called TTR. It is argued that in generic stationaryaxisymmetric spacetime it is natural to consider also the non-closed TTSs of the geometry of sphericalcups, satisfying the same conditions (“partial” TTS or PTTS), which fill the three-dimensional re-gions, PTTR. We then show that PR, TTR and PTTR together with the corresponding anti-trappingregions constitute the complete set of regions filling the entire three-space (where timelike surfacesare defined) of Kerr-like spacetimes. This construction provides a novel optical description of suchspacetimes without recurring to explicit solution of the geodesic equations. Applying this analysisto Kerr-Newman metrics (including the overspinning ones) we reveal four different optical types fordifferent sets of the rotation and charge parameters. To illustrate their properties we extend Syngeanalysis of photon escape in the Schwarzschild metric to stationary spacetimes and construct densitygraphs describing escape of photons from all the above regions.
PACS numbers: 04.20.Dw,04.25.dc,04.40.Nr,04.70.s,04.70.Bw ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] J a n . INTRODUCTION Photon spheres are important characteristics of spherically symmetric spacetimes. Theyare closely related to gravitational lensing [1] and the BH shadows [2]. In the case of theSchwarzschild metric studied in detail by Virbhadra and Ellis [1], closed circular photon orbits(often called photon rings) are located at r = 3 M and form a photon sphere due to sphericalsymmetry. It is essential that the photon rings cover the photon sphere densely, i.e. thephotons can have any direction tangent to the sphere. The photon rings in Schwarzchildspacetime are unstable (so do the photon spheres); their Lyapunov exponents define the highfrequency spectrum of the quasi-normal modes [3] (see also [4, 5]). The photon sphere also hasan interesting property to be a separatrice where the centrifugal force acting on an orbitingbody changes sign, becoming inward directed inside it [6, 7]. In spacetime, a photon sphere forma timelike hypersurface marking the closest distance of approach of a scattered light ray wherethe bending angle become infinitely large. An existence of a photon sphere thus may indicate onemergency of multiple images. (In what follows we use the same term PS for a two-dimensionalsphere in space and the corresponding three-dimensional hypersurface in spacetime, assumingthat the correct meaning is clear from the context.)Obviously, existence of the photon sphere is related to spherical symmetry of spacetime.It is worth noting, that the photon sphere is not destroyed by the Newman-Unti-Tamburino(NUT) parameter, in which case the so (3) algebra still holds locally, though metric is alreadynon-static. With this exception, stationary metrics with true rotation do not admits photonspheres or more general photon surfaces. In static spacetime various uniqueness theorems wereformulated in which an assumption of the existence of a regular horizon was replaced by anassumption of existence of a photon sphere [8–13]. No such general results are available forstationary spacetimes. So the problem of optical characterization of stationary metrics which wediscuss in this paper remains actual. Mention an interesting relation between the separabilityof spacetime and properties of the circular and the spherical photon orbits discovered recently.Namely, a spacetime is non-separable, if there exist an equatorial circular orbit and, at thesame time, there are no spherical orbits beyond the equatorial plane [14, 15]. This propertymay serve a diagnostic of the non-Kerr nature of spacetime.In certain cases the circular photon orbits may be stable. Normally this does not happenin the BH spacetimes outside the event horizon, but often happens inside the inner horizon or2n the field of naked singularities and wormholes. In spherically symmetric static spacetimes,stable circular geodesics form ”anti-photon” surfaces, a term suggested by Cvetic, Gibbonsand Pope [16]. In a spacetime with separable geodesic equations, the stable photon orbitscorrespond to a minimum of the effective radial potential. Above the minimum, the boundedlight orbits with two radial turning points are located. Their existence serves an indicationthat the solution may be unstable [17–20]. Stable photon rings are often encountered near(hypothetical) horizonless ultra-compact objects, thus indicating on their instability [21]. Theywere shown to exist also on the horizons of extreme static black holes [22, 23].In non-spherical static spacetimes, properties of the photon spheres are shared by the photonsurfaces of non-spherical form. In [24], the photon surface is defined as a closed timelikehypersurface S , such that any null geodesic initially tangent to S continues to be included in S .Several examples of spacetime have been found that allow non-spherical photon surfaces, whichare not necessarily asymptotically flat (vacuum C-metric, Melvin’s solution of Einstein-Maxwelltheory and its generalizations including the dilaton field [25]).Mathematically, an important property of the photon surfaces is established by the theoremasserting that these are conformally invariant and totally umbilical hypersurfaces in spacetime[26, 27]. This means that their second fundamental form is pure trace, i.e. is proportional tothe induced metric. This property is especially useful in the cases when the geodesic equationsare non-separable, so no analytic solution can be found, while the umbilical hypersurfaces stillcan be described analytically.Recently two other interesting characteristic surfaces in the strong gravitational field weresuggested. One is the loosely trapped surface (LTS) [28]. As was shown long ago by Synge [2], theregion between the horizon and the photon sphere in the Schwarzschild metric is approximately“half” trapped: if photons are isotropically distributed in this region, more than half of themwill be absorbed by the black hole. In a sense, these photons are “loosely” trapped. Once thehorizon is approached, they become fully trapped. Considering two-spheres between the horizonand the photon sphere one can associate the radial derivative of the trace of their extrinsiccurvature as a measure of the gravity strength: while this derivative is negative outside thephoton sphere, it becomes positive and when moving towards the horizon. This gives rise tothe definition of the loosely trapped surface in a non-vacuum spacetime as a compact 2-surfacewith positive trace of its extrinsic curvature and the non-negative radial derivative of the trace.3hen for such a surface a Penrose-like inequality holds involving the area of the photon surface,provided the scalar curvature is positive [28].In rotating spacetimes the photon orbits with constant Boyer-Lindquist radius may exist aswell (e.g. spherical orbits in Kerr [29, 30]), but they do not fill densely the photon spheres, sincetheir existence requires certain relation between the constants of motion. Such orbits fill thethree-dimensional volumes — the photon regions [31, 32]. In more general spacetimes with twocommuting Killing vectors, the orbits which fill some compact region were called fundamentalphoton orbits [19]. In the Kerr case such orbits (outside the horizon) are unstable. But in moregeneral case they may also be stable — a signal of spacetime instability.Another generalization of the notion of the photon surface is the transversely trapped surface (TTS) [33]. Its relation to LTS is discussed in [33]. TTS generalize PS allowing for the initiallytangent photons to leave the closed two-surface but only in the inward direction. Similarly toPS, the TTS can be defined in geometric terms using inequalities involving the second quadraticform of the corresponding hypersurfaces in spacetime. This can be also useful in the cases whenthe geodesic equations are non-separable, such as certain Weyl spacetimes. Here we argue thatTTS may be reasonably generalized further to partial TTS, or PTTS, allowing for non-closedtwo-sections. In the Boyer-Lindquist coordinates such two-surfaces are spherical caps. ThePTTS can be left by photons in both directions, but only at certain restricted angles. We hopethat this concept may be useful for analysis of strong gravitational lensing. Finally, to completethe list of characteristic surfaces in a way to ensure foliation of most of the spacetime manifold,we introduce the anti-TTS (ATTS) and anti-PTTS (APTTS) hypersurfaces such that theirtwo-sections can be left by the initially tangential photons in the outward direction. Using theKerr-Newman metric as an example, it will be shown that the set of surfaces presented aboveis sufficient to fully characterize the null geodesic structure of Kerr-like spacetimes.The plan of the paper is as follows. In Section 2, we define the characteristic hypersurfacesin space-time based on the properties of the second fundamental form. In Section 3, the neces-sary and sufficient conditions for the surfaces PS, TTS and ATTS are obtained for stationaryspace-times parametrized by Boyer-Lindquist type coordinates. Section 4 is devoted to the ap-plications of our formalism to the Kerr-Newman spacetime. The critical values of the rotationparameter and the charge separating the four optical types of KN are found. In Section 5, westudy the output of photons from different characteristic regions to spatial infinity depending4n the angle of Singe and present numerous density graphs illustrating the picture of the escapefrom different regions. In conclusion, the main results are summarized and some perspectivesare discussed.
