Appearance of Keplerian discs orbiting Kerr superspinars

# Appearance of Keplerian discs orbiting Kerr superspinars

## Abstract

We study optical phenomena related to appearance of Keplerian accretion discs orbiting Kerr superspinars predicted by the string theory. The superspinar exterior is described by the standard Kerr naked singularity geometry breaking the black hole limit on the internal angular momentum (spin). We construct local photon escape cones for a variety of orbiting sources that enable to determine the superspinars silhouette in the case of distant observers. We show that the superspinar silhouette depends strongly on the assumed edge where the external Kerr spacetime is joined to the internal spacetime governed by the string theory and significantly differs from the black hole silhouette. The appearance of the accretion disc is strongly dependent on the value of the superspinar spin in both their shape and frequency shift profile. Apparent extension of the disc grows significantly with growing spin, while the frequency shift grows with descending spin. This behavior differs substantially from appearance of discs orbiting black holes enabling thus, at least in principle, to distinguish clearly the Kerr superspinars and black holes. In vicinity of a Kerr superspinar the non-escaped photons have to be separated to those captured by the superspinar and those being trapped in its strong gravitational field leading to self-illumination of the disc that could even influence its structure and causes self-reflection effect of radiation of the disc. The amount of trapped photons grows with descending of the superspinar spin. We thus can expect significant self-illumination effects in the field of Kerr superspinars with near-extreme spin .

## 1 Introduction

Kerr superspinars with mass and angular momentum violating the general relativistic bound on the spin of rotating black holes () could be primordial remnants of the high-energy phase of very early period of the evolution of the Universe when the effects of the string theory were relevant ([24]). The spacetime outside the superspinar, where the stringy effects are irrelevant, is assumed to be described by the standard Kerr geometry. It is expected that extension of the internal region is limited to covering thus the region of causality violations (naked time machine) and still allowing for the presence of the relevant astrophysical phenomena related to the Kerr naked singularity spacetimes. There is an expectation that the pathological naked time machine is replaced by a correctly behaving stringy solution [24], being motivated by the resolution of the problems of 4+1 SUSY black hole solution [11] where the pathological time machine region is replaced by a portion of the Godel universe [23, 10]. Being assumed remnants of the early phases of the evolution of the Universe, Kerr superspinars can be considered as a modern alternative to the ideas of naked singularities (white holes, retarded cores of expansion) discussed as a model of quasars in early times of relativistic astrophysics [36, 37, 38, 81, 44, 32].

It should be stressed that the assumed existence of Kerr superspinars is not in contradiction with the Penrose cosmic censorship hypothesis [42] according to which the spacetime singularities generated by a gravitational collapse have to be hidden behind an event horizon. On the other hand, the cosmic censorship hypothesis that forbids the existence of naked singularities is far from being proved and, in fact, even existence of spherically symmetric naked singularity spacetimes caused by a self-similar gravitational collepse [40, 41] or a non-self-similar collapse [31] is frequently discussed (see also [76, 16, 29, 15, 21]) and the related optical phenomena are studied, e.g., in [79, 80]. The electrically charged Reissner-Nordstrom naked singularity spacetimes are also considered [22] even with the presence of the cosmological constant [64]. Formation of naked singularities due to collapse of rotating matter is modelled too [52]. Therefore, it is of some relevance and importance to consider even the complete Kerr naked singularity spacetimes and compare their (observationally relevant) properties to those of Kerr superspinars having the naked singularity and causality violating parts of the Kerr geometry removed. (The possibility of over-spinning black holes is discussed, e.g., in [28].)

The Kerr superspinars should have extremely strong gravity in their vicinity and some properties substantially differing from those of the standard Kerr black holes [18, 17, 19, 56, 57, 78, 49, 24, 25, 4]. The differences could be related both to the accretion phenomena and optical effects and their cooperation is strongly and most clearly reflected by the optical phenomena related to appearance of the Keplerian accretion discs that will be studied in the present paper. Both optical and accretion phenomena were studied even in Kerr-de Sitter naked singularity spacetimes [3, 59, 62, 63, 67, 53, 60]. Here we restrict our attention to the optical phenomena in Kerr naked singularity spacetimes with zero cosmological constant.

