Eur. Phys. J. C manuscript No. (will be inserted by the editor)
Observers in Kerr spacetimes: the ergoregion on the equatorial plane
D. Pugliesea,1, H. Quevedo2,3,4 1Institute of Physics, Faculty of Philosophy & Science, Silesian University in Opava, Bezrucovoˇ námestíˇ 13, CZ-74601 Opava, Czech Republic 2Dipartimento di Fisica, Università di Roma “La Sapienza", I-00185 Roma, Italy 3Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, AP 70543, México, DF 04510, Mexico 4Department of Theoretical and Nuclear Physics, Kazakh National University, Almaty 050040, Kazakhstan
Received: date / Accepted: date
Abstract We perform a detailed analysis of the properties viewpoint of stationary observers and their frequencies, as- of stationary observers located on the equatorial plane of suming the invariance of the system symmetries (axial sym- the ergosphere in a Kerr spacetime, including light-surfaces. metry and time independence). One of the goals of this work This study highlights crucial differences between black hole is to explore the existence of spin transitions in very weak and the super-spinner sources. In the case of Kerr naked sin- naked singularities [2], which are characterized by a spin gularities, the results allow us to distinguish between “weak” parameter a/M ≈ 1. If the collapse of a stellar object or and “strong ” singularities, corresponding to spin values close the merging of several stellar or BH attractors lead to the to or distant from the limiting case of extreme black holes, formation of a naked singularity, then a total or partial de- respectively. We derive important limiting angular frequen- struction of the horizon may occur which should be accom- cies for naked singularities. We especially study very weak panied by oscillations of the spin-to-mass ratio. Naked sin- singularities as resulting from the spin variation of black gularities can also appear in non-isolated BH configurations holes. We also explore the main properties of zero angular as the result of their interaction with the surrounding mat- momentum observers for different classes of black hole and ter, i. e., in some transient process of the evolution of an naked singularity spacetimes. interacting black hole. Indeed, the interaction can lead to modifications of characteristic BH parameters, for instance, through a spin-up or spin-down process which can also alter 1 Introduction the spacetime symmetries. The details of such spin transi- tions, leading possibly to the destruction of the horizon, and The physics of black holes (BHs) is probably one of the their consequences are still an open problem. most complex and still controversial aspects of Einstein’s geometric theory of gravitation. Many processes of High In this work, keeping the Kerr spacetime symmetries un- Energy Astrophysics are supposed to involve singularities changed, we focus on the variation of the dimensionless spin and their formation from a stellar progenitor collapse or from parameter in the region within the static limit on the equato- the merging of a binary BH system. The interaction of these rial plane of the attractor, this being the plane of symmetry arXiv:1801.06149v1 [gr-qc] 18 Jan 2018 sources with the matter environment, which can lead to ac- of the Kerr solution. This special plane of the axisymmetric cretion and jets emission, is the basis for many observed geometry has many interesting properties; for instance, con- phenomena. As a consequence of this interaction, the singu- stants of motion emerge due to the symmetry under reflec- larity properties, determined generally by the values of their tion with respect to this plane; the geometry has some pecu- intrinsic spin, mass or electric charge parameters, might be liarities that make it immediately comparable with the limit- modified, leading to considerable changes of the singularity ing static Schwarzschild solution, in particular, the location itself. In this work, we concentrate our analysis on the er- of the outer ergoregion boundary is independent of the spin goregion in the naked singularity (NS) and BH regimes of value, and coincides with the location of the Schwarzschild the axisymmetric and stationary Kerr solution. We are con- horizon. There is also a clear astrophysical interest in the ex- cerned also about the implications of any spin-mass ratio ploration of such a plane, as the large majority of accretion oscillation between the BH and the NS regimes from the disks are considered to be located on the equatorial plane of ae-mail:[email protected] their attractors. 2
From a methodological viewpoint, our analysis repre- can be relevant in connection to superspinars, as discussed sents a comparative study of stationary and static observers in [44]. The stability of Kerr superspinars has been analyzed in Kerr spacetimes for any range of the spin parameter. The quite recently in [46], assenting the importance of boundary findings in this work highlight major differences between conditions in dealing with perturbations of NSs. the behavior of these observers in BH and NS geometries. An interesting perspective exploring duality between el- These issues are clearly related to the most general and widely ementary particles and black holes, pursuing quantum black discussed problem of defining BHs, their event horizon and holes as the link between microphysics and macrophysics, their intrinsic thermodynamic properties [3–8]. Further, it can be found in [47–50]– see also [51]. A general discussion seems compelling to clarify the role of the static limit and on the similarities between characteristic parameter values of the ergoregion in some of the well-analyzed astrophysi- of BHs and NSs, in comparison with particle like objects, is cal processes such as the singularity formation, through the addressed also in [52–55]. Quantum evaporation of NSs was gravitational collapse of a stellar “progenitor” or the merg- analyzed in [56], radiation in [57], and gravitational radia- ing of two BHs. Similarly, it is interesting to analyze the role tion in [58–60]. of the frame-dragging effect in driving the accretion pro- Creation and stability of naked singularities are still in- cesses. In fact, the ergosphere plays an important role in the tensively debated [61–66]. A discussion on the ergoregion energetics of rotating black holes. stability can be found in [67, 68]. However, under quite gen- The dynamics inside the ergoregion is relevant in Astro- eral conditions on the progenitor, these analysis do not ex- physics for possible observational effects, since in this re- clude the possibility that considering instability processes a gion the Hawking radiation can be analyzed and the Penrose naked singularity can be produced as the result of a grav- 1 energy extraction process occurs [9–13] . For the actual itational collapse. These studies, based upon a numerical state of the Penrose process, see [15]. Another interesting integration of the corresponding field equations, often con- effect connected directly to the ergoregion is discussed in sider the stability of the progenitor models and investigate [16]. The mechanism, by which energy from compact spin- the gravitational collapse of differentially rotating neutron ning objects is extracted, is of great astrophysical interest stars in full general relativity [69]. Black hole formation is and the effects occurring inside the ergoregion of black holes then associated with the formation of trapped surfaces. As a are essential for understanding the central engine mecha- consequence of this, a singularity without trapped surfaces, nism of these processes [17, 18]. Accreting matter can even as the result of a numerical integration, is usually consid- get out, giving rise, for example, to jets of matter or ra- ered as a proof of its naked singularity nature. However, the diation [17, 19] originated inside the ergoregion. Another non existence of trapped surfaces after or during the gravi- possibility is the extraction of energy from a rotating black tational collapse is not in general a proof of the existence of hole through the Blandford-Znajek mechanism (see, for in- a naked singularity. As shown in [70], in fact, it is possible stance, [20–29]). An interesting alternative scenario for the to choose a very particular slicing of spacetime during the role of the Blandford-Znajek process in the acceleration of formation of a spherically symmetric black hole where no jets is presented in[30]. Further discussions on the Penrose trapped surfaces exist (see also [71]). Eventually, the pro- and Blandford-Znajek processes may be found in [31, 32]. cess of gravitational collapse towards the formation of BHs In general, using orbits entering the ergosphere, energy can (and therefore, more generally, the issues concerning the for- be extracted from a Kerr black hole or a naked singularity. mation or not of a horizon and hence of NSs) is still, in spite On the other hand, naked singularity solutions have been of several studies, an open problem. There are transition pe- studied in different contexts in [33–42, 44, 45]. Kerr naked riods of transient dynamics, possibly involving topological singularities as particle accelerators are considered in [43]– deformations of the spacetime, in which we know the past see also [44, 45]. More generally, Kerr naked singularities and future asymptotic regions of the spacetime, but it is still in fact largely unclear what happens during that process. The 1 The Hawking process is essentially due to the vacuum fluctuation hap- problem is wide and involves many factors as, especially in pening in the regions close to the BH horizon; it is not related to the properties of the ergoregion itself. The Hawking radiation is the (spon- non-isolated systems, the role of matter and symmetries dur- taneous) emission of thermal radiation which is created in the vacuum ing collapse. Another major process that leads to black hole regions surrounding a BH, and leads to a decrease of the mass. Con- formation is the merging of two (or more) black holes, re- nected in many ways to the Unruh effects, it generally leads to the pro- cently detected for the first time in the gravitational waves duction of pairs of particles, one escaping to infinity while the other is trapped by the BH horizon. On the other hand, the Penrose energy ex- sector [72]. See also [73, 74] for the first observation of the traction, or its wave-analogue of super-radiance, is related essentially probable formation of a BH from the coalescence of two to a classical (i.e. non quantum) phenomenon occurring in the ergore- neutron stars. An interesting and detailed analysis of Kerr + gion, ]r+,rε [, due to the frame-dragging of the spinning spacetime. In and Kerr-Newman naked singularities in the broader con- this way, energy can be extracted from the source, lowering its angular momentum. For a study of the Hawking radiation in Kerr and Kerr- text of braneworld Kerr-Newman (B-KN) spacetimes can be Newman spacetimes see also [14] found in [75], where a new kind of instability, called mining 3 instability, of some B-KN naked singularity spacetimes was established that the motion of test particles on the equatorial found. In there, the exploration of the “causality violation plane of black hole spacetimes can be used to derive infor- region” is also faced. This is the region where the angular mation about the structure of the central source of gravita- coordinate becomes timelike, leading eventually to closed tion; moreover, typical effects of repulsive gravity were ob- timelike curves. Details on the relation between this region served in the naked singularity ergoregion (see also [34, 84– and the Kerr ergoregion can be found in the aforementioned 86]). In addition, it was pointed out that there exists a dra- reference. matic difference between black holes and naked singulari- In [2, 52–55, 76], we focused on the study of axisymmet- ties with respect to the zero and negative energy states in cir- ric gravitational fields, exploring different aspects of space- cular orbits (stable circular geodesics with negative energy times with NSs and BHs. The results of this analysis show a were for the first time discussed in [87]). The static limit clear difference between naked singularities and black holes would act indeed as a semi-permeable membrane separat- from the point of view of the stability properties of circu- ing the spacetime region, filled with negative energy parti- lar orbits2. This fact would have significant consequences cles, from the external one, filled with positive energy parti- for the extended matter surrounding the central source and, cles, gathered from infinity or expelled from the ergoregion hence, in all processes associated with energy extraction. In- with impoverishment of the source energy. The membrane deed, imagine an accretion disk made of test particles which is selective because it acts so as to filter the material in tran- are moving along circular orbits on the equatorial plane of sient between the inner region and outside the static limit. a Kerr spacetime. It