II. THE SETUP
Recall the basic concepts of the geometry of hypersurfaces [26]. Let ˆ M be a 4-dimensionalspacetime. Consider an embedded three-dimensional timelike hypersurface M represented inthe parametric form F ( M ) : M → ˆ M as x µ = f µ ( σ A ) , µ = 0 , ..., , A = 0 , ..., , (2.1)where σ A are the local coordinates on M . We denote by f µA = ∂f µ /∂σ A the linearly independenttangent vectors at each point of F ( M ). The induced metric g AB and the unit spacelike normalvector field n µ on M are defined by g AB = ˆ g µν f µA f νB , ˆ g µν f µA n ν = 0 , ˆ g µν n µ n ν = 1 . (2.2)The components of the second fundamental tensor H AB are obtained from the Gauss decom-position: H AB = ˆ g µν ( ˆ ∇ f A f µB ) n ν , (2.3)where ˆ ∇ covariant derivative in ˆ M .Consider a null affinely parameterized geodesic ˆ γ with the tangent vector ˙ˆ γ µ ( s ) emittedtangentially to M at some point F ( P ) on F ( M ):ˆ ∇ ˙ˆ γ ˙ˆ γ µ ( s ) = 0 . (2.4)Let us introduce another null curve γ starting from the point P with the tangent vector ˙ γ A ( s ),which is assumed to be a null geodesic on the hypersurface M . Suppose that at an initialmoment the tangent vectors to both geodesics coincide. Thus at the point P , we can choose˙ˆ γ µ (0) = ˙ γ µ (0) ≡ f µA ˙ γ A (0). Rewriting the equation ∇ ˙ γ ˙ γ A ( s ) = 0 for γ in terms of the four-dimensional quantities, we find ˆ ∇ ˙ γ ˙ γ µ ( s ) = ( H AB ˙ γ A ( s ) ˙ γ B ( s )) n µ . (2.5)5here are the following basic possibilities. The first is that the two trajectories ˆ γ and γ locallycoincide, in which case H AB ˙ γ A ( s ) ˙ γ B ( s ) = 0. Suppose that for some timelike hypersurface thiscondition is satisfied for every null tangent vector ˙ σ A in M : H AB ˙ σ A ˙ σ B = 0 . (2.6)Then any initially tangent null geodesic remains in the hypersurface at any time. This is awell-known property of a photon sphere and its generalization – a photon surface (PS) [24].Equivalently, the Eq. (2.6) can be rewritten as statement that the surface is totally umbilic[24, 26]: H AB = 13 Hg AB . (2.7)The last statement is fairly restrictive both on the geometry of the hypersurface and the space-time allowing for its existence. Photon surfaces may exist in the gravitational field of blackholes, wormholes, naked singularities and other ultracompact objects, being a useful tool al-lowing to discriminate their different optical behavior [34]. It was proved that in the vacuumasymptotically flat case the Schwarzschild PS at r = 3 M can not be deformed, though it is notforbidden in presence of matter [35].However, PS do not exist in rotating spacetimes. Though the so-called spherical orbits inKerr metric exist with constant value of the Boyer-Lindquist radial coordinate r [29, 30, 36],they correspond to a discrete set of tangential directions on the sphere r = const. Sphericalorbits with different r then fill the three-dimensional domain – the photon region (PR) [31]which is an important feature of rotating spacetimes.Both PS and PR may be unstable (UPR) or stable (SPR) [31], in the latter case signallingon instability of spacetime [17–20]. The photon region can be considered purely geometricallyin a manner similar to the photon surface. In fact, it is possible to use the condition (2.6),but demanding its validity only for some null vectors and, correspondingly, for some directionsin the tangent space [30]. In the case of rotating space, from this we can derive exactly thephoton region conditions. For more general spaces, the photon regions do not exist, but somegeneralizations like fundamental photon orbits are possible [19].A completely different situation arises when the trajectories ˆ γ and γ do not coincide. Thecurves ˆ γ propagate opposite to the outward normal n µ if and only if H AB ˙ γ A ( s ) ˙ γ B ( s ) >
0. Ifthis condition is satisfied for each null tangent vector on some entire closed surface, then it is6 trapping surface, allowing tangential photons to leave it only in the inward direction. Thisproperty underlies the notion of the transversely trapping surface (TTS) [33]. The necessaryand sufficient condition for M to be a TTS is that it is timelike, and for each point on M thecondition H AB ˙ σ A ˙ σ B ≥ , (2.8)holds for any null tangent vector ˙ σ A on M . For an axially symmetric space, using 3+1 splitting,this condition can be formulated purely in terms of the 2-dimensional external curvature andthe covariant derivative of the lapse function [33]. (Note that in [33] the sign of the inequalityis opposite because of different conventions.)In the original definition of [33] this surface was assumed to be closed. However, it turns outthat already Kerr-Newman spacetime can not be foliated by closed TTS only, since the solutionof the Eq. (2.8 exist with non-closed two-section. To achieve a more complete characterizationof the null geodesic structure in a stationary gravitational field it is thus useful to extend theabove definition of TTS to non-closed two-surfaces. Such incomplete TTS will be called partialtransversely trapping surfaces (PTTS). They have similar confining property (2.8) locally, butthey still allow for the initially tangent photons to escape from the inner part in some restrictedrange of directions passing across the boundary. However, as we will see later, in Kerr-Newmanspace the initially tangent to PTTS null geodesics cannot escape to spatial infinity, so thetrapping properties of TTS and PTTS are not much different globally. But to prove this, theseparability of the geodesic equations is essential, and one can expect that in non-separablespacetimes the situation can be different.The third possibility is that the initially tangent trajectories ˆ γ propagate onward with respectto the outer normal n µ . The corresponding condition is H AB ˙ σ A ˙ σ B ≤ . (2.9)We will call such surfaces anti -TTS (ATTS). In the Schwarzchild metric the ATTSs fill theregion outside the photon surface up to infinity. But ATTSs may occur also inside the innerhorizon describing the region which can be left by photons into an analytically continuedsector of spacetime. The antitrapping surfaces can also be non-closed, in which case we callthem APTTS. All the surfaces considered above are not isolated, but densely fill the three-dimensional volume regions which we will call the transversely trapping region (TTR), the anti a) (b) FIG. 1: Characteristic regions in Kerr spacetime. Red color – the unstable photon regions, yellow –the stable photon regions, dark blue – the region between the horizons ∆ r ≤