The energy efficiency of accretion in Keplerian discs around superspinars can be extremely high. In the field of Kerr naked singularities with spin very close to the minimal value of the efficiency overcomes substantially even the annihilation efficiency, being and is much larger than the efficiency that could be approached in the field of near-extreme Kerr black holes ([56]). Therefore, spectral profiles and appearance of Keplerian discs have to be strongly affected due to both the energy efficiency of the accretion process and the strong gravitational redshift of radiation coming from the deepest parts of the gravitational potential. Further, appearance of the Keplerian discs, their spectral continuum, and profiled spectral lines related to the innermost parts of the Keplerian discs, should reveal important differences between their character in the field of black holes and naked singularities due to the existence of bound photon orbits that appears in the vicinity of a superspinars with sufficiently small spin [56, 57] influencing thus both the optical phenomena and even the character of the innermost parts of the Keplerian disc that can extend down to the region of bound photon orbits. (Of course, no region of bound photons is possible in the field of Kerr black holes because of the presence of the event horizon.)

Geometrically thick, toroidal accretion discs orbiting a Kerr superspinar are relevant in situations where the pressure gradients are important for the accretion discs structure. 1 The inner edge of the accretion disc is then closer to the superspinar surface as compared with the edge of the Keplerian disc. Here we restrict attention to the Keplerian discs only, postponing the more complex study of optical effects related to toroidal discs to future studies.

The Kerr superspinars could appear in Active Galactic Nuclei (AGN) where supermassive black holes are usually assumed, or in Galaxy Black Hole Candidates (GBHC) observed in some X-ray binary systems [45, 50, 55]. In both classes of the black hole candidates some objects are reported with the spin estimates extremely close to the limit value and these could serve as superspinar candidates. The high spin estimates are implied by the X-ray observations in AGN MGC-6-30-15 [74] or the GBHC GRS1915+105 [51] and are related to the spectral continuum models [33, 51], profiled iron spectral lines [27, 46], and high-frequency quasi-periodic oscillations (QPOs) explained by the orbital resonant models applied to near-extreme black holes [77, 70, 69, 68, 54]. The standard most common twin peak high-frequency QPOs observed in a variety of black hole candidate binary systems with frequency ratio that are in some cases quite well explained by resonant phenomena between the radial and vertical epicyclic oscillation modes in the accretion disc, giving estimates of the black hole mass and spin [77], could equally be explained by the resonances of the epicyclic oscillations in the field of a Kerr superpsinar with different values of the mass and spin [78]. Therefore, it is quite interesting to check, if it is possible to recognize an optical effect that could, at least in principle, distinguish its demonstration in the black hole and superspinar field, or if it is necessary to combine the results of observations of different origin in order to distinguish black hole and superspinar systems. Recall that quite recently it was shown that the observed spectrum of Keplerian accretion discs cannot serve as a single indicator of the presence of a Kerr superspinar, because for a black hole with any spin an identical spectrum can be generated by properly chosen superspinar with spin in the range [72].

Due to the accretion process, the superspinar spin can be rapidly reduced and conversion of a Kerr superspinar into a near-extreme black hole is possible [57, 13]. The conversion due to counterrotating accretion discs is extremely effective and the time necessary for the conversion can be much shorter than the characteristic time of the black hole evolution [65]. Moreover, the process of converting the Kerr superspinar into a near-extreme black hole due to corotating accretion discs is very dramatic and could be interesting with connection to the most extreme cases of Gamma Ray Bursts (GBR) since a large part of he accretion disc becomes to be dynamically unstable after the transition of the superspinar into a near-extreme black hole state [57].

Observations of a hypothetical Kerr superspinar could thus be expected at high redshift AGN and quasars. If such an extremely compact object violating the standard Kerr spin bound will be observed in the Universe, it could quite well be naturally interpreted in the framework of the string theory. The Kerr superspinars can thus represent one of the most relevant experimental tests of the string theory [23]. On the other hand, one has to be very careful in making any definite conclusion based on the general-relativistic Kerr spin bound, since its breaching is also allowed in modifications of General Relativity, e.g., for braneworld Kerr black holes having quite regular event horizon when the tidal charge braneworld parameter, reflecting the braneworld black hole interaction with the external bulk space, takes negative values, and [1, 2, 66, 48, 47].

