Disclosing connections between black holes and naked singularities: horizon remnants, Killing throats and bottlenecks
Abstract
We study the properties of black holes and naked singularities by considering stationary observers and light surfaces in Kerr spacetimes. We reconsider the notion of Killing horizons from a special perspective by exploring the entire family of Kerr metrics. To this end, we introduce the concepts of extended plane, Killing throats and bottlenecks for weak (slowly spinning) naked singularities. Killing bottlenecks (or horizon remnants in analogy with the corresponding definition of throats in black holes) are restrictions of the Killing throats appearing in special classes of slowly spinning naked singularities. Killing bottlenecks appear in association with the concept of prehorizon regime introduced in de Felice (Mon Not R Astron Soc 252:197–202, 1991) and de Felice and UsseglioTomasset (Class Quantum Gravity 8:1871–1880, 1991). In the extended plane of the Kerr spacetime, we introduce particular sets, metric bundles, of metric tensors which allow us to reinterpret the concept of horizon and to find connections between black holes and naked singularities throughout the horizons. To evaluate the effects of framedragging on the formation and structure of Killing bottlenecks and horizons in the extended plane, we consider also the Kerr–Newman and the Reissner–Norström spacetimes. We argue that these results might be significant for the comprehension of processes that lead to the formation and eventually destruction of Killing horizons.
1 Introduction
 (i)

The metric (1) is invariant under the application of any two different transformations of the form \({{\varvec{\mathcal {P}}}}_{\mathbf {Q}}:\mathbf {Q}\rightarrow \mathbf {Q}, \) where \(\mathbf {Q}\) is one of the coordinates \((t,\phi )\) or the metric parameter a: a single transformation leads to a spacetime with an opposite rotation with respect to the unchanged metric.
 (ii)

The Kerr solution is stationary and axisymmetric due to the presence of the Killing fields \(\xi _{t}=\partial _{t} \) and \(\xi _{\phi }=\partial _{\phi } \), respectively.
 (iii)

As the metric is invariant under reflections with respect to the equatorial hyperplane \(\theta =\pi /2\), equatorial trajectories are confined in the equatorial geodesic plane.
 (iv)

In the limiting case of the Schwarzschild spacetime (\(a=0\)), \(r=2M\) is the lilling horizon with respect to the Killing vector \(\partial _t\). In general, in the special case of static (and spherically symmetric) BH spacetimes, the event, apparent, and Killing horizons with respect to the Killing field \(\xi _t\) coincide.
 (v)

The event horizons of a spinning BH are Killing horizons with respect to the Killing field \({\mathcal {L}}_H=\partial _t +\omega _H \partial _{\phi }\), where \(\omega _H\) is the angular velocity of the horizon.
In this work, we extensively discuss the properties of the Killing vector \({\mathcal {L}}=\partial _t +\omega \partial _{\phi }\) in the case of NS geometries. In BH spacetimes, this vector plays a crucial role in defining thermodynamic variables. As we will see below, the velocity \(\omega \) (and its limit \(\omega _H\)) and the vector \({\mathcal {L}}\) (and its limit \({\mathcal {L}}_H\)) are important for defining horizons and establishing relations between black holes and extreme black holes. In fact, it can be shown that: (a) in the context of the rigidity theorem, \(\omega _H\) represents the BH rigid rotation. Stated differently, the (strong) rigidity theorem connects the event horizon with a Killing horizon. In fact, under certain conditions, the event horizon of a stationary asymptotically flat solution (with matter satisfying suitable hyperbolic equations) is a Killing horizon. (b) The BH event horizon of this stationary solution is moreover a Killing horizon with constant surface gravity (zeroth BH lawarea theorem – the surface gravity is constant on the horizon of stationary black holes) [3, 4]. (c) Finally, the surface area of the BH event horizon is nondecreasing in time, which is the content of the second BH law (the laws state also the impossibility to achieve by a physical process a BH state with surface gravity \(\kappa =0\).)
We note here that the surface gravity of a BH may be defined as the rate at which the norm of the Killing vector vanishes from the outside. (The surface gravity is related to the acceleration of a particle corotating with the BH at the horizon and it can be written as (\({\mathcal {S}}{\mathcal {G}}_{Kerr}= (r_+r_)/2(r_+^2+a^2)\)). It is, therefore, a conformal invariant of the metric).
Possibly, we could isolate the contribution of the rotation in the expression of the surface gravity by comparing it with the static (and spherically symmetric) metric of Schwarzschild. In fact, the Kerr BH surface gravity can be written as the combination \(\kappa =\kappa _s\gamma _a\), where \(\kappa _s\equiv { {1}/{4M}}\) is the Schwarzschild surface gravity, while \(\gamma _a=M\omega _{H}^{2}\) is the contribution due to the additional component of the BH intrinsic spin; \(\omega _{H}\) is, therefore, the angular velocity (in units of 1 / M) on the event horizon.
These laws, which depend also on the horizon angular velocity, impose important constraints on any physical process in the BH spacetime, but they also allow to distinguish the static solution, \(a=0\), from the Kerr BH solution. The first law of BH thermodynamics, applied to a Kerr BH spacetime, actually relates the variation of the BH mass, horizon area and angular momentum, including the surface gravity and angular velocity on the horizon, i.e., \(\delta M = (1/8\pi )\kappa \delta A + \omega _H \delta J\). In here, the term dependent on the BH angular velocity represents the “work term” of the first law, while the fact that the surface gravity is constant on the BH horizon, together with other considerations, allows us to associate it with the concept of temperature. This aspect tends to emphasize the difference (also topological) between Kerr’s BH and its extreme solution: in the extreme case, where (\(r_{\pm }=M\)), it is easy to see that the surface gravity is zero and, considering the association with the temperature, there is \(T_H = 0\), with consequences also with respect to the stability against Hawking radiation. Nevertheless, the entropy (or BH area) of an extremal BH is not null [3, 4, 5, 6, 7, 8, 9, 10]. (An analogue implication of the third law it is said that a nonextremal BH cannot reach an extremal case in a finite number of steps.)
We investigate the properties of Kerr BHs and NSs from the point of view of stationary observers. In particular, we explore the characteristics of light surfaces, which correspond to the limiting frequencies of stationary observers. From the analysis of these orbital frequencies (and associated orbits), we introduce the concept of Killing throats, arising in NS spacetimes as the “opening” and disappearance of Killing horizons. More precisely, the Killing throat is a region bounded by a particular set of curves that we identify with the frequency of a stationary observer, which depends on the radial distance and the spin parameter a of the source. We define a Killing bottleneck as a particular case of a of Killing throat that appears in weak naked singularities (WNS). Thus, bottlenecks can be interpreted as throats “restrictions” that characterize WNS . The concept of strong and weak NSs depends on the value of the spin parameter and has been explored in several works [11, 12, 13, 14, 15, 16, 17, 18]. However, in general, they are also differently defined as strong curvature singularities, for example, in [19]. Regarding various NSs properties and characteristics of the gravitational collapse, possible formation and stability of naked singularities as well as other analysis concerning observational phenomena related to possible NSs existence, we refer also to [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 43, 44]. In this work, WNSs are characterized by spinmass ratios close to the value of the extreme BH. To explore these NS effects and to compare BHs with NSs, it is convenient to introduce the concept of “metric bundles” and “extended planes”. A metric bundle is a curve on the extended plane, i.e., a family of spacetimes defined by one characteristic photon orbital frequency \(\omega \) and characterized by a particular relation between the metrics parameters. This turns out to establish a relation between BHs and NSs in the extended plane. All the metric bundles are tangent to the horizon curve in the extended plane. Then, the horizon curve emerges as the envelope surface of the set of metric bundles. As a consequence, WNSs turn out to be related to a part of the inner horizon, whereas strong naked singularities (SNSs) with \(a>2M\) are related to the outer horizon.
Lookup table with the main symbols and relevant notations used throughout the article
\( \omega _{\pm }\)  Limiting frequencies for stationary observers: Eqs. (7), (8) 
\(\omega _0=M/a\)  Limiting frequency \(\omega _{\pm }\) at the singularity and frequency of the metric bundle: Eq. (9)–Sect. 4 
\({\mathcal {L}}_{\pm }\)  Null Killing vector (generators of Killing event horizons): Eq. (10) 
\(r_{s}^{\pm }\)  Light surfaces radii: Eq. (11) 
\(\omega _H^{\pm }\)  Frequencies at the horizons \(r_{\pm }\): Eq. (12) 
\(r^{\mp }_{\mp }\)  Photon orbits with frequencies \(\omega _H^{\pm }\) at the horizons: Eq. (13)–Fig. 1 
\(g_{\omega }^{\pm }\)  Metric bundles in the extended plane \(\pi _a\): Sect. 4 
\(a_{\omega }^{\pm }(r,\omega ;M)\)  \(g_{\omega }^{\pm }\) in terms of the bundle frequency \(\omega \): Eq. (15) 
\(a_{\pm }\)  Horizon curve in the extended plane: Fig. 8 
\(r_{\partial }^{\pm }(\omega )\)  closing radii of the metric bundle in \(\pi _a^+\) (=\(\pi _a\) for \(a>0\)): Eq. (16) 
\(a_g\)  Spin of metric bundle tangent to the horizons in \(\pi _a^+\): Eq. (17)–Fig. 13 
\(a_p\)  Bundle origin, i.e., \(a_g(a_0)=a_g(a_0^{\prime })\) with \(a^{\prime }_0=a_p\equiv 4M^2/a_0\): Figs. 14, 15, 16; Tables 2 and 3 
Horizons relations I  \(\omega _0^{1}\equiv a_0^{\pm }/M=\frac{2 r_{\pm }(a_g)}{a_g}\equiv \omega _H^{1}(a_g)\), \(\omega _H^+(r_g,a_g)=\omega _0=Ma_0^{1}\), \(\omega _H^(r^{\prime }_g,a_g)=\omega ^{\prime }_0=M/a_0^{\prime }\) \(r^{\prime }_g\in r_\) (\(r_+=r_g\), \(r_=r_g^{\prime }\)): Fig. 13 
Horizons relations II  \(\omega ^{\prime }_0=\frac{1}{4 \omega _0}\), \( \omega _H^+\omega _H^=\frac{1}{4}\), (\(a_0^+(a_g)a_0^(a_g)=4M^2\)), \(a_0^{\pm }/M=\frac{2 r_{\pm }(a_g)}{a_g}\) where \(a=a_0\) and \(a=a_p\): Fig. 14 
SNS  (= \(\mathbf{SNS}^+\cup \mathbf{SNS}^\)) strong naked singularities \(a_0>2M\), \(\mathbf{SNS}^+\) for \(a_0>4M\) \(\mathbf{SNS}^\) for \(a_0\in [2M,4M[\): Fig. 13 
WNS  Weak naked singularities \(a_0\in ]M,2M[\): Fig. 13 
\(\mathbf{BH}=\mathbf{BH}^+\cup \mathbf{BH}^\)  \(\mathbf{BH}^+\) for \(a\in [a_{g}^1,M]\), \(a_{g}^1=3/4 M\) and \(\mathbf{BH}^\) for \(a\in [0,a_{g_1}]\): Fig. 13 
Left region Fig. 13  \(a_0\in [0,2M]\) 
Right regionFig. 13  \(a_0>2M\) 
Upsector Fig. 13  \(a_g>a_{g}^1\) 
Downsector Fig. 13  \(a_g<a_{g}^1\) 
\(\varpi _\pm \)  Second frequency of a metric bundle: Eq. (22) 
\(a_{\omega }^{\mathbf {(\natural )}}(\omega _H^{\mathbf {\flat }})\)  Metric bundles parameterized for the tangent point \(a_g\): Eq. (23) 
\(a_{tangent}(r)\)  Tangent curve to the horizon in terms of \(r_g\): Eq. (24) 
\((r^{real}_g, r^{\checkmark }_g, r_g^\mp )\)  Solution of the tangency condition \(a_g=a_{\pm }\): Eq. (25), Fig. 17 
\( (Q_{\omega }^{\pm })^2\)  Metric bundles in terms of the charge Q: Eq. (35) 
2 Stationary observers and light surfaces
3 Killing throats and bottlenecks
The concept of Killing throats emerges through the analysis of the radii \(r_{s}^{\pm }(\omega ,a)\) (the frequencies \(\omega _{\pm }(r,a)\)) with respect to the orbital frequency \(\omega \) (the radius r) of lightlike particles – see Figs. 2 and 3. A Killing throat in NS geometries is a connected region in the \(r\omega \) plane, which is bounded by \(r_{s}^{\pm }(\omega ,a)\) or equivalently \(\omega _{\pm }(r,a)\), and contains all the stationary observers allowed within the limiting frequencies \(]\omega _, \omega _+[\). Conversely, a BH Killing throat is either a disconnected region in the Kerr spacetime (in a sense similar to the concept of a pathconnected space) or a region bounded by nonregular surfaces in the extreme Kerr BH spacetime. A Killing bottleneck in a NS spacetime is a narrowing of the Killing throat which appears only for specific naked singularities and involves a narrow range of frequencies and orbits – Fig. 1. The limiting case of a Killing bottleneck occurs in the extreme Kerr spacetime, as seen in the BL frame, where the narrowing actually closes^{3} at the BH horizon \(r=M\).
 (1)