turns out that in the case of a black This membrane wraps and selectively isolates the horizon hole the accretion disk is continuous whereas in the case in Kerr black holes and the singularity in superspinning so- of a naked singularity it is discontinuous. This means that lutions, partially isolating it from the outer region by let- we can determine the values of intrinsic physical parameters ting selectively rotating infalling or outgoing matter to cross of the central attractor by analyzing the geometric and topo- the static limit. As mentioned above, the ergoregion is in- logical properties of the corresponding Keplerian accretion volved in the BH spin-up and spin-down processes leading disk. In addition, these disconnected regions, in the case of a to a radical change of the dynamical structure of the region naked singularities, are a consequence of the repulsive grav- closest to the source and, therefore, potentially could give ity properties found also in many other black hole solutions rise to detectable effects. It is possible that, during the evo- and in some extensions or modifications of Einstein’s the- lutionary phases of the rotating object interacting with the ory. The effects of repulsive gravity in the case of the Kerr orbiting matter, there can be some evolutionary stages of geometry were considered in [80] and [81]. Analogies be- spin adjustment, for example, in the proximity of the ex- tween the effects of repulsive gravity and the presence of a treme value (a . M) where the speculated spin-down of the cosmological constant was shown also to occur in regular BH can occur preventing the formation of a naked singu- black hole spacetimes or in strong gravity objects without larity with a & M (see also [40, 63, 65, 88–95]). The study horizons [82, 83]. of extended matter configurations in the Kerr ergoregion is Several studies have already shown that it is necessary faced for example in [2, 96]. In [96–100], a model of multi- to distinguish between weak (a/M ≈ 1) and strong naked accretion disks, so called ringed accretion disks, both coro- singularities (a/M >> 1). It is also possible to introduce tating and counterrotating on the equatorial plane of a Kerr a similar classification for black holes; however, we prove BH, has been proposed, and a model for such ringed accre- here that only in the case of naked singularities there are ob- tion disks was developed. Matter can eventually be captured vious fundamental distinctions between these classes which by the accretion disk, increasing or removing part of its en- are not present among the different black hole classes. Our ergy and angular momentum, therefore prompting a shift of focus is on strong BHs, and weak and very weak NSs. This its spin [64, 87, 101–104]. A further remarkable aspect of analysis confirms the distinction between strong and weak this region is that the outer boundary on the equatorial plane NSs and BHs, characterized by peculiar limiting values for of the central singularity is invariant for every spin change, the spin parameters. Nevertheless, the existence and mean- and coincides with the radius of the horizon of the static ing of such limits is still largely unclear, and more investi- case. In the limit of zero rotation, the outer ergosurface co- gation is due. However, there are indications about the ex- alesces with the event horizon. The extension of this region istence of such limits in different geometries, where weak increases with the spin-to-mass ratio, but the outer limit is and strong singularities could appear. In [2, 52–55], it was invariant. Although on the equatorial plane the ergoregion is invariant with respect to any transformation involving a 2Test particle motion can be used to determine the topological prop- erties of general relativistic spacetimes [77–79]. Moreover, we proved change in the source spin (but not with respect to a change that in certain NS geometries different regions of stable timelike circu- in the mass M), the dynamical structure of the ergoregion lar orbits are separated from each other by empty regions; this means is not invariant with respect to a change in the spin-to-mass that an accretion disk made of test particles will show a particular ring- ratio. Nevertheless, concerning the invariance of this region like structure with specific topological properties. 4 with respect to spin shifts it has been argued, for example The Kerr solution has several symmetry properties. The in [105], that the ergoregion cannot indeed disappear as a Kerr metric tensor (1) is invariant under the application of consequence of a change in spin, because it may be filled any two different transformations: PQ : Q → −Q, where by negative energy matter provided by the emergence of a Q is one of the coordinates (t,φ) or the metric parameter Penrose process3 [13]. The presence of negative energy par- a while a single transformation leads to a spacetime with ticles, a distinctive feature of the ergoregion of any spinning an opposite rotation with respect to the unchanged metric. source in any range of the spin value, has special proper- The metric element is independent of the coordinate t and ties when it comes to the circular motion in weakly rotating the angular coordinate φ. The solution is stationary due to naked singularities. The presence of this special matter in the presence of the Killing field ξt = ∂t and the geometry an “antigravity” sphere, possibly filled with negative energy is axisymmetric as shown by the presence of the rotational formed according to the Penrose process, and bounded by Killing field ξφ = ∂φ . orbits with zero angular momentum, is expected to play an An observer orbiting, with uniform angular velocity, along important role in the source evolution. In this work, we clar- the curves r =constant and θ =constant will not see the ify and deepen those results, formulate in detail those con- spacetime changing during its motion. As a consequence siderations, analyze the static limit, and perform a detailed of this, the covariant components pφ and pt of the particle study of this region from the point of view of stationary ob- four–momentum are conserved along the geodesics4 and we servers. In this regards, we mention also the interesting and can introduce the constants of motion recent results published in [106] and [107]. α β α β E ≡ −gαβ ξt p , L ≡ gαβ ξφ p . (3) In detail, this article is organized as follows: in Sec. 2 The constant of motion (along geodesics) L is interpreted we discuss the main properties of the Kerr solution and the as the angular momentum of the particle as measured by an features of the ergoregion in the equatorial plane of the Kerr observer at infinity, and we may interpret E , for timelike spacetimes. Concepts and notation used throughout this work geodesics, as the total energy of a test particle coming from are also introduced. Stationary observers in BH and NS ge- radial infinity, as measured by a static observer located at ometries are introduced in Sec. 3. Then, in Sec. 4, we in- infinity. vestigate the case of zero angular momentum observers and As a consequence of the metric tensor symmetry under find all the spacetime configurations in which they can exist. reflection with respect to the equatorial hyperplane θ = π/2, Finally, in Sec. 5, we discuss our results. the equatorial (circular) trajectories are confined in the equa- torial geodesic plane. Several remarkable surfaces character- ize these geometries: For black hole and extreme black hole 2 Ergoregion properties in the Kerr spacetime spacetimes the radii p 2 2 rr The Kerr metric is an axisymmetric, stationary (nonstatic), r± ≡ M ± M − a : g = 0 (4) asymptotically flat exact solution of Einstein’s equations in are the event outer and inner (Killing) horizons5, whereas vacuum. In spheroidal-like Boyer–Lindquist (BL) coordi- p r± ≡ M ± M2 − a2cos2θ : g = 0 (5) nates, the line element can be written as ε tt 4We adopt the geometrical units c = 1 = G and the signature ρ2 ds2 = −dt2 + dr2 + ρ2dθ 2 + (r2 + a2)sin2 θdφ 2 (−,+,+,+), Greek indices run in {0,1,2,3}. The four-velocity sat- α ∆ isfy u uα = −1. The radius r has units of mass [M], and the an- 2 t r 2M 2 2 gular momentum units of [M] , the velocities [u ] = [u ] = 1 and + r(dt − asin θdφ) , (1) φ θ −1 φ t −1 ρ2 [u ] = [u ] = [M] with [u /u ] = [M] and [uφ /ut ] = [M]. For the sake of convenience, we always consider a dimensionless energy and 2 2 2 2 2 2 ∆ ≡ r − 2Mr + a , and ρ ≡ r + a cos θ . (2) effective potential [Ve f f ] = 1 and an angular momentum per unit of mass [L]/[M] = [M]. The parameter M ≥ 0 is interpreted as the mass parameter, 5 A Killing horizon is a null surface, S0, whose null generators coin- while the rotation parameter a ≡ J/M ≥ 0 (spin) is the spe- cide with the orbits of an one-parameter group of isometries (i. e., there cific angular momentum, and J is the total angular momen- is a Killing field L which is normal to S0). Therefore, it is a light- tum of the gravitational source. The spherically symmetric like hypersurface (generated by the flow of a Killing vector) on which the norm of a Killing vector goes to zero. In static BH spacetimes, (static) Schwarzschild solution is a limiting case for a = 0. the event, apparent, and Killing horizons with respect to the Killing A Kerr black hole (BH) geometry is defined by the range field ξt coincide. In the Schwarzschild spacetime, therefore, r = 2M of the spin-mass ratio a/M ∈]0,1[, the extreme black hole is the Killing horizon with respect to the Killing vector ∂t . The event case corresponds to a = M, whereas a super-spinner Kerr horizons of a spinning BH are Killing horizons with respect to the Killing field Lh = ∂t + ωh∂φ , where ωh is defined as the angular ve- compact object or a naked singularity (NS) geometry occurs locity of the horizon. In this article we shall extensively discuss this when a/M > 1. special vector in the case of NS geometries. We note here that the sur- face gravity of a BH may be defined as the rate at which the norm 3We note that the wave analog of the Penrose process is the superradi- of the Killing vector vanishes from the outside. The surface gravity, 2 2 ant scattering. SG Kerr = (r+ − r−)/2(r+ + a ), is a conformal invariant of the met- 5
6 − + are the outer and inner ergosurfaces, respectively , with rε ≤ trate the surface r = rε , in general, in a finite time t. For this + r− ≤ r+ ≤ rε . In an extreme BH geometry, the horizons co- reason, the ergoregion boundary is not a surface of infinite ± incide, r− = r+ = M, and the relation rε = r± is valid on the redshift, except for the axis of rotation where the ergoregion rotational axis (i.e., when cos2 θ = 1). coincides with the event horizon [2, 109]. This means that an In this work, we will deal particularly with the geomet- observer at infinity will see a non-zero emission frequency. + + ric properties of the ergoregion Σ : ]r+,r ]; in this region, In the spherical symmetric case (a = 0), however, as gtφ = 0 ε ε p we have that gtt > 0 on the equatorial plane (θ = π/2) and the proper time interval dτ = |gtt |dt goes to zero as one + − + also rε |π/2 = r+|a=0 = 2M and rε = 0. The outer bound- approaches r = r+ = rε . For a timelike particle with pos- + itive energy (as measured by an observer at infinity), it is ary rε is known as the static (or also stationary) limit [108]; it is a timelike surface except on the axis of the Kerr source possible to cross the static limit and to escape towards in- where it matches the outer horizon and becomes null-like. finity. In Sec. 3, we introduce stationary observers in BH On the equatorial plane of symmetry, ρ = r and the space- and NS geometries. We find the explicit expression for the time singularity is located at r = 0. In the naked singular- angular velocity of stationary observers, and perform a de- ity case, where the singularity at ρ = 0 is not covered by a tailed analysis of its behavior in terms of the radial distance + to the source and of the angular momentum of the gravity horizon, the region Σε has a toroidal topology centered on the axis with the inner circle located on the singularity. On source. We find all the conditions that must be satisfied for the equatorial plane, as a → 0 the geometry “smoothly" re- a light-surface to exist. + sembles the spherical symmetric case, r+ ≡ rε |π/2, and the frequency of the signals emitted by an infalling particle in motion towards r = 2M, as seen by an observer at infinity, 3 Stationary observers and light surfaces goes to zero. + We start our analysis by considering stationary observers In general, for a 6= 0 and r ∈ Σε , the metric component + 2 which are defined as observers whose tangent vector is a gtt changes its sign and vanishes for r = rε (and cos θ ∈ α spacetime Killing vector; their four-velocity is therefore a ]0,1]). In the ergoregion, the Killing vector ξt = (1,0,0,0) linear combination of the two Killing vectors ξφ and ξt , i.e., becomes spacelike, i.e., g ξ α ξ β = g > 0. As the quantity αβ t t tt the coordinates r and θ are constants along the worldline of , introduced