We shall discuss appearance of the corotating Keplerian discs orbiting Kerr superspinars and the silhouette shape of the superspinar surface. We shall focus our attention on the innermost parts of the Keplerian discs near the geodetical innermost stable circular orbit (ISCO) when the relativistic phenomena are strongest and the signatures of the spin effects related to the superspinars are most profound. We assume the superspinar boundary surface to be located at which guarantees that the surface is well under the ISCO. Further, we assume that the boundary of the superspinar has properties similar to those of the black hole horizon and its one-way membrane property, i.e., it is not radiating and eats the accreted matter with no optical response. Of course, the boundary properties has to be given by the exact solutions of the string theory that has to be joined to the standard Kerr geometry at the boundary and has to determine physical properties of the boundary. However, such solutions are not known at present time, and we follow the simplest possibility that is commonly used in most of the studies of the phenomena taking place in the field of Kerr superspinars [24, 4, 5]. In section 2 we shortly summarize properties of the photon geodetical motion in the Kerr naked singularity spacetimes. In section 3 we construct the local photon escape cones of a variety of relevant astrophysical observers, namely the locally non-rotating observers and the equatorial circular geodesic observers that are relevant for the Keplerian accretion discs. In section 4 the Kerr superspinar silhouette as seen by a distant static observer is constructed in dependence on the spacetime parameters and the inclination angle of the observer. We compare the cases of superspinar with and the Kerr naked singularity. Possibilities to determine the superspinar parameters from expected observational data (e.g. from the Galaxy central object Sgr ) are briefly discussed. In section 5 we integrate the equations of photon motion and investigate the optical phenomena relevant for the appearance of a Keplerian disc, namely the frequency shift of its radiation and deformations of the isoradial curves of direct and indirect images. The Keplerian discs appearances modified by the presence of a Kerr superspinar are confronted with those modified by the presence of a near-extreme black hole. In section 6 we briefly discuss the results and study separation of non-escaping photons into the photons captured by the superspinar surface and the trapped photons that could influence the accretion disc. In section 7 we present conclusions.

## 2 Photon motion

### 2.1 Geometry and its null geodesics

Kerr superspinars are described by the Kerr geometry that is given in the standard Boyer-Lindquist coordinates and geometric units () in the form

 ds2=−(1−2MrΣ)dt2+ΣΔdr2+Σdθ2+AΣsin2θdφ2−4Marsin2θΣdtdφ, (1)

where

 Δ=r2−2Mr+a2,Σ=r2+a2cos2θ,andA=(r2+a2)2−Δa2sin2θ, (2)

denotes spin and mass of the spacetimes that fulfill condition in the superspinar case. In the following, we put , i.e., we use dimensionless radial coordinate and spin .

In order to study the optical effects we have to solve equations of motion of photons given by the null geodesics of the spacetime under consideration. The geodesic equation reads

 Dkμdw=0, (3)

where is the wave vector tangent to the null geodesic and is the affine parameter. The normalization condition reads . Since the components of the metric tensor do not depend on and coordinates, the conjugate momenta

 kφ=gφνkν≡Φ, (4) kt=gtνkν≡−E, (5)

are the integrals of motion. Carter found another integral of motion as a separation constant when solving Hamilton-Jacobi equation

 gμν∂S∂xμ∂S∂xν=0, (6)

where he assumed the action in separated form

 S=−Et+Φφ+Sr(r)+Sθ(θ). (7)

The equations of motion can be integrated and written separately in the form

 Σdrdw = ±√R(r), (8) Σdθdw = ±√W(θ), (9) Σdφdw = −PWsin2θ+aPRΔ, (10) Σdtdw = −aPW+(r2+a2)PRΔ, (11)

where

 R(r) = P2R−ΔK, (12) W(θ) = K−(Pwsinθ)2, (13) PR(r) = E(r2+a2)−aΦ, (14) PW(θ) = aEsin2θ−Φ. (15)

It is useful to introduce integral of motion by the formula

 Q=K−(E−aΦ)2. (16)

Its relevance comes from the fact that in the case of astrophysically most important motion in the equatorial plane () there is for both photons and test particles with non-zero rest energy.

### 2.2 Radial and latitudinal motion

The photon motion (with fixed constants of motion , , ) is allowed in regions where and . The conditions and determine turning points of the radial and latitudinal motion, respectively, giving boundaries of the region allowed for the motion. Detailed analysis of the -motion can be found in [9, 20], while the radial motion was analysed (with restrictions implied by the -motion) in [57] and [58]. Here we summarize the analysis relevant for the naked singularity spacetimes.

 Σ2(drdw′)2 = [r2+a2−aλ]2−Δ[L−2aλ+a2], (17) Σ2(dθdw′)2 = L+a2cos2θ−λ2sin2θ (18)

where we have introduced impact parameters

 λ = ΦE, (19) L = LE2=Q+Φ2E2=q+λ2, (20)

and rescaled the affine parameter by . We assume and . (The special case of photon motion with is treated in [57].)

#### Latitudinal motion

The turning points of the latitudinal motion are determined by the condition

 λ2=λ2t≡sin2θ(L+a2cos2θ) (21)

The extrema of the function are determined by

 L+a2cos2θ=0. (22)

The character of the regions allowed for the latitudinal motion in dependence on the impact parameters and is represented by Fig.1 At the maxima of the function , there is

 λ2t=(L+a2)24a2. (23)

The impact parameter can be negative when and is small enough. The latitudinal motion can be divided into two subclasses, namely of the orbital motion (with ) which reaches the equatorial plane and the vortical motion (with ) when the motion is limited to the region above or under the equatorial plane. Clearly, only photons of the orbital type could be radiated from the equatorial Keplerian accretion discs considered in our study.