Coalescence of the Killing horizons, which occurs in the extreme Kerr BH solution;
 (2)

Formation of the Killing throat and emergence of the bottleneck in weak NSs;
 (3)

Disappearance of the Killing bottleneck in strong NSs.
There are several notable radii associated with the frequencies \(\omega _{\pm }\) and light surfaces \(r^{\pm }_s\), which are related to the extremes and saddle points of the curves in the region \(\Sigma _{\epsilon }^+\) (see [18]). The function \(\Delta \omega \equiv \Delta ^\pm \omega =\omega _+\pm \omega _\) is considered in Figs. 6 and 7. The radius \(r_{\Delta }\), solution of \(\partial _{r}\Delta \omega =0\), clearly shows the presence of closed surfaces for \(r_{III}^{\pm }\) and \(r_{II}^{\pm }\), and provides a characterization of the Killing bottleneck. An analysis of \(\omega _{\pm }\) and \(\Delta \omega \) in Figs. 6 and 7 shows the emergence of horizons as the envelope surface in the plane \(ra\) of the limiting frequencies \(\omega _{\pm }\). This important aspect will be addressed in details in Sect. 4, revealing the role played by the Killing horizons in BHs–NSs connections.
4 Unveiling BHs–NSs connections
In this section, we explore the entire parameterized family of Kerr solutions with \(a/M\ge 0\). To this end, we introduce the concept of extended plane \(\pi _a\), where the entire collection of metrics of a parametrized family of solutions can be considered.
Metric bundles \(g_{\omega }^{\pm }\)
The metric bundles \(g_{\omega }\equiv g_{\omega }^{\pm }\) are defined by the curves \(\left. a_{\omega }^{\pm }\right _{\omega }\) of constant frequency \(\omega \) in \(\pi _a\)– Fig. 8. Each \(g_{{\bar{\omega }}}=\left. a_{\omega }^{\pm }\right _{{\bar{\omega }}}\) for a fixed frequency \({\bar{\omega }}\) is represented by closed and bounded curves which are continuous almost everywhere in \(\pi _a\). Below, we will discuss extensively the properties of these curves.
The bundles \(g_{{\bar{\omega }}}\) contain an (almost) continuum and infinite set of metric parameter values a / M. Each value of a / M sets a specific Kerr geometry. From Fig. 8, it is clear that eventually some bundles contain both BHs and NSs spacetimes, others define only BHs, while none of the bundle is constituted by NS only. In fact, all the metric bundles are tangent at least in one point to the horizons \(a_{\pm }\). Thus, from the quantities (15) an alternative definition of BHs Killing horizons \(r_{\pm }\) emerge. Indeed, from Fig. 8, it follows that the horizons \(a_{\pm }\) arise as the envelope surfaces of the curves \(a_{\omega }^{\pm }(r)\), i.e., the metric bundles \(g_{\omega }=\left. a_{\omega }^{\pm }\right _{\omega }\) in \(\pi _a\). This will be a crucial property of the metric bundles with significant consequences that allow us to connect NSs to BHs in the extended plane. In fact, since the curve \(a_{\pm }\) is tangent to all metric bundles \(g_{\omega }\), the (inner and outer) horizons contain all the frequencies \(\omega \) defining each metric bundle and, therefore, describing both NSs and BHs in the extended plane.
We note that \(r_{\partial }^\) is negative for positive values of the frequency. As we are restricting our analysis in this work to the case \(\omega >0\), we shall not consider \(r_{\partial }^\); nevertheless, an analysis of the case \(\omega <0\) would, in fact, provide additional information about the spacetime structure even in the equatorial plane. A very small \(\omega _{\partial }^+\), on the other hand, corresponds to a very large (origin) spin \(a_0=M/\omega \). Note that the spin \(a_0\) corresponds to the frequency \(\omega _0=M/a\), which was introduced in Eq. (8) by considering the behavior of the stationary observer frequencies near the singularity \(r=0\); this is of importance in NS geometries as described in Sect. 3. The properties of this special frequency \(\omega _0\) have also been extensively discussed in [18]. More generally, as noted in Sects. 2 and 3, the dimensionless spin parameter a / M is related to the quantity \(M/\omega \) though the frequencies of the light surfaces (see [18]). Then, the function \(a_{\omega _\partial }^+(r)\) in Eq. (16) is obtained as \(a_{\omega }^{\pm }(r,{\omega _{\partial }^+})\). As shown in Fig. 10, the condition \(a_{\omega _\partial }^+(r)=a_{\pm }\) is valid only at the spacetime singularity \(r=0\); otherwise, \(a_{\pm }<a_{\omega _\partial }^+(r)<2M\), while the condition \( a_{\omega _\partial }^+(r)= 2M\) (a NS) is reached asymptotically for large values of r / M.
Note that the asymptotic limit of \( a_{\omega _\partial }^+(r)=2M\) is relevant as it corresponds to a metric bundle \(g_{\omega }^{\pm }\) at constant frequency \(\omega =0.5\), which defines the point of the envelope corresponding to the extreme Kerr BH solution \(a=M\). Figure 12 refers to the analysis of Fig. 8 and sketch the correspondence between BHs and NSs derived from the analysis of \(a_{\omega }^{\pm }\). The envelope \(a_{\pm }\) of the \(a_{\omega }^{\pm }\) curves is defined as the set of points (a / M, r / M) for which \(\partial _{\omega }a_{\omega }^{\pm }=0\), i.e., as the curve tangent to all \(a_{\omega }^{\pm }\) or also as the boundary of the region filled by the curves \(a_{\omega }^{\pm }\). Then, small changes of a and shifts along the orbit radius r leave \(\omega \) almost constant as \(a_{\omega }^{\pm }\) are continuum functions.
The relation between the radius \(r_{\partial }^{\pm }\), the spin \(a_{\partial }^{\pm }\) and frequencies \(\omega _{\pm }\) has to be confronted with Eq. (12) for the frequencies \(\omega _{H}^{\pm }\) at the horizons and radii \(r_\pm \) of the BH Killing horizons. These quantities play an important role also for the Killing bottleneck emergence considered in Sect. 3, as the surfaces \(a_{\omega }^{\pm }(\omega )=\) constant are related to the solutions \(\omega _{\pm }(a)\) of Eq. (8) and shown in Fig. 11. The relation \(r_{\partial }^{\pm }\omega =a_{\omega }^{\pm }/2\) of Eq. (16) is also used in Fig. 12 to unveil some BHs and NSs properties: The points on the lines \(a_{0}=\) constant for \(a_0\in ]M,a_{\bullet }]\) lead to the Killing bottleneck emergence of Fig. 11.
BHs and NSs in metric bundles \(g_{\omega }\)
As discussed above, a metric bundle \(g_{\omega }\) can comprise only BHs or BHs and NSs. Moreover, the horizons describe BHs and NSs in the extended plane. In the remaining of this section, we describe this last aspect and the BHsNSs relation more closely.
Firstly, Fig. 10left represents BHs and NSs in the \(\omega r\) plane. The plane is divided into regions, where the metric bundles \(g_{\omega }\) include BHs or NSs; there are regions with only BHs or NSs and transition regions that cross different sections and are connected to Killing horizons and bottlenecks – see also Fig. 11. A special transition region is, for example, around the extreme BH horizon \(r=M\) with frequency value \(\omega =1/2\), which corresponds to the extreme Kerr BH.
