#### Observers in Kerr spacetimes: the ergoregion on the equatorial plane

Eur. Phys. J. C
Observers in Kerr spacetimes: the ergoregion on the equatorial plane
D. Pugliese 2
H. Quevedo 0 1 3
0 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México , AP 70543, 04510 Mexico, DF , Mexico
1 Dipartimento di Fisica, Università di Roma “La Sapienza” , 00185 Rome , Italy
2 Faculty of Philosophy and Science, Institute of Physics, Silesian University in Opava , Bezrucˇovo námeˇstí 13, 74601 Opava , Czech Republic
3 Department of Theoretical and Nuclear Physics, Kazakh National University , Almaty 050040 , Kazakhstan
We perform a detailed analysis of the properties of stationary observers located on the equatorial plane of the ergosphere in a Kerr spacetime, including light-surfaces. This study highlights crucial differences between black hole and the super-spinner sources. In the case of Kerr naked singularities, the results allow us to distinguish between “weak” and “strong ” singularities, corresponding to spin values close to or distant from the limiting case of extreme black holes, respectively. We derive important limiting angular frequencies for naked singularities. We especially study very weak singularities as resulting from the spin variation of black holes. We also explore the main properties of zero angular momentum observers for different classes of black hole and naked singularity spacetimes.
1 Introduction
The physics of black holes (BHs) is probably one of the most
complex and still controversial aspects of Einstein’s
geometric theory of gravitation. Many processes of High Energy
Astrophysics are supposed to involve singularities and their
formation from a stellar progenitor collapse or from the
merging of a binary BH system. The interaction of these sources
with the matter environment, which can lead to accretion and
jets emission, is the basis for many observed phenomena. As
a consequence of this interaction, the singularity properties,
determined generally by the values of their intrinsic spin,
mass or electric charge parameters, might be modified,
leading to considerable changes of the singularity itself. In this
work, we concentrate our analysis on the ergoregion in the
naked singularity (NS) and BH regimes of the axisymmetric
and stationary Kerr solution. We are concerned also about
the implications of any spin-mass ratio oscillation between
the BH and the NS regimes from the viewpoint of
stationary observers and their frequencies, assuming the invariance
of the system symmetries (axial symmetry and time
independence). One of the goals of this work is to explore the
existence of spin transitions in very weak naked singularities
[
2
], which are characterized by a spin parameter a/M ≈ 1. If
the collapse of a stellar object or the merging of several stellar
or BH attractors lead to the formation of a naked singularity,
then a total or partial destruction of the horizon may occur
which should be accompanied by oscillations of the
spinto-mass ratio. Naked singularities can also appear in
nonisolated BH configurations as the result of their interaction
with the surrounding matter, i. e., in some transient process
of the evolution of an 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 the spacetime symmetries. The details of
such spin transitions, leading possibly to the destruction of
the horizon, and their consequences are still an open problem.
In this work, keeping the Kerr spacetime symmetries
unchanged, we focus on the variation of the
dimensionless spin parameter in the region within the static limit on
the equatorial plane of the attractor, this being the plane
of symmetry of the Kerr solution. This special plane of the
axisymmetric geometry has many interesting properties; for
instance, constants of motion emerge due to the
symmetry under reflection with respect to this plane; the geometry
has some peculiarities that make it immediately comparable
with the limiting static Schwarzschild solution, in particular,
the location of the outer ergoregion boundary is
independent of the spin value, and coincides with the location of
the Schwarzschild horizon. There is also a clear
astrophysical interest in the exploration of such a plane, as the large
majority of accretion disks are considered to be located on
the equatorial plane of their attractors.
From a methodological viewpoint, our analysis represents
a comparative study of stationary and static observers in Kerr
spacetimes for any range of the spin parameter. The findings
in this work highlight major differences between the
behavior of these observers in BH and NS geometries. These issues
are clearly related to the most general and widely discussed
problem of defining BHs, their event horizon and their
intrinsic thermodynamic properties [
3–8
]. Further, it seems
compelling to clarify the role of the static limit and of the
ergoregion in some of the well-analyzed astrophysical processes
such as the singularity formation, through the gravitational
collapse of a stellar “progenitor” or the merging of two BHs.
Similarly, it is interesting to analyze the role of the
framedragging effect in driving the accretion processes. In fact,
the ergosphere plays an important role in the energetics of
rotating black holes.
The dynamics inside the ergoregion is relevant in
Astrophysics for possible observational effects, since in this region
the Hawking radiation can be analyzed and the Penrose
energy extraction process occurs [
9–13
].1 For the actual state
of the Penrose process, see [
15
]. Another interesting effect
connected directly to the ergoregion is discussed in [
16
]. The
mechanism, by which energy from compact spinning objects
is extracted, is of great astrophysical interest and the effects
occurring inside the ergoregion of black holes are essential
for understanding the central engine mechanism of these
processes [
17,18
]. Accreting matter can even get out, giving rise,
for example, to jets of matter or radiation [
17,19
] originated
inside the ergoregion. Another possibility is the extraction
of energy from a rotating black hole through the Blandford–
Znajek mechanism (see, for instance, [
20–29
]). An
interesting alternative scenario for the role of the Blandford–Znajek
process in the acceleration of jets is presented in [30]. Further
discussions on the Penrose and Blandford–Znajek processes
may be found in [
31,32
]. In general, using orbits entering
the ergosphere, energy can be extracted from a Kerr black
hole or a naked singularity. On the other hand, naked
singularity solutions have been studied in different contexts in
1 The Hawking process is essentially due to the vacuum fluctuation
happening in the regions close to the BH horizon; it is not related to
the properties of the ergoregion itself. The Hawking radiation is the
(spontaneous) emission of thermal radiation which is created in the
vacuum regions surrounding a BH, and leads to a decrease of the mass.
Connected in many ways to the Unruh effects, it generally leads to the
production of pairs of particles, one escaping to infinity while the other
is trapped by the BH horizon. On the other hand, the Penrose energy
extraction, or its wave-analogue of super-radiance, is related essentially
to a classical (i.e. non quantum) phenomenon occurring in the
ergoregion, ]r+, rε+[, due to the frame-dragging of the spinning spacetime. In
this way, energy can be extracted from the source, lowering its angular
momentum. For a study of the Hawking radiation in Kerr and Kerr–
Newman spacetimes see also [14].
[
33–42,44,45
]. Kerr naked singularities as particle
accelerators are considered in [43] – see also [
44,45
]. More
generally, Kerr naked singularities can be relevant in connection
to superspinars, as discussed in [44]. The stability of Kerr
superspinars has been analyzed quite recently in [
46
],
assenting the importance of boundary conditions in dealing with
perturbations of NSs.
An interesting perspective exploring duality between
elementary particles and black holes, pursuing quantum black
holes as the link between microphysics and macrophysics,
can be found in [
47–50
] – see also [51]. A general discussion
on the similarities between characteristic parameter values
of BHs and NSs, in comparison with particle like objects, is
addressed also in [
52–55
]. Quantum evaporation of NSs was
analyzed in [56], radiation in [
57
], and gravitational radiation
in [
58–60
].
Creation and stability of naked singularities are still
intensively debated [
61–66
]. A discussion on the ergoregion
stability can be found in [
67,68
]. However, under quite general
conditions on the progenitor, these analysis do not exclude
the possibility that considering instability processes a naked
singularity can be produced as the result of a gravitational
collapse. These studies, based upon a numerical integration of
the corresponding field equations, often consider the
stability of the progenitor models and investigate the gravitational
collapse of differentially rotating neutron stars in full
general relativity [
69
]. Black hole formation is then associated
with the formation of trapped surfaces. As a consequence
of this, a singularity without trapped surfaces, as the result
of a numerical integration, is usually considered as a proof
of its naked singularity nature. However, the non existence
of trapped surfaces after or during the gravitational collapse
is not in general a proof of the existence of a naked
singularity. As shown in [
70
], in fact, it is possible to choose a
very particular slicing of spacetime during the formation of
a spherically symmetric black hole where no trapped surfaces
exist (see also [
71
]). Eventually, the process of gravitational
collapse towards the formation of BHs (and therefore, more
generally, the issues concerning the formation or not of a
horizon and hence of NSs) is still, in spite of several
studies, an open problem. There are transition periods of transient
dynamics, possibly involving topological deformations of the
spacetime, in which we know the past and future asymptotic
regions of the spacetime, but it is still in fact largely unclear
what happens during that process. The problem is wide and
involves many factors as, especially in non-isolated systems,
the role of matter and symmetries during collapse. Another
major process that leads to black hole formation is the
merging of two (or more) black holes, recently detected for the first
time in the gravitational waves sector [
72
]. See also [
73,74
]
for the first observation of the probable formation of a BH
from the coalescence of two neutron stars. An interesting and
detailed analysis of Kerr and Kerr–Newman naked
singularities in the broader context of braneworld Kerr–Newman
(BKN) spacetimes can be found in [75], where a new kind of
instability, called mining instability, of some B-KN naked
singularity spacetimes was found. In there, the exploration
of the “causality violation region” is also faced. This is the
region where the angular coordinate becomes timelike,
leading eventually to closed timelike curves. Details on the
relation between this region and the Kerr ergoregion can be found
in the aforementioned reference.