The reality conditions and lead to the restrictions on the impact parameter

 Lmin≤L≤Lmax, (24)

where

 Lmax≡(aλ−2r)2Δ+r2+2r, (25)

and

 Lmin≡{λ2for|λ|≥a,2a|λ|−a2for|λ|≤a. (26)

The upper(lower) constraint, (), comes from the radial-motion (latitudinal-motion) reality condition. The properties of the photon motion are determined by the behaviour of the surface , as given by (25). The extrema of the surface (giving spherical photon orbits) are determined by

 λ=λ+ ≡ r2+a2a, (27) λ=λ− ≡ r2−a2−rΔa(r−1). (28)

The values of at these extreme points are given by

 Lmax(λ+)≡L+ = 2r2+a2, (29) Lmax(λ−)≡L− = 2r(r3−3r)+a2(r+1)2(r−1)2. (30)

The character of the extrema follows from the sign of . One finds that

 ∂2Lmax∂r2 = 8r2Δ,forλ=λ+, (31) ∂2Lmax∂r2 = 8r2Δ−8r(r−1)2,forλ=λ−. (32)

Clearly, there are only minima of along for , corresponding to unstable spherical orbits.

Further, we have to determine where the restrictions given by the latitudinal motion meet the restrictions on the radial motion . We find that (for ) is fullfilled where

 λ=~λ±≡a(−2r±r2√Δ)r2−2r, (33)

while (for ) is fullfilled where

 λ=¯λ≡1Δ[4r−r2−a2+2√Δ(−2r)]. (34)

Clearly, the last function is defined only in the region of negative values of and is thus irrelevant in the case of the Kerr superspinars with surface located at .

The extreme points of curves , which are also the intersection points of these curves with , are determined by the equation

 f(r;a)≡r3−6r2+9r−4a2=0. (35)

The equation determines loci of the photon equatorial circular orbits; in an implicit form the radii are given by the condition

 a2=a2ph(r)=r(r−3)24. (36)

In the field of Kerr superspinars with , there exists only one (counterrotating) equatorial photon orbit at radius given by

 rph=2+[a+√a2−1]2/3+[a+√a2−1]−2/3. (37)

The maximum of the function is located at

 rm=1+(a2−1)1/3 (38)

and the value of this maximum is

 λm=a−1[3−a2−3(a2−1)2/3]. (39)

The maxima of the curve (and common points with the curve ) are located at satisfying the equation

 2r3−3r2+a2=0. (40)

These extremal points are irrelevant for our discussion being located at negative values of the radial coordinate, while we restrict our study to the region of .

The Kerr superspinar spacetimes can be classified due to the properties of the photon motion as determined by the behaviour of the functions , , . The behaviour of these functions is represented in Fig.2(left) where five characteristic values of the impact parameter , crucial for their classification and dependent on the spin parameter of the Kerr superspinars, are introduced. Dependence of characteristic values of on the spin parameter of the spacetime is given in Fig.2(right).

In the case of Kerr naked singularity spacetimes, we can distinguish four qualitatively different classes of the photon motion in dependence on the values of the spin parameter. These are given by the relative position of the characteristic values of the impact parameter .

• .

The impact parameters satisfy the relation

 λd>λa>λb>λc>λe. (41)
• .

The impact parameters satisfy the relation

 λa>λd>λb>λc>λe. (42)
• .

The impact parameters satisfy the relation

 λa>λb>λd>λc>λe. (43)
• The impact parameters satisfy the relation

 λa>λb>λc>λd>λe. (44)

In the case of Kerr superspinars, when we consider the photon motion in the region of , it is enough to distinguish only two classes of the spacetimes, namely those with and .

For each interval of as determined by the sequence of - introduced in Fig.3, there exists a characteristic type of behaviour of the restricting ”radial” function and its relation to the ”latitudinal” restricting function , see [58] for details. Here we give the sequence of the ”effective” potential sections in Fig.3 for the Kerr naked singularity spacetimes with . For the naked singularity spacetimes with , some characteristic sections are given in Fig.4 - notice the case of and when two stable spherical photon orbits exist. Further, it should be stressed the for the Kerr superspinars the effective potential sections of are relevant only in the region of .