We concentrate on the restricted \(\pi _a^+\subset \pi _a\) plane determined by \(a\ge 0\) in Fig. 8. The restriction of the extended plane \(\pi _a\) to \(\pi _a^+\) exploits the symmetry by reflection around the axis \(a=0\), where negative origins of metric bundles build up the horizons \(a_{\pm }<0\). The plane has the following remarkable sections: \(\mathcal {P}_\mathcal {S}\), \(\mathcal {P}_{\mathcal {L}}\), \(\mathcal {P}_\mathcal {\odot }\), \(\mathcal {P}_\mathcal {H}\), and \(\mathcal {P}_ {\mathcal {\otimes }}\). The line \(\mathcal {P}_\mathcal {S}=(r=0, a=\text{ constant })\) includes BHs and NSs. \(\mathcal {P}_\mathcal {S}\) represents the collection of all origins \(a_0\). A variation of the dimensionless spin of the singularity (at \(r=0\)) corresponds to a variation of \(a_0\) in \(\mathcal {P}_\mathcal {S}\). This line also represents the singularity frequencies \(\omega _0=M/a_0\). Moreover, we define \(\mathcal {P}_{\mathcal {L}}=(r_{\partial },a_{\partial })\) and the set \(\mathcal {P}_\mathcal {H}=(a_{\pm }, r)\) for the Killing horizons. Curves \(a_{\omega }^{\pm }\) at constant \(\omega \) closes on \(\mathcal {P}_{{\mathcal {L}}}\) and have origins in \(\mathcal {P}_{\mathcal {S}}\). The line \(\mathcal {P}_\mathcal {\odot }=(a=0, r)\) describes the limiting case of the Schwarzschild solution. \(\mathcal {P}_\mathcal {\odot }\) crosses \(\mathcal {P}_\mathcal {H}\) in \(r=0\) and \(r=2M\). Finally, the set \(\mathcal {P}_ {\mathcal {\otimes }}=(a=M, r)\) describes the extreme Kerr spacetime and crosses \(\mathcal {P}_\mathcal {H}\) at \(a_{\pm }=M\) and \(r=M\). The collection of all the points \(a_{\omega }=\) constant generates the lightcurves shown in Fig. 11, where the Killing bottleneck and Killing throat emerge at \(a_{\omega }/M>1\).
According to Fig. 8, metric bundles can be classified in two classes. (1) The first class includes the curves \(a_{\omega }^{\pm }\) tangent to \(r_\), including those bundles which are “entirely” contained in the BH sector of \(\pi _a^+\), i.e., \(a\in [0,M]\) and \(r\in [0,r_]\). (2) The second class includes metric bundles tangent to the outer horizon \(r_+\), containing BHs and NSs in the same bundle. These two classes are separated by the limiting bundle \(g_{\omega }^{\pm }\) with origin \(a_0=2M\) and tangent to the maximum of \(a_{\pm }\); that is, the point in \(\pi _a^+\) with \(r=M\), for \(a=M\) and frequency \(\omega =0.5\), describing the extreme Kerr spacetime. Bundles with origins in the BH sector are completely contained in an “inaccessible” region between the singularity, \(r=0\), and the inner horizon \(r_\). Bundles with origin in the the naked singularity sector \(a>M\) comprise BH and NS geometries, which are, therefore, related because they are contained in the same metric bundle. (Notice that the definition of \(g_{\omega }^{\pm }\), through the stationary observes definition, provides a BHNS relation). The case of very strong NSs (very large a / M, asymptotic value \(a\approx +\infty \)) corresponds to the limiting point \(r_+\approx 2M\) and \( a\approx 0\) – see Fig. 8. This suggests that the NS sector is closely related to the BH sector. We explore this connection more deeply below.
The horizons as an envelope surface of the metric bundles
An important consequence of the approach presented here is that it allows us to establish a relation between BHs and NSs. In fact, the outer horizon in \(\pi _a^+\) emerges as an envelope surface of metric bundles with only NSs origins. That is, the BH outer Killing horizon \(r_+\) relates a BH with \(a\in [0,M]\) with a NS with \(a\in ]2M,\infty ]\). The inner horizon in \(\pi _a^+\) can be constructed by metric bundles with NS and BH origins. Viceversa, the BH horizons are tangent to all the metric bundles. All the BHs and NSs frequencies \(\omega _{\pm }\) are, therefore, related to the horizon frequency \(\omega _H\), and all horizon frequencies are the limiting orbital photon frequencies in NS and BH spacetimes – Fig. 8. As the horizon is tangent to all the metric bundles, each metric bundle is defined by one frequency \(\omega \), which coincides with the horizon frequency \(\omega _H\) on the tangent point in the extended plane. Frequencies \(\omega \) in (15) are actually the limiting frequencies \(\omega _\pm \). It follows that the metric bundles, connected the singularities frequencies \(\omega _0\), are defined by the origins of the metric bundles and the horizon frequencies \(\omega _H\).
Remarkably, the construction of the inner horizon \(r_\) is confined to metric bundles contained entirely in the BH sector. This fact could lead to important consequences, when considering the collapse towards a BH and the process of formation of the inner horizon. These metric bundles are all confined in the \(\pi _a^+\) region \(r\le r_\) and \(a\le M\). This implies that the inner horizon \(r_({\bar{a}})\) of a BH with spin \({\bar{a}}\) is related to a metric bundle with origin \(a_0\) in the BH or WNS (\(a\in [M,2M]\)) regions, while the outer horizon \(r_+({\bar{a}})\) is related to a NS metric bundle.
Before continuing, it is convenient to return to the concept of metric bundle, as shown in Fig. 8, and analyze three particular curves in detail: (1) the horizontal lines \(a=\) constant of \(\pi _a\); (2) the vertical lines \(r=\) constant and; (3) the curves \(a_{\pm }\) corresponding to horizons.
(1) The horizontal lines a = constant
For a BH or a NS with \(a_0\in \mathcal {P}_\mathcal {S}\), there are two curves \(a_{\omega }^{\pm }\) of the bundle, which are tangent at a point \(p\in \mathcal {P}_\mathcal {H}\) on the horizons. Each metric bundle \(g_{\omega }^{\pm }(a_0)\) is associated to a constant frequency, \(\omega _0=M/a_0\), defined by the bundle origin \(a_0\). Considering a bundle, there is a pair of points \(p_1(a)=(a,r_1(a))\) and \(p_2(a)=(a,r_2(a))\) with \(a=\) constant and \(r_1(a)<r_2(a)\), which are located respectively on the two curves of the bundle. A special case is the pair of points present on the origin line, \(P_{\mathcal {S}}\), where \(r_1(a_0)=r_2(a_0)=0 \). Also the horizons \(P_{\mathcal {H}}\) for the extreme Kerr spacetime are special. Note that, in general, the condition \(r_1(a)<r_2(a)\) on the horizon for \(a=a_H\in a_{\pm }(r)\) leads to Eq. (13). On the orbits \((r_1(a),r_2(a))\), lightlike orbital frequencies \(\omega _{\pm }\) are equal to \(\omega _a(r_1)=\omega _a(r_2)=M/a\), where \(\omega _a\in (\omega _+,\omega _)\). There are two special geometries associated to the closure points \(P_{\mathcal {S}}\) and \(P_{\mathcal {H}}\): \( P_{\mathcal {S}}\) represents the singularity \(r=0\) with \(\omega _a(0)=\omega _a(0)=M/a\), corresponding to a spacetime with spin a. Moreover, the second special geometry is always a BH, whose (inner or outer) horizon has the frequency \(\omega _H=M/a\), i.e., the frequency of the bundle. We investigate the spin \(a_g\) of this specific spacetime below.
We note that metric bundles cross each other in \(\pi _a^+\). This means that, in a fixed spacetime with a fixed radius, there are two limiting frequency values \(\omega _{\pm }\). Therefore, the maximum number of crossing points between metric bundles is two. Consequently, there are two crossing metric bundles with origins at \(a=1/\omega _{\pm }\), respectively.