0, mesh – the ergoregion,blue mesh – the causality violating region g ϕϕ <
0, dash denote the throat at r = 0, green - the(P)TTR, aqua – the A(P)TTR. The dotted circles with arrows are examples of (P)TTS. The arrowsindicate directions of outgoing photons. transversely trapping region (ATTR) and, in the case of non-closed two-surfaces, the PTTR andthe APTTR. Together with the photon regions PR (if it is defined), they all cover an entire ormost of the space (where null geodesics and timelike hypersurfaces are defined) opening a wayto give a complete optical characterization of many known exact solutions.In Fig. (1) we illustrate this for the Kerr spacetime. Our picture can be compared withthe picture of photon regions only in [31]. The plots represent slices y = 0 in Cartesian typecoordinates. The regions are filled with different colors; the arrows indicate directions of theoutgoing photons. III. STATIONARY AXIALLY SYMMETRIC SPACETIMES
To proceed further, we now specify the general conditions (2.6-2.9) to the case of spacetimeswith two commuting Killing vectors ∂ t , ∂ ϕ . We take ˆ M to be a 4-dimensional axially symmetricstationary spacetime endowed with the Boyer-Lindquist type coordinates x µ = ( t, r, θ, φ ) definedas follows: ds = − α ( dt − ωdφ ) + λdr + βdθ + γdφ , (3.1)where α, ω, λ, β, γ are functions of r, θ only.8et’s consider the hypersurface r = r T = const. As a consequence of the uniqueness theoremsof [33], the closed surfaces with topology different from spherical do not exist. This hypersurfaceis timelike, and its outer normal for r > n µ = (0 , / √ λ, , , (3.2)is well defined outside the horizon. Note that in the analytically continued region r < − n µ , this leads to interchange of the definitions of (P)TTS and A(P)TTS.From the Eq. (2.3) we obtain:2 √ λH AB ˙ σ A ˙ σ B = ∂ r α ( ˙ t − ω ˙ φ ) − ∂ r β ˙ θ − ∂ r γ ˙ φ − α∂ r ω ( ˙ t − ω ˙ φ ) ˙ φ, (3.3) g AB ˙ σ A ˙ σ B = − α ( ˙ t − ω ˙ φ ) + β ˙ θ + γ ˙ φ = 0 . (3.4)Like in [33], the component ˙ θ can be excluded via the normalization condition:˙ θ = αβ ( ˙ t − ω ˙ φ ) − γβ ˙ φ ≥ , (3.5)2 √ λH AB ˙ σ A ˙ σ B = α ( ˙ t − ω ˙ φ ) (cid:18) ∂ r αα − ∂ r ββ (cid:19) + ˙ φ (cid:18) ∂ r ββ − ∂ r γγ (cid:19) − α ˙ φ ( ˙ t − ω ˙ φ ) ∂ r ω. (3.6)Then dividing the second expression by α ( ˙ t − ω ˙ φ ) , we find from (2.8) the following necessaryand sufficient conditions for (P)TTS: β > , α > , γ > , ξ = γ ˙ φ α ( ˙ t − ω ˙ φ ) ≤ , (3.7) ξ (cid:18) ∂ r ββ − ∂ r γγ (cid:19) − (cid:114) αγ ∂ r ωξ + (cid:18) ∂ r αα − ∂ r ββ (cid:19) ≥ . (3.8)or else for ergo region β > , α < , γ < , ξ = γ ˙ φ α ( ˙ t − ω ˙ φ ) ≥ , (3.9) ξ (cid:18) ∂ r ββ − ∂ r γγ (cid:19) − (cid:114) αγ ∂ r ωξ + (cid:18) ∂ r αα − ∂ r ββ (cid:19) ≤ . (3.10)In a similar way one obtains the necessary and sufficient conditions for A(P)TTS from(2.9). Thus, the problem has been reduced to analyzing the negative/positive definitness ofthe quadratic functions (3.8, 3.10) on a given interval ξ ≤ ξ ≥
1. The analysis isnot complicated, but rather cumbersome. It is possible to obtain explicit restrictions on thecoefficients of quadratic forms (see appendix A). Some more details of the computation can be9
ABLE I: (P)TTS, A(P)TTS and PR conditionsType Region Quadratic conditions Linear conditionsTTS Outer( α > γ >
0) ( i ) b > c ( ii ) b > c , a > c ( iii ) b ≤ c , a > c Ergo( α < γ <
0) ( i ) b < c ( ii ∗ ) b < c , a < c ATTS Outer( α > γ >
0) ( i ) b < c ( ii ) b < c , a < c ( iii ) b ≥ c , a < c Ergo( α < γ <
0) ( i ) b > c ( ii ∗ ) b > c , a > c PR - ( − × ( iii ) - found in [33] (the Eqs. 38a-c, appendix B). The only difference is that here we do not use theabsolute values in the final inequalities but add the appropriate linear conditions.The table I represents the detailed form of conditions on (P)TTS and A(P)TTS (2.8, 2.9)in the outer and the ergo regions of spacetime, where a ≡ ∂ r α/α, b ≡ ∂ r β/β, c ≡ ∂ r γ/γ, d ≡ (cid:112) α/γ∂ r ω, (3.11)and the quadratic inequalities are defined as( i ) ( a − b )( b − c ) ≥ d , (3.12)( ii ) ( a − c ) ≥ d > b − c ) , (3.13)( ii ∗ ) ( a − c ) ≥ d < b − c ) , (3.14)( iii ) ( a − c ) ≥ d . (3.15)It is worth noting that the conditions for PS (and PR) for Plebanski-Demianski class ofsolutions [31] are based on the same quadratic condition (iii) as (P)TTS (with opposite sign):( a − c ) ≤ d . (3.16)10or (P)TTS, however, one has to take into account the linear conditions together with (iii).This is not a coincidence, but can be obtained directly from the Eq. (2.6). For this, asalready noticed, it is necessary to require the fulfillment of the Eq. (2.6) only for some isotropicvectors. The easiest way to do this is to express the null vector components in terms of theenergy and the azimuthal orbital momentum and substitute them into th Eq.