The allowed values of the impact parameter lie between the limiting functions and . If the minimum of the limiting function is less than the value of the limiting function , an incoming photon () travelling from infinity will return back for all values of . If , the incoming photon () travelling from infinity returns back if its impact parameter satisfies the condition and is captured by the Kerr superspinar if . The minimum determines (with the particular value of ) an unstable photon spherical orbit, i.e., a sphere where photons move with but with varying latitude (and, of course, varying ). When the condition is satisfied simultaneously, the spherical photon orbit is transformed to an (unstable) equatorial photon circular orbit. Photons coming from distant regions or regions close to the superpsinar surface will wind up around the photon sphere when for a given value of the impact parameter . The maxima of the limiting function correspond to the stable photon spherical orbits that are central to the region of photons trapped by the strong gravitational field of Kerr superspinars.

### 2.3 Keplerian discs

We summarize shortly properties of the equatorial circular geodesic motion in the field of Kerr superspinars that are relevant for Keplerian, thin accretion discs, and in principle are important even for structure of thick or slim accretion discs. The Carter equations imply the specific energy and specific angular momentum of the circular geodesics to be given by the relations [8, 56]

 EKm=r3/2−2r1/2±ar3/4√r3/2−3r1/2±2a (45)
 ΦKm=±r2+a2∓2ar1/2r3/4√r3/2−3r1/2±2a. (46)

The angular velocity with respect to static observers at infinity is given by the relation

 ΩK=±1r3/2±a (47)

while the specific angular momentum related to the conservative energy parameter is given by the relation

 lK=ΦKEK=±r2+a2∓2ar1/2r3/2−2r1/2±a. (48)

In theory of Keplerian discs a crucial role is devoted to the innermost stable circular orbit (ISCO) in the field of a given spacetime that is usually considered to correspond to the inner edge of Keplerian discs [39]. In Kerr spacetimes the ISCO is determined by the relation [8]

 rms=3+Z2−√(3−Z2)(3+Z1+2Z2), (49)

where

 Z1=1+(1−a2)1/3[(1+a)1/3(1−a)1/3], (50)
 Z2=√3a2+Z21. (51)

The marginally bound orbits with have radii given by

 rmb=2+a∓2(1+a)1/2. (52)

In all presented relations the upper sign corresponds to the so called 1-st family orbits that are corotating relative to distant observers () - such orbits are locally corotating () in regions distant from superspinars, but could be locally counterrotating () in vicinity of superspinars with the spin parameter . For superspinars with spin the 1-st family orbits with could have negative energy (), while located close enough to the superspinar boundary [56] see Fig.5. The lower sign in all presented relations corresponds to the 2-nd family orbits that are both counterrotating relative to distant observers () and locally counterrotating () everywhere for all superspinars. The Keplerian energy and angular momentum radial profiles are illustrated in Fig.5. for typical values of the superspinar spin. The Keplerian angular velocity and specific angular momentum profiles are illustrated in Fig.6 for the same representative values of the superspinar spin. Notice the discontinuity of the specific angular momentum profile occuring for superspinars with spin due to decline of energy to negative values in such spacetimes. Its physical implications for toroidal discs are discussed in [53, 60].

## 3 Light escape cones

The optical phenomena related to accretion processes in the field of Kerr superspinars can be efficiently studied by using the notion of light escape cones of local observers (sources) that determine which portion of radiation emitted by a source could escape to infinity and, complementary, which portion is trapped by the superspinar [48, 47, 73, 49]. Here we focus our attention to the family of observers (sources) that are of direct physical relevance to Keplerian discs, namely the circular geodetical observers. For comparison, we construct the escape cones in the locally non-rotating frames. Later we focus our attention to the corotating frames .

### 3.1 Local frames of stationary observers

We consider two families of stationary frames, namely (Locally Nonrotatig Frames) and (Circular Geodesic Frames). The are of fundamental physical importance since the physical phenomena take the simplest form when expressed in such frames, because the rotational spacetime effects are maximally suppressed there [7, 34]. The are directly related to Keplerian accretion discs in the equatorial plane of the spacetime, both corotating and counterrotating. The are, of course, geodetical frames, while are generally accelerated frames.

The radial and latitudinal 1-forms of the stationary frame tetrads are common for both families of frames and read

 ω(r) = {0,√Σ/Δ,0,0}, (53) ω(θ) = {0,0,√Σ,0}. (54)

correspond to observers with (zero angular momentum observers). Their time and azimuthal 1-forms read

 ω(t) = {√ΔΣA,0,0,0}, (55) ω(φ) = {−ΩLNRF√AΣsinθ,0,0,√AΣsinθ}. (56)

where

 ΩLNRF=2aMrA (57)

is the angular velocity of as seen by observers at infinity.