(2) Vertical lines \(r=\) constant
Let us now focus on the vertical lines in \(\pi _a^+\) and the intersections on each metric bundle. For a fixed orbit r, there are, in general, two Kerr geometries corresponding to the spins \(a_1(r)<a_2(r)\) of the same bundle. In addition, there are the following limiting cases: 1. At \(r=0\) there is an infinite number of origins, where \(a_1(0)=a_2(0)\) on a bundle. 2. The point \(r_t\) is the tangent point of the vertical line to the bundle, satisfying the condition \(a_{\omega }^+(r_t)=a_{\omega }^(r_t)\) – see Eq. (16) and Fig. 10. The condition \(r_{\partial }^+=r_{\pm }\) is satisfied only in special geometries \(a_{R}\). In general, for \(r\in ]0,r_{\partial }^+[\), there are two geometries \(a_1<a_2\), corresponding to two BHs or one BH and one NS. This implies that at a fixed r, there are two geometries \((a_1,a_2)\) with frequencies \(\omega =1/a\). The case of the geometries identified in the extended plane by the vertical lines will be clarified at the end of this section because it is necessary to consider the tangency conditions as shown in Fig. 17. With respect to this property, \(a_1<a_2\), there are two exceptions represented by the metric bundles with vertical lines tangent to the horizon. There are two asymptotic cases, where \(a_1=a_2\); these cases with respect to the horizon points \((a=0, r=0)\) and (\(a=0,r=2M\)) correspond to the limit of the Schwarzschild spacetime – Fig. 17. In general, a vertical line \(r={\bar{r}}\) on a metric bundle \(g_{\omega _0}\) defines two geometries with \(\omega _+=\omega _0\) and \(\omega _=\omega _0\). More details on this point will be addressed at the end of this section. In other words, this property relates limiting frequencies of different spacetimes. This result is in agreement also with the results presented in Sect. 3.
(3) The horizon surfaces \(a_{\pm }\)
All the metric bundles have the frequencies \(\omega _{H}=1/a_{\pm }\) at the horizon in the extended plane. As the metric bundle frequency \(\omega _0\) is also a limiting photon orbital frequency \(\omega _+\) or \(\omega _\) and as all frequencies \(\omega _0\) represent also the horizon frequency \(\omega _{H}\), then the limiting photon frequencies \(\omega _{\pm }\) on an orbit r in all the spacetimes \(a\in [0,\infty [\) are the horizons frequencies \(\omega _H\) in the extended plane. Viceversa, the set of the horizon frequencies \(\omega _H\) in the extended plane is the collection of all the limiting orbital photon frequencies \(\omega _{\pm }\) in any BH or NS spacetime. This issue will be discussed in detail below.
BHs–NSs correspondence
Furthermore, it is immediate to see that from the expression \(a_g(a_0)\) of Eq. (17), we obtain a relation between the horizon tangent spin (horizon frequency) and the bundle origin (bundle frequency), namely, \(\omega _0^{1}\equiv a_0^{\pm }/M=\frac{2 r_{\pm }(a_g)}{a_g}\equiv \omega _H^{1}(a_g)\); in particular, from \(\omega _0^{1}\equiv a_0^{\pm }/M=\omega _H^{1}(a_g)\), it is seen that the bundle frequencies \(\omega _0\) represent all (and only) the horizon frequencies on the tangent point, i.e., \(\omega _H(a_g(a_0))=\omega _0\). This implicitly relates also the horizons frequencies with the singularity frequencies; there are then particular BH spacetimes, where the outer horizon \(r_+\) or the inner and outer horizon \(r_{\pm }\) are defined by bundles with NSs. We detail this aspect below. Here we note that there is only one fixed point for the transformation \(a_g(a_0)\), namely \(a_0>a_g\), and \(a_0=a_g\) for \(a_0=0\).
We now introduce the concept of couples of related metric bundles, say, BD and BC (see Fig. 14). In the first couple, the first bundle has its origin in the NS region (tangent to the outer horizon) and the second bundle is completely or partially contained in the BH region (tangent to the inner horizon). The two bundles with origins (\(a_0,a_0^{\prime }\)) share equal tangent spin \(a_g\). That is, if \(a_0>a_0^{\prime }\), then the bundle with origin \(a_0\), frequency \(\omega _0=Ma_0^{1}\), has the contact spin \(a_g\) in \(r_g\in r_+\) with horizon frequency \(\omega _H^+(r_g,a_g)=\omega _0=Ma_0^{1}\). The second bundle of the couple has its origin at \(a^{\prime }_0\), frequency \(\omega ^{\prime }_0=M/a_0^{\prime }\), tangent spin \(a_g\) in \(r^{\prime }_g\in r_\) and horizon frequency \(\omega _H^(r^{\prime }_g,a_g)=\omega ^{\prime }_0=M/a_0^{\prime }\). Bundles \(g_{\omega _0}\) and \(g_{\omega _0^{\prime }}\) determine, in the sense of the envelope surface, respectively, the outer horizon \(r_+=r_g\) and the inner horizon \(r_=r_g^{\prime }\) of the BH spacetime with spin \(a_g\). The relation between the tangent points \((a_g, r_g)\) and the origin \(a_0\) and the relations between BHs and NSs through the bundles will be addressed in full details below. Importantly, the condition of the bundle correspondence, i.e., \(a_g(a_0)=a_g(a_0^{\prime })\), leads to the nontrivial solution \(a^{\prime }_0=a_p\equiv 4M^2/a_0\) (see DD and BB model bundles of Figs. 14, 15, and 16 and Tables 2 and 3). In terms of the horizon frequency (equal to the bundles frequencies), there is \(\omega ^{\prime }_0=\frac{1}{4 \omega _0}\); in fact, using Eq. (12), this relation can be written in compact form as \( \omega _H^+\omega _H^=\frac{1}{4}\) (or we can write \(a_0^+(a_g)a_0^(a_g)=4M^2\)), which is independent of the spin a, in general. In Fig. 14, the solutions \(a_0^{\pm }/M=\frac{2 r_{\pm }(a_g)}{a_g}\) correspond to \(a=a_0\) and \(a=a_p\).
In the second couple, BC, the origin spin of one bundle \(a_0^{\prime }\) is the tangent point \(a_g(a_0)\) of the second bundle with origin \(a_0\) (therefore, \(a_0^{\prime }\) is always a BH and the other bundles are all BHs). An example of these couples are the models BB and CC of Figs. 14, 15, and 16 and Tables 2 and 3.
To enlighten some properties of the BC and BD bundles, we consider the difference \(G[a_0]\equiv (a_0a_g)\) as a function of \(a_0\) and the recurrence relation for \(a_g\), i.e., \(a_g[n+1]=\frac{4 a[n]M^2}{a[n]^2+4M^2}\) where \(a[0]=a_0\). Then, \(a_g[n+1]\) decreases with the cycle order n (see Fig. 14). In fact, in BC bundles, the cycles are confined to the inner horizon in the extended plane (apart from the starting point \(a_0\)). Naturally, the only fixed point, \(a_p=a_0\), of this relation is in \(a_0=2M\), corresponding to the extreme BH.
Recurrence relation \(a_g[n]\) for different starting points, see also Fig. 14. BD bundles with coincident cycles (BB and DD) are clearly denoted (excluding the initial point) and partially coincident cycles in BC bundles are also shown. Cycles (with the exclusion of the initial point \(a_0\)) are entirely confined in the BH region. Models \({\mathbf {XX}}\) where \(\mathbf {X}=\{\mathbf {A,B,C,D}\}\) are defined in Fig. 15
\(a_0 \)  \(a_g[1]\)  \(a_g[2] \)  \( a_g[3]\)  \( a_g[4] \)  \( a_g[5]\)  \(a_g[6] \)  \( a_g[7]\) 

