Can accretion disk properties observationally distinguish black holes from naked singularities?

Can accretion disk properties observationally distinguish black holes from naked singularities?

Z. Kovács zkovacs@hku.hk Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong, P. R. China    T. Harko harko@hkucc.hku.hk Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong, P. R. China
July 12, 2019
Abstract

Naked singularities are hypothetical astrophysical objects, characterized by a gravitational singularity without an event horizon. Penrose has proposed a conjecture, according to which there exists a cosmic censor who forbids the occurrence of naked singularities. Distinguishing between astrophysical black holes and naked singularities is a major challenge for present day observational astronomy. In the context of stationary and axially symmetrical geometries, a possibility of differentiating naked singularities from black holes is through the comparative study of thin accretion disks properties around rotating naked singularities and Kerr-type black holes, respectively. In the present paper, we consider accretion disks around axially-symmetric rotating naked singularities, obtained as solutions of the field equations in the Einstein-massless scalar field theory. A first major difference between rotating naked singularities and Kerr black holes is in the frame dragging effect, the angular velocity of a rotating naked singularity being inversely proportional to its spin parameter. Due to the differences in the exterior geometry, the thermodynamic and electromagnetic properties of the disks (energy flux, temperature distribution and equilibrium radiation spectrum) are different for these two classes of compact objects, consequently giving clear observational signatures that could discriminate between black holes and naked singularities. For specific values of the spin parameter and of the scalar charge, the energy flux from the disk around a rotating naked singularity can exceed by several orders of magnitude the flux from the disk of a Kerr black hole. In addition to this, it is also shown that the conversion efficiency of the accreting mass into radiation by rotating naked singularities is always higher than the conversion efficiency for black holes, i.e., naked singularities provide a much more efficient mechanism for converting mass into radiation than black holes. Thus, these observational signatures may provide the necessary tools from clearly distinguishing rotating naked singularities from Kerr-type black holes.

pacs:
04.20. Cv, 04.20. Dw, 04.70. Bw, 04.80.Cc

I Introduction

Investigating the final fate of the gravitational collapse of an initially regular distribution of matter, in the framework of the Einstein theory of gravitation, is one of the most active fields of research in contemporary general relativity. One would like to know whether, and under what initial conditions, gravitational collapse results in black hole formation. One would also like to know if there are physical collapse solutions that lead to naked singularities. If found, such solutions would be counterexamples of the cosmic censorship hypothesis, which states that curvature singularities in asymptotically flat space-times are always shrouded by event horizons.

Roger Penrose Pe69 () was the first to propose the idea, known as cosmic conjecture: does there exist a cosmic censor who forbids the occurrence of naked singularities, clothing each one in an absolute event horizon? This conjecture can be formulated in a strong sense (in a reasonable space-time we cannot have a naked singularity) or in a weak sense (even if such singularities occur they are safely hidden behind an event horizon, and therefore cannot communicate with far-away observers). Since Penrose’ s proposal, there have been various attempts to prove the conjecture (see Jo93 () and references therein). Unfortunately, no such attempts have been successful so far.

Since, due to the complexity of the full Einstein equations, the general problem appears intractable, metrics with special symmetries are used to construct gravitational collapse models. One such case is the two-dimensional reduction of general relativity obtained by imposing spherical symmetry. Even with this reduction, however, very few inhomogeneous exact nonstatic solutions have been found Va51 ()-Or98 ().

Within the framework of various physical models, the spherical gravitational collapse has been analyzed in many papers SiJo96 ()-No99 (). The idea of probing naked space-times singularities with waves rather than with particles has been proposed in IsHo99 (). For some space-times the classical singularity becomes regular if probed with waves, while stronger classical singularities remain singular.

In order to obtain the energy-momentum tensor for the collapse of a null fluid an inverted approach was proposed in Hu96 (). In the framework of the same approach a large class of solutions, including Type II fluids, and which includes most of the known solutions of the Einstein field equations, has been derived WaWu98 (), Ro02 (). The Vaidya radiating metric has been extended to include both a radiation field and a string fluid in GlKr98 (); GlKr99 (); GoGo03 (). The collapse of the quark fluid, described by the bag model equation of state , with constant, has been studied in HaCh00 (), and the conditions for the formation of a naked singularity have been obtained. The obtained solution has been generalized to arbitrary space-time dimensions and to a more general linear equation of state in GhDh02 (); GhDh03 (). The gravitational collapse of a high-density null charged matter fluid, satisfying the Hagedorn equation of state, was considered in the framework of the Vaidya geometry in ha ().