In [
2,52–55,76
], we focused on the study of
axisymmetric gravitational fields, exploring different aspects of
spacetimes with NSs and BHs. The results of this analysis show a
clear difference between naked singularities and black holes
from the point of view of the stability properties of circular
orbits.2 This fact would have significant consequences for the
extended matter surrounding the central source and, hence,
in all processes associated with energy extraction. Indeed,
imagine an accretion disk made of test particles which are
moving along circular orbits on the equatorial plane of a
Kerr spacetime. It turns out that in the case of a black hole
the accretion disk is continuous whereas in the case of a
naked singularity it is discontinuous. This means that we can
determine the values of intrinsic physical parameters of the
central attractor by analyzing the geometric and topological
properties of the corresponding Keplerian accretion disk. In
addition, these disconnected regions, in the case of a naked
singularities, are a consequence of the repulsive gravity
properties found also in many other black hole solutions and in
some extensions or modifications of Einstein’s theory. The
effects of repulsive gravity in the case of the Kerr
geometry were considered in [80] and [
81
]. Analogies between
the effects of repulsive gravity and the presence of a
cosmological constant was shown also to occur in regular black
hole spacetimes or in strong gravity objects without horizons
[
82,83
].
Several studies have already shown that it is necessary
to distinguish between weak (a/M ≈ 1) and strong naked
singularities (a/M >> 1). It is also possible to introduce a
similar classification for black holes; however, we prove here
that only in the case of naked singularities there are obvious
fundamental distinctions between these classes which are not
present among the different black hole classes. Our focus is
on strong BHs, and weak and very weak NSs. This analysis
confirms the distinction between strong and weak NSs and
BHs, characterized by peculiar limiting values for the spin
parameters. Nevertheless, the existence and meaning of such
limits is still largely unclear, and more investigation is due.
2 Test particle motion can be used to determine the topological
properties of general relativistic spacetimes [
77–79
]. Moreover, we proved
that in certain NS geometries different regions of stable timelike circular
orbits are separated from each other by empty regions; this means that
an accretion disk made of test particles will show a particular ring-like
structure with specific topological properties.
However, there are indications about the existence of such
limits in different geometries, where weak and strong
singularities could appear. In [
2,52–55
], it was established that the
motion of test particles on the equatorial plane of black hole
spacetimes can be used to derive information about the
structure of the central source of gravitation; moreover, typical
effects of repulsive gravity were observed in the naked
singularity ergoregion (see also [
34,84–86
]). In addition, it was
pointed out that there exists a dramatic difference between
black holes and naked singularities with respect to the zero
and negative energy states in circular orbits (stable
circular geodesics with negative energy were for the first time
discussed in [87]). The static limit would act indeed as a
semi-permeable membrane separating the spacetime region,
filled with negative energy particles, from the external one,
filled with positive energy particles, gathered from infinity
or expelled from the ergoregion with impoverishment of the
source energy. The membrane is selective because it acts so
as to filter the material in transient between the inner region
and outside the static limit. This membrane wraps and
selectively isolates the horizon in Kerr black holes and the
singularity in superspinning solutions, partially isolating it from
the outer region by letting selectively rotating infalling or
outgoing matter to cross the static limit. As mentioned above,
the ergoregion is involved in the BH spin-up and spin-down
processes leading to a radical change of the dynamical
structure of the region closest to the source and, therefore,
potentially could give rise to detectable effects. It is possible that,
during the evolutionary phases of the rotating object
interacting with the orbiting matter, there can be some evolutionary
stages of spin adjustment, for example, in the proximity of
the extreme value (a M ) where the speculated spin-down
of the BH can occur preventing the formation of a naked
singularity with a M (see also [
40,63,65,88–95
]). The
study of extended matter configurations in the Kerr
ergoregion is faced for example in [2,96]. In [96–100], a model of
multi-accretion disks, so called ringed accretion disks, both
corotating and counterrotating on the equatorial plane of a
Kerr BH, has been proposed, and a model for such ringed
accretion disks was developed. Matter can eventually be
captured by the accretion disk, increasing or removing part of its
energy and angular momentum, therefore prompting a shift
of its spin [
64,87,101–104
]. A further remarkable aspect of
this region is that the outer boundary on the equatorial plane
of the central singularity is invariant for every spin change,
and coincides with the radius of the horizon of the static
case. In the limit of zero rotation, the outer ergosurface
coalesces with the event horizon. The extension of this region
increases with the spin-to-mass ratio, but the outer limit is
invariant. Although on the equatorial plane the ergoregion
is invariant with respect to any transformation involving a
change in the source spin (but not with respect to a change
in the mass M ), the dynamical structure of the ergoregion
is not invariant with respect to a change in the spin-to-mass
ratio. Nevertheless, concerning the invariance of this region
with respect to spin shifts it has been argued, for example
in [105], that the ergoregion cannot indeed disappear as a
consequence of a change in spin, because it may be filled
by negative energy matter provided by the emergence of a
Penrose process3 [13]. The presence of negative energy
particles, a distinctive feature of the ergoregion of any spinning
source in any range of the spin value, has special
properties when it comes to the circular motion in weakly rotating
naked singularities. The presence of this special matter in
an “antigravity” sphere, possibly filled with negative energy
formed according to the Penrose process, and bounded by
orbits with zero angular momentum, is expected to play an
important role in the source evolution. In this work, we clarify
and deepen those results, formulate in detail those
considerations, analyze the static limit, and perform a detailed study
of this region from the point of view of stationary observers.
In this regards, we mention also the interesting and recent
results published in [106] and [
107
].
In detail, this article is organized as follows: in Sect. 2
we discuss the main properties of the Kerr solution and the
features of the ergoregion in the equatorial plane of the Kerr
spacetimes. Concepts and notation used throughout this work
are also introduced. Stationary observers in BH and NS
geometries are introduced in Sect. 3. Then, in Sect. 4, we
investigate the case of zero angular momentum observers
and find all the spacetime configurations in which they can
exist. Finally, in Sect. 5, we discuss our results.
2 Ergoregion properties in the Kerr spacetime
The Kerr metric is an axisymmetric, stationary (nonstatic),
asymptotically flat exact solution of Einstein’s equations in
vacuum. In spheroidal-like Boyer–Lindquist (BL)
coordinates, the line element can be written as
ds2 = −dt 2 + ρΔ2 dr 2 + ρ2dθ 2 + (r 2 + a2) sin2 θ dφ2
2M
+ ρ2 r (dt − a sin2 θ dφ)2,
Δ ≡ r 2 − 2Mr + a2, and ρ2 ≡ r 2 + a2 cos2 θ .
The parameter M ≥ 0 is interpreted as the mass
parameter, while the rotation parameter a ≡ J /M ≥ 0 (spin) is
the specific angular momentum, and J is the total angular
momentum of the gravitational source. The spherically
sym3 We note that the wave analog of the Penrose process is the superradiant
scattering.
metric (static) Schwarzschild solution is a limiting case for
a = 0.
A Kerr black hole (BH) geometry is defined by the range
of the spin-mass ratio a/M ∈]0, 1[, the extreme black hole
case corresponds to a = M , whereas a super-spinner Kerr
compact object or a naked singularity (NS) geometry occurs
when a/M > 1.
The Kerr solution has several symmetry properties. The
Kerr metric tensor (1) is invariant under the application of
any two different transformations: PQ : Q → −Q, where
Q is one of the coordinates (t, φ) or the metric parameter
a while a single transformation leads to a spacetime with
an opposite rotation with respect to the unchanged metric.
The metric element is independent of the coordinate t and
the angular coordinate φ. The solution is stationary due to
the presence of the Killing field ξt = ∂t and the geometry
is axisymmetric as shown by the presence of the rotational
Killing field ξφ = ∂φ .