1 (BB)  0.8  0.689655  0.616366  0.562903  0.521586  0.48837  0.460889 
2 (CC)  1.  0.8  0.689655  0.616366  0.562903  0.521586  0.48837 
4 (DD)  0.8  0.689655  0.616366  0.562903  0.521586  0.48837  0.460889 
1/2 (AA)  0.470588  0.445902  0.424787  0.406451  0.39033  0.376008  0.363172 
\(a_p\)(1/2) = 8  0.470588  0.445902  0.424787  0.406451  0.39033  0.376008  0.363172 
\(a_g\)(1/2) = 0.470588  0.445902  0.424787  0.406451  0.39033  0.376008  0.363172  0.351579 
Models \({\mathbf {XX}}\), where \(\mathbf {X}=\{\mathbf {A,B,C,D}\}\) as defined in Fig. 15
Models  AAmodel  BBmodel 

Bundle frequency:  \(\mathbf {\omega _0=2}\)  \(\mathbf {\omega _0=1}\) 
Tangent point \(r_g\):  \(r_g = 2/17\)  \(r_g = 2/5\in r_ \) 
Bundle spin \(a_g\) at \(r_g\):  \(a_g=0.470\)  \(a_g=\underline{a_ {\omega }^} (r_g) = 4/5\) (D) 
\({a_ {\omega }^+} (r_g) =0.474\) (\(A^{o}>A^i\))  \({a_ {\omega }^+} (r_g) = 13/15\) (\(B^{o}>B^i\))  
Horizon frequency \(\omega _H\) at \(r_g\):  \(\underline{{\varvec{\omega }}_\mathbf{H}^(\mathbf{r}_{\mathbf{g}}, \mathbf{a}_{\mathbf{g}})} = \mathbf{2}\equiv {\varvec{\omega }}_\mathbf{0}\)  \({\underline{{\varvec{\omega }}_\mathbf{H}^(\mathbf{r}_\mathbf{g}, \mathbf{a}_\mathbf{g})} = \mathbf{1}\equiv {\varvec{\omega }}_\mathbf{0}}\) 
Bundle frequency for \(r_ +\) or \(r_\):  \({\omega _H^+(r_g, a_g)} = 1/8\)  \({\omega _H^+(r_g, a_g)} = 1/4\) (\(D_i\)) 
Horizon frequency (\(X^{o}\)):  \(\omega _H^(r_g, {a_ {\omega }^+}(r_g)) =1.984\) \(\omega _H^+(r_g, a_ {\omega }^+(r_g)) =0.126\)  \(\omega _H^(r_g, {a_ {\omega }^+}(r_g)) =0.865\) \(\omega _H^+(r_g, a_ {\omega }^+(r_g)) =0.289\) 
Models  CCmodel  DDmodel 

Bundle frequency:  \(\mathbf {\omega _0=1/2} \)  \(\mathbf {\omega _0=1/4} \) 
Tangent point \(r_g\):  \( r_g =1 \)  \( r_g =8/5 \) 
Bundle spin \(a_g\) at \(r_g\):  \(a_g=1 \)  \(a_g=4/5\) (B) 
\({a_ {\omega }^+} (r_g) =5/3\) \((C^o>C^i)\)  \({a_ {\omega }^+} (r_g)=3.644\)  
Horizon frequency \(\omega _H\) at \(r_g\):  \({\underline{{\varvec{\omega }}_\mathbf{H}^(\mathbf{r}_\mathbf{g}, \mathbf{a}_\mathbf{g})} = \mathbf{1}/\mathbf{2}\equiv {\varvec{\omega }}_\mathbf{0}}\)  \({\underline{{\varvec{\omega }}_\mathbf{H}^+(\mathbf{r}_\mathbf{g}, \mathbf{a}_\mathbf{g})} = \mathbf{1}/\mathbf{4}\equiv {\varvec{\omega }}_\mathbf{0}}\) (\(B_i\)) 
Bundle frequency for \(r_ +\) or \(r_\):  \({\omega _H^(r_g, a_g)} = 1\) (B)  
Horizon frequency (\(X^{o}\)):  \(\emptyset \)  \(\emptyset \) 
To conclude this analysis, we note that it is possible to find a linear relation between the horizons curves (in the extended plane) by reexpressing the curve of the inner horizon as a function of the outer horizon (on the lines \(a=\) costant) and, viceversa, i.e., \(r_(r_+)\) and \(r_+(r_)\) – Fig. 14.
BH–NS correspondence: onetoone relation
There is a onetoone correspondence between the points of the horizons, the horizon frequencies, and the spins \(a\in [0,\infty [\) for NSs or BHs. BHs are related to a part of the \(r_\) curve, SNSs (\(a>2M\)) correspond to \(r_+\), and WNSs (\(a\in [M,2M]\)) are in correspondence with the envelope surfaces of the inner horizon, i.e., \(a_\in \pi _a^+\) for \(r\in [0.8M,M]\). The limiting cases of Schwarzschild, \(a=0\), and extreme Kerr BH, \(a=M\), are connected with the limit \(a=\infty \) and the double point \(a_0=2M\), respectively.
In fact, we can say that \(\forall {\bar{a}}\in ]0,+\infty [\), \(\exists !\, {\bar{\omega }}_0\equiv M/{\bar{a}}\) and, viceversa, \( \forall {\bar{\omega }}_0\) there is one and only one \(a_0\in ]0,M]: {\bar{\omega }}_0=\omega _H\), where \(\omega _H\) is the horizon frequency, i.e., we connect points of \(\mathcal {P}_\mathcal {S}\) to points of \(\mathcal {P}_\mathcal {H}\) of the horizons by considering that each metric bundle relates univocally an origin \(a_0\) of \(\mathcal {P}_\mathcal {S}\) with a tangent point of \(\mathcal {P}_\mathcal {H}\). (\(a_g\) is solution of \(a_0=\omega _H^{1}\)). Any origin of the metric bundle \(g_{\omega _0}^{\pm }\), associated to a frequency \(\omega _0 =\) constant, is associated to one and only one point \(p_{\pm }:\; p_\in r_\) or the outer \(p_+\in r_+ \), according to the origin \(a_0\).
A special case is the extreme Kerr BH solution, where the origin spin \(a_0=2M\), associated to the metric bundle \(g_{\omega _0}\) of constant frequency \(\omega _0=1/2\), crosses the horizons at \(r_\pm =M\) and \(a_\pm =M\). In general, for a fixed origin \(a_0\), there are two frequencies \(\omega _H\) and \(\omega _0\equiv M/a_0\ne \omega _H\), respectively, where \(\omega _0=M/a_0\) is the frequency of the metric bundle \(g_{\omega _0}\), and \(\omega _H\ne \omega _0\) is the horizon frequency defined by the spin \(a_{\pm }\) defined by the tangent point. \(p_{\pm }\equiv (a_{\pm },r_{\pm })\in \pi _a^{+}\).
The onetoone BHs–NSs correspondence is described by the function \(a_g\) as given in Eq. (17) and illustrated in Fig. 13. We can say that each BH solution is connected to one and only one \({\mathbf {NS}}\) in \(\pi _{a}^+\), as it emerges from the analysis of the Killing horizons and light frequencies on the equatorial plane and, viceversa, each NS is related to one BH. These considerations include the limiting cases of the Kerr extreme spacetime, where the associated metric bundle has origin \(a_0=2M\), and the limiting case of the static Schwarzschild BH, which is connected to the NS with \(a=+\infty \). The BH–NS relation allows us to consider a spin shift from an initial \({\mathbf {BH}}_1\) (\({\mathbf {NS}}_1\)) source as corresponding to a shift of the respective \({\mathbf {NS}}_1\) (\({\mathbf {BH}}_1\)); therefore, the pair \({\mathbf {BH}}_1{\mathbf {NS}}_1\) shifts to the pair \({\mathbf {BH}}_2{\mathbf {NS}}_2\). The segment \(a_{{\mathbf {BH}}_1}a_{{\mathbf {NS}}_1}\) of \(\mathcal {P_S}\in \pi _a^+\) transforms into \(a_{{\mathbf {BH}}_2}a_{{\mathbf {NS}}_2}\), becoming larger or smaller depending on the curve \(a_g\) as shown in Fig. 13.
To conclude this section, we analyze the relation between the tangent radius \(r_g\) on the horizon and the bundle origin \(a_0\). We start with a description of Fig. 13.
Analysis of figure 13