The observers move along -direction in the equatorial plane with velocity (+…corotating, -…counterrotating) relative to the and with angular velocity relative to the static observers at infinity given by [8]

 Ω±=±1r3/2±a. (58)

The velocity is given by

 VGF±=±(r2+a2)∓2ar1/2√Δ(r3/2±a). (59)

The standard Lorentz transformation of the tetrad gives the tetrad of in the form

 ω(t)± = {r2−2r±ar1/2Z±,0,0,∓(r2+a2)r1/2∓2arZ±}, (60) ω(φ)± = ⎧⎪ ⎪⎨⎪ ⎪⎩∓√Δr1/2Z±,0,0,√Δ(r2±ar1/2)Z±,⎫⎪ ⎪⎬⎪ ⎪⎭ (61)

where

 Z±=r√r2−3r±2ar. (62)

Note that the family is restricted to the equatorial plane, while are defined at any .

### 3.2 Construction of escape cones

The analysis of the turning points of the radial motion of photons is crucial in determining the local escape cones as the boundary of the escape cones is given by directional angles related to unstable spherical photon orbits. For each direction of emission in the local frame of a source, there is a corresponding pair of values of the impact parameters and which can be related to the directional cosines of the photon trajectory in the local frame at the position of the source and we have to find those corresponding to the unstable spherical photon orbits. Notice that in the case of Kerr black holes the inversion of the local escape cone about the symmetry axis of the spacetime represents silhouette of the black hole as observed from the corresponding local frame [47, 71], but it is not the case of the Kerr superspinar spacetimes, as we have to select photons captured by the superspinars and those trapped in their gravitational field without being captured by the superspinar surface.

Projection of a photon 4-momentum onto the local tetrad of an observer is given by the formulae

 k(t) = −k(t)=1, (63) k(r) = k(r)=cosα0, (64) k(θ) = k(θ)=sinα0cosβ0, (65) k(φ) = k(φ)=sinα0sinβ0, (66)

where , are directional angles of the photon in the local frame (see [47]) and

 cosγ0=sinα0sinβ0. (67)

In terms of the local tetrad components of the photon 4-momentum and the related directional angles, the conserved quantities, namely, the azimutal momentum , energy and read

 Φ = kφ=−ω(t)(t)φk(t)+ω(r)(r)φk(r)+ω(θ)(θ)φk(θ)+ω(φ)(φ)φk(φ), (68) E = −kt=ω(t)(t)tk(t)−ω(r)(r)tk(r)−ω(θ)(θ)tk(θ)−ω(t)(φ)φk(φ), (69) K = 1Δ{[E(r2+a2)−aΦ]2−(Σkr)2}. (70)

The impact parameters and defined by relations (19) and (20) are thus fully determined by any double, , of angles from the set .

In a given source frame, with fixed coordinates , , we can construct light escape cones using the following procedure:

• for given , say , we calculate ,

• determines the behaviour of ,

• from the analysis presented in the previous section we calculate minimum of , which reads ,

• we search for such a double which satisfies equation .

Here, we present in detail the construction of light escape cones in particular case of the . The procedure is analogous for the . Notice that for large distances from the superspinar the LNRF are almost identical to the frames of static observers.

Photons radiated in the field of a Kerr superspinar by a given point-like source that are not able escape to infinity can be separated into two parts - the first one consists from the photons captured by the superspinar surface, the second one consists from those that are trapped in vicinity of the superspinar. Recall that in the field of a Kerr black hole all such photons are captured by the black hole. First we restrict our attention to the construction of the escape cones (see also [47]); the trapped photons will be discussed later.

The impact parameter expressed in terms of the angle , related to the , reads

 λ0=1ΩLNRF0+Σ0√Δ0A0sinθ0cosγ0, (71)

where index ’’ refers to the frame with coordinates . The minimum of is located at

 rmin=⎧⎨⎩√aλ−a2forλ≥a1−k1k2+k23forλ

where

 k1 = a2+aλ−3, (73) k2 = {27(1−a2)+2√3√27(1−a2)2+k31}1/3. (74)

The relevant values of lie between and determined by Eqs (25) and (26). The intersections of functions and give the relevant interval of angles (see figure Fig.9). For each from we calculate minimal value of the photon impact parameter for which the photon reaches the turning point and escapes to infinity. This minimal value is the minimum of which is located at , eg. , where is given by (72). Now we can calculate the value of using equation

 cosα0=k(r)k(t)=ω(r)LNRFμkμω(t)LNRFμkμ. (75)