in Eq. (3), is associated to the Killing field E a stationary observer [110]. As a consequence of this prop- = , then the particle energy can be also negative inside ξt ∂t erty, a stationary observer does not see the spacetime chang- +. For stationary spacetimes (a 6= 0) in +, the motion Σε Σε ing along its trajectory. It is convenient to introduce the (uni- with φ = const is not possible and all particles are forced form) angular velocity ω as to rotate with the source, i.e., φ˙a > 0. This fact implies in φ t α α α particular that an observer with four-velocity proportional dφ/dt = u /u ≡ ω, or u = γ(ξt + ωξφ ), (6) to α so that ˙ = r˙ = ˙ = 0, (the dot denotes the derivative ξt θ φ which is a dimensionless quantity. Here, γ is a normalization with respect to the proper time τ along the trajectory), can- factor not exist inside the ergoregion. Therefore, for any infalling −2 2 matter (timelike or photonlike) approaching the horizon r+ γ ≡ −κ(ω gφφ + 2ωgtφ + gtt ), (7) + in the region Σε , it holds that t → ∞ and φ → ∞, implying α β where gαβ u u = −κ. The particular case ω = 0 defines that the world-lines around the horizon, as long as a 6= 0, are static observers; these observers cannot exist in the ergore- subjected to an infinite twisting. On the other hand, trajec- gion. r = const r > tories with and ˙ 0 (particles crossing the static The angular velocity of a timelike stationary observer limit and escaping outside in the region r ≥ r+) are possible. ε (κ = +1) is defined within the interval Concerning the frequency of a signal emitted by a source q + 2 2 in motion along the boundary of the ergoregion rε , it is clear ω ∈]ω−,ω+[ where ω± ≡ ωZ ± ωZ − ω∗ , (8) that the proper time of the source particle is not null7. Then, g gtt g 2 ≡ tt = , ≡ − φt , for an observer at infinity, the particle will reach and pene- ω∗ φφ ωZ gφφ g gφφ ric, but it rescales with the conformal Killing vector. Therefore, it is as illustrated in Figs. 1 and 2-right, where the frequencies not the same on all generators (but obviously it is constant along one ω are plotted for fixed values of r/M and as functions of specific generator because of the symmetries). ± 6 the spacetime spin a/M and radius r/M, respectively. In par- In the Kerr solution, the Killing vector ∂t , representing time trans- lations at infinity, becomes null at the outer boundary of the ergore- ticular, the combination + + gion, rε , which is however a timelike surface; therefore, rε is not a Killing horizon. More precisely, on the ergosurfaces the time transla- L± ≡ ξt + ω±ξφ (9) tional Killing vector becomes null. α β 7 ± defines null curves, g = 0, and, therefore, as we However, since gtt (rε ) = 0, it is also known as an infinity redshift αβ L± L± surface; see, for example, [108]. shall see in detail below, the frequencies ω± are limiting 6 angular velocities for physical observers, defining a family The constant limit ω0 ≡ M/a plays an important role be- of null curves, rotating with the velocity ω± around the axis cause it corresponds to the asymptotic limit for very small of symmetry. The Killing vectors L± are also generators of values of r and R ≡ r/a. Note that, on the equatorial plane, β 2 Killing event horizons. The Killing vector ξt +ωξφ becomes g α = R , where ≡ | . The asymptotic be- αβ L0 L0 L0 L± ω0 null at r = r+. At the horizon ω+ = ω− and, consequently, havior of these frequencies may be deeper investigated by stationary observers cannot exist inside this surface. considering the power series expansion for the spin parame- ter and the radius determined by the expression M M 3.1 The frequencies ω± for r → ∞ : ω = ± 1 − + o[r−3] , (11) ± r r We are concerned here with the orbits r =const and ω =const, which shows a clear decreasing as the gravitational field di- which are eligible for stationary observers. This analysis en- minishes. For large values of the rotational parameter, we lightens the differences between NS and BH spacetimes. In- obtain side the ergoregion, the quantity in parenthesis in the r.h.s. M 2M ± r ω = + (12) of Eq. (7) is well defined for any source. However, it be- ± a 2M + r comes null for photon-like particles and the rotational fre- M r2 1 quencies ω . On the equatorial plane, the frequencies ω ∓2M2 − 2Mr ∓ r2 + o[a−5] , ± ± a3 (2M + r)2 2 are given as √ so that for extreme large values of the source rotation, the 2 2 2aM ± M r ∆ frequencies vanish and no stationary observers exist, thought ω± ≡ 3 2 (10) r + a (2M + r) differently for the limiting frequencies ω± (see Figs. 2). It is a M therefore convenient to introduce the dimensionless radius with ω±(r+) = ωZ(r+) = ωh ≡ ≡ , 2r+ 2ω0r+ R ≡ r/a, for which we obtain the limit M 2 2 3 and lim ω± = 0, lim ω± = ω0 ≡ . M MR M R 3 r→∞ r→0 a R → 0 : ω = − − + o[R ]; (13) + a 2a 4a2 Moreover, for the case of very strong naked singularities M M2 + a2R2 a M, we obtain that ω → 0. ω− = − R + − (14) ± a 2Ma The above quantities are closely related to the main black a4 + 4M2a2 − M4R3 hole characteristics, and determine also the main features 3 2 2 + o[R ]; that distinguish NS solutions from BH solutions. The con- 4a M (∓M2 + 4Ma ∓ a2)M M2 M stant ωh plays a crucial role for the characterization of black −3 R → ∞ : ω± = 3 3 ∓ 2 2 ± + o[R ]. holes, including their thermodynamic properties. It also de- 2a R a R aR termines the uniform (rigid) angular velocity on the horizon, (15) representing the fact that the black hole rotates rigidly. This Equations (12), (13) and (14) show the particularly differ- quantity enters directly into the definition of the BH surface ent behavior of ω± with respect to the asymptote ω0. The gravity and, consequently, into the formulation of the rigid- behavior of the frequencies for fixed values of the radial co- ity theorem and into the expressions for the Killing vector ordinate r and varying values of the specific rotational pa- (6). More precisely, the Kerr BH surface gravity is defined rameter a/M is illustrated in Fig. 1. We see that the region as κ = κs − γa, where κs ≡ 1/4M is the Schwarzschild sur- of allowed values for the frequencies is larger for naked sin- 2 face gravity, while γa = Mωh (the effective spring constant, gularities than for black holes. In fact, for certain values of according to [111]) is the contribution due to the additional the radial coordinate r, stationary observers can exist only component of the BH intrinsic spin; ωh is therefore the an- in the field of naked singularities. This is a clear indica- gular velocity (in units of 1/M) on the event horizon. The tion of the observational differences between black holes (strong) rigidity theorem connects then the event horizon and naked singularities. The allowed values for the fre- with a Killing horizon stating that, under suitable conditions, quencies are bounded by the limiting value ω0 = M/a; for a the event horizon of a stationary (asymptotically flat solution broader discussion on the role of the dimensionless spin pa- with matter satisfying suitable hyperbolic equations) BH is rameter a/M in Kerr geometries, see also [96]9 Moreover, a Killing horizon8. 9For simplicity we use here dimensionless quantities. We introduce the 8 Assuming the cosmic censorship validity, the gravitational collapse rotational version of the Killing vectors ξt and ξφ , i.e., the canonical 2 2 2 should lead to BH configurations. The surface area of the BH event vector fields V˜ ≡ (r + a )∂t + a∂φ and W˜ ≡ ∂φ + aσ ∂t . Then, the horizon is non-decreasing with time (which is the content of the sec- contraction of the geodesic four-velocity with W˜ leads to the (non- ond law of black hole thermodynamics). The BH event horizon of this conserved) quantity L − E aσ 2, which is a function of the conserved stationary solution is a Killing horizon with constant surface gravity quantities (E ,L ), the spacetime parameter a and the polar coordinate (zeroth law) [4, 112–114]. θ; on the equatorial plane, it then reduces to L − E a. When we con- 7
ω0 = M/a and from below by the null value ω± = 0. Thus, at fixed radius r, for very strong naked singularities a/M 1, we have that ω0 ≈ 0 and the range of possible frequencies for stationary observers becomes smaller. This effect will be discussed more deeply in Sec. 3.2, where we shall focus specifically on the frequency ω0. On the other hand, con- sidering the limits (10), together with Eqs (11)– (15), we find that the range of possible frequencies shrinks also in the following situations: when moving outwardly with re- spect to the singularity (at fixed a), very close to the source, approaching the horizon rh according to Eq. (10), or also for very large or very small R = r/a. The last case points out again the importance of the scaled radius r/a. Essentially, stationary observers can be near the singu- Fig. 1 Plot of the limit frequencies ω± for fixed values of r/M. Fre- + larity only at a particular frequency. The greater is the NS quencies ω±, on r = rγ ∈ Σε , photon circular orbit in the BHs ergore- gion are also plotted–see Table 1 and [2]. dimensionless spin, the lower is the limiting frequency ω±, with the extreme limit at ω+ = ω−. In other words, the fre- quency range, ]ω−,ω+[, for stationary observers vanishes as for a given value of ω±, the corresponding radius is located at a certain distance from the source, depending on the value the value r = 0 is approached. The singularity at r = 0 in the of the rotational parameter a. The following configuration of NS regime is actually related to the characteristic constant frequencies, radii and spin determines the location structure frequency ω = ω0 in the same way as in BH-geometries the of stationary observers: outer horizon r = r+ is related to the constant frequency ωh (cf. Eq. (10)). Consequently, a NS solution must be charac- ω+ ∈]0,ω0[, for a ∈]0,M[ in r ∈]0,r−] ∪ [r+,+∞[ (16) terized by the frequency ω0 and a BH solution by the fre- and for a ≥ M in r > 0 quency ωh. Therefore, the frequency ω0 may be seen actu- + ω− ∈]0,ω0[ for a ∈]0,M[ in r ∈]0,r−] ∪ [r+,rε [ (17) ally as the NS counterpart of the BH horizon angular fre- and for a ≥ M in r ∈]0,r+[ . quency ωh (see Fig. 4). For r > r+, it holds that ω+ > ω−. ε Then, in general, for BHs and NSs in the static limit Thus, we see that in the interval ]0,M/a[ observers can ex- + rε = 2M, we obtain that ist with frequencies ω±; moreover, the frequency ω− is al- + ε + aM + lowed in r ∈ Σε , while observers with ω− < 0 can exist in ω+ ≡ ω+(rε ) = with ω−(rε ) = 0. (18) + 2M2 + a2 r > rε . Moreover, it is possible to show that, in BH geome- Moreover, ω < 0 for r > r+, and ω > 0 inside the ergore- tries, the condition ω± 1/2 must be satisfied outside the − ε − + > outer horizon (r > r+). The particular value ω± = ωh = 1/2 gion Σε , while ω+ 0 everywhere. is therefore the limiting angular velocity in the case of an In general, any frequency value should be contained within extreme black hole, i.e., for a = M so that r = r+ = r− = M the range ω+ − ω−; therefore, it is convenient to define the in Eq.(10). The behavior of the special frequency ω± = 1/2 frequency interval is depicted in Fig. 3 and in Figs. 4, 2, 5, and 6, where other q ∆ ≡ ω − ω = 2 ω2 − ω2, (19) relevant frequencies are also plotted. ω± + − Z ∗ Eqs (16) enlighten some important properties of the light which is a function of the radial distance from the source surfaces (frequencies ω±) and of stationary observers, asso- and of the attractor spin. Figs. 7 show the frequency interval ciated with frequencies ω ∈]ω−,ω+[ in the regime of strong ∆ω± as a function of r/M and a/M. singularities. Eqs(16) also enlighten the dependence of the An analysis of this quantity makes it possible to derive frequencies on the dimensionless spin a/M and radius R = some key features about the eligible frequencies. For con- r/a. It is clear that when the frequency interval ]ω−,ω+[ venience, we present in Table 1 some special values of the shrinks, depending on the singularity spin a/M or the dis- spin-mass ratio, which we will consider in the following tance from the source r/M, the range of possible frequen- analysis. We summarize the obtained results in the following cies for stationary observers reduces. This occurs in gen- way: eral when ω+ ≈ ω−. According to Eqs (16), the frequen- Firstly, for any NS source with a > a∆ ≡ 1.16905M, the cies ω± are bounded from above by the limiting frequencies interval ∆ω± increases as the observer (on the equatorial plane) moves inside the ergoregion Σ + towards the static −1 ˜ ε sider the principal null congruence γ± ≡ ±∂r +∆ V, the angular mo- limit. mentum L = aσ 2, that is, `¯ = 1 (and E = +1, in proper units), every principal null geodesic is then characterized by `¯ = 1. On the horizon, Secondly, in the case of NS geometries with a ∈]M,a∆ [, it is L = E = 0 [96, 115] i.e., belonging partially to the class of NSI spacetimes, the 8
± Fig. 2 Left and central panels: Plot of the limit radii rs as functions of the spacetime spin a/M and frequencies ω-see also Figs. 4. Right panel: Plot of the limit frequencies ω± as functions of the spacetime spin a/M and radius r/M–see also Fig. 5.