(2.6). In the caseof Plebanski-Demianski space one then finds a θ -independent relation between the energy andthe angular momentum. Let’s do this for the Kerr solution (see below (4.3), (5.1-5.2)). From(2.6) and (3.6) we get equality Er ( r − m ) + a E ( r + m ) + aL ( r − m ) = 0 , (3.17)accurate to nonzero multipliers. Obviously, the relationship between E and L does not dependon the θ . Solving this equation for L and substituting in (3.5) we find exactly the condition(3.16).Note that from the Table I it is clear that the quadratic inequalities distinguish betweenA(P)TTS and (P)TTS regions. Moreover, every region of Kerr-like space (where timelikesurfaces r = const are defined) to be related to one or another optical type. In the initialformulation, the TTS and the PTTS have different optical properties in the sense that lightcan leave a compact region after touching a PTTS across its boundary. However, as we will seein the case of spaces admitting separation of variables in the geodesic equations, this differenceis diminished by the existence of a radial potential. Still, absence of the closed TTSs, suggeststhat the strong field region is more accessible for observation. With this in mind, we presentin the next section new classification of optical types of Kerr-Newman solution depending onparameters.Note that in non-separable spacetimes, the photon regions most probably do not exist, whilethe (P)TTSs and A(P)TTSs do exist. Instead of the photon region one may encounter a regionwhere the the projection (2.6) of the external curvature does not have a definite sign. IV. KERR-NEWMAN OPTICAL TYPES
In this section, we illustrate how it is possible to classify optical types of the Kerr-Newmanspace based on our complete set of characteristic surfaces. This classification can be consideredas generalization to stationary spaces of the scheme of [34] applicable only in the static case.11
ABLE II: Optical typesType TTR PTTR (PS)PR I + ± + II − + + III − − + IV − − − In what follows we classify both the black holes and the naked singularity configurationslike overextreme Kerr-Newman space. We make the following definitions. We call the type I such spaces in which the closed TTSs exist in the outer domain. The second type II includessolutions with the presence of only PTTSs and photon regions. The third type III includesmetrics in which there are only photon regions but no PTTSs. Finally, in spaces IV , none of theabove structures is present. The naked singularity of the type IV is called strongly naked [34].These definitions are summarized in the (table II). The optical meaning of this classificationwill be explained in section 5.Now let’s analyze a concrete example of a Kerr-Newman solution using Boyer-Lindquistcoordinates. The metric has the form (3.1) with the following metric functions (see [37, 38]): α = ∆ r − a sin θ Σ , ω = − a (2 mr − q ) sin θ ∆ r − a sin θ , (4.1) λ = Σ∆ r , β = Σ , γ = ∆ r Σ sin θ ∆ r − a sin θ , (4.2)∆ r = r − mr + q + a , Σ = r + a cos θ. (4.3)Here a is the rotation parameter, q = e + g comprises the electric and magnetic charges.Basically, the coordinates t and r may range over the whole R , while θ and φ are the standardcoordinates on the unit two-sphere. The horizons (∆ r = 0), the ergosphere ( α = 0) and thering singularity in the equatorial plane θ = π/ r h ± = m ± (cid:112) m − q − a , r e ± = m ± (cid:112) m − q − a cos θ, r s = 0 . (4.4)The horizon disappears if | a | > a e = (cid:112) m − q . Since the normal vector n µ to the hypersurface r = r T is spacelike only for ∆ r > i ) has a simple form − a r sin θ ∆ r Σ ≥ , (4.5)and obviously can not hold if ∆ r >
0. Although the calculation is more tedious, one can showthat the second conditions in ( ii ) and ( ii ∗ ) is satisfied automatically in outer and ergo regionsrespectively (see [33]) if other ones holds. The condition ( iii ) and the first one in ( ii ) and ( ii ∗ )reads: 16 r a ∆ r sin θ ≤ (4 r ∆ r − Σ∆ (cid:48) r ) . (4.6)Recall that for the photon region, this condition has a reversed sign (3.16). A detailed analysisof the photon regions as well as their stability in the subextreme and overextreme Plebanski-Demianski spaces can be found in [31, 39]. The latter also provides classification of solutionsbased on the properties of the spherical photon orbits. Previous analysis of the trapping setsin the Kerr metric can be also found in [36].In the case of closed TTSs or ATTSs, the Eq. (4.6) should be satisfied for all θ what leadsto the condition (cid:0) r − mr + 2 q (cid:1) ≥ a ( mr − q ) . (4.7)In order to distinguish between TTS and ATTS, we must add the linear conditions (table I).For example, the linear (P)TTS conditions ( a > c ) reduce to the strongest one for θ = 0: r ( r − mr + 2 q ) + a ( r + m ) < , (4.8)which determines, in particular, the existence of the polar PTTS. In our case, it simply se-lects one of the regions defined by the inequality (4.7). In the case of a sub-extremal space,an expression describing the boundary of a closed TTR can be obtained using the Cardanoformulas: 2 r maxT = 3 m + √ F − (cid:115) v − F − a m √ F , F = v + v − a uQ / + Q / ,Q = 216 a m + v − a vu + 24 √ a (cid:112) ( m − a − q )( v q − a m ) , (4.9) v = 9 m − q , u = 3 m − q . In the particular case of a vanishing charge q = 0, this was found in [33] and [36] as thelower boundary of the trapping set. Solutions that allow the existence of a closed TTS in13ur classification will be called an optical type I . Such solutions contain a compact regionof a strong gravitational field, completely covered with transversely trapping surfaces and,accordingly, strongly hidden for an external observer. At the same time, the closed photonregion, which is also present in this case, can create a set of relativistic images [1].Inequalities (4.7, 4.8) can have a common solution only for certain values of the parameters( a, q ). Accordingly, the closed TTS domain exists exists only for the following values of a and q : | a | < a C = √ q (cid:18) − q m (cid:19) , a C > a e , q m > q > q C , (4.10) | a | ≤ a e , q ≤ q C , q C = 3 / m , q m = 9 / m . (4.11)In this expression, an important quantity is the critical charge q C . For values q ≤ q C , thesolution resembles pure Kerr. In particular, closed TTSs do not exist in the overextreme case | a | > a e [33]. Respectively a region containing the singularity and a stable photon regionbecomes observable along the majority of null geodesics coming from infinity. In the oppositecase, q > q C , closed TTSs exist even in the superextreme case, limited by the critical valueof the rotation parameter | a | < a C , and, accordingly the area is hidden for most isotropicgeodesics. In the plots (2a-2c) we illustrate