the origin \(a_0^=M\) distinguishes NSs from BHs;

the spin \(a_0=2M\) defines the left region where \(a_0\in [0,2M]\) (for BH and WNS) and right region where \(a_0>2M\) (for \(\mathbf {SNSs}=\mathbf {SNS}^\cup \mathbf {SNS}^+\));

the spin \(a_g^1\equiv a_{g}(a_0^)=0.8M\) defines the upsector where \(a\in [a_g^1,M]\) (for \({\mathbf {BH}}^+\)) and the down sector where \(a\in [0,a_g^1[\) (for \({\mathbf {BH}}^\));

the spin \(a_0^+=4M: \;a_g=a_g^+\) distinguishes strong naked singularities \(\mathbf {SNS}^+\) and \(\mathbf {SNS}^\).

a bundle origin \(a_0\) in the BHregion corresponds through the horizon curve to \({\mathbf {BH}}^\) singularities (\(a_g:\quad a_0\in \mathbf{BH}\mapsto a_g \in \mathbf{BH}^\));

origins \(a_0\) in the WNSregion correspond to \(a_g \in {\mathbf {BH}}^+\).

the origin \(a_0\) in \(\mathbf {SNS}^\) corresponds to \(a_g\in {\mathbf {BH}}^+\);

the origin \(a_0\) in \(\mathbf {SNS}^+\) corresponds to \(a_g\in {\mathbf {BH}}^\).
Because \(\partial _{a_0}a_{g}\ge 0\), increasing the origin spin \(a_0\in \mathbf{BH}\cup \mathbf{WNS}\) and the tangent spin \(a_g\in \mathbf{BH}^\cup \mathbf{BH}^+\) increases the left region. Let us consider the BHs and NSs correspondence determined by the tangent point spin \(a_g\). Note that a fixed origin \(a_0\in \mathbf {SNS}^{+}\) (downright region) corresponds to the outer horizon tangent point (we identify \(a_g(a_0)\in {\mathbf {BH}}\)). This is sufficient for the determination of the spin \(a_0^\in \mathbf{BH}^{}\) and the origin of the bundle metric tangent to the inner horizon in \((a_g,r^_g)\); therefore, this defines the inner horizon \(r^_g\) of the \(\mathbf{BH}^\) spacetime with spin \(a_g\).
Similarly, in the rightup region there are couples following related geometries \((a_g,a_0)\): (1) \(\mathbf {BH^+}\mathbf {WNS}\) and (2) \(\mathbf {BH^+}\mathbf {SNS}^{}\).
We note that the origins \(\mathbf {SNS}^{\pm }\) (\(a_0\)) are in correspondence with a \(a_g\in \mathbf{BH}\); also the \(a_0\in \mathbf {WNS}\) and \(a_0 \in {\mathbf {BH}}\) are in correspondence with a \(a_g\in \mathbf{BH}\).
Therefore, in general there is a correspondence \(a_0\mapsto a_g\in {\mathbf {BH}}\mapsto a^{\prime }_0\), i.e, the origin of the bundle defines a tangent point to the horizon \(a_g\) and \(r_+\) or \(r_\) in the spacetime with \(a_g\); correspondingly, there is the bundle with origin \(a^{\prime }_0\) whose tangent point to the horizon is \(a_g \)and \(r_\) or \(r_+\), respectively.
The black hole area, delimited by the (outer) horizon, is a crucial quantity, determining the thermodynamic properties of BHs. Given the relevance of this concept, we study in Sect. 1 some properties of the areas of the regions delimited by metric bundles and compare them with the horizon area \(\mathcal {A}_{r_{\pm }}^{+}=\pi /2\) in the extended plane \(\pi _a^+\), i.e., the region in the plane \(\pi _a^+\) bounded by the horizon curve \(a_{\pm }\).
To conclude, we show that the horizons frequency \(\omega _H\) defines the bundles frequencies. We will resume part of the discussion carried out in relation with Eq. (17). We also investigate the limiting frequencies obtained from the bundles crossing and some properties of the tangents to the horizons in the extended plane.
On the frequencies
By definition, the frequency \(\omega \) is constant along the bundle; thus, the bundle frequency \(\omega \) is the bundle origin frequency \(\omega _0=M/a_0\) and, particularly, the frequency at \((a_g,r_g)\), where \(a_g\) is the bundle spin at the tangent point \(r_g\) on the horizon curve. We can show that the bundle frequency coincides with the horizon frequency at the tangent point \(r_g\), that is, \(\omega _0=\omega _H(a_g,r_g)\).
First, note that the bundle frequency \(\omega =\omega _0\) is defined in \([0,\infty ]\). The analysis of the frequency variation domain gives us indications that both the inner \(\omega _H^\) and outer horizon frequency \(\omega _H^+\) define the bundle frequency, as there is \(\omega _H^+\in [0,1/2]\) and \(\omega _H^\in [1/2,+\infty [\) (for \(a\in [0,M]\)) – see also Fig. 1. Then, condition \(\omega _0=\omega _H\), where \(\omega _H\) is the union \(\omega _H^+\cup \omega _H^\), leads to \(a_g(a_0)\), relating the tangentpointspin \(a_g\) and the bundle origin \(a_0=\omega _H^{1}(a_g)\).
Figure 15 and Table 3 show some notable numerical examples, proving that the bundle frequencies are, in fact, the horizon frequency at the tangent point to the horizons. Connections between the two bundles \(g_{\omega _1}\) and \(g_{\omega _2}\) at equal \(a_g\), which are related to the horizons points \(r_{\pm }(a_g)\), are also shown in Table 3. The tangent spin \(a_g\) can be also obtained from the relation \(a_{\omega }(r_\pm ,\omega _0)=a\) by solving for \(a_0(a)\) and representing the result as \(a_g(a_0)\) in Eq. (17).
On the tangent lines
The bundlehorizon tangent line at the point \((r_g,a_g)\) is provided by the relation \(\partial _r a_{\omega }^{\pm }=\partial _r a_{\pm } \), where \(a_{\pm }\) is the horizon curve in the extended plane. These solutions are shown in Fig. 16, where some properties of the tangents are highlighted.
It is clear that \(r_g^{real}\) is a combination of \(r_g^{\pm }\) and provides the tangent point \(r_g(a_0)\) for the origin \(a_0\). In fact, by fixing \(a_0\) there is only one tangent point in \(r_g^{real}\) equal to \(r_g^\) for \(a_0<2M\), or else equal to \(r_g^+\) for \(a_0>2M\); however, the second point at \(a_0\) on the curve \(r_g^{\checkmark }\) (shaded region) has no immediate meaning with respect to the bundle \(g_{\omega _0}\); on the other hand, the point \(r_g^{\checkmark }(a_0)\) provides the second horizon (\(r_\) or \(r_+\)) in the spacetime with \(a=a_g\). Therefore, the connecting bundle \(g_{\omega _0}\) with \(g_{\omega _1}\) is tangent to the horizon at \(r_g^{\checkmark }(a_g)\). This case has been also represented in Table 3 with respect to the \({\mathbf {BB}}\) and DD models. Note that the CC model, extreme Kerr spacetime, corresponds to \(a_0=2M\) and \(r_{g}^{real}=r_g^{\checkmark }=M\) (\(r_g^{+}=r_{g}^=M\)).
We return to the analysis of \((r_{g}^{real},r_g^{\checkmark })\) (\(r_g^{\pm }\)) in Fig. 17. Let us consider as an example the BB and DD models. For \(a_0<2M\) the bundles are tangent to the inner horizon (note also the saddle point at \(a_0/M={2}/{\sqrt{3}}\approx 1.1547\)). According to the DD model \(a_0=4M\), the correct tangent point is on \(r_g^{real}=r_g^+\). The second point, for \(a_0=4M\), on the \(r_g^{\checkmark }=r_g^\) is the tangent point \(r_g(B)\) in the BB model with \(a_0=M\). The BB and DD models share same tangent spin \(a_g\).
5 The Kerr–Newman geometries
The investigation of Sects. 3 and 4 is performed here for the case of Kerr–Newman (KN) and Reissner–Nordström (RN) spacetimes and in the region outside the equatorial plane of the Kerr spacetime. This further analysis will allow us to better evaluate the role of the framedragging. The analysis of Fig. 8 is presented in Fig. 20 for electrically charged geometries with \(a=0\). We prove that the closure of the metric bundles is a consequence of the rotation of the singularity: the correspondent curves, defining the BHs horizons for the static RN case, are open; the analysis of the KN case represented in Fig. 20 better shows the influence of the spin in the bending and separation into the two families of closed curves on the equatorial plane.