− − Table 1 Classes of BH and NS geometries according to their specific spins. The radii (rγ ,rmso) corresponds to the photon circular orbit (or also last circular orbit) and the marginally stable circular orbit, respectively, for corrotating orbits in BH geometries. The NS case is characterized by (NS)− + the zero angular momentum radii (L (rˆ±) = 0) and the radius of the marginally stable circular orbit rmso ∈ Σε . The explicit expressions for these radii can be found in [52–55, 76]
Black hole classes: BHI : [0,a1[; BHII : [a1,a2[, BHIII : [a2,M] √ √ − + − + a1/M ≡ 1/ 2 ≈ 0.707107 : rγ (a1) = rε , a2/M ≡ 2 2/3 ≈ 0.942809 : rmso(a2) = rε . Naked singularity classes:NSI : ]M,a3], NSII : ]a3,a4], NSIII : ]a4,+∞] √ √ (NS)− + a3/M ≡ 3 3/4 ≈ 1.29904 :r ˆ+(a3) = rˆ−(a3), a4/M ≡ 2 2 ≈ 2.82843 : rmso (a4) = rε
Fig. 3 Plot of the limiting frequency ω± = 1/2. The spin a+(r) ≡ Fig. 4 Stationary observers: The angular velocities ωε (gray curve), p + r(2M − r), solution of r = r+, and aγ , solution of r = rγ where rγ ∈ ωh (black curve), ωn (dot-dashed curve), ω0 (dashed curve), ω¯n > ωn > + ε Σε is the photon orbit in the ergoregion in a Kerr BH, are also plotted. ωh (black thick curve). Here ω¯n = ωn = ωh = 1/2 at a = M, ω+ = ωh = . a = a ε = = . a = a 0 321797 at s,√ and ω+ ωn 0 282843 at 1. The maximum ε + of ω+, at a = a = 2M (dashed line) where a : re = rε –see Eq. (31), is marked with a point. See also Fig. 2. The angular velocities ω± on situation is very articulated. There is a region of maximum + the BH photon orbit rγ ∈ Σε are also plotted. Note that ωn it is an and a minimum frequencies, as the observer moves from extension of ω+(rγ ) for a < a1–see Table 2. the source towards the static limit. This phenomenon in- volves an orbital range partially located within the interval ]rˆ−,rˆ+[, which is characterized by the presence of counter- range of possible frequencies for stationary observers van- rotating circular orbits with negative orbital angular momen- ± ishes. The points r∆ (a) represent the extrema of the interval tum L = −L− (cf. Fig. 8, where the radiir ˆ± are plotted.). ∆ω± , i.e., the solutions of the equation ∂r∆ω± = 0 – Fig. 8. This property is present only in the case of NS geometries. In + − + − ± For the maximum spin, a = a∆ , we obtain ω = ω fact, there are the two critical orbits r∆ > r∆ and r = r∆ (a∆ ), ± on the radius r ≡ r∆ (a∆ ) = 0.811587M and, therefore, the which are the boundaries of a closed region, whose exten- 9
− + Fig. 6 The radii r± versus the frequency ω for different values of the Fig. 5 Upper panel: Plot of the curves rs =constant and rs =constant s (inside panel) in the plane (ω,a/M). The numbers denote the constant spin a/M (numbers close to the curves). The gray region is the only ± ± region allowed for the case of BH spacetimes. The surfacesr ˆ± at a = M radii rs /M (light cylinders). Bottom panel: The radii rs versus the spin a/M, for different values of the velocity ω (numbers close to the (extreme-BH-case) are shown in black-thick. curves), the gray region is a ∈ [0,M] (BH-spacetime). The black region corresponds to r < r+. The dashed lines denote a1 < a2 < a3 < a4. The ± angular momentum and the velocity (a,ω) for rs (a,ω) = 0 are related by ω = M/a. See also Figs. 2. of a photon or null-like particle corotating with the source. + Such a kind of orbit, contained in Σε , is a feature of the BHII-III spacetimes [2], this is also know as marginally or sion reaches a maximum in the case of the extreme Kerr last circular orbit as no circular particle motion is possible in − + the region r < r−. We close this section with a brief discus- geometry a = M, and is zero for a = a∆ . For r ∈]r∆ ,r∆ [, γ sion on the variation of the frequency interval ∆ , follow- the separation parameter ∆ω± decreases with the orbital dis- ω± − ing a spin transition with a > 0. In the case of a singularity tance, then on the inner radius r∆ it reaches a maximum + spin-transition, there are two extreme radii for the frequency value, whereas on the outer radius r∆ it reaches a minimum. + interval In the outer regions, at r > rε , the separation parameter in- creases with the distance from the source. This feature con- stitutes therefore a major difference in the the behavior of 1 8a2 r+ ≡ η cos arccos − , (20) stationary observers within and outside the ergoregion of a 3 η3 naked singularity spacetime. However, a deeper analysis of √ 1 8a2 2 8M2 − a2 the equatorial plane, outside the static limit, shows the exis- − √ r ≡ η sin arcsin 3 , η ≡ , tence of a second region for light surfaces in the NS case. 3 η 3 s 2 2 On the other hand, the angular velocity ω− decreases r (r − 8M ) √ or alternatively a = − for r ∈]0,2 2M[ , with the orbit in the Kerr spacetime. The maximum fre- r + 2M quency ω+ also decreases in the NS spacetimes. In the BH cases, the angular velocity is always increasing for sources ± where r : ∂ ∆ ± = 0 are maximum points– see Figs. 8 of the class BHI, while for the other sources there is a maxi- a ω± r − mum for the velocity ω+ at r = rγ , which is the circular orbit and 7. 10
Fig. 7 Upper panel: Plot of the frequency interval ∆ω± = ω+ −ω− as a function of the radius r/M and the BH and NS spin a/M. The extrema r± and r± are solutions of ∂ ∆ω = 0 and ∂ ∆ω = 0, respectively. ∆ r ± a ± Lower panel: The frequency interval ∆ω± = ω+ −ω− as a function of a/M for selected values of the orbit radius r/M; the maximum points are for the radii r± or r+–see Figs. 8. ∆
3.2 Light surfaces
In this section, we briefly study the conditions for the exis- tence of light surfaces and and their morphology. The con- dition (8), for the definition of a stationary observer, can be restated in terms of the solutions r±, considering as a fixed s ω Fig. 8 Upper panel: The effective potential Ve f f for the ZAMOS ± Z parameter. Therefore, we now consider the solutions rs of L = 0,for BH and NS sources as a function of the source spin a/M the equation for the light surfaces defined in Eq. (9) in terms and the radius r/M. The effective potential function is the value of of the Killing null generator , as functions of the fre- E /µ at which the (radial) kinetic energy of the particle vanishes. L± Black planes represent the spin values a = M, extreme Kerr BH, and quency . We obtain √ ω a3 ≡ 3 3/4M, a NS geometry, wherer ˆ− = rˆ+. The orbitsr ˆ− ≤ rˆ+, − 1 + 1 gray surfaces, are for a < M (BH-case) inside the horizon (r < r+). r 2β1 sin arcsinβ0 r 2β1 cos arccos(−β0) s ≡ √3 , s ≡ 3 √ The inner black surface is the horizon r+. Central panel: The radius M 3 M 3 r(a), solution of ∂r∆ω± = 0, i.e., it represents the critical points of the √ separation parameter ≡ ( − )| on the equatorial plane s 2 ∆ω± ω+ ω− π/2 1 1 3 3β1ω ± θ = π/2. The radius rυ , where the orbital energy E = 0, and the orbits where β1 ≡ 2 − 2 , β0 ≡ 2 , (21) ω ω ω rˆ±, for which L = 0, are also plotted. Dashed lines represent the spins 0 + 1 p ω0 a = . M a = / / M ≈ . M a = . M σ 1√064306 , µ 4 2 3 3 1 08866 , ∆ 1 16905 and a = 3 3/4M. The black region corresponds to r < r . Bottom panel: where ω ≡ M/a (cf. Eq. (10) and Fig. 2). For ω = 1/2, in 3 + 0 The radii r± : ∂ ∆ = 0 are plotted as functions of a/M–see also a ω± the limiting case of a = M, we have that ωn = ω¯n = ωh = Figs. 7. ± 10 1/2 and rs = M–see Figs. 4, 6 and 2 . Thus, there are
10 + − More precisely, it is rs = rs = 0 for a > 0 and ω = ω0. Also, r+ = r− > r for a = 0 and ω = ± √1 . In the extreme Kerr spacetime geometry, we have that r+ = r− > 0 for a = M, ω = 1/2 for r = M, s s + 3 3 s s 11
+ − solutions rs = rs = 0 for a ∈]0,M[ if ω ∈ (ωn,ω¯n) where clude a part of BHII-sources and the condition C1 ap- (for simplicity we use a dimensionless spin a → a/M) plies. √ 9 − a2 + 6 9 − 5a2 cos 1 arccosα For spacetimes with rotation a = as, the conditions C1 3 ε ω¯n ≡ (22) and C apply. Then, in the special case ω = ω or ω = a(a2 + 27) 2 + h ωε , there is a solution with r = r+. √ + s 1 9 − a2 − 6 9 − 5a2 sin arcsinα Finally, for spacetimes with a ∈]as,M[, which belong to ω ≡ 3 (23) n a(a2 + 27) the class of BHII and BHIII sources, the condition C1 holds, whereas the condition C applies for frequencies a4 − 36a2 + 27 2 ≡ (24) within the interval ω < ω . Finally, in the special case α 3/2 h n (9 − 5a2) + ω = ωh, there is one solution at r = rs , and for ω = ωn − The situation is summarized in Table 2. We see that ωn = we have the solution r = rs . ε 11 ε ω+ for a = a1, ωn = ω¯n = ωh = 1/2 at a = M, ω+ = ωh at r A summary and comparison of these two cases is pro- √ ± a = as ≡ 2 2 − 1 M ≈ 0.91017M and a = 0 (the static posed also in Figs. 5 and 6, where the surfaces rs are studied as functions of a/M and ω. It is evident that the extreme so- solution). Moreover, we have that ω > ω¯ > ω > ωε and 0 n n + lution a/M = 1 is a limiting case of both surfaces r±, vary- ω > ω for BH-sources, where ω > ωε for a ∈]a ,M]. In s n h h + s ing both in terms of the spin and the angular velocity ω. the NS case, there are no crossing points for the radii r± s Thus, the difference between the regions where stationary and ω > ωε (see Fig. 4). The shrinking of the frequency 0 + observers can exist in the BH case (gray regions in Figs. 6) interval ]ω ,ω [ is also shown in Figs. 6, 9 and 10, where − + and in the NS case are clearly delineated. In BH spacetimes, the radii r± are also plotted as functions of the frequencies. s the surfaces r± are confined within a restricted radial and Figures 4, 5, and 6 contain all the information about the s frequency range. On the other hand, in the naked singularity differences between black holes with a < M, and the case of case, the orbits and the frequency range is larger than in the naked singularities with a > M. We summarize the situation black hole case. Moreover, the surfaces r± can be closed in in the following statements: s the case of NS spacetimes, inside the ergoregion, for suffi- Naked singularities spacetimes: For a > M, the solutions for ciently low values of the spin parameter, namely a ∈]M,a4]. the equation of the light surfaces in the limiting case ω = Furthermore, in any Kerr spacetime, there is a light surface 0 (static observer) are located at r = r+. While for any ± + ε at rs = r with ω = M/a4. In Sec. 4, we complete this ε ε frequency within the range ω ∈]0,ω+[ there is one solu- analysis by investigating the special case of zero angular − ε tion rs , for larger frequencies in the range ω ∈ [ω+,ω0[ momentum observers, and we find all the spacetime con- ± + there are two solutions rs . In the ergoregion Σε of a figurations in which they can exist. naked singularity, there exists a limit ω0 ≡ M/a for the angular frequency. Extreme black hole spacetime: For a = M, we obtain the fol- 4 Zero Angular Momentum Observers + lowing set of solutions (ω = 0, r = rε ), (ω ∈]0,1/3[, r = − ± This section is dedicated to the study of Zero Angular Mo- rs ), and (ω ∈ [1/3,1/2[, r = rs ). Black hole spacetimes: We consider first the class BHI with mentum Observers (ZAMOs) which are defined by the con- a ∈]0,a1]. In the limit ω = 0, there exists a solution for dition the light surface with r = r+. More generally, the solu- α α β ˙ ε L ≡ uα ξ(φ) = gαβ ξφ p = gtφ t˙+ gφφ φ = 0. (28) tions are constrained by the following set of conditions: In terms of the particle’s four–velocity, the condition = 0 ε − L C1 : ω ∈]0,ω ] ∪ ω 6= ωh with solution r = r (25). + s is equivalent to dφ/dt = −gφt /gφφ ≡ ωZ = (ω+ + ω−)/2, ε ± C2 : ω ∈ [ω+,ωn[ with solution r = rs , (26) where the quantity ωZ is the ZAMOs angular velocity intro- − duced in Eq. (8), and the frequency of arbitrary stationary ω = ωn, with solution r = rs . (27) observers is written in terms of ω [2]. The sign of ω is in Then, we consider BH spacetimes with spin a ∈]a ,a [, Z Z 1 s concordance with the source rotation. The ZAMOs angular r √ where as ≡ 2 2 − 1 M < a2. These spacetimes in- velocity is a function of the spacetime spin (see Figs. 11 and 12, where constant ZAMOs frequency profiles are shown). and ω = −1/7 for r = 4M. For a Kerr geometry, where a/M ∈]0,1[, In the plane θ = π/2, we find explicitly + − it is rs = rs > r+ for ω = ωn or ω¯n (one positive and one negative 2 value solution), while in the naked singularity case where a > M, the 2aM + − ωZ|(θ=π/2) = 3 2 . (29) condition rs = rs > 0 is valid only for one negative frequency – see r + a (r + 2M) Figs. 5 and 6. As discussed in [2, 54, 55], ZAMOs along circular orbits 11For a closer look at the role of this special frequency we note that r ω¯n = ωn = ωh = 1/2 at a = M and, clearly, ω0 = 1/2 for a = 2M. We with radii ˆ± are possible only in the case of “slowly ro- refer then to Figs. 3, 4, 5, 6, and 9. tating” naked singularity spacetimes of class NSI. This is 12
Table 2 Existence of stationary observers in BH and NS spacetimes, respectively. The spin/mass ratio a/M, angular frequencies ω and orbital ranges r are listed. See also Fig. 4. Black holes: Naked singularities: ε − + ε − + a ∈]0,a1] ω ∈]0,ω+[ r ∈]rs ,rε ] a > M ω ∈]0,ω+[ r ∈]rs ,rε ] ε − + ε − + a ∈]a1,M] ω ∈]0,ω+[ r ∈]rs ,rε ] ω = ω+ r ∈]rs ,rε [ ε − + ε − + ω = ω+ r ∈]rs ,rε [ ]ω+,ω0[ r ∈]rs ,rs [ ε − + ω ∈]ω+,ωn[ r ∈]rs ,rs [
± Fig. 9 Plots of the surfaces rs (in units of mass) versus the frequency ω for different spin values a/M, including BH and NS geometries–see ± ± also Figs. 6. The surfaces rs are represented as revolution surfaces with height rs (vertical axes) and radius ω (horizontal plane). Surfaces are ± ± generated by rotating the two-dimensional curves rs around an axis (revolution of the function curves rs around the “z” axis). Thus, r =constant + with respect to the frequency ω is represented by a circle under this transformation. The disks in the plots are either r = M, r = r+ or r = rε = 2M. ± ± The surfaces rs are green and pink colored, respectively (as mentioned in the legend). In the last panel (a = 0.7M), both radii rs are green colored a characteristic of naked singularities which is interpreted For the NS regime it is interesting to investigate the vari- ± generically as a repulsive effect exerted by the singularity ation of ZAMO frequency ωZ on the orbitsr ˆ . These spe- 2 2 [52–55, 76]. On the other hand, ωZ = ω∗ for r = r±, while cial radii of the NS geometries do not remain constant under 2 2 ωZ > ω∗ in the region r > r+ for BH spacetimes, and in the a spin-transition of the central singularity. We shall consider region r > 0 for NS spacetimes (see also Fig. 12). this aspect focusing on the curvesr ˆ±(a) of the plane r −a as ZAMOs angular velocity and orbital regions illustrated in Fig. 8. This will enable us to evaluate simulta- neously the frequency variation on these special orbits, fol- The ZAMOs angular velocity ωZ is always positive for a > 0, and vanishes only in the limiting case a = 0. This lowing a spin variation of the naked singularity in the rage ± means that the ZAMOs rotate in the same direction as the of definition ofr ˆ , and to evaluate the combined effects of source (dragging of inertial frames). a variation in the orbital distance from the singularity and a change of spin. A similar analysis will be done, from a As can be seen from Eq.(29), the frequency ωZ for a fixed mass and a 6= 0 is strictly decreasing as the radius r/M different point of view, also for stationary observers. increases. 13
Fig. 10 Plots of frequency surfaces ω±(r,θ) as functions of the radial distance r in Cartesian coordinates (x,y) for different spin values a, including BHs and NSs –see also Figs. 6.
± − In Σε , the velocity ωZ = ωˆ (in NSs) always decreases Change in the intrinsic spin The angular velocity of − − + with the orbital radiir ˆ , i.e. ∂rˆ− ωˆ < 0, when the spin in- the ZAMOs inside Σε varies according to the source spin. − + creases, i.e. ∂arˆ > 0 (see Figs. 8 and 12). Asr ˆ monoton- This might be especially important in a possible process of ically decreases with the spin during a NS spin-up process spin-up or spin-down as a result of the interaction, for ex- + + (see Fig. 8), the frequency ωˆ = ωZ(rˆ ) decreases in the ample, with the surrounding matter. In [2], this phenomenon spin-range a ∈ [M,aω [, and increases in the range ]aω ,a3]; and its implications were investigated, considering different therefore, the special value aω = 1.1987M is a minimum regions close to the singularity. For a fixed orbital radius r, + point of the ZAMOs frequency ωˆZ – see Fig. 12. Viceversa, the ZAMOs angular velocity strongly depends on the value asr ˆ− increases after a NS spin-up, the corresponding ZAMOs of the spacetime spin-mass ratio. In particular, depending on − − frequency ωˆ = ωZ(rˆ ) decreases as the observer moves the value of the ratio a/M, there can exist a radius of maxi- − along the curver ˆ (a). Thus, we can say that, if the NS mum frequency re given by spin increases, the frequency ωˆ + decreases, approaching, √ but never reaching, the singularity, i. e., rˆ+ < 0 for a ∈ 3 2 2 r ∂a 3a +ϒ 3 √ q r ≡ , ≡ 9Ma2 + 3 a4 (27M2 − a(31)2) [M,aω [. Viceversa, increasing the NSs spin in spacetimes e 2/3 ϒ + 3 ϒ with a ∈]aω ,a3[, the frequency ωˆ increases again and the that are solutions of the equation ∂aωZ| = 0 at which orbitr ˆ+ moves towards the central singularity. On the other π/2 the frequency is denoted by ωe ≡ ωZ(re) (see Figs. 11 and hand, the frequency ωˆ − monotonically decreases with the 12). A detailed analysis of the expression for the radius re r− > naked singularity spin, i.e. ∂a ˆ 0; therefore, for a fixed shows that in can exist in spacetimes that belong to the class NS spin, the frequency interval decreases, i.e. ωˆ − > ωˆ +. In BHII with spin a = as, where re(as) = r+(as), and to the fact, the velocity ωZ is strictly decreasing with the radius r classes√BHIII, NSI, and NSII with the limiting value a = in the BH and NS regimes with a 6= 0 (i.e. ∂rωZ < 0). More- + a = 2M, where a : re = rε (see Figs. 11 and 2). Space- over, in general ωZ increases as the observer approaches the times with spin as belong to the class BHII, as defined in black hole at fixed spin, and it decreases as the observer Table 2, and have been analyzed in the context of stationary moves far away from the center of rotation. observers in Sec. 3.2 and Sec. 3.1 (Figs. 4, 11, 12 and 13 (r+) = ε / In the static limit, we have that ωZ ε ω+ 2. In fact, show the behavior of several quantities related to ZAMOs the asymptotic behavior of the frequency is determined by in relation to other frequencies.). In this particular case, we the relations have that lim ωZ = lim ω± = ωh, lim ωZ = 0, lim ωZ = ω0. (30) ε r→r+ r→r+ r→+∞ r→0 ω+ = ωh = 0.321797 and re(as) = r+(as). (32) 14
+ We focus our attention on ergoregion Σε , bounded from ZAMOs energy + above by the radius rε and from below by r = 0 and r = r+ The circular motion of test particles can be described for NSs and BHs, respectively. We consider the role of the easily by using the effective potential approach [116]. The radius re, as the maximum point of the ZAMO frequency, exact form of such an effective potential in the Kerr space- as a function of the source spin-mass ratio. Thus, for black time is well known in the literature (see, for example, [54, holes with a ∈ [0,as], the frequency ωZ increases with a/M + 55]). The effective potential function Ve f f represents the always inside the ergoregion; this holds for any orbit inside value of E /µ that makes r into a turning point (Ve f f = + + Σε (i.e. for a fixed valuer ¯ ∈ Σε , if a BH spin-up shift occurs E /µ), µ being the particle mass; in other words, it is the in the range [0,as], the function ωZ(r¯,a) increases withr ¯). value of E /µ (in the case of photons, µ shall depend on an For spins a ∈]as,M], instead, the frequency ωZ grows with affine parameter and the impact parameter ` ≡ L /E is rel- + the spin only forr ¯ ∈]re,rε [; on the contrary, for radii located evant for the analysis of trajectories) at which the (radial) close to the horizon,r ¯ ∈]r+,re], ωZ(r¯,a) decreases follow- kinetic energy of the particle vanishes. This can easily be ing a spin up in the range ∈]as,M] (i.e, ∂ar+ < 0 and ∂are > obtained from the geodesic equations with the appropriate 0). In the case