the size of the TTR depending on the magnitude ofthe rotation. In the last image, there is a small TTR bump in superextreme case, correspondingto what was said.The structure of a closed ATTR can determine the presence of a stable photon region [31].It exists if | a | > a C = √ q (cid:18) − q m (cid:19) , a C < a e , q < q C , (4.12) | a | > a e , q m > q > q C , q C = 3 / m , q m = 9 / m . (4.13)In particular, a stable photon region inside the inner horizon exists only at sufficiently smallvalues of the charge q < q C .If the Eq. (4.8) is fulfilled for some values of the metric parameters, while the conditionsof existence of closed TTS (4.10, 4.11) are violated, one can still have the pure polar PTTSdefined by the inequalities (4.6, 4.8). Such configurations belong to an optical type II in ourclassification. In this case, the inner region of space containing the singularity is still hiddenby the closed photon region, but not by the TTS, and, accordingly, is observable for a certain14 .0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0 a r (a) q = 0 a r (b) q < q C a r (c) q > q C FIG. 2: Closed TTR and ATTR for various a . Green color – TTR, aqua – ATTR, blue – the horizon.Dotted lines – a C set of isotropic geodesics coming from infinity. Such configurations are still capable to create aset of relativistic images [1, 34].However, with certain values of the metric parameters, even the polar PTTS disappear. Wepresent a constraint on the charge and the rotation parameter for type II solution explicitly: a < a c = 6 m (cid:112) m + 2 q sin( ϑ + π/ − m + q ) , q < q m ,ϑ = 13 arccos (cid:18) m + 14 m q + q m (5 m + q ) / (cid:19) , q m = 9 / m . (4.14)We note that the critical value of the rotation parameter also determines the condition for theexistence of polar spherical photon orbits [39], which allows us to relate various classificationsof the overextreme spaces. In the particular case q = 0, we get the previously known value: a c = m (cid:113) √ − . (4.15)The spaces with | a | > a c contain only a photon region which become non-closed at thepoles, and goes into APTTS rather than PTTS to which PR was connected for smaller | a | .Accordingly, they belong to an optical type III . In these spaces, there is no well-definedcompact trapping region. The singularity is strongly naked for geodesic passing near the poles.However, it is still hidden for some geodesics passing in the vicinity of the equatorial plane frominfinity.The value q m is maximal in the sense that for larger charges q > q m both the photon regionsand the PTTSs completely disappear. So q m defines the strong naked singularity condition.Such solutions have radically different optical properties, in particular, they cannot create15elativistic images [34]. These represent the fourth class of solutions, IV . Interestingly, thiscondition accurately duplicates a similar condition for the Reissner-Nordstrom space [24].In order to illustrate the above classification, we use the visualization method proposed in[31]. The Figs. (3-5) represents slices y = 0 of Cartesian type coordinates where each regionof space is filled with a certain color depending on the properties of the corresponding photonsurface r = const. Schematic representation of the geodetic behavior for each area is shown inthe Fig. (1).Let’s start with the Kerr metric ( q = 0). In absence of rotation ( a = 0), there is a uniquephoton sphere and TTSs filling the region between the photon sphere and the horizon (3a). Theinner region (4a) behind the singularity is filled by ATTSs, which correspond to weak gravityor antigravity. The solution obviously belongs to an optical type I in our classification.For weaker rotation, | a | < a e , the photon sphere spreads into the photon region. There isan area with PTTSs and ATTSs in the vicinity of the poles and the singularities, respectively(3d,3g). The presence of the latter in these cases corresponds to antigravity effects occurringin the vicinity of the singularity, while the PTTSs characterize some properties of the stronggravitational lensing. The closed TTR persists also and decreases with increasing rotation, asshown in the Fig. (2a). A stable photon region emerges directly from the ring singularity andis always located inside the inner horizon. Behind the throat, a new PTTR appears, whichcharacterizes strong gravitational field in this accessible region, separated from the ATTR byan unstable photon region (4d, 4g). These solutions also belong to the type I .In the extreme case | a | = a e , the closed TTSs and ATTSs completely disappear, while thePTTSs survives in the vicinity of the poles. The APTTS region passes exactly into the stablephoton region (3j). In the overextreme case, the stable photon region is no longer hidden by thehorizon. As usual for the horizonless ultracompact objects, this indicates on their instability[21]. These solutions belong to the type II . At critical rotation a c , even the PTTSs disappearcompletely and the ATTR merges with the region of a weak gravitational field (5c) which istypical for the type III solutions.The difference from Kerr in the case of the Kerr-Newman metric ( q < q C ) can alreadybe seen from the Fig. (2b). First, there is a large antigravity area characterized by closedATTSs. Second, the region of stable photons is separated from the singularity and arises assmall islands in APTTR (4h) only for certain values of rotation ( | a | > a C ) (see 4d, 4e and16g,4h). In the extreme case, closed TTSs completely disappear again, but the stable photonregion is separated from the singularity by the ATTR (3k). However, as in the case of Kerr,the solution is unstable in an overextreme mode and falls into the type II .In contrast to this, when q > q C , i.e. even in the overextreme regime, the TTSs still existprovided | a | < a C (2c). At the same time, the stable photon regions completely disappearand reappear only in the superextreme space, which indicates on their instability in the sameway, but now this region is hidden by closed TTSs (3l, 5e) right up to the rotation value a C ,which may indicate some improvement in the observability properties as being type I . Fromthe observational point of view, this solution mimics a subextreme one.With an extreme value of the charge q e , the horizon disappears at any value of rotation, atthe same time a stable photon region appears (5h) and it is still hidden by the TTS area rightup to the rotation a C .In the overextreme case q > q e , there are both photon and antiphoton spheres that limitthe TTR (5j). Rotation leads to smearing of these surfaces into the corresponding stable andunstable photon regions separated by (P)TTS (5k). Full classification is presented in the TableIV and Figs. (3-5). We will provide more motivation for the above classification in the nextsection. V. PHOTON ESCAPE