and are represented in Fig. 19. In this section, we consider the problem faced in the Sects. 3 and 4 and analyze the entire range \(Q^2\ge 0\) and \(a^2\ge 0\). We test the conjectures presented in Sect. 4 and reproduce the analysis of Sect. 3, in particular, for the case of spherical symmetry, when the framedragging due to the source’s spin is absent, isolating the effects of the electric charge from the rotation component of the total charge \(\mathcal {Q}_T\). In doing so, we generalize the extended plane \(\pi _a^+\) used in Sects. 3 and 4, considering a twoparameter family of solutions and passing from the \(\mathbf{(1+1)}\) dimensional problem of the Kerr spacetimes to a \(\mathbf{(1+2)}\) problem of the KN solutions. Fixing one of the two components of the total charge, we can obtain an entire parametrized family of Einstein solutions. The offequatorial case will also be briefly addressed.
This analysis confirms the results of Sects. 3 and 4 and the role of the framedragging (Figs. 24, 25, 26).
6 Concluding remarks and future perspectives
In this work, specially in Sect. 3, we explored the Killing throats and bottlenecks, arising from the properties of stationary observers in the Kerr geometries. In the case of WNSs (\(a/M\in ]1,2]\)), the Killing throats show “restrictions”, which we identify as Killing bottlenecks (Figs. 27, 28). To explore the properties of the bottlenecks, we introduced in Sect. 4 the concept of extended plane, which is a graph relating a particular characteristic of a spacetime in terms of the parameters entering the corresponding spacetime metrics. More precisely, the analysis of some peculiar characteristics of the bottlenecks, defined as horizons remnants, indicated some links between BHs and NSs (and in particular WNSs). To compare the BH and NS characteristics, it was convenient to introduce in Sect. 4 the concept of metric bundles, i.e., curves in the extended plane \(\pi _a^+\), representing a collection of metrics defined by a particular photon orbital frequency, named metric bundle frequency. The bundle frequencies (and the orbital limiting frequencies in any point on the equatorial plane of any spacetime of the family) are all and only the frequencies of the horizon defined in the extended plane. This analysis has been done mainly on the equatorial plane of axisymmetric geometries. Metric bundles show, in fact, the remarkable property to be tangent to the horizon curve in the extended plane, the space where the curves representing the metric bundles are defined. In the case of the the equatorial plane of the Kerr metric, the extended plane is essentially equivalent to the function that relates the frequency with the spin. Notably, the metric bundle associated to the extreme Kerr BH corresponds to a regular curve tangent to the horizon with bundle origin \(a_0=2M\). As proved in Sect. 4, all the metric bundles are tangent to the horizon in the extended plane \(\pi _a^+\) and the horizon emerges as the envelope surface of the metric bundles in \(\pi _a^+\). A bundle frequency corresponds to one limiting photon orbital frequency for all BHs or some BHs and NSs geometries. The bundle frequency coincides with the frequency at the (inner or outer) horizon curve, which is tangent to the metric bundle. On the other hand, the metric bundles are all defined by all and only the frequency of the horizon in \(\pi _a^+\). Viceversa, all the horizon frequencies in the extended plane are metric bundle frequencies. In Sect. 5, we consider the static and electrically charged Reissner–Nordström spacetime and the Kerr–Newman axisymmetric electrovacuum solution and show that the bending (closing) of the curves of the metric bundles is due to the rotation of the gravitational source. Therefore, we can say that the horizons frequencies determine the BHs and NSs limiting photon orbital frequencies. This fact establishes a connection between BHs and NSs: the inner BH horizon is connected to BH and WNS origin bundles, whereas the outer horizon sets the BHs–SNSs correspondence. In the extended plane, NSs are associated with portions of the horizon. In this sense, the inner horizon is partially constructed by BH metric bundles. The inner horizon is associated with WNS origins. This last property turns out to be related to the Killing bottlenecks appearing in the light surfaces. Interestingly, the outer horizon in \(\pi _a^+\) is generated by SNSs metric bundles only. This fact has the interesting consequence that only the horizon frequencies determine the frequencies \(\omega _{\pm }\) at each point, r, on the equatorial plane of a Kerr BH or NS geometry: all the frequencies \(\omega _{\pm }(r)\) on the equatorial plane are only those of the horizon in \(\pi _a^+\), the horizon in \(\pi _a^+\) contains information about all limiting photon orbits also in NS spacetimes. In Secs. 3 and 4, we have also introduced the concept of inner horizon confinement. In this sense, NSs are “necessary” for the construction of horizons. The outer horizon is associated with a NS (the bundle origin) in the extended plane and the inner horizon to a WNS or a BH. Therefore, we believe that this result could be of interest for the investigation of gravitational collapses in which connections between NS and BH solutions are expected and emerge [58, 59, 60].
Some further aspects of these properties are currently under investigation. Firstly, it would be necessary to further analyze the offequatorial case and test the results of Sect. 4 in other kinds of geometries. In a future analysis, we shall analyze other axisymmetric spacetimes admitting Killing horizons and consider the possible thermodynamic implications of the results discussed here, particularly, in relation to the possibility of formulating the BH thermodynamic laws in terms of metric bundles. We also point out that metric bundles and horizons remnants seem to be related to the concept of prehorizon regimes. There is a prehorizon regime in the spacetime when there are mechanical effects allowing circular orbit observers, which can recognize the close presence of an event horizon. This concept was introduced in [1] and detailed for the Kerr geometry in [61, 62]. The prehorizon was identified and analyzed in [2], leading to the conclusion that a gyroscope would observe a memory of the static case in Kerr metric. Clearly, this aspect could be of relevance during the gravitational collapse [23, 63, 64, 65].
 (i)

We identified bottlenecks essentially as horizon remnants. Similar interpretations have been presented in [1, 2, 61, 62], by using the concept of prehorizons, and in [34], by analyzing the socalled whale diagrams. These structures could play an important role for describing the formation of black holes and for testing the possible existence of naked singularities. Notably, the concept of remnants, as expressed here, refers to and evokes a sort of spacetime “plasticity”, which naturally led to the introduction in Sect. 4 of the concept of the extended plane. In this new frame, we found several properties emerging from and affecting the spacetime geometries when we consider an entire family of solutions as a unique geometric object.
 (ii)

Considering the Kerr family as a single object, the geometric quantities (for example, the horizons) defined for a single solution acquire a completely different significance when considered for the entire family.
 (iii)

Considering metric bundles with WNS, we found that the inner horizon is confined in the metric bundle framework.
 (iv)

We proved the existence of a connection between black holes and naked singularities. To each BH corresponds the pair (WNS, SNS) or the pair (BH, SNS). This correspondence is important for the definition of the Killing horizons.
 (v)

We proved that WNSs (SNSs) are necessary for the construction of the inner (outer) Killing horizon. This result could shed light on the physical meaning of NSs solutions.