of NS-spacetimes, the frequency ωZ(r¯,a) is constraints or through the normalization conditions of the an increasing function of the dimensionless spin in the NS four-velocities, taking into account the constraints and the + spin range ]M,a[ and on the orbitr ¯ ∈]re,rε ]. Moreover, the constants of motion [116]. Here we consider specifically an frequency ωZ(r¯,a) decreases with the spin in the range of effective potential associated to the ZAMOs. values a ∈]M,a[ and onr ¯ ∈]0,re]. This situation is distinctly The orbitsr ˆ± are critical points of the effective potential, different for NS with a > a, for which in the ergoregion an 2 i. e.,r ˆ± : ∂r Ve f f = 0. Here we consider for the ZAMO increase of the spin corresponds to a decrease of ωZ. This is Z 2 ˜ 2 2 ˜ an important distinction between different NS regimes. Ve f f Z = κgφφ [ω∗ − ωZ] where κ is a factor related to the r √ normalization condition of the ZAMO four-velocity (κ˜ = We note that re(as) = r+(as) for the spin as = 2 2 − 1 M φ t t 2 −1 for timelike ZAMOs, where u = −ωZu and u = −εE /gφφ [ω∗ − (see Fig. 11). Moreover, in NSII naked singularity space- 2 2 √ ωZ], ε = 1 according to Eq. (3); in the ergoregion Ve f f Z > + ε 2 times with spin a = 2M : re = rε , we obtain that ωZ = 0, but Ve f f Z = 0 for r = 0 and r = r+). The energy E of ωe–Fig. 12. Remarkably, the spin a is the maximum point the ZAMOs is always positive for both BH and NS space- ε ε−Max of the frequency ωZ (a) = ωe(a) ≡ ωZ = 0.176777 times, and it grows with the source spin; in fact, solutions ε and also the maximum point of the frequency ω+ (see Fig. 2). for Ve f f Z = 0 are not possible because this would corre- In other words, in naked singularity spacetimes with a = a, spond to the case of a null angular momentum with null en- + where re = rε , the ZAMOs frequency at the ergosurface ergy. The energy on the orbitsr ˆ where = 0 is always ε ± L ωZ reaches a maximum value which is equal to ωe, defined positive. In BH geometries, the potential , at = 0, in- ε Ve f f L through the radius in Eq. (31); moreover, the frequency ω+ creases with the distance from the source and has no critical reaches its maximum value at the ergosurface. points as a function of r/M. The most interesting case is then for the slow naked singularity spacetimes of the first class, NSI with a ∈]M,a1], where there is a closed and connected orbital region of circular orbits with r ∈]rˆ−,rˆ+[. The radii rˆ± are ZAMOs orbits, and in this region the potential de- creases with the orbital radius. However, in the outer region r ∈]r+,rˆ−[∪]rˆ+,2M[, the potential increases with the radius. This implies that the radiir ˆ± are possible circular ZAMOs orbits. In fact,r ˆ− is an unstable orbit andr ˆ+ is a stable orbit. Thus, in any geometry of this set, there is a stable orbit for ± the ZAMOs with angular velocity ωˆZ ≡ ωZ(rˆ±) different − + from zero, where ωˆZ < ωˆZ (see Fig. 12). In [2], we investigated the orbital nature of the static ε limit. Here, in Fig. 13, the velocity ωZ and the ratio R ≡ ε ε E−/L− (that is, the inverse of the specific angular momen- tum defined as uφ /ut ) are considered as functions of the Fig. 11 The plot shows the orbits (gray curves) of constant ZAMOs source spin at the static limit. We explore the relation be- velocity ωZ =constant in the BH and NS regions. The radius re and tween the ZAMOs and the stationary observes, where ωZ = the spin as : re = r+ are marked by dashed lines. The arrows show the increasing of the angular velocity. (ω+ + ω−)/2, for NSI sources at the static limit. A maxi- mum value, Rε = 0.853553M, is reached at a = 2M ∈ NSII. ε−Max Also, a maximum value ωZ = 0.176777 exists for the 15
ZAMOs angular velocity at a = a ∈ NSII. This ratio is al- the frequency ranges are larger than for black holes. Again, ways greater than the angular momentum of the ZAMOs at we conclude that light surfaces can be found more often in the static limit. naked singularities. The observation and measurement of the In BH spacetimes, the angular velocity for stationary ob- physical parameters of a particular light surface is sufficient servers is limited by the value ωh which occurs for the ra- to determine the main rotational properties of the spinning dius r+. We can evaluate the deviation of this velocity in gravitational source. We believe that the study of light sur- a neighborhood of the radius r+, since the four-velocity of faces (defining the “throat” discussed in Sec. 3) has impor- a the observers rotating with ω (where u ≡ ξt + ωξφ ) must tant applications regarding the possibility of directly observ- be timelike outside the horizon and therefore it has to be ing a black hole in the immediate vicinity of an event hori- R = E /L > ωh in that range (the event horizon of a Kerr zon (within the region defined by the static limit), as this black hole rotates with angular velocity ωh [1]). This limit seems to be possible in the immediate future through, for cannot be extended to the case of naked singularities. How- example, the already active Event Horizon Telescope (EHT) 12 ever, one can set similarly the threshold E > ωaL in the projects . case of circular orbits, where the frequency limit is restricted We also analyzed the conditions under which a ZAMO −1 to the values ωa ∈ [1,aµ M[ as ωa ∈ [ω0(a = M),ω0(aµ )[. can exist in a Kerr spacetime. In particular, we computed the orbital regions and the energy of ZAMOs. The frequency of the ZAMOs is always positive, i.e., they rotate in the 5 Summary and Conclusions same direction of the spinning source as a consequence of the dragging of inertial frames. The energy is also always In this work, we carried out a detailed analysis of the phys- positive. The most interesting case is that of slowly rotating ical properties of stationary observers moving in the ergore- naked singularities (NSI) where there exists a closed and gion along equatorial circular orbits in the gravitational field disconnected orbital region. This particular property could, of a spinning source, described by the stationary and ax- in principle, be used to detect naked singularities of this isymmetric Kerr metric. We derived the explicit value of class. We derived the particular radius at which the frequency the angular velocity of stationary observers and analyzed of the ZAMOs is maximal, showing that the measurement of all possible regions where circular motion is allowed, de- this radius could be used to determine whether the spinning pending on the radius and the rotational Kerr parameter. We source is a black hole or a naked singularity and its class, found that in general the region of allowed values for the according to the classification scheme formulated here. To frequencies is larger for naked singularities than for black be more specific, from Table 2 we infer that the existence holes. In fact, for certain values of the radius r, stationary of stationary observers in black hole spacetimes is limited observers can exist only in the field of naked singularities. from above by the frequency ωε , which is the highest fre- We interpret this result as a clear indication of the observa- + quency on the static limit, implying the frequency lower tional differences between black holes and naked singulari- bound ω = 0 – see also Fig. 4. In this figure, we also show ties. Given the frequency and the orbit radius of a stationary the maximum frequency, ω+, at the static limit for a naked observer, it is always possible to determine the value of the √ε singularity with a = a = 2M ∈ NSII. This spin plays an rotational parameter of the gravitational source. Our results important role for the variation of the ZAMOs frequency show that in fact the probability of existence of a stationary in NSs in terms of the singularity dimensionless spin – see observer is greater in the case of naked singularities that in Fig. 12 and Fig. 13. On the other hand, for strong BHs, with the case of black holes. Moreover, it is possible to introduce a > a , the frequency is bounded from below by ω = ω+ and a classification of rotating sources by using their rotational 1 ε from above by ω , as the radial upper bound is r+. A similar parameter which, in turn, determines the properties of sta- n s situation occurs for NSs, provided that ω is replaced with tionary observers. Black holes and naked singularities turn n the limiting frequency ω . The special role of the BH spin out to be split each into three different classes in which sta- 0 a is related to the presence of the photon circular orbit in tionary observers with different properties can exist. In par- 1 the BH ergoregion, which is absent in NS geometries; con- ticular, we point out the existence of weak (NSI) and strong sequently, as seen in Table 2, there is no distinction between (NSIII) naked singularities, corresponding to spin values the naked singularities classes. However, the analysis of the close to or distant from the limiting case of an extreme black frequencies in Fig. 4 shows differently that there are indeed hole, respectively. distinguishing features in the corresponding ergoregions. In Light surfaces are also a common feature of rotating the case of naked singularities, the frequency range of sta- gravitational configurations. We derived the explicit value tionary observers has as a boundary the outer light-surface, of the radius for light surfaces on the equatorial plane of r = r+, then it narrows as the spin increases, and finally van- the Kerr spacetime. In the case of black holes, light surfaces s ishes near the static limit. are confined within a restricted radial and frequency range. On the contrary, in the naked singularity case, the orbits and 12 http://www.eventhorizontelescope.org/ 16