An observer orbiting, with uniform angular velocity, along
the curves r =constant and θ =constant will not see the
spacetime changing during its motion. As a consequence of
this, the covariant components pφ and pt of the particle
fourmomentum are conserved along the geodesics4 and we can
introduce the constants of motion
E ≡ −gαβ ξtα pβ ,
L ≡ gαβ ξφα pβ .
The constant of motion (along geodesics) L is interpreted
as the angular momentum of the particle as measured by an
observer at infinity, and we may interpret E , for timelike
geodesics, as the total energy of a test particle coming from
radial infinity, as measured by a static observer located at
infinity.
As a consequence of the metric tensor symmetry under
reflection with respect to the equatorial hyperplane θ = π/2,
the equatorial (circular) trajectories are confined in the
equatorial geodesic plane. Several remarkable surfaces
characterize these geometries: for black hole and extreme black hole
spacetimes the radii
r± ≡ M ±
M 2 − a2 : grr = 0
(3)
(4)
(1)
(2)
4 We adopt the geometrical units c = 1 = G and the signature
(−, +, +, +), Greek indices run in {0, 1, 2, 3}. The four-velocity
satisfy uαuα = −1. The radius r has units of mass [M], and the
angular momentum units of [M]2, the velocities [ut ] = [ur ] = 1 and
[uφ] = [uθ ] = [M]−1 with [uφ/ut ] = [M]−1 and [uφ/ut ] = [M]. For
the sake of convenience, we always consider a dimensionless energy
and effective potential [Vef f ] = 1 and an angular momentum per unit
of mass [L]/[M] = [M].
are the event outer and inner (Killing) horizons,5 whereas
rε± ≡ M ±
M 2 − a2 cos2 θ : gtt = 0
(5)
are the outer and inner ergosurfaces, respectively,6 with rε− ≤
r− ≤ r+ ≤ rε+. In an extreme BH geometry, the horizons
coincide, r− = r+ = M , and the relation rε± = r± is valid
on the rotational axis (i.e., when cos2 θ = 1).
In this work, we will deal particularly with the geometric
properties of the ergoregion Σε+ : ]r+, rε+]; in this region, we
have that gtt > 0 on the equatorial plane (θ = π/2) and also
ε π/2 = r+|a=0 = 2M and rε− = 0. The outer boundary
r +
rε+ is known as the static (or also stationary) limit [
108
];
it is a timelike surface except on the axis of the Kerr source
where it matches the outer horizon and becomes null-like. On
the equatorial plane of symmetry, ρ = r and the spacetime
singularity is located at r = 0. In the naked singularity case,
where the singularity at ρ = 0 is not covered by a horizon, the
region Σε+ has a toroidal topology centered on the axis with
the inner circle located on the singularity. On the equatorial
plane, as a → 0 the geometry “smoothly” resembles the
spherical symmetric case, r+ ≡ r +
ε 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, goes to zero.
In general, for a = 0 and r ∈ Σε+, the metric component
gtt changes its sign and vanishes for r = rε+ (and cos2 θ ∈
]0, 1]). In the ergoregion, the Killing vector ξtα = (1, 0, 0, 0)
becomes spacelike, i.e., gαβ ξtα ξtβ = gtt > 0. As the quantity
E , introduced in Eq. (3), is associated to the Killing field ξt =
∂t , then the particle energy can be also negative inside Σε+.
For stationary spacetimes (a = 0) in Σε+, the motion with
φ = const is not possible and all particles are forced to rotate
with the source, i.e., φ˙ a > 0. This fact implies in particular
5 A Killing horizon is a null surface, S0, whose null generators coincide
with the orbits of an one-parameter group of isometries (i. e., there is
a Killing field L which is normal to S0). Therefore, it is a lightlike
hypersurface (generated by the flow of a Killing vector) on which the
norm of a Killing vector goes to zero. In static BH spacetimes, the
event, apparent, and Killing horizons with respect to the Killing field
ξt coincide. In the Schwarzschild spacetime, therefore, r = 2M is
the Killing horizon with respect to the Killing vector ∂t . The event
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
velocity of the horizon. In this article we shall extensively discuss this
special vector in the case of NS geometries. 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,
S G K err = (r+ − r−)/2(r +2 + a2), is a conformal invariant of the
metric, but it rescales with the conformal Killing vector. Therefore, it
is not the same on all generators (but obviously it is constant along one
specific generator because of the symmetries).
6 In the Kerr solution, the Killing vector ∂t , representing time
translations at infinity, becomes null at the outer boundary of the
ergoregion, rε+, which is however a timelike surface; therefore, rε+ is not a
Killing horizon. More precisely, on the ergosurfaces the time
translational Killing vector becomes null.
that an observer with four-velocity proportional to ξtα so that
θ˙ = r˙ = φ˙ = 0, (the dot denotes the derivative with respect
to the proper time τ along the trajectory), cannot exist inside
the ergoregion. Therefore, for any infalling matter (timelike
or photonlike) approaching the horizon r+ in the region Σε+,
it holds that t → ∞ and φ → ∞, implying that the
worldlines around the horizon, as long as a = 0, are subjected
to an infinite twisting. On the other hand, trajectories with
r = const and r˙ > 0 (particles crossing the static limit and
escaping outside in the region r ≥ rε+) are possible.
Concerning the frequency of a signal emitted by a source
in motion along the boundary of the ergoregion rε+, it is clear
that the proper time of the source particle is not null.7 Then,
for an observer at infinity, the particle will reach and
penetrate the surface r = rε+, in general, in a finite time t . For this
reason, the ergoregion boundary is not a surface of infinite
redshift, except for the axis of rotation where the ergoregion
coincides with the event horizon [
2, 109
]. This means that an
observer at infinity will see a non-zero emission frequency.
In the spherical symmetric case (a = 0), however, as gtφ = 0
the proper time interval dτ = √|gtt |dt goes to zero as one
approaches r = r+ = rε+. For a timelike particle with
positive energy (as measured by an observer at infinity), it is
possible to cross the static limit and to escape towards
infinity. In Sect. 3, we introduce stationary observers in BH and
NS geometries. We find the explicit expression for the
angular velocity of stationary observers, and perform a detailed
analysis of its behavior in terms of the radial distance to the
source and of the angular momentum of the gravity source.
We find all the conditions that must be satisfied for a
lightsurface to exist.
3 Stationary observers and light surfaces
We start our analysis by considering stationary observers
which are defined as observers whose tangent vector is a
spacetime Killing vector; their four-velocity is therefore a
linear combination of the two Killing vectors ξφ and ξt , i.e.,
the coordinates r and θ are constants along the worldline of a
stationary observer [
110
]. As a consequence of this property,
a stationary observer does not see the spacetime changing
along its trajectory. It is convenient to introduce the (uniform)
angular velocity ω as
dφ/dt = uφ /ut ≡ ω,
or uα = γ (ξtα + ωξφα ),
which is a dimensionless quantity. Here, γ is a normalization
factor
γ −2 ≡ −κ (ω2 gφφ + 2ωgtφ + gtt ),
(6)
(7)
7 However, since gtt (rε±) = 0, it is also known as an infinity redshift
surface; see, for example, [
108
].
where gαβ uα uβ = −κ . The particular case ω = 0 defines
static observers; these observers cannot exist in the
ergoregion.
The angular velocity of a timelike stationary observer (κ =
+1) is defined within the interval
ω ∈]ω−, ω+[
where ω± ≡ ωZ ±
ω∗2 ≡ ggφtφt
gtt
= gφφ ,
gφt ,
ωZ ≡ − gφφ
ω2Z − ω∗2,
as illustrated in Figs. 1 and 2-right, where the frequencies
ω± are plotted for fixed values of r/M and as functions of
the spacetime spin a/M and radius r/M , respectively. In
particular, the combination
L± ≡ ξt + ω±ξφ
defines null curves, gαβ L±α L±β = 0, and, therefore, as we
shall see in detail below, the frequencies ω± are limiting
angular velocities for physical observers, defining a family
of null curves, rotating with the velocity ω± around the axis
of symmetry. The Killing vectors L± are also generators of
Killing event horizons. The Killing vector ξt + ωξφ becomes
(8)
(9)
null at r = r+. At the horizon ω+ = ω− and, consequently,
stationary observers cannot exist inside this surface.
3.1 The frequencies ω±
We are concerned here with the orbits r = const and ω =
const, which are eligible for stationary observers. This
analysis enlightens the differences between NS and BH
spacetimes. Inside the ergoregion, the quantity in parenthesis in
the r.h.s. of Eq. (7) is well defined for any source. However,
it becomes null for photon-like particles and the rotational
frequencies ω±. On the equatorial plane, the frequencies ω±
are given as
ω± ≡ 2ra3 M+2a±2(2MM + r )
√r 2Δ
with
and
a
ω±(r+) = ωZ (r+) = ωh ≡ 2r+
rl→im∞ ω± = 0,
rl→im0 ω± = ω0 ≡
M
≡ 2ω0r+
M
.
a
Moreover, for the case of very strong naked singularities
a M , we obtain that ω± → 0.