We arrive to the formula

 cosα0=±√A0√(r20+a2−aλ0)2−Δ0(Lminmax−2aλ0+a2)−a(asin2θ0−λ0)Δ0+(r20+a2)(r20+a2−aλ0), (76)

where , and . The angle can be calculated from the formula (66). In this way we obtain angles from the arc . The remaining arc can be obtained by turning the arc around the symmetry axis determined by angles and . This procedure can be done because photons released under angles and have the same constants of motion. Clearly, for sources under the radius corresponding to the (counterrotating) equatorial photon circular orbit, only outward directed photons with no turning point of the -motion can escape. With radius of the source approaching the Kerr superspinar surface (at ), the escape cone shrinks, but its extension remains finite, contrary to the case of the Kerr black holes when approaching the black hole horizon, the escape cone shrinks to infinitesimal extension, with exception of the extreme black holes [7]. For the , the procedure of the related light escape cone construction can be directly repeated, but with the relevant tetrad 1-form components being used in the procedure. (If the region of trapped photons in the vicinity of the superspinar surface has to be constructed, we have to consider the stable spherical photon orbits and the related impact parameters that will determine the trapped-photon region.)

In order to reflect properly the effect of the superspinar spin on the escape cone structure, we shall give the cones for two sequences, namely for and observers located at the ISCO-radius of the spacetimes with appropriately chosen values of the spin.

Behaviour of the escape cones in dependence on the superspinar spin is represented in Fig.7. The complementary local cones, corresponding to non-escaping photons trapped in the field of the superspinar or captured by its surface, are shaded.

For the circular (corotating) geodesic frames the local escape cones are presented in the Fig.8 for the same values of spin as in the case of LNRF. In both the and local escape cones the latitude is kept to relevant for the Keplerian discs. For comparison, we include in the first position for both of the sequences ( and ) the local escape cone constructed for a near-extreme Kerr black hole with the canonical value introduced by Thorne [75].

The local escape cones at the ISCO radius grow with the value of the spin growing in its whole considered range and for the Kerr superspinars are always larger than the one corresponding to the canonical black hole - this is the picture given for local frames (of observers or sources) that maximally restrict the rotational effects of the Kerr spacetimes [8]. The local escape cones at the ISCO also demonstrate growing as the superspinar spin grows, but for sufficiently small values of the spin () they are smaller as compared with the local escape cone of the orbiting a canonical black hole with while they are larger for .

Further, it should be noticed that at the ISCO radius of the canonical black hole the local escape cone is substantially smaller in comparison with the one, demonstrating thus the influence of the Keplerian rotation. The rotational peculiarity of the Kerr superspinars is demonstrated by the fact that the ISCO-radius escape cones are larger (smaller) in comparison with the ISCO escape cones for (). This kind of behaviour reflects the influence of the rotational state of the orbiting matter that is locally counterrotating () at ISCO for Kerr superspinars with .

## 4 Silhouette of Kerr superspinars

In principle, it is of astrophysical importance to consider a Kerr superspinar (a Kerr naked singularity or a black hole [7, 48, 47]) being located in front of a source of illumination whose angular size is large as compared with the angular size of the superspinar. A distant observer will see a silhouette of the superspinar in the larger bright source. The superspinar silhouette is determined by photons that reach its surface and finish their travel there, contrary to the case of the rim of a black hole silhouette that corresponds to photon trajectories spiralling near the unstable spherical photon orbit around the black hole many times before they reach the observer. The spiralling photons concentrated around unstable spherical photon orbits will create an additional arc characterizing the superspinar (or a Kerr naked singularity) [26]. Of course, the shape of the silhouette and the arc enables, in principle, determination of the superspinar (or black hole) spin. But we have to be aware of the strong dependence of the silhouette shape on the observer viewing angle; clearly, the shape will be circular for observers on the superspinar rotation axis, and its deformation grows with observer approaching the equatorial plane.

Assuming that distant observers measure photon directions relative to the symmetry center of the gravitational field, the component of the angular displacement perpendicular to the symmetry axis is given by (for superspinar rotating anticlockwise relative to distant observers), while for angular displacement parallel to the axis it is given by . These angles are proportional to , therefore, it is convenient to use the impact parameters in the form independent of [7]

 ~α=−r0p(φ)p(t)=−λsinθ0, (77)

and

 ~β = r0p(θ)p(t)=[q+a2cos2θ0−λ2cot2θ0]1/2 (78) = [L+a2cos2θ−λ2sin2θ0]1/2.

Photon trajectories reaching the observer are represented by points in the plane covering a small portion of the celestial sphere of the observer.

The shape of the superspinar silhouette (arc) is the boundary of the no-turning-point region, i.e., it is the curve expressed in the plane of the impact parameters. For observers in the equatorial plane , , .