The frequency of the orbits on the static limit, in fact, includes the extreme Kerr solution. This region has been converges to the limit ω0 = M/a, which is an important enlarged in Fig. 6-bottom. Figures 9 and 10 show from a frequency threshold for the NS regime. The presence of a different perspective the transition between the BH region, maximum for the special NS geometry with a = a on the gray region in Figs. 6, and the NS region for different spins. static limit is symptomatic for the nature of this source – Any spin oscillation in that region generates a tunnel in the ± 13 see Figs. 4, 6 and 13. The study of the surfaces rs on light-surface . The transition region is around ω± ≈ 1/2, the plane (r,ω), for different values of the spin-mass ra- which is a special value related to the spin a = 2M of strong tio, shows a clear difference between the allowed regions in naked singularities–see Figs. 3 and 2. In this region, as in naked singularities and black holes (gray region in Fig. 6). the neighborhood of the ring singularity (r = 0), the orbital 14 There is an open “throat” between the spin values a . M range reaches relatively small values . This shows the ex- (strong BHs) and a ' M (very weak NSs), with an opening istence of limitations for a spin transition in the parameter of the cusp (at r = 0 in these special coordinates) for the fre- region of very weak naked singularities, pointed out also in quency ω = 0.5. We note a change in the situation for spins [52–55, 76]. in a/M ∈]1,1.0001]; this region is in fact extremely sensi- On the other hand, in the strong NSs regimes, a spin ± tive to a change of the source spin; the throat of rs has, in threshold emerges at a = 2M and a = M (see Figs. 3, 4 this special spin range, a saddle point around (r = M,ω = and 6). In Fig. 13, we analyze the properties at the static 1/2) between [a ,a ], which is not present in stronger sin- + µ 3 limit rε . The maximum value of E−/L− is then reached in 15 gularities. The spins in this range are related to the neg- the ergoregion of the NSII class . Around a = a3 the throat ± ative state energy and the radii rυ , where the orbital en- width becomes more or less constant. The situation is dif- ergy is E = 0 – Fig. 8. Particularly, we point out the spin ferent for a > a3 and a > 2M and then for a4, where the fre- a = a = 1.064306M, where r− = r− = 0.5107M, for which + σ υ ∆ quencies range narrows, and near r = rε becomes restricted ± at r∆ there is a critical point of the frequency amplitude to a small range of a few mass units in the limit of large ∆ω±. In BH geometries, the frequencies increase with the spin a/M. In strong and very strong NSs, the wide region is spin and with the decrease of the radius towards the hori- inaccessible for stationary observers, whereas it is accessi- ± zon. The curves rs continue to increase with the presence ble in the BH case. This significantly separates strong and of a transition throat at r = M that increases, stretching and weak NSs, and distinguish them from the BH case. Inter- widening. This throat represents a “transition region” be- estingly, the saddle point around r = M, which narrows the tween BH and super-spinning sources from the viewpoint throat of frequencies even in the case of NS geometries for of stationary observes. The regions outlined here play a dis- a ∈]M,aσ [, could perhaps be viewed as a trace of the pres- tinct role in the collapse processes with possible spin os- ence of r+, which is absent for a > M. For a = aσ , where the ± cillations and different behaviors for weak, very weak, and saddle point disappears, the shape of the rs tube is different. strong naked singularities. As the spin increases, the fre- This, on the other hand, would suggest that the existence of quencies of NSs observes move to lower values, widening the flex in the case of very weak NSs would prevent a fur- the throat. This trend, however, changes with the spin, en- ther increasing of the spin. This does not hold for a transition lightening some special thresholds. to stronger NSs, a ≥ aσ , where no saddle point is present– This analysis shows firstly the importance of the limit- 13 ing frequency ω0 = M/a, determining the main properties Since any simulation of stellar collapse returns to the BH regime, of both frequencies ω and the radii r±; it is also relevant in there must be some (retroactive) mechanism that closes the observer ± s tunnel, as even light does not run away in the forbidden region at r < M. relation to ZAMOs dynamics in NS geometries. In this way, Moreover, hypothetical super-luminary matter would violate the bond- we may see ω0 as an extension of the frequency ωh at the ing of the tunnel wall. horizon for BH solutions–Fig. 4. In the NS regime, all the 14It is worth to mention that predicted quantum effects close to the sin- curves r± converge to the same “focal point” r = 0, regard- gularities could play a major role in this region. However, we recall that s the extreme limit a = M in this model is never faced, as we continue to less of the type of naked singularity, but as ωh is the limiting see the spacetime for all NSs using a Boyer-Lindquist frame. It is well frequency at the BH horizon, each source is characterized known that approaching the horizon at a = M, the radial coordinate ve- by only one ω0 6= 0 frequency. The greater is the spin, the locity appears as never penetrating the black hole, spiraling as t goes to infinity. This is the consequence of a coordinate singularity which can lower is the frequency ω± at fixed radius, and particularly be avoided by using Kerr coordinates or Eddington-Finkelstein coordi- in the neighborhood of the singularity ring, according to the nates. limiting value ω0. The frequency range at fixed r/M nar- 15 The throat depth in the region would lead to an immediate change rows for higher dimensionless NS spin a/M. This feature of the observers properties and it is reasonable to ask if this may imply distinguishes between strong, weak and very weak naked an activation instead of a “positive feedback” phenomenon. We recall singularities. From Figs. 6 it is clear also that the throat of that in this scenario, we are not considering a change of symmetries ± which would have an essential role. Then it is important to emphasize the light-surfaces rs , in the plane r − ω, for different spins that in these hypothetical spin transitions, the external boundary of the a/M closes for a ≈ M, which is a spin transition region that ergoregion remains unchanged, but not the frequency at the static limit. 17
Fig. 8-bottom. Obviously, the consequences of the hypothet- References ical transition processes should also take into account the transient phase times. Very weak naked singularities show 1. R. M. Wald, General Relativity, (The University of a “rippled-structure” in the frequency profiles of ω with re- Chicago Press, Chicago, 1984). spect to r/M and a/M, as appears in Figs. 2, 6, 9, and 10. 2. D. Pugliese & H. Quevedo, Eur. Phys. J. C 75, 5, 234 The significance of this structure is still to be fully inves- (2015). tigated, but it may be seen perhaps as a fingerprint-remnant 3. J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973). of the BH horizon. This may open an interesting perspective 4. R. M. Wald, Living Rev. Relativ. 4(1), 6 (2001). for the study of NS geometries. 5. T. M. Fiola, J. Preskill, A. Strominger and S. P. Trivedi, An interesting application of our results would be re- Phys. Rev. D 50, 3987 (1994). lated to the characterization of the optical phenomena in the 6. H. Nikolic, Phys. Lett. B 678, 218 (2009). Kerr naked singularity and black hole geometries, such as 7. K. Bradler & C. Adami, JHEP 1405, 095 (2014). the BH raytracing and the determination of the BH silhou- 8. L. Susskind, Scientific American 276, 4, 52-57 (1997). ette (shadow). The light escape cones are a key element for 9. S.W.Hawking, Comm. Math. Phys. 43, 199 (1975) Er- such phenomena. Light escape cones of local observers (in- ratum - ibidem 46, 206 (1976). tended as sources) determine the portion of radiation emitted 10. S.W.Hawing, Mon. Not. R. astr. Soc.152, 75-78 (1971). by a source that could escape to infinity and the one which 11. R. Penrose, Revista del Nuovo Cimento (1969). is trapped. This is related to the study of the radial motion of 12. J.D. Bekenstein, Phys. Rev. D, 12, 3077 (1975). photons because the boundary of the escape cones is given 13. R. Penrose, and R. M. Floyd, Science 229, 177U179˝ by directional angles associated to unstable spherical photon (1971). orbits. Light escape cones can be identified in locally non- 14. F. Gray, S. Schuster, A. VanUBrunt˝ and M. Visser, rotating frames, in frames associated to circular geodesic Class. Quant. Grav. 33, 11, 115003, (2016). motion and in radially free-falling observers [42, 117–120]. 15. M. Bejger, T. Piran, M. Abramowicz and F. Hakanson, We want to point out, however, that light escape cones do not Phys. Rev. Lett. 109, 121101 (2012). define the properties of the light-cone causal structure, and 16. Z. Stuchlik, P. Slany, G. Torok and M. A. Abramowicz, are not directly related to stationary observers; they rather Phys. Rev. D 71, 024037 (2005). depend on the photon orbits. A thoroughout analysis of the 17. D. L. Meier, Black Hole Astrophysics The Engine photon circular motion in the region of the ergoregion can Paradigm, (Springer-Verlag, Berlin Heidelberg, 2012). be found in [2]. In Figs. 1, 3, 4 and 12, we show the pho- 18. V. P. Frolov, A. Zelnikov, Introduction to Black Hole ton orbit rγ and the limiting frequencies crossing this radius; Physics, (Oxford University Press, 488, 2011). this enlightens the relation with the frequency ωn. We con- 19. J. Gariel, G. Marcilhacy and N. O. Santos, Astrophys. sider there in more detail the relation between the quantities J. 774, 109 (2013). ωZ ω∗, the constants of motion L and E and the effective 20. R. Znajek, Nature, 262, 270 (1976). potential, briefly addressed also in Sec. 4. 