The above quantities are closely related to the main black
hole characteristics, and determine also the main features that
distinguish NS solutions from BH solutions. The constant ωh
plays a crucial role for the characterization of black holes,
including their thermodynamic properties. It also determines
the uniform (rigid) angular velocity on the horizon,
representing the fact that the black hole rotates rigidly. This quantity
enters directly into the definition of the BH surface
gravity and, consequently, into the formulation of the rigidity
theorem and into the expressions for the Killing vector (6).
More precisely, the Kerr BH surface gravity is defined as
κ = κs − γa , where κs ≡ 1/4M is the Schwarzschild
surface gravity, while γa = M ωh2 (the effective spring
constant, according to [
111
]) is the contribution due to the
additional component of the BH intrinsic spin; ωh is therefore the
angular velocity (in units of 1/M ) on the event horizon. The
(strong) rigidity theorem connects then the event horizon
with a Killing horizon stating that, under suitable conditions,
(10)
the event horizon of a stationary (asymptotically flat solution
with matter satisfying suitable hyperbolic equations) BH is
a Killing horizon.8
The constant limit ω0 ≡ M/a plays an important role
because it corresponds to the asymptotic limit for very small
values of r and R ≡ r/a. Note that, on the equatorial plane,
gαβ L0α L0β = R2, where L0 ≡ L±|ω0 . The asymptotic
behavior of these frequencies may be deeper investigated by
considering the power series expansion for the spin parameter
and the radius determined by the expression
for r →
M
∞ : ω± = ± r
M
1 − r
+ o[r −3],
(11)
which shows a clear decreasing as the gravitational field
diminishes. For large values of the rotational parameter, we
obtain
ω± =
M 2M ± r M r 2
a 2M + r + a3 (2M + r )2
×
∓2M 2 − 2Mr ∓ 21 r 2
+ o[a−5],
so that for extreme large values of the source rotation, the
frequencies vanish and no stationary observers exist, thought
differently for the limiting frequencies ω± (see Fig. 2). It is
therefore convenient to introduce the dimensionless radius
R ≡ r/a, for which we obtain the limit
M
a −
M R2
M 2 R3
−
4a2
+ o[ R3];
R → 0 : ω+ =
ω− =
−
2a
M M 2 + a2 R2
a − R + 2M a
a4 + 4M 2a2 − M 4 R3
R →
∞ : ω± =
4a2 M 2
(∓M 2 + 4M a∓a2)M
2a3 R3
M 2 M
∓ a2 R2 ± a R + o[ R−3].
+ o[ R3];
Equations (12), (13) and (14) show the particularly different
behavior of ω± with respect to the asymptote ω0. The
behavior of the frequencies for fixed values of the radial coordinate
r and varying values of the specific rotational parameter a/M
is illustrated in Fig. 1. We see that the region of allowed
values for the frequencies is larger for naked singularities than
for black holes. In fact, for certain values of the radial
coordinate r , stationary observers can exist only in the field of naked
singularities. This is a clear indication of the observational
8 Assuming the cosmic censorship validity, the gravitational collapse
should lead to BH configurations. The surface area of the BH event
horizon is non-decreasing with time (which is the content of the
second law of black hole thermodynamics). The BH event horizon of this
stationary solution is a Killing horizon with constant surface gravity
(zeroth law) [
4,112–114
].
(12)
(13)
(14)
(15)
differences between black holes and naked singularities. The
allowed values for the frequencies are bounded by the
limiting value ω0 = M/a; for a broader discussion on the role of
the dimensionless spin parameter a/M in Kerr geometries,
see also [96].9 Moreover, for a given value of ω±, the
corresponding radius is located at a certain distance from the
source, depending on the value of the rotational parameter
a. The following configuration of frequencies, radii and spin
determines the location structure of stationary observers:
ω+ ∈]0, ω0[, for a ∈]0, M [ in r ∈]0, r−] ∪ [r+, +∞[
and for a ≥ M in r > 0 (16)
ω− ∈]0, ω0[ for a ∈]0, M [ in r ∈]0, r−] ∪ [r+, rε+[
and for a ≥ M in r ∈]0, rε+[ .
(17)
Thus, we see that in the interval ]0, M/a[ observers can exist
with frequencies ω±; moreover, the frequency ω− is allowed
in r ∈ Σε+, while observers with ω− < 0 can exist in r >
rε+. Moreover, it is possible to show that, in BH geometries,
the condition ω± 1/2 must be satisfied outside the outer
horizon (r > r+). The particular value ω± = ωh = 1/2
is therefore the limiting angular velocity in the case of an
extreme black hole, i.e., for a = M so that r = r+ = r− = M
in Eq. (10). The behavior of the special frequency ω± = 1/2
is depicted in Fig. 3 and in Figs. 2, 4, 5, and 6, where other
relevant frequencies are also plotted.
Equation (16) enlighten some important properties of the
light surfaces (frequencies ω±) and of stationary observers,
associated with frequencies ω ∈]ω−, ω+[ in the regime of
strong singularities. Equation (16) also enlighten the
dependence of the frequencies on the dimensionless spin a/M and
radius R = r/a. It is clear that when the frequency interval
]ω−, ω+[ shrinks, depending on the singularity spin a/M
or the distance from the source r/M , the range of possible
frequencies for stationary observers reduces. This occurs in
general when ω+ ≈ ω−. According to Eq. (16), the
frequencies ω± are bounded from above by the limiting frequencies
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 Sect. 3.2, where we
shall focus specifically on the frequency ω0. On the other
9 For simplicity we use here dimensionless quantities. We introduce
the rotational version of the Killing vectors ξt and ξφ , i.e., the canonical
vector fields V˜ ≡ (r 2 + a2)∂t + a∂φ and W˜ ≡ ∂φ + aσ 2∂t . Then,
the contraction of the geodesic four-velocity with W˜ leads to the
(nonconserved) quantity L − E aσ 2, which is a function of the conserved
quantities (E , L ), the spacetime parameter a and the polar coordinate
θ ; on the equatorial plane, it then reduces to L − E a. When we
consider the principal null congruence γ± ≡ ±∂r + Δ−1V˜ , the angular
momentum L = aσ 2, that is, ¯ = 1 (and E = +1, in proper units),
every principal null geodesic is then characterized by ¯ = 1. On the
horizon, it is L = E = 0 [
96,115
].
Fig. 4 Stationary observers: the angular velocities ωε (gray curve),
+
ωh (black curve), ωn (dot-dashed curve), ω0 (dashed curve), ω¯ n >
ωn > ωh (black thick curve). Here ω¯ n = ωn = ωh = 1/2 at a = M,
ω+ε = ωh = 0.321797 at a = as , and ω+ε = ωn = 0.282843 at
a = a1. The maximum of ω+ε, 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 the BH photon orbit rγ ∈ Σε+ are also plotted.
Note that ωn it is an extension of ω+(rγ ) for a < a1 – see Table 2
hand, considering 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
respect 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
singularity only at a particular frequency. The greater is the NS
dimensionless spin, the lower is the limiting frequency ω ,
±
with the extreme limit at ω+ = ω−. In other words, the
frequency range, ]ω−, ω+[, for stationary observers vanishes as
the value r = 0 is approached. The singularity at r = 0 in
the NS regime is actually related to the characteristic constant
Fig. 5 Upper panel: plot of the curves rs− =constant and rs+ =constant
(inside panel) in the plane (ω, a/M). The numbers denote the constant
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
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 Fig. 2
frequency ω = ω0 in the same way as in BH-geometries the
outer horizon r = r+ is related to the constant frequency ωh
(cf. Eq. (10)). Consequently, a NS solution must be
characterized by the frequency ω0 and a BH solution by the frequency
ωh . Therefore, the frequency ω0 may be seen actually as the
NS counterpart of the BH horizon angular frequency ωh (see
Fig. 4). For r > r+, it holds that ω+ > ω−.
Then, in general, for BHs and NSs in the static limit rε+ =
2M , we obtain that
ω+ε ≡ ω+(rε+) = 2Ma2M+ a2
Moreover, ω− < 0 for r > rε+, and ω
ergoregion Σε+, while ω
−
+ > 0 everywhere.