To clarify physical meaning of the geometric structures introduced above, we consider escapeto infinity of the photons emitted from different types of characteristic surfaces in the Kerr-Newman spacetime, generalizing the work of Synge [2]. A detailed analysis of null geodesicsbehavior in the Kerr-Newman metric as well as its shadow can be found, e.g., in [40]. The goalhere is to relate the escape properties to our set of characteristic surfaces including (P)TTS. Wewill restrict by the outer region r > r h + (∆ r >
0) of the (sub)extreme Kerr-Newman metric.17
ABLE III: Different optical types of Kerr-Newman q a
Horizon TTS PTTS UPR SPR Type q < q C a = 0 + + − + − I < | a | < a C + + + + − Ia C < | a | < a e + + + + + Ia e < | a | < a c − − + + + II | a | > a c − − − + + IIIq C < q < q e a = 0 + + − + − I < | a | ≤ a e + + + + − Ia e < | a | ≤ a C − + + + + Ia C < | a | < a c − − + + + II | a | > a c − − − + + IIIq e < q ≤ q m a = 0 − + − + − I < | a | ≤ a C − + + + + Ia C < | a | < a c − − + + + II | a | > a c − − − + + IIIq > q m − − − − − IV Let’s start by listing the equations of motion in an explicit form [31]:Σ ˙ φ = L − Ea sin θ sin θ + a (( r + a ) E − aL )∆ r , (5.1)Σ ˙ t = a ( L − Ea sin θ ) + ( r + a )(( r + a ) E − aL )∆ r (5.2)(Σ ˙ r ) = (( r + a ) E − aL ) − K ∆ r ≡ R ( r ) , (5.3)(Σ ˙ θ ) = K − (cid:18) L − Ea sin θ sin θ (cid:19) ≡ Θ( θ ) , (5.4)where K , L and E are the Carter constant, the azimuthal orbital moment and the energy,respectively.We want to find the conditions under which a photon emitted at a certain point r o , θ o ona spherical surface with a certain angle of inclination ψ relative to an outward normal vector18oes to infinity. The angular parameter ψ is defined following Synge [2]:cot ψ = 1∆ r (cid:18) drdθ (cid:19) , (5.5)so that 0 ≤ ψ ≤ π/
2, with ψ = 0 for the normally emitted photons and ψ = π/ L/E = ρ which has ameaning of the azimuthal impact parameter. Assuming that the orbit starts at r o , θ o , one canexpress ψ via the Eqs. (5.3), (5.4) and (5.5) through ρ, K, r o , θ o . Alternatively, the Carterconstant K can be expressed in terms on ψ , defining the function K o : K o /E = ( ρ − a sin θ o ) csc θ o cos ψ + ( a + r o − aρ ) sin ψ ∆ r . (5.6)Substituting this (and the impact parameter) into the radial potential divided by E , we obtainthe function R o ( r ) = E − R ( r, K o , ρ ) , (5.7)which depends on the initial point on the sphere r o , θ o and the inclination angle ψ as parameters.The specified initial conditions are suitable if and only if R o ( r o ) ≥ , Θ o ( θ o ) ≥ . (5.8)Now we can formulate the escape condition as the absence of a turning point of the radialvariable for all r > r o , which means: R o ( r ) > , ∀ r > r o . (5.9)Note that instead of ψ we could use the second (polar) impact parameter as in [40].One can find the following sufficient condition for photons escape in terms of the initialconditions: R (cid:48) o ( r o ) > , (5.10)where prime denotes a derivative and we keep (5.8) in mind. Indeed, the right-hand side ofthe Eq. (5.3) is the difference of two polynomials of the forth and the second degree. Bothof these polynomials monotonously increase for values of r > m if additionally aρ < a + r .Moreover, the polynomial of the fourth degree grows faster at large r . Accordingly, if initially19 (cid:48) o ( r o ) >
0, the condition R (cid:48) o ( r ) > r > r o , which means that R o ( r )grows monotonously. Since R o ( r o ) ≥
0, this would mean that the condition (5.9) is satisfied.For tangentially emitted photons ( ψ = π/ R o ( r o ) = 0 and the condition (5.9) is violatedalready in a small neighborhood of r o if R o ( r o ) < R o ( r o ) >
0, and we can find another sufficient condition.Indeed, suppose the condition (5.10) is violated. This means that the second-degree polynomialgrows faster than the fourth-degree one, and R o ( r ) monotonously decreases. Then at some point r (cid:48) o > r o the growth rate will be the same. If at this point still R o ( r (cid:48) o ) >
0, there will be noturning point and the photon can escape to infinity. In this way we obtain the second sufficientcondition: R (cid:48) o ( r (cid:48) o ) = 0 , R o ( r (cid:48) o ) > , R o ( r o ) > , r (cid:48) o > r o , (5.11)If the condition (5.11) is violated, the light cannot leave the compact area and will be trapped.Thus for the validity of the Eq. (5.9), it is necessary that at least one of the conditions (5.10,5.11) holds. In fact, the first sufficient conditions (5.10) are stronger and gives an overestimatedescape angle.Using the Eqs. (5.3), (5.6) and (5.7) it is easy to find an explicit form of the inequalities(5.8) and (5.10): ( a + r o − aρ ) − ∆ r ( ρ − a sin θ o ) csc θ o ≥ , (5.12)4 r o ( a + r o − aρ )∆ r − ( a + r o − aρ ) ∆ (cid:48) r + ∆ (cid:48) r (cid:0) ( a + r o − aρ ) − ∆ r ( ρ − a sin θ o ) csc θ o (cid:1) cos ψ > , (5.13)accurate to the non-negative multipliers cos ψ , sin ψ/ ∆ r and 1 / ∆ r respectively. In whatfollows, we will omit the index o . The roots of the equality (5.12) determine the allowablevalues of the impact parameter for a given emission point. For example, if we are outside theergosphere, the coefficient at ρ will be negative and the expression (5.12) is non-negative onlyfor ρ min ≤ ρ ≤ ρ max , where: ρ max,min = sin θ ( a + r ± a sin θ √ ∆ r ) a sin θ ± √ ∆ r . (5.14)From the condition (5.12) it is clear that the coefficient at cos ψ in (5.13) is positive for r > m and therefore the escape area monotonously decreases with ψ and is maximal for the20ormally emitted photons, as could be expected. For a photon emitted tangentially to thesphere ψ = π/