21. A. R. Prasanna and R. K. Varma, Pramana 8, 3 229 In general, we see that it is possible to detect black holes (1977). and naked singularities by analyzing the physical proper- 22. A. R. Prasanna and C. V. Vishveshwara, Pramana 11, ties (orbital radius and frequency) of stationary observers 359 (1978). and ZAMOs. Moreover, the main physical properties (mass 23. A. N. Aliev and N. Özdemir, Mon. Not. Roy. Astron. and angular momentum) of the spinning gravitational source Soc. 336, 241 (2002). can be determined by measuring the parameters of station- 24. G. Preti, Class. Quantum Grav. 21, 3433 (2004). ary observers. This is certainly important for astrophysical 25. P. Bakala, E. Šrámková, Z. Stuchlík, and G. Török, purposes since the detection and analysis of compact as- Class. Quantum Grav. 27, 045001 (2010). trophysical objects is one of the most important issues of 26. V. P. Frolov and A. A. Shoom, Phys. Rev. D 82, 084034 modern relativistic astrophysics. In addition, the results pre- (2010). sented in this work are relevant especially for investigating 27. T. Igata, T. Harada, M. Kimura, Phys.Rev. D 85, 104028 non-isolated singularities, the energy extraction processes, (2012). according to Penrose mechanism, and the gravitational col- 28. R. Shiose, M. Kimura, T. Chiba, Phys. Rev. D 90, lapse processes which lead to the formation of black holes. 124016 (2014). 29. S. Hussain, I. Hussain, M. Jamil, Eur. Phys. J. C 74 12, Acknowledgements D.P. acknowledges support from the Junior GACR 3210 (2014). grant of the Czech Science Foundation No:16-03564Y. This work was 30. G. Pei, S. Nampalliwar, C. Bambi and M. J. Middleton, partially supported by UNAM-DGAPA-PAPIIT, Grant No. 111617. Eur. Phys. J. C 76, 10, 534 (2016). D.P. thanks Dr. Jan Schee for interesting discussion on the light escape 31. S. S. Komissarov, J. Korean Phys. Soc. 54, 2503 (2009). cone and stationary observers. 18
32. J.-P. Lasota, E. Gourgoulhon, M. Abramowicz, 59. H. Iguchi, K. i. Nakao and T. Harada, Phys. Rev. D 57, A. Tchekhovskoy and R. Narayan, Phys. Rev. D 89, 2, 7262 (1998). 024041 (2014). 60. H. Iguchi, T. Harada and K. I. Nakao, Prog. Theor. Phys. 33. Z. Stuchlik & J. Kovar, Class. Quant. Grav. 23, 3935 103, 53 (2000). (2006). 61. S. L. Shapiro & S. A. Teukolsky, Phys. Rev. Lett. 66, 34. Z. Stuchlik, D. Pugliese, J. Schee and H. Kucáková, Eur. 994 (1991). Phys. J. C 75, 9, 451 (2015). 62. T. A. Apostolatos & K. S. Thorne,Phys. Rev. D 46, 2435 35. A. Kotrlová, G. Török, E. Sramková & Z. Stuchlík, (1992). A&A 572, A79 (2014). 63. T. Jacobson& T. P. Sotiriou, Phys. Rev. Lett. 103, 36. K. Boshkayev, E. Gasperín, A.C. Gutiérrez-Piñeres, H. 141101 (2009). Quevedo, & S. Toktarbay Phys. Rev. D 93, 024024 64. T. Jacobson & T. P. Sotiriou, J. Phys. Conf. Ser. 222, (2015). 012041 (2010). 37. M. Kolos & Z. Stuchlík, Phys. Rev. D 88 065004, 65. E. Barausse, V. Cardoso, and G. Khanna. Phys. Rev. (2013). Lett. 105, 261102 (2010). 38. J. Schee & Z. Stuchlik, JCAP 1304, 005 (2013). 66. B. Giacomazzo, L. Rezzolla and N. Stergioulas, Phys. 39. G. Torok & Z. Stuchlik, Astron. Astrophys. 437, 775 Rev. D 84, 024022 (2011). (2005). 67. V. Cardoso, P. Pani, M. Cadoni and M. Cavaglia, Phys. 40. Z. Stuchlik, Bull. Astron. Inst. Czech, 32, 2, 68-72 Rev. D 77, 124 044 (2008). (1981). 68. N. Comins & B. F. Schutz, Proc. R. Soc. A 364, 1717 41. Z. Stuchlík, S. Hledík, K. Truparová, Class. Quant. 211-226 (1978). Grav. 28, 15 155017 (2011). 69. A. Helou, I. Musco and J. C. Miller, arXiv:1601.05109 42. Z. Stuchlik & J. Schee, Class. Quant. Grav. 27, 215017 [gr-qc]. (2010). 70. R. M. Wald & V. Iyer, Phys. Rev. D 44, 3719 (1991). 43. M. Patil & P. S. Joshi, Class. Quant. Grav. 28, 235012 71. P. S. Joshi, Gravitational Collapse and Spacetime Sin- (2011). gularities , Cambridge Monographs on Mathematical 44. Z. Stuchlik & J. Schee, Class. Quant. Grav. 29 065002, Physics, New York 2007. (2012). 72. B. P. Abbott et al. (LIGO Scientific Collaboration 45. Z. Stuchlik & J. Schee, Class. Quant. Grav. 30, 075012 and Virgo Collaboration) Phys. Rev. Lett. 116, 061102 (2013). (2016). 46. K. I. Nakao, P. S. Joshi, J. Q. Guo, P. Kocher- 73. B. P. Abbott et al. [LIGO Scientific and Virgo Collabo- lakota, H. Tagoshi, T. Harada, M. Patil and A. Krolak, rations], Phys. Rev. Lett. 119, 16, 161101 (2017). arXiv:1707.07242 [gr-qc]. 74. B. P. Abbott et al. [LIGO Scientific and Virgo and 47. M. J. Lake & B. Carr, JHEP 1511, 105 (2015). Fermi-GBM and INTEGRAL Collaborations], Astro- 48. B. J. Carr, arXiv:1703.08655 [gr-qc]. phys. J. 848, 2, L13 (2017). 49. B. J. Carr, J. Mureika and P. Nicolini, JHEP 1507, 052 75. M. Blaschke & Z. Stuchlík, Phys. Rev. D 94, 8, 086006 (2015). (2016). 50. B. J. Carr, Springer Proc. Phys. 170, 159 (2016). 76. D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D 51. Y. Prok et al. [CLAS Collaboration], Phys. Lett. B 672 83, 024021 (2011). (2009) 12 77. D. B. Malament, J. Math. Phys., 18, 1399 (1977). 52. D. Pugliese, H. Quevedo and R. Ruffini, Eur. Phys. J. C 78. S. W. Hawking, A. R. King, and P. J. McCarthy, J. Math. 77, 4, 206 (2017). Phys., 17, 174 (1976) 53. D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D 79. R. P. Geroch, J. Math. Phys., 8, 782 (1967) 83, 104052 (2011). 80. F. de Felice, A&A 34, 15 (1974). 54. D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D 81. F. de Felice, Nature 273, 429-431 (1978). 88, 024042 (2013). 82. Z. Stuchlík & J. Schee, Int. J. Mod. Phys. D 24, 1550020 55. D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D (2015). 84, 044030 (2011). 83. J. Schee & Z. Stuchlik, JCAP 1506 048, (2015). 56. R. Goswami, P. S. Joshi and P. Singh, Phys. Rev. Lett. 84. J. Gariel, N. O. Santos and J. Silk, Phys. Rev. D 90, 96, 031302 (2006). 063505 (2014). 57. C. Vaz & L. Witten, Phys. Lett. B 442, 90 (1998). 85. N. Pelavas, N. Neary and K. Lake, Class. Quant. Grav. 58. H. Iguchi, T. Harada and K. i. Nakao, Prog. Theor. Phys. 18, 1319 (2001). 101, 1235 (1999). 86. C. Herdeiro & E. Radu, Phys. Rev. D 89, 124018 (2014). 19
87. Z. Stuchlik, Bull. Astron. Inst. Czech 31, 129 (1980). 106. C. Chakraborty, M. Patil, P. Kocherlakota, S. Bhat- 88. S. Gao & Y. Zhang, Phys. Rev. D 87, 4 044028 (2013). tacharyya, P. S. Joshi and A. Krolak, Phys. Rev. D 95, 89. Z. Stuchlik, S. Hledik and K. Truparova, Class. Quant. 8, 084024 (2017). Grav. 28, 155017 (2011). 107. A. V. Frolov & V. P. Frolov, Phys. Rev. D 90, 12, 90. M. H. P. M. van Putten, Prog. Theor. Phys. 127, 2 331- 124010 (2014). 354 (2012). 108. J. B. Griffiths & J. Podolský, Exact space-times in 91. C. F. Gammie, S. L. Shapiro and J. C. McKinney, As- Einstein’s General Relativity, (Cambridge University trophys. J. 602, 312 (2004). Press, Cambridge, 2009). 92. P. Abolmasov Mon. Not. R. Astron. Soc. 432, 761-768 109. L.D. Landau& E.M. Lifshitz, The Classical Theory of (2013). Fields, (Oxford, Pergamon Press, 1980). 93. M. Kesden, Phys. Rev. D 83, 104011 (2011). 110. E. Poisson, A Relativist’s Toolkit: The Mathematics 94. R. M. Wald, Ann. Phys. 82, 548 (1974). of Black-Hole Mechanics, 1st Edition 2004, Cambridge 95. P. Pradhan & P. Majumdar, Eur. Phys. J. C 73, 6 2470 University Press. (2013). 111. M. R. R. Good & Y. C. Ong, Phys. Rev. D 91, 4, 96. D. Pugliese & G. Montani, Phys. Rev. D 91, 8, 083011 044031 (2015). (2015). 112. P. T. Chrusciel, J. Lopes Costa and M. Heusler, Living 97. D. Pugliese & Z. Stuchlik, Astrophys. J. Suppl. 221, 25 Rev. Rel. 15, 7 (2012). (2015). 113. B. Carter, gr-qc/9712038. 98. D. Pugliese & Z. Stuchlik, Astrophys. J. Suppl. 229, 2, 114. R. M. Wald, Class. Quant. Grav. 16, A177 (1999). 40 (2017). 115. S. Chandrasekhar, The Mathematical Theory of Black 99. D. Pugliese & Z. Stuchlik, Astrophys. J. Suppl. 223, 2, Holes, Oxford University Press (1983). 27 (2016). 116. C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravita- 100. D. Pugliese & Z. Stuchlik, JHEAp 17, 1 (2018). tion, (Freeman, San Francisco, 1973). 101. Z. Li & C. Bambi, Phys. Rev. D 87, 124022 (2013). 117. J. Schee, Z. Stuchlik & J. Juran, Proceedings of RAG- 102. J. Biéak, Z. Stuchlík, and V. Balek, Bull. Astron. Inst. time 6/7: Workshops on black holes and neutron stars, Czech 40, 2 65-92 (1989). Opava, 16-18/18-20 September, 2004/2005 103. V. Balek, J.Bicak, Z. Stuchlik, Bull. Astron. Inst. 118. J. Schee & Z. Stuchlik, Int. J. Mod. Phys. D 18, 983 Czech 40, 3 133-165 (1989). (2009). 104. J. Kovar, O. Kopacek, V. Karas and Z. Stuchlik, Class. 119. R. Takahashi, M. Takahashi, A&A, 513, A77, 7 Quant. Grav. 27, 135006 (2010). (2010). 105. S. W. Hawking & G. F. R. Ellis, The large scale struc- 120. J. Schee & Z. Stuchlik, Gen. Rel. Grav. 41, 1795 ture of space-time, (Cambridge University Press, Cam- (2009). bridge, 1973). 20
ε ε Fig. 13 Upper panel: The ratio E−/L− and the angular momentum of ε + the ZAMOs ωZ as a function of a/M in the static limit r = rε . The ε + angular momentum ω+ ≡ ω+(rε ) which is a boundary frequency for the stationary observer (outer light surface) is plotted (gray curve). The + + radius rε is defined by the condition ω−(rε ) = 0, ωh is the ZAMOs angular velocity on r = r+, i.e. ω±(r±) = ωh. The maxima are denoted by points. The NSII region is in light-gray. A zoom of this plot in the BH region is in the bottom panel.
Fig. 12 Upper panel: The angular velocity ωe ≡ ωZ(re) as a func- ε + tion of a/M. The angular velocities ωZ ≡ ωZ(rε ) (dashed curve), ± ωh ≡ ω±(r+) = ωZ(r+) (dot-dashed curve), ωˆZ ≡ ωZ(rˆ±) as functions of the spacetime rotation a/M for different BH and NS classes. Dotted a ≈ . M = ˆ + a ≈ . M = lines are κ 0 3002831060√ : ωe ωZ , s 0 91017√ : ωe ωh, + − ε a3 : ωˆZ = ωˆZ = 8/9 3, and finally the spin a = 2M : ωZ = ω2 ε (dashed line) which is a maximum for ωZ (the maximum point is marked with a point). The inset plot is a zoom. The radius re/M is a maximum for ωe. The angular velocities ω± on the BH photon orbit + ± rγ ∈ Σε are also plotted (colored lines). Center panel: ωˆZ ≡ ωZ(rˆ±) as functions of a/M for different NS classes. The minimum point of the + ZAMOs frequency ωˆZ is marked with a point at spin aω = 1.19866M. Bottom panel: The ZAMOs angular velocity ωZ is plotted as a function of the spin a/M and the radius r/M. The plane a = M and the horizon surface r = r+ are black surfaces. The gray surface denotes the orbit re. For both NS and BH spacetimes, the ZAMOs have a maximum fre- quency which is a function of a/M. The black thick curve corresponds to E = 0. The black region denotes the region inside the outer horizon r < r+.