In general, any frequency value should be contained within
the range ω+ − ω−; therefore, it is convenient to define the
frequency interval
> 0 inside the
Δω
± ≡ ω+ − ω− = 2
ω2Z − ω2,
∗
(19)
with
ω−(rε+) = 0.
(18)
which is a function of the radial distance from the source and
of the attractor spin. Figure 7 show the frequency interval
Δω± as a function of r/M and a/M .
An analysis of this quantity makes it possible to derive
some key features about the eligible frequencies. For
convenience, we present in Table 1 some special values of the
spinmass ratio, which we will consider in the following analysis.
We summarize the obtained results in the following way:
Firstly, for any NS source with a > aΔ ≡ 1.16905M ,
the interval Δω± increases as the observer (on the equatorial
plane) moves inside the ergoregion Σε+ towards the static
limit.
Secondly, in the case of NS geometries with a ∈]M, aΔ[,
i.e., belonging partially to the class of NSI spacetimes, the
situation is very articulated. There is a region of maximum
and a minimum frequencies, as the observer moves from the
source towards the static limit. This phenomenon involves
an orbital range partially located within the interval ]rˆ−, rˆ+[,
which is characterized by the presence of counterrotating
circular orbits with negative orbital angular momentum L =
−L− (cf. Fig. 8, where the radii rˆ± are plotted.).
Fig. 7 Upper panel: plot of the frequency interval Δω± = ω+ − ω− as
a function of the radius r/M and the BH and NS spin a/M. The extrema
rΔ± and r ± are solutions of ∂r Δω± = 0 and ∂a Δω± = 0, respectively.
Lower panel: the frequency interval Δω± = ω+ − ω− as a function of
a/M for selected values of the orbit radius r/M; the maximum points
are for the radii rΔ± or r + – see Fig. 8
For the maximum spin, a = aΔ, we obtain ω+ = ω−
on the radius r ≡ rΔ±(aΔ) = 0.811587M and, therefore,
the range of possible frequencies for stationary observers
vanishes. The points rΔ±(a) represent the extrema of the
interval Δω± , i.e., the solutions of the equation ∂r Δω± = 0
– Fig. 8. This property is present only in the case of NS
geometries. In fact, there are the two critical orbits rΔ+ > r −
Δ
and r = rΔ±(aΔ), which are the boundaries of a closed
region, whose extension reaches a maximum in the case of
the extreme Kerr geometry a = M , and is zero for a = aΔ.
For r ∈]rΔ−, rΔ+[, the separation parameter Δω± decreases
with the orbital distance, then on the inner radius rΔ− it
reaches a maximum value, whereas on the outer radius r +
Δ
it reaches a minimum. In the outer regions, at r > rε+, the
separation parameter increases with the distance from the
source. This feature constitutes therefore a major difference
in the the behavior of stationary observers within and
outside the ergoregion of a naked singularity spacetime.
However, a deeper analysis of the equatorial plane, outside the
static limit, shows the existence of a second region for light
surfaces in the NS case.
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 : rm−so(a2) = rε+
Naked singularity classes: NSI : ]M, a3], NSII : ]a3, a4], NSIII : ]a4, +∞]
a3/M ≡ 3√3/4 ≈ 1.29904 : rˆ+(a3) = rˆ−(a3), a4/M ≡ 2√2 ≈ 2.82843 : r m(NsoS)−(a4) = rε+
acterized by the zero angular momentum radii (L (rˆ±) = 0) and the
radius of the marginally stable circular orbit r m(NsoS)− ∈ Σε+. The explicit
expressions for these radii can be found in [
52–55,76
]
On the other hand, the angular velocity ω− decreases with
the orbit in the Kerr spacetime. The maximum frequency ω+
also decreases in the NS spacetimes. In the BH cases, the
angular velocity is always increasing for sources of the class
BHI, while for the other sources there is a maximum for
the velocity ω+ at r = rγ−, which is the circular orbit of a
photon or null-like particle corotating with the source. Such
a kind of orbit, contained in Σε+, is a feature of the
BHIIIII spacetimes [
2
], this is also know as marginally or last
circular orbit as no circular particle motion is possible in the
region r < rγ−. We close this section with a brief discussion
on the variation of the frequency interval Δω± , following
a spin transition with a > 0. In the case of a singularity
spin-transition, there are two extreme radii for the frequency
interval
r + ≡ η cos
r − ≡ η sin
where r ± : ∂a Δω± r ± = 0 are maximum points – see Figs. 7
and 8.
3.2 Light surfaces
In this section, we briefly study the conditions for the
existence of light surfaces and and their morphology. The
condition (8), for the definition of a stationary observer, can be
restated in terms of the solutions rs±, considering ω as a fixed
parameter. Therefore, we now consider the solutions rs± of the
equation for the light surfaces defined in Eq. (9) in terms of
the Killing null generator L±, as functions of the frequency
ω. We obtain
rs−
M ≡
rs+
M ≡
2β1 sin 13 arcsin β0
√3 ,
2β1 cos 13 arccos(−β0)
√3
(21)
(24)
where β1 ≡
where ω0 ≡ M/a (cf. Eq. (10) and Fig. 2). For ω = 1/2, in
the limiting case of a = M , we have that ωn = ω¯ n = ωh =
1/2 and rs± = M – see Figs. 2, 4 and 6.10 Thus, there are
solutions rs+ = rs− = 0 for a ∈]0, M [ if ω ∈ (ωn , ω¯ n ) where
(for simplicity we use a dimensionless spin a → a/M )
9 − a2 + 6√9 − 5a2 cos 13 arccos α
ω¯ n ≡ a(a2 + 27) (22)
9 − a2 − 6√9 − 5a2 sin 13 arcsin α
ωn ≡ a(a2 + 27) (23)
α ≡
static solution). Moreover, we have that ω0 > ω¯ n > ωn >
ω+ε and ωn > ωh for BH-sources, where ωh > ω+ε for
a ∈]as , M ]. In the NS case, there are no crossing points for
the radii rs± and ω0 > ω+ε (see Fig. 4). The shrinking of the
frequency interval ]ω−, ω+[ is also shown in Figs. 6, 9 and
10, where the radii rs± are also plotted as functions of the
frequencies.
Figures 4, 5, and 6 contain all the information about the
differences between black holes with a < M , and the case of
naked singularities with a > M . We summarize the situation
in the following statements:
10 More precisely, it is rs+ = rs− = 0 for a > 0 and ω = ω0. Also,
rs+ = rs− > r+ for a = 0 and ω = ± 3 √13 . In the extreme Kerr spacetime
geometry, we have that rs+ = rs− > 0 for a = M, ω = 1/2 for r = M,
and ω = −1/7 for r = 4M. For a Kerr geometry, where a/M ∈]0, 1[,
it is rs+ = rs− > r+ for ω = ωn or ω¯ n (one positive and one negative
value solution), while in the naked singularity case where a > M, the
condition rs+ = rs− > 0 is valid only for one negative frequency – see
Figs. 5 and 6.
11 For a closer look at the role of this special frequency we note that
ω¯ n = ωn = ωh = 1/2 at a = M and, clearly, ω0 = 1/2 for a = 2M.
We refer then to Figs. 3, 4, 5, 6, and 9.
Naked singularities spacetimes: For a > M , the solutions
for the equation of the light surfaces in the limiting case
ω = 0 (static observer) are located at r = rε+. While
for any frequency within the range ω ∈]0, ω+ε[ there is
one solution rs−, for larger frequencies in the range ω ∈
[ω+ε, ω0[ there are two solutions rs±. In the ergoregion
Σε+ of a naked singularity, there exists a limit ω0 ≡ M/a
for the angular frequency.
Extreme black hole spacetime: For a = M , we obtain the
following set of solutions (ω = 0, r = rε+), (ω ∈
]0, 1/3[, r = rs−), and (ω ∈ [1/3, 1/2[, r = rs±).
Black hole spacetimes: We consider first the class BHI with
a ∈]0, a1]. In the limit ω = 0, there exists a solution for
the light surface with r = rε+. More generally, the
solutions are constrained by the following set of conditions:
C1 :
C2 :
ω ∈ [ω+ε, ωn [ with solution r = rs±,
ω = ωn, with solution r = rs−.
ω ∈]0, ω+ε] ∪ ω = ωh with solution r = rs−.
(25)
(26)
(27)
Then, we consider BH spacetimes with spin a ∈]a1, as [,
where as ≡ 2 √2 − 1 M < a2. These spacetimes
include a part of BHII-sources and the condition C1
applies.
For spacetimes with rotation a = as , the conditions C1
and C2 apply. Then, in the special case ω+ε = ωh or
ω = ωε , there is a solution with r = rs+.