Analysis of Killing throats and definition Killing bottlenecks for particular naked singularities – Sect. 3. To define Killing throats, we study the limiting photons surfaces \(r_{s}^{\pm }\) and frequencies \(\omega _{\pm }\) which are defined in Eq. (11) and Eq. (8), respectively. We also interpret bottlenecks as horizons remnants in weak naked singularities.

Inner horizon confinement – Sect. 3. In Eq. (13), we identify the photon orbits characterized by the horizons orbital frequencies. The radii \(r^{\mp }_{\mp }\) represent the set of photon orbits with frequencies \(\omega _H^{\pm }\) at the \(\mathbf{BH}\) horizons – Eq. (13) – Fig. 1. The inner horizon confinement, according to the constraints on \(r^{}_{}\), is in agreement with the confinement of the metric bundles containing BHs and WNSs (see Sect. 4).

Definition of the extended plane \(\pi _a\) and metric bundles \(g_{\omega }\) in Sect. 4. Metric bundles are curves in \(\pi _a\) tangent to the horizons characterized by the limiting photon frequency \(\omega _+\) or \(\omega _\).

Tangency condition and horizon construction in the extended plane. We demonstrate that the horizon curve corresponds to the envelope surface of the metric bundles. All the metric bundles are tangent to the horizon curve and all the points of the horizons are associated to one and only one metric bundle tangent to that point – Figs. 14, 15, 16 and Tables 2 and 3. Consequently, all the limiting photon orbital frequencies (on the equatorial plane) are all and only the frequencies of the horizon curve in \(\pi _a\), related to two metric bundles, with frequencies \(\omega _0\) and \(\varpi _\pm \), respectively. Particularly, through the relationship between metric bundles and horizons in \(\pi _a^+\), a NS or BH metric can be parameterized in terms of the horizon frequency identified by the corresponding metric bundle. The inner and outer horizons in \(\pi _a^+\) correspond to envelope surfaces. We analyze the lines \(a=\) constant (i.e. a single spacetime of the metric bundle) and \(r=\) constant in the extended plane – Fig. 8 and classify the singularities according to the horizon construction as shown in Fig. 13, i.e., strong naked singularities, \(\mathbf{SNS}=\mathbf{SNS}^+\cup \mathbf{SNS}^\), having \(a_0>2M\) with \(\mathbf{SNS}^+\) for \(a_0>4M\) and \(\mathbf{SNS}^\) for \(a_0\in [2M,4M[\); WNS – weak naked singularities with \(a_0\in ]M,2M[\) and \(\mathbf{BH}=\mathbf{BH}^+\cup \mathbf{BH}^\) with \(\mathbf{BH}^+\) for \(a\in [a_{g}^1,M]\), \(a_{g}^1=3/4 M\) and \(\mathbf{BH}^\) for \(a\in [0,a_{g_1}]\).

Demonstration of the confinement of metric bundles with origin in BHs and WNSs and identification and study of the corresponding bundles. In Table 2, we proved the confinement of the bundles in the extended plane delimited by the inner horizon curve. The horizon and bundle frequencies are related by the relation \(\omega _0^+\omega _0^=1/4\).

Properties of horizons and bundles. For the entire family of Kerr geometries, we established the relations between metric bundles and horizons. We found two relations which can be specified in detail as follows. Relations I: \(\omega _0^{1} \equiv a_0^{\pm }/M=({2 r_{\pm }(a_g)}/{a_g})\equiv \omega _H^{1}(a_g)\), \(\omega _H^+(r_g,a_g)=\omega _0=Ma_0^{1}\), \(\omega _H^(r^{\prime }_g,a_g)=\omega ^{\prime }_0=M/a_0^{\prime }\) where \(r^{\prime }_g\in r_\) (\(r_+=r_g\), \(r_=r_g^{\prime }\)) – Fig. 13. Relation II: \(\omega ^{\prime }_0=({4 \omega _0})^{1}\), \( \omega _H^+\omega _H^={1}/{4}\), (or equivalently \(a_0^+(a_g)a_0^(a_g)=4M^2\)), \(a_0^{\pm }/M=(2 r_{\pm }(a_g))/{a_g}\) where \(a=a_0\) and \(a=a_p\) – Fig. 14.

Properties of specific spacetimes. We studied the metric bundles corresponding to a single BH spacetime (with equal tangent spin \(a_g\)). We analyzed the Kerr spacetime in terms of metric bundles – Sect. 4.

Proof of the BH–NS relation through the properties of the corresponding metric bundles.

Analysis of the frequencies \(\varpi _\pm \) of the spacetime by using the maximum crossing of two metric bundles – Eq. (22). We proved he existence of the corresponding metric bundles and analyzed its significance for the horizon construction and properties of a BH spacetime – Tables 2 and 3; Figs. 13, 14, 15, 16,17 and Eqs. (19) and (27).

Identification of the Killing throats and bottlenecks and metric bundles in the static and charged Reissner Nordström solution and in the axisymmetric, electrically charged, Kerr–Newmann spacetime. We proved that the bending (curvature) of the Kerr metric bundles in \(\pi _a\) is related to the framedragging of the spinning spacetimes. We noticed the different roles played by the electric and rotational charges – Eq. (28). We studied the metric bundles \( (Q_{\omega }^{\pm })^2\) in terms of the electric charge Q – Eq. (35). In Sect. 1, we analyzed the offequatorial case of the Kerr and Kerr–Newman geometries. In Sect. 1, we considered the relations between the areas of the horizon and of the metric bundles regions in the extended plane.
Footnotes
 1.
We use geometrical units with \(c=1=G\) and the signature \((,+,+,+)\), Greek indices run in \(\{0,1,2,3\}\). The fourvelocity satisfies the condition \(u^\alpha u_\alpha =1\). The radius r has units of mass [M], and the angular momentum units of \([M]^2\), the velocities \([u^t]=[u^r]=1\) and \([u^{\phi }]=[u^{\theta }]=[M]^{1}\) with \([u^{\phi }/u^{t}]=[M]^{1}\) and \([u_{\phi }/u_{t}]=[M]\). For the sake of convenience, we consider a dimensionless energy and an angular momentum per unit of mass \([L]/[M]=[M]\).
 2.
This equation corrects a typo in Eq. (8) of Ref. [18].
 3.
 4.
Note that the Killing bottleneck and Killing throat inherit some of the properties of the Killing vectors, particularly, regarding conformal transformations of the metric or vectors. Consider a Killing throat defined by two Killing vectors \((\xi _i,\xi _j)\). The linear combination \(a_\alpha \xi ^\alpha \) defines a Killing vector and we can define a Killing field up to a conformal transformation. In other words, \({\mathcal {L}}=\xi _t+a_{\phi }\xi _\phi \) identifies a Killing throat up to a conformal transformation. The simplest case is when one considers a conformal expanded (or contracted) spacetime where \(\tilde{\mathbf {\xi }}^2\equiv \tilde{\mathbf {g}}(L,L)=\Xi ^2\mathbf {g}(L,L)\). This holds also for a conformal expanded Killing tensor \(\tilde{{\mathcal {L}}}\equiv \Xi {\mathcal {L}}\).
 5.
The definition of metric bundle, tangent to the horizon in the extended plane presented in this work, should not be confused with the definition of bundle metric and of the tangent bundle of a differentiable manifold in differential geometry. A metric on a vector bundle is a choice of smoothly varying inner products on the fibers. While the a tangent bundle TM could be defined as the disjoint union of tangent vectors in M. Although different in their definitions, it is clear that the concepts introduced in this analysis can be read in terms of tangent bundles in a differential manifold, providing a deep insight on the properties considered here. While it is not the goal of this work, it is worth noting that we may consider the horizon as a onedimensional surface embedded in the extended plane considered as \(R^2\) (including the reflection \(\pi _a^\)). For a general and smooth curve c(r) in \(R^2\), the associated tangent bundle may be seen as a regular surface in \(R^4\), written as \(T(r, \epsilon )=[c(r), \epsilon c'(r)]\) with \( \epsilon \in [1,1]\). The metric bundles identify at every point on the horizon a tangent vector. We study the tangent to the horizons and the metric bundles ending Sect. 4.
Notes
Acknowledgements
D.P. acknowledges partial support from the Junior GACR grant of the Czech Science Foundation no:1603564Y. D.P. is grateful to Donato Bini, Fernando de Felice and Andrea Geralico for discussing many aspects of this work. This work was partially supported by UNAMDGAPAPAPIIT, Grant no. 111617, and by the Ministry of Education and Science of RK, Grant no. BR05236322 and AP05133630.
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