We consider the superspinar being observed by static distant observers that are related to the LNRF with zero angular velocity. Therefore, we use static frames at far distances, with the tetrad given by

 ω(t) = {√ΔΣA,0,0,0} (79) ω(r) = {0,√Σ/Δ,0,0} (80) ω(θ) = {0,0,√Σ,0} (81) ω(φ) = {−ΩLNRF√AΣsinθ,0,0,√AΣ} (82)

The silhouette of the superspinar is quite naturally related to the trapped (escape) light cones of the static frames.

The marginal values of impact parameters and (resp ) are obtained from the light escape cone. Using the stationary of the Kerr spacetime we “shoot out“ virtual photons from observer (static frame at very large distance ) and we are looking for the light escape cone of this virtual source (using the results of the previous section). The trapped light cone of this virtual source is constructed from the light escape cone of the virtual source by transformations of directional angle to and directional angle to . In this way we get marginal directions for received photons from bright background behind the superspinar. Then we can use the formulas (68), (69) and (70) to calculate the marginal values of and () in order to obtain the silhouette of the Kerr superspinar in the plane , i.e., the set of doubles from equations (77) and (78). Here we plotted the silhouette directly from the trapped light cone on the observer’s sky . Note that the angle is the radial coordinate and the angle is the polar coordinate in the polar graph of the silhouette.

We shall give the silhouette of the superspinar for observers located at fixed radius M that corresponds to the angular size of arcsec; for higher distances the angular size falls accordingly to the dependence. We systematically compare the superspinar silhouette, when the photons defining its shape are those captured by the superspinar surface, with the silhouette of the Kerr naked singularity having the same spin when its shape is given by the photons that escape to the parallel asymptotic infinity corresponding to . Additionally, an arc corresponding to the unstable photon spherical orbits defines both the superspinars and naked singularities equivalently. (For Kerr naked singularities viewed under the inclination angle , the silhouette shape is deformed into a line lying in the equatorial plane and terminated in the arc.) The silhouette of the superspinars and Kerr naked singularities is determined by both spin and inclination angle, while for superspinars it is further determined by the radius of its surface. In the case of Kerr naked singularities some characteristics of the silhouette could be defined properly in order to determine the spin and the inclination angle [25]. Here we demonstrate that for superspinars with the additional parameter giving their radius the situation with defining the characteristics enabling the superspinar parameters is more complicated. The rotational effect on the shape of the silhouette grows with inclination angle growing and becomes strongest when .

We give an illustrative picture of the spin influence on the silhouette properties for two characteristic inclination angles (Fig.9) and (Fig.10) when the superspinar (naked singularity) rotational effects are very strong and mediate.

The silhouette shape can be efficiently used to determine the parameters of Kerr naked singularities [25] and black holes [47, 25]. In order to characterize the influence of the spin on the silhouette of a Kerr superspinar or naked singularity we define here two quantities in principle measurable by distant observers. The angle parameter of the silhouette arc

 χ=arctan(βMαM), (83)

and the silhuette ellipticity

 ϵ=2βmaxαmax, (84)

where are coordinates of the edge point of the silhouette arc on the observers sky and and are maximal width and height of the silhouette. The definition of the angle parameter and ellipticity is illustrated in Fig.11. (For an analogous definition of parameters characterizing black-hole silhouettes see [47, 25, 73].) We calculated angle parameter and ellipticity of the Kerr naked singularity silhouette as a function of its spin taken from the interval (see Fig.12). These are quantities that could be measured and used for a Kerr naked singularity spin and the inclination angle of the observer estimates, if observational techniques could be developed to the level enabling the silhouette detailed measuring. Such a possibility seems to be realistic in near future for the case of the supermassive object (black hole or superspinar) predicted in the Galaxy Centre (Sgr ) [12, 47]. Since the ellipticity parameter is not giving unique predictions in dependence on the inclination angle (see Fig.12), probably some more convenient silhouette parameter has to be introduced.

Clearly, we have to introduce three properly chosen characteristics of the superspinar silhouette and its arc in order to determine three parameters related to observed superspinars - namely spin, inclination angle of the observer and superspinar surface radius. For Kerr naked singularities only the spin and inclination angle have to be determined using two characteristics of the silhouette and its arc as shown in [25]. In the case of Kerr superspinars, an additional silhouette-characteristic parameter related to the superspinar surface radius has to be introduced. Such a parameter has to reflect relative positions of the silhouette and the arc, however, it is not clear yet if such a parameter could be uniquely related to the superspinar parameters and the observer inclination angle.

It is evident that the problem of defining the three superspinar silhouette and its arc parameters and relating them to the three parameters is much more complex in comparison with the problem of relating two Kerr naked singularity parameters and two silhouette and arc characteristics as presented in [25]. It will be discussed in a future paper.