+
Finally, for spacetimes with a ∈]as , M [, which belong to
the class of BHII and BHIII sources, the condition C1
holds, whereas the condition C2 applies for frequencies
within the interval ωh < ωn. Finally, in the special case
ω = ωh , there is one solution at r = rs+, and for ω = ωn
we have the solution r = rs−.
A summary and comparison of these two cases is proposed
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 solution
a/M = 1 is a limiting case of both surfaces rs±, varying both
in terms of the spin and the angular velocity ω. Thus, the
difference between the regions where stationary observers can
exist in the BH case (gray regions in Fig. 6) and in the NS
case are clearly delineated. In BH spacetimes, the surfaces rs±
are confined within a restricted radial and frequency range.
On the other hand, in the naked singularity case, the orbits
and the frequency range is larger than in the black hole case.
Moreover, the surfaces rs± can be closed in the case of NS
spacetimes, inside the ergoregion, for sufficiently low values
of the spin parameter, namely a ∈]M, a4]. Furthermore, in
any Kerr spacetime, there is a light surface at rs± = rε+ with
ω = M/a4. In Sect. 4, we complete this analysis by
investigating the special case of zero angular momentum observers,
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
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 Fig. 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,
and we find all the spacetime configurations in which they
can exist.
4 Zero angular momentum observers
This section is dedicated to the study of zero angular
momentum observers (ZAMOs) which are defined by the condition
L ≡ uα ξ(αφ) = gαβ ξφα pβ = gtφ t˙ + gφφ φ˙ = 0.
(28)
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
In terms of the particle’s four-velocity, the condition L
is equivalent to dφ/dt = −gφt /gφφ ≡ ωZ = (ω+ + ω−)/2,
where the quantity ωZ is the ZAMOs angular velocity
introduced in Eq. (8), and the frequency of arbitrary stationary
observers is written in terms of ωZ [
2
]. The sign of ωZ is in
concordance with the source rotation. The ZAMOs angular
velocity is a function of the spacetime spin (see Figs. 11 and
12, where constant ZAMOs frequency profiles are shown).
In the plane θ = π/2, we find explicitly
= 0
As discussed in [
2, 54, 55
], ZAMOs along circular orbits with
radii rˆ± are possible only in the case of “slowly rotating”
naked singularity spacetimes of class NSI. This is a
characteristic of naked singularities which is interpreted generically
as a repulsive effect exerted by the singularity [
52–55, 76
].
On the other hand, ω2Z = ω∗2 for r = r±, while ω2Z > ω∗2
in the region r > r+ for BH spacetimes, and in the region
r > 0 for NS spacetimes (see also Fig. 12).
ZAMOs angular velocity and orbital regions
The ZAMOs angular velocity ωZ is always positive for a >
0, and vanishes only in the limiting case a = 0. This means
that the ZAMOs rotate in the same direction as the source
(dragging of inertial frames).
As can be seen from Eq. (29), the frequency ωZ for a
fixed mass and a = 0 is strictly decreasing as the radius r/M
increases.
For the NS regime it is interesting to investigate the
variation of ZAMO frequency ωZ on the orbits rˆ±. These special
radii of the NS geometries do not remain constant under a
spin-transition of the central singularity. We shall consider
this aspect focusing on the curves rˆ±(a) of the plane r − a as
illustrated in Fig. 8. This will enable us to evaluate
simultaneously the frequency variation on these special orbits,
following a spin variation of the naked singularity in the rage
of definition of rˆ±, and to evaluate the combined effects of
a variation in the orbital distance from the singularity and
a change of spin. A similar analysis will be done, from a
different point of view, also for stationary observers.
In Σε±, the velocity ωZ = ωˆ − (in NSs) always decreases
with the orbital radii rˆ−, i.e. ∂rˆ− ωˆ − < 0, when the spin
increases, i.e. ∂arˆ− > 0 (see Figs. 8 and 12). As rˆ+
monotonically decreases with the spin during a NS spin-up process
(see Fig. 8), the frequency ωˆ + = ωZ (rˆ+) decreases in the
spin-range a ∈ [M, aω[, and increases in the range ]aω, a3];
ω±(r+) = ωZ (r+) (dot-dashed curve), ωˆ ±Z ≡ ωZ (rˆ±) as functions of
the spacetime rotation a/M for different BH and NS classes. Dotted
lines are aκ ≈ 0.3002831060M : ωe = ωˆ +Z, as ≈ 0.91017M : ωe =
√
ωh, a3 : ωˆ +Z = ωˆ −Z = 8/9√3, and finally the spin a = 2M : ωεZ =
ω2 (dashed line) which is a maximum for ωεZ (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: ωˆ ±Z ≡
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 frequency 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+
therefore, the special value aω = 1.1987M is a minimum
point of the ZAMOs frequency ωˆ +Z – see Fig. 12.
Viceversa, as rˆ− increases after a NS spin-up, the corresponding
ZAMOs frequency ωˆ − = ωZ (rˆ−) decreases as the observer
moves along the curve rˆ−(a). Thus, we can say that, if the
NS spin increases, the frequency ωˆ + decreases,
approaching, but never reaching, the singularity, i. e., ∂arˆ+ < 0 for
a ∈ [M, aω[. Viceversa, increasing the NSs spin in
spacetimes with a ∈]aω, a3[, the frequency ωˆ + increases again and
the orbit rˆ+ moves towards the central singularity. On the
other hand, the frequency ωˆ − monotonically decreases with
the naked singularity spin, i.e. ∂arˆ− > 0; therefore, for a fixed
NS spin, the frequency interval decreases, i.e. ωˆ − > ωˆ +. In
fact, the velocity ωZ is strictly decreasing with the radius
r in the BH and NS regimes with a = 0 (i.e. ∂r ωZ < 0).
Moreover, in general ωZ increases as the observer approaches
the black hole at fixed spin, and it decreases as the observer
moves far away from the center of rotation.
In the static limit, we have that ωZ (rε+) = ωε /2. In fact,
+
the asymptotic behavior of the frequency is determined by
the relations
lim ωZ = rl→imr+ ω± = ωh ,
r→r+
lim
r→+∞
ωZ = 0, lim ωZ = ω0.
r→0
Change in the intrinsic spin (30)
The angular velocity of the ZAMOs inside Σε+ varies
according to the source spin. This might be especially important in
a possible process of spin-up or spin-down as a result of
the interaction, for example, with the surrounding matter. In
[
2
], this phenomenon and its implications were investigated,
considering different regions close to the singularity. For a
fixed orbital radius r , the ZAMOs angular velocity strongly
depends on the value of the spacetime spin-mass ratio. In
(31)
ZAMOs energy
particular, depending on the value of the ratio a/M , there
can exist a radius of maximum frequency re given by
re ≡
Υ ≡
,
3 9Ma2 + √3 a4 27M 2 − a2
that are solutions of the equation ∂a ωZ |π/2 = 0 at which
the frequency is denoted by ωe ≡ ωZ (re) (see Figs. 11,
12). A detailed analysis of the expression for the radius re
shows that in can exist in spacetimes that belong to the class
BHII with spin a = as , where re(as ) = r+(as ), and to
the classes BHIII, NSI, and NSII with the limiting value
a = a = √2M , where a : re = rε+ (see Figs. 2, 11).
Spacetimes with spin as belong to the class BHII, as defined
in Table 2, and have been analyzed in the context of stationary
observers in Sects. 3.1 and 3.2 (Figs. 4, 11, 12 and 13 show the
behavior of several quantities related to ZAMOs in relation
to other frequencies.). In this particular case, we have that
ω+ε = ωh = 0.321797 and re(as ) = r+(as ).
(32)
We focus our attention on ergoregion Σε+, bounded from
above by the radius rε+ and from below by r = 0 and r = r+
for NSs and BHs, respectively. We consider the role of the
radius re, as the maximum point of the ZAMO frequency,
as a function of the source spin-mass ratio. Thus, for black
holes with a ∈ [0, as ], the frequency ωZ increases with a/M
always inside the ergoregion; this holds for any orbit inside
Σε+ (i.e. for a fixed value r¯ ∈ Σε+, if a BH spin-up shift occurs
in the range [0, as ], the function ωZ (r¯, a) increases with
r¯). For spins a ∈]as , M ], instead, the frequency ωZ grows
with the spin only for r¯ ∈]re, rε+[; on the contrary, for radii
located close to the horizon, r¯ ∈]r+, re], ωZ (r¯, a) decreases
following a spin up in the range ∈]as , M ] (i.e, ∂ar+ < 0
and ∂are > 0). In the case of NS-spacetimes, the frequency
ωZ (r¯, a) is an increasing function of the dimensionless spin
in the NS spin range ]M, a [ and on the orbit r¯ ∈]re, rε+].