2, the first condition will be independent of the angle θ and is simply given by4 r ∆ r − ( a + r − aρ )∆ (cid:48) r > , (5.15)implying r ( r − mr + 2 q ) + a ( r + m ) + a ( r − m ) ρ > . (5.16)This differs from the expression (4.8) in one term proportional to ρ .The simplest case is that of a photon emitted at the pole, in which case ρ = 0. It is easy tosee that for ψ = 0 (normal photons) the escape region reaches the horizon, while for ψ = π/ ρ which defines the boundary of the escape zone, i.e. the photon orbits of the constantradius [40]: ρ c = ( r + a )∆ (cid:48) r − r ∆ r a ∆ (cid:48) r , (5.17)and substitute the solution into the Eq. (5.12): (cid:0) r a ∆ r sin θ − (4 r ∆ r − Σ∆ (cid:48) r ) (cid:1) / ( a ∆ (cid:48) r sin θ ) ≥ . (5.18)Thus we get the condition for the photon region. Let us explain the meaning of the conditionswe obtained. Set for definiteness a >
0. Then the condition (5.15) is satisfied for all ρ > ρ c .Suppose that the point P ( r o , θ o ) is located inside the photon region. The condition (5.18)means that ρ c is limited by ρ min < ρ c < ρ max . Respectively, for all ρ c < ρ < ρ max , P is thepoint of escape for tangent photons. Consequently, the tangent ray can leave the photon region,but only with sufficiently large values of the impact parameter. If the condition of the photonregion is violated, there may be two situations. The first corresponds to the case4 r ∆ r − ( a + r )∆ (cid:48) r > , (5.19)than ρ c < ρ c < ρ min . So the tangent geodesics with all possible values ρ min < ρ < ρ max escape from the region. This happens in the the weak field domain. It isobvious that the condition (5.19) corresponds to our previously defined A(P)TTR.Another opportunity is 4 r ∆ r − ( a + r )∆ (cid:48) r < , (5.20)21han ρ c > ρ c > ρ max . So tangent geodesics with all possible values ρ min <ρ < ρ max can not escape from the region what corresponds to a strong field area, or (P)TTR.Thus, we established an explicit correspondence between the geometric structure of the(P)TTSs and behavior of geodesics with different value of the impact parameter. It is worthnoting that in the case of metrics admitting separation of variable in the geodesic equations,there is no big difference between the properties of TTSs and PTTSs for a distant observer.Of course, the tangent light can leave the PTTR across its boundary. However, such geodesicscannot reach the spatial infinity. Nevertheless, photons escape is possible for other emissionangles ψ .From (5.10) it is easy to get an overestimated escape angle ψ :cos ψ = ( a + r − aρ ) ([ a + r − aρ ]∆ (cid:48) r − r ∆ r )∆ (cid:48) r (cid:0) ( a + r − aρ ) − ∆ r ( ρ − a sin θ ) csc θ (cid:1) . (5.21)The exact value of the escape angle can be obtained applying two conditions (5.10, 5.11)simultaneously. The Fig. (6a-6e) represents density graphs for the escape angle for given ρ superimposed on previously obtained images (3) with a = 0 . a e related to the type I. Darkblue color corresponds to the escape angle ψ = π/
2, and yellow corresponds to the escapeangle ψ = 0. The progressive colours cover the intermediate angles. As expected, in general,the (P)TTS region becomes permeable for some emission angles that tend to normal as weapproach the horizon.The other Figs. (6f-6i) represent discrete density graphs of escape impact parameter ( ρ =0 , ± . m, ... ) with a fixed angle of emission ψ = π/
2. Dark blue color corresponds to the ρ = ± m , and yellow corresponds to the ρ = 0. From the figures it can be derived that the(P)TTR is completely inaccessible for tangent geodesics going from infinity. So the (P)TTSsthemselves cut half of all possible emission angles for arbitrary values of the impact parameter,and in this sense are darkest, while the photon region is lighter. The outer region correspondsto the escape of all possible rays as it could be expected [2]. The difference between the types Iand II is visible from the Figs. (6f, 6h): in the latter case, even the tangent geodesics can leavethe vicinity of the horizon. More detailed analysis of the properties of null geodesic as well astheir relationship to the shadow can be found, e.g., in [31, 40].22 I. SUMMARY AND DISCUSSION
In this paper, we argued that one can reasonably extend the notion of transversely trappedphoton surfaces (TTS) in stationary spacetimes originally intended for closed two-sections tothe non-closed case. The TTS is defined as a surface such that an initially tangent photoneither remains in it forever or leaves it in the inward direction. In Boyer-Lindquist coordinates,such surfaces have a constant radial coordinate, but their closure is not guaranteed due to thepresence of an additional restriction on the polar angle. As a consequence, the conditions onTTSs do not necessarily hold for all θ . Such TTSs therefore are not enough to ensure filling(together with the photon region) of the full space manifold in Kerr-like spacetimes. For thispurpose one has also to consider non-closed TTSs, whose two-sections cover only a sphericalcap. The initially tangent photons in principle could leave the inner region across the boundaryof the cap, but, as we have seen, such photons do not escape to spatial infinity in Kerr andKerr-Newman metrics.We also found that it is useful to introduce the anti-trapping surfaces (ATTS), so thatinitially tangent photons can leave them on the outside. For them, the necessary conditionsare also θ -dependent, so this concept should be extended to the non-closed surfaces (APTTS).The (P)TTSs and A(P)TTSs