Moreover, the frequency ωZ (r¯, a) decreases with the spin in
the range of values a ∈]M, a [ and on r¯ ∈]0, re]. This
situation is distinctly different for NS with a > a , for which
in the ergoregion an increase of the spin corresponds to a
decrease of ωZ . This is an important distinction between
different NS regimes.
We note that re(as ) = r+(as ) for the spin as =
2 √2 − 1 M (see Fig. 11). Moreover, in NSII naked
sin√
gularity spacetimes with spin a = 2M : re = rε+, we
obtain that ωεZ = ωe – Fig. 12. Remarkably, the spin a is
the maximum point of the frequency ωεZ (a ) = ωe(a ) ≡
ωεZ−Max = 0.176777 and also the maximum point of the
frequency ω+ε (see Fig. 2). In other words, in naked
singularity spacetimes with a = a , where re = rε+, the ZAMOs
frequency at the ergosurface ωεZ reaches a maximum value
which is equal to ωe, defined through the radius in Eq. (31);
moreover, the frequency ω+ε reaches its maximum value at
the ergosurface.
The circular motion of test particles can be described easily
by using the effective potential approach [
116
]. The exact
form of such an effective potential in the Kerr spacetime is
well known in the literature (see, for example, [
54,55
]). The
effective potential function Ve +ff represents the value of E /μ
that makes r into a turning point (Ve f f = E /μ), μ being the
particle mass; in other words, it is the value of E /μ (in the
case of photons, μ shall depend on an affine parameter and
the impact parameter ≡ L /E is relevant for the analysis of
trajectories) at which the (radial) kinetic energy of the particle
vanishes. This can easily be obtained from the geodesic
equations with the appropriate constraints or through the
normalization conditions of the four-velocities, taking into account
the constraints and the constants of motion [116]. Here we
consider specifically an effective potential associated to the
ZAMOs.
The orbits rˆ± are critical points of the effective
poten2
tial, i. e., rˆ± : ∂r Ve f f Z = 0. Here we consider for the
ZAMO Ve f f 2Z = κ˜ gφφ [ω∗2 − ω2Z ] where κ˜ is a factor related
to the normalization condition of the ZAMO four-velocity
(κ˜ = −1 for timelike ZAMOs, where uφ = −ωZ ut and
ut = −εE /gφφ [ω∗2 −2 ω2Z ], ε = 1 according to Eq. (3); in
2
the ergoregion Ve f f Z > 0, but Ve f f Z = 0 for r = 0
and r = r+). The energy E of the ZAMOs is always
positive for both BH and NS spacetimes, and it grows with the
source spin; in fact, solutions for Ve f f Z = 0 are not possible
because this would correspond to the case of a null angular
momentum with null energy. The energy on the orbits rˆ±
where L = 0 is always positive. In BH geometries, the
potential Ve f f , at L = 0, increases with the distance from
the source and has no critical 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 decreases with the orbital
radius. However, in the outer region r ∈]r+, rˆ−[∪]rˆ+, 2M [,
the potential increases with the radius. This implies that the
radii rˆ± are possible circular ZAMOs orbits. In fact, rˆ− is an
unstable orbit and rˆ+ is a stable orbit. Thus, in any geometry
of this set, there is a stable orbit for the ZAMOs with angular
velocity ωˆ ±Z ≡ ωZ (rˆ±) different from zero, where ωˆ −Z < ωˆ +Z
(see Fig. 12).
In [
2
], we investigated the orbital nature of the static limit.
Here, in Fig. 13, the velocity ωZ and the ratio Rε ≡ E−ε /L−ε
(that is, the inverse of the specific angular momentum defined
as uφ /ut ) are considered as functions of the source spin at
the static limit. We explore the relation between the ZAMOs
and the stationary observes, where ωZ = (ω+ + ω−)/2,
for NSI sources at the static limit. A maximum value,
Rε = 0.853553M , is reached at a = 2M ∈ NSII. Also, a
maximum value ωεZ−Max = 0.176777 exists for the ZAMOs
angular velocity at a = a ∈ NSII. This ratio is always
greater than the angular momentum of the ZAMOs at the
static limit.
In BH spacetimes, the angular velocity for stationary
observers is limited by the value ωh which occurs for the
radius r+. We can evaluate the deviation of this velocity in
a neighborhood of the radius r+, since the four-velocity of
the observers rotating with ω (where ua ≡ ξt + ωξφ ) must
be timelike outside the horizon and therefore it has to be
R = E /L > ωh in that range (the event horizon of a Kerr
black hole rotates with angular velocity ωh [
1
]). This limit
cannot be extended to the case of naked singularities.
However, one can set similarly the threshold E > ωa L in the case
of circular orbits, where the frequency limit is restricted to
the values ωa ∈ [1, aμ−1 M [ as ωa ∈ [ω0(a = M ), ω0(aμ)[.
5 Summary and conclusions
In this work, we carried out a detailed analysis of the physical
properties of stationary observers moving in the ergoregion
along equatorial circular orbits in the gravitational field of a
spinning source, described by the stationary and
axisymmetric Kerr metric. We derived the explicit value of the angular
velocity of stationary observers and analyzed all possible
regions where circular motion is allowed, depending on the
radius and the rotational Kerr parameter. We found that in
general the region of allowed values for the frequencies is
larger for naked singularities than for black holes. In fact, for
certain values of the radius r , stationary observers can exist
only in the field of naked singularities. We interpret this result
as a clear indication of the observational differences between
black holes and naked singularities. Given the frequency and
the orbit radius of a stationary observer, it is always
possible to determine the value of the rotational parameter of
the gravitational source. Our results show that in fact the
probability of existence of a stationary observer is greater
in the case of naked singularities that in the case of black
holes. Moreover, it is possible to introduce a classification of
rotating sources by using their rotational parameter which,
in turn, determines the properties of stationary observers.
Black holes and naked singularities turn out to be split each
into three different classes in which stationary observers with
different properties can exist. In particular, we point out the
existence of weak (NSI) and strong (NSIII) naked
singularities, corresponding to spin values close to or distant from the
limiting case of an extreme black hole, respectively.
Light surfaces are also a common feature of rotating
gravitational configurations. We derived the explicit value of the
radius for light surfaces on the equatorial plane of the Kerr
spacetime. In the case of black holes, light surfaces are
confined within a restricted radial and frequency range. On the
contrary, in the naked singularity case, the orbits and the
frequency ranges are larger than for black holes. Again, we
conclude that light surfaces can be found more often in naked
singularities. The observation and measurement of the
physical parameters of a particular light surface is sufficient to
determine the main rotational properties of the spinning
gravitational source. We believe that the study of light surfaces
(defining the “throat” discussed in Sect. 3) has important
applications regarding the possibility of directly observing
a black hole in the immediate vicinity of an event horizon
(within the region defined by the static limit), as this seems to
be possible in the immediate future through, for example, the
already active Event Horizon Telescope (EHT) projects.12
We also analyzed the conditions under which a ZAMO
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
same direction of the spinning source as a consequence of
the dragging of inertial frames. The energy is also always
positive. The most interesting case is that of slowly rotating
naked singularities (NSI) where there exists a closed and
disconnected orbital region. This particular property could, in
principle, be used to detect naked singularities of this class.
We derived the particular radius at which the frequency of
the ZAMOs is maximal, showing that the measurement of
this radius could be used to determine whether the spinning
source is a black hole or a naked singularity and its class,
according to the classification scheme formulated here. To
be more specific, from Table 2 we infer that the existence of
stationary observers in black hole spacetimes is limited from
above by the frequency ωε , which is the highest frequency on
+
the static limit, implying the frequency lower bound ω = 0
– see also Fig. 4. In this figure, we also show the maximum
frequency, ωε+, at the static limit for a naked singularity with
a = a = √2M ∈ NSII. This spin plays an important role
for the variation of the ZAMOs frequency in NSs in terms of
the singularity dimensionless spin – see Figs. 12 and 13. On
the other hand, for strong BHs, with a > a1, the frequency
is bounded from below by ω = ωε+ and from above by ωn,
as the radial upper bound is rs+. A similar situation occurs
for NSs, provided that ωn is replaced with the limiting
frequency ω0. The special role of the BH spin a1 is related to the
presence of the photon circular orbit in the BH ergoregion,
which is absent in NS geometries; consequently, as seen in
Table 2, there is no distinction between the naked
singularities classes. However, the analysis of the frequencies in Fig. 4
shows differently that there are indeed distinguishing features
in the corresponding ergoregions. In the case of naked
singularities, the frequency range of stationary observers has as a
boundary the outer light-surface, r = rs+, then it narrows as
the spin increases, and finally vanishes near the static limit.
The frequency of the orbits on the static limit, in fact,
converges to the limit ω0 = M/a, which is an important
frequency threshold for the NS regime. The presence of a
maximum for the special NS geometry with a = a on the
static limit is symptomatic for the nature of this source – see
Figs. 4, 6 and 13. The study of the surfaces rs± on the plane
(r, ω), for different values of the spin-mass ratio, shows a
clear difference between the allowed regions in naked
singularities and black holes (gray region in Fig. 6). There is
an open “throat” between the spin values a M (strong
12 http://www.eventhorizontelescope.org/.
BHs) and a M (very weak NSs), with an opening of
the cusp (at r = 0 in these special coordinates) for the
frequency ω = 0.5. We note a change in the situation for spins in
a/M ∈]1, 1.0001]; this region is in fact extremely sensitive to
a change of the source spin; the throat of rs± has, in this special
spin range, a saddle point around (r = M, ω = 1/2) between
[aμ, a3], which is not present in stronger singularities. The
spins in this range are related to the negative state energy
and the radii rυ±, where the orbital energy is E = 0 – Fig. 8.
Particularly, we point out the spin a = aσ = 1.064306M ,
where rυ− = rΔ− = 0.5107M , for which at rΔ± there is a
critical point of the frequency amplitude Δω±. In BH
geometries, the frequencies increase with the spin and with the
decrease of the radius towards the horizon. The curves rs±
continue to increase with the presence of a transition throat
at r = M that increases, stretching and widening. This
throat represents a “transition region” between BH and
superspinning sources from the viewpoint of stationary observes.
The regions outlined here play a distinct role in the collapse
processes with possible spin oscillations and different
behaviors for weak, very weak, and strong naked singularities. As
the spin increases, the frequencies of NSs observes move
to lower values, widening the throat. This trend, however,
changes with the spin, enlightening some special thresholds.
This analysis shows firstly the importance of the limiting
frequency ω0 = M/a, determining the main properties of
both frequencies ω± and the radii rs±; it is also relevant in
relation to ZAMOs dynamics in NS geometries. In this way,
we may see ω0 as an extension of the frequency ωh at the
horizon for BH solutions – Fig. 4. In the NS regime, all the
curves rs± converge to the same “focal point” r = 0,
regardless of the type of naked singularity, but as ωh is the limiting
frequency at the BH horizon, each source is characterized
by only one ω0 = 0 frequency. The greater is the spin, the
lower is the frequency ω± at fixed radius, and particularly
in the neighborhood of the singularity ring, according to the
limiting value ω0. The frequency range at fixed r/M
narrows for higher dimensionless NS spin a/M . This feature
distinguishes between strong, weak and very weak naked
singularities. From Fig. 6 it is clear also that the throat of
the light-surfaces rs±, in the plane r − ω, for different spins
a/M closes for a ≈ M , which is a spin transition region
that includes the extreme Kerr solution. This region has been
enlarged in Fig. 6-bottom. Figures 9 and 10 show from a
different perspective the transition between the BH region,
gray region in Fig. 6, and the NS region for different spins.
Any spin oscillation in that region generates a tunnel in the
light-surface.13 The transition region is around ω± ≈ 1/2,
13 Since any simulation of stellar collapse returns to the BH regime,
there must be some (retroactive) mechanism that closes the observer
tunnel, as even light does not run away in the forbidden region at r <
M. Moreover, hypothetical super-luminary matter would violate the
bonding of the tunnel wall.
which is a special value related to the spin a = 2M of strong
naked singularities – see Figs. 3 and 2. In this region, as in
the neighborhood of the ring singularity (r = 0), the orbital
range reaches relatively small values.14 This shows the
existence of limitations for a spin transition in the parameter
region of very weak naked singularities, pointed out also in
[
52–55,76
].
On the other hand, in the strong NSs regimes, a spin
threshold emerges at a = 2M and a = M (see Figs. 3, 4, 6). In
Fig. 13, we analyze the properties at the static limit rε+. The
gmiaoxni mofumthevNalSueIIocflaEs−s./1L5A−roisu nthdena
r=eacah3etdheinththroeaterwgoidrtehbecomes more or less constant. The situation is different for
a > a3 and a > 2M and then for a4, where the
frequencies range narrows, and near r = rε+ becomes restricted to a
small range of a few mass units in the limit of large spin a/M .
In strong and very strong NSs, the wide region is
inaccessible for stationary observers, whereas it is accessible in the
BH case. This significantly separates strong and weak NSs,
and distinguish them from the BH case. Interestingly, the
saddle point around r = M , which narrows the throat of
frequencies even in the case of NS geometries for a ∈]M, aσ [,
could perhaps be viewed as a trace of the presence of r+,
which is absent for a > M . For a = aσ , where the saddle
point disappears, the shape of the rs± tube is different. This,
on the other hand, would suggest that the existence of the
flex in the case of very weak NSs would prevent a further
increasing of the spin. This does not hold for a transition
to stronger NSs, a ≥ aσ , where no saddle point is present
– Fig. 8-bottom. Obviously, the consequences of the
hypothetical transition processes should also take into account the
transient phase times. Very weak naked singularities show a
“rippled-structure” in the frequency profiles of ω with respect
to r/M and a/M , as appears in Figs. 2, 6, 9, and 10. The
significance of this structure is still to be fully investigated, but
it may be seen perhaps as a fingerprint-remnant of the BH
horizon. This may open an interesting perspective for the
study of NS geometries.
14 It is worth to mention that predicted quantum effects close to the
singularities could play a major role in this region. However, we recall that
the extreme limit a = M in this model is never faced, as we continue to
see the spacetime for all NSs using a Boyer–Lindquist frame. It is well
known that approaching the horizon at a = M, the radial coordinate
velocity appears as never penetrating the black hole, spiraling as t goes
to infinity. This is the consequence of a coordinate singularity which can
be avoided by using Kerr coordinates or Eddington-Finkelstein
coordinates.
15 The throat depth in the region would lead to an immediate change
of the observers properties and it is reasonable to ask if this may imply
an activation instead of a “positive feedback” phenomenon. We recall
that in this scenario, we are not considering a change of symmetries
which would have an essential role. Then it is important to emphasize
that in these hypothetical spin transitions, the external boundary of the
ergoregion remains unchanged, but not the frequency at the static limit.
An interesting application of our results would be related
to the characterization of the optical phenomena in the Kerr
naked singularity and black hole geometries, such as the
BH raytracing and the determination of the BH silhouette
(shadow). The light escape cones are a key element for such
phenomena. Light escape cones of local observers (intended
as sources) determine the portion of radiation emitted by a
source that could escape to infinity and the one which is
trapped. This is related to the study of the radial motion of
photons because the boundary of the escape cones is given
by directional angles associated to unstable spherical
photon orbits. Light escape cones can be identified in locally
non-rotating frames, in frames associated to circular geodesic
motion and in radially free-falling observers [
42,117–120
].
We want to point out, however, that light escape cones do not
define the properties of the light-cone causal structure, and
are not directly related to stationary observers; they rather
depend on the photon orbits. A thoroughout analysis of the
photon circular motion in the region of the ergoregion can
be found in [
2
]. In Figs. 1, 3, 4 and 12, we show the
photon orbit rγ and the limiting frequencies crossing this radius;
this enlightens the relation with the frequency ωn. We
consider there in more detail the relation between the quantities
ωZ ω∗, the constants of motion L and E and the effective
potential, briefly addressed also in Sect. 4.
In general, we see that it is possible to detect black holes
and naked singularities by analyzing the physical
properties (orbital radius and frequency) of stationary observers
and ZAMOs. Moreover, the main physical properties (mass
and angular momentum) of the spinning gravitational source
can be determined by measuring the parameters of stationary
observers. This is certainly important for astrophysical
purposes since the detection and analysis of compact
astrophysical objects is one of the most important issues of modern
relativistic astrophysics. In addition, the results presented in this
work are relevant especially for investigating non-isolated
singularities, the energy extraction processes, according to
Penrose mechanism, and the gravitational collapse processes
which lead to the formation of black holes.
Acknowledgements DP acknowledges support from the Junior GACR
Grant of the Czech Science Foundation No: 16-03564Y. This work was
partially supported by UNAM-DGAPA-PAPIIT, Grant no. 111617. DP
thanks Dr. Jan Schee for interesting discussion on the light escape cone
and stationary observers.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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