foliate the corresponding volume regions (P)TTR andA(P)TTR, which together with the photon regions (PR) cover the entire space manifold inKerr-like spacetimes. This was illustrated explicitly in the cases of Kerr and Kerr-Newmanmetrics. Although the separability of geodesic equations in these cases makes it possible toexplicitly describe all photon orbits, therefore our characteristic of the null geodesic structuremay seem excessive, but this does not apply to metrics (for example, the Weyl class) for whichthe geodetic equations are non-separable. Our method does not appeal to solving the geodesicequations while it still remains a purely analytic tool.Using the Kerr-Newman solution as an example, we have demonstrated how this descriptioncan be applied to classify optical properties of strong gravitational fields. The critical valuesof the charge parameter q C = 3 / m , q m = 9 / m and the rotation parameter a C , a c , markingthe qualitative changes in the optical structure, are found. In particular, it was shown thatfor q > q C , the singularity in an overextreme regime can be weakly naked in the sense of theexistence of a closed TTS, and it is strongly naked when q > q m , according to classificationintroduced in [34]. 23o clarify physical significance of different regions, we examined their escape properties,extending the work of Synge [2, 40]. In this description, the rays starting at certain anglesto a normal to a chosen characteristic surface are traced to infinity. The escape angle valuesare visualized associating to them different colors on the density graphs. We found that, asexpected, the TTSs capture all tangential geodesics for any values of the impact parameter,while the photon regions absorb only a fraction of such geodesics. The (P)TTSs are stillpermeable to some non-tangential ray starting angles. However, now the angle is different fordifferent values of the impact parameter.In the case of the Kerr-Newman metric, the system admits separation of variables, andthe difference between properties of TTS and PTTS is reduced. But the situation may bedifferent in the case of non-separable spaces and non-integrable geodesic equations [41]. Itshould be noted that in such cases the geometric structure of the transversely trapping surfacesis preserved, while other characteristics like the photon region are not applicable, which can beseen, for example, for Zipoy-Voorhees or Tomimatsu-Sato [42] solutions. But even such spacescan be still classified with the help of (P)TTS. It turns out that in these cases the differencesbetween the TTSs and the PTTSs become significant. A ray that starts at a certain angle inthe region bounded by PTTS may leave this area at a different angle. In any case, TTSs willretain their main feature - to characterize the compact trapping region of a strong gravitationalfield, while the PTTSs characterize the strong lensing [43–45]. Acknowledgement
We thank G´erard Cl´ement for careful reading of the manuscript and valuable comments. Thework was supported by the Russian Foundation for Basic Research on the project 17-02-01299a,and by the Government program of competitive growth of the Kazan Federal University.
APPENDIX A
Here we give derivation of the conditions listed in Table I from the inequalities (3.8, 3.10).First, the roots of a polynomial P ( ξ ) = ( b − c ) ξ − dξ + ( a − b ) = 0 (6.22)24ead ξ ± = d ± (cid:112) d − ( a − b )( b − c ) b − c . (6.23)Let’s start with the PTTS conditions P ( ξ ) ≥ ξ ≤
1. The simplest opportunity is b > c, d ≤ ( a − b )( b − c ) , (6.24)meaning that P ( ξ ) is non-negative everywhere and corresponds to the first condition in TableI. Another possibility is that the both roots (6.23) are located to the left or to the right of theinterval ξ ≤ b > c, ξ − ≥ or ξ + ≤ − . (6.25)This condition reduces to the following b > c, a − c ≥ | d | , | d | > ( b − c ) . (6.26)Squaring this we get the second conditions in Table I. The last opportunities are: b < c or b = c, P ( ± ≥ , (6.27)which means that the roots (6.23) of the polynomial P ( ξ ) are located on the opposite sides ofthe interval ξ ≤ P ( ξ ) | b = c is above the axis ξ respectively. Explicitly, itis the third condition in Table I.Inside the ergoregion, the PTTS conditions become P ( ξ ) ≤ ξ ≥
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0, dashed – throat at r = 0, mesh – ergoregion,blue mesh – causality violation, green – (P)TTR, aqua – A(P)TTR. a) a = 0, q = 0 (b) a = 0, q < q C (c) a = 0, q > q C (d) 0 < a < a e , q = 0 (e) 0 < a < a C , q < q C (f) 0 < a < a e , q > q C (g) 0 < a < a e , q = 0 (h) a C < a < a e , q < q C (i) 0 < a < a e , q > q C (j) a = a e , q = 0 (k) a = a e , q < q C (l) a = a e , q > q C FIG. 4: The inner part of (3). Red color – unstable photon region, yellow – stable photon region,dark blue – region with ∆ r ≤
0, dashed – throat at r = 0, mesh – ergoregion, blue mesh – causalityviolation, green – (P)TTR, aqua – A(P)TTR. a) a = 0, q = q C (b) a = a C , q = q C (c) a = a c , q = q C (d) a = 0, q > q C (e) a = a C , q > q C (f) a = a c , q > q C - - - - x z (g) a = 0, q = q e (h) a = a C , q = q e (i) a = a c , q = q e - - - - x z (j) a = 0, q > q e (k) a = a C , q > q e (l) a = a c , q > q e FIG. 5: Optical types of Kerr and Kerr-Newman spaces. Red color – unstable photon region, yellow– stable photon region, dark blue – the region ∆ r ≤
0, dashed – the throat at r = 0, mesh – theergoregion, blue mesh – the causality violating region, green – (P)TTR, aqua - A(P)TTR. a) ρ = 2 m (b) ρ = 0 (c) ρ = − m (d) ρ = 4 m (e) ρ = − m (f) a = 0 . a e , ρ ≥ a = 0 . a e , ρ ≤ a = a e , ρ ≥ a = a e , ρ ≤ FIG. 6: Density graphs for the escape angle and the impact parameter. For (6a-6e) dark blue corre-sponds to the escape angle ψ = π/ ψ = 0. For (6f-6i)dark blue corresponds to the ρ = ± m while yellow corresponds to the ρ = 0. The progressive colourscover the intermediate angles or impact parameters.= 0. The progressive colourscover the intermediate angles or impact parameters.