Since theoretical studies alone cannot give an answers about the existence or non-existence of naked singularities in nature, the differences in the observational properties of the black holes and naked singularities could be used to discriminate between these two classes of objects. Such a distinctive observational feature is represented by the lensing properties, and a research project that uses gravitational lensing as a tool to differentiate between naked singularities and black holes was initiated in vir1 (). This pioneering work was extended in vir2 (), where the distinctive lensing features of black holes and naked singularities were studied in detail. The gravitational lensing due to a strongly naked singularity is qualitatively different from that due to a Schwarzschild black hole; a strongly naked singularity gives rise to either two or nil Einstein ring(s), and one radial critical curve. The gravitational lensing (particularly time delay, magnification centroid, and total magnification) for a Schwarzschild black hole and for a Janis-Newman-Winicour naked singularities jnw () were studied in vir3 (). The lensing features are qualitatively similar (though quantitatively different) for Schwarzschild black holes, weakly naked, and marginally strongly naked singularities. However, the lensing characteristics of strongly naked singularities are qualitatively very different from those due to Schwarzschild black holes. Gravitational lensing by rotating naked singularities was considered in bul (), and it was shown that the shift of the critical curves as a function of the lens angular momentum decreases slightly for the weakly naked, and vastly for the strongly naked singularities, with the increase of the scalar charge.

It is the purpose of the present paper to consider another observational possibility that may distinguish naked singularities from black holes, namely, the study of the properties of the thin accretion disks around rotating naked singularities and black holes, respectively. Thus, in the following we consider a comparative study of the physical properties of thin accretion disks around a particular type of rotating naked singularity, and rotating black holes, described by the Kerr metric, respectively. In particular, we consider the basic physical parameters describing the disks, like the emitted energy flux, the temperature distribution on the surface of the disk, as well as the spectrum of the emitted equilibrium radiation. Due to the differences in the exterior geometry, the thermodynamic and electromagnetic properties of the disks (energy flux, temperature distribution and equilibrium radiation spectrum) are different for these two classes of compact objects, thus giving clear observational signatures, which may allow to discriminate, at least in principle, naked singularities from black holes. We would like to point out that the proposed method for the detection of the naked singularities by studying accretion disks is an indirect method, which must be complemented by direct methods of observation of the ”surface” of the considered black hole/naked singularity candidates, and/or by the study of the lensing properties of the central compact object.

The mass accretion around rotating black holes was studied in general relativity for the first time in NoTh73 (). By using an equatorial approximation to the stationary and axisymmetric spacetime of rotating black holes, steady-state thin disk models were constructed, extending the theory of non-relativistic accretion ShSu73 (). In these models hydrodynamical equilibrium is maintained by efficient cooling mechanisms via radiation transport, and the accreting matter has a Keplerian rotation. The radiation emitted by the disk surface was also studied under the assumption that black body radiation would emerge from the disk in thermodynamical equilibrium. The radiation properties of thin accretion disks were further analyzed in PaTh74 (); Th74 (), where the effects of photon capture by the hole on the spin evolution were presented as well. In these works the efficiency with which black holes convert rest mass into outgoing radiation in the accretion process was also computed. More recently, the emissivity properties of the accretion disks were investigated for exotic central objects, such as wormholes, non-rotating or rotating quark, boson or fermion stars, brane-world black holes, type gravity models, and Horava-Lifshitz gravity Harko (). In all these cases it was shown that particular signatures can appear in the electromagnetic spectrum, thus leading to the possibility of directly testing different physical models by using astrophysical observations of the emission spectra from accretion disks.

The present paper is organized as follows. The geometry of the considered naked singularity is presented in Section II. In Section III we obtain the main physical parameters (specific energy, the specific angular momentum, and angular velocity) for massive test particles in stable circular orbits in stationary and axisymmetric spacetimes. The motion of test particles around rotating naked singularities is considered in Section IV. The frame dragging effect is analyzed in Section V. The properties of standard thin accretion disks are reviewed in Section VI. The energy flux, temperature distribution, and radiation spectrum from thin disks around naked singularities and Kerr black holes are discussed in Section VII. Some observational implications of our results are considered in Section VIII. We discuss and conclude our results in Section IX.

Ii Spacetime geometry of the naked singularity

As an example of a rotating naked singularity geometry we will consider in the following the Kerr-like solution of the Einstein gravitational field equations with a massless scalar field, , where is the scalar field, obtained in sol (). In the coordinate system , the line element, adapted to the axial symmetry, is given by

(1)

where

and is a constant.

The energy-momentum tensor of the massless scalar field generating the naked singularity is given by

(2)

The scalar field satisfies the equation , which follows from the conservation of the energy-momentum tensor, , and is given by

(3)

For this metric describes the spacetime geometry of a naked singularity, with a total mass , and an angular momentum . Here is the dimensionless spin parameter. It can be shown that the scalar curvature of this geometry diverges at a radius where the condition holds, which indicates the existence of a curvature singularity at the radius

(4)

Considering the angular dependence of this expression, we see that the minimal radial coordinate of the singularity is at the pole, , whereas it has a maximal radius in the equatorial plane, . The bigger root of the equation is

(5)

Since for any value of , is always less than or equal to , the singularity is not hidden behind the event horizon. It can also be shown that there is at least one null geodesic in the equatorial plane connecting the singularity with the future null infinity, i.e., the singularity is naked proof ().

For the line element describes the spacetime geometry of the Kerr black hole, where the vanishing horizon function, , has the roots . Then gives the horizon radius for . In the case of rotating black holes, the condition does not provide any singular surface, but the ergosphere located at .

We note that since for the metric (1) and , respectively, the frame dragging frequency of this rotating solution can be written as

(6)

For the Kerr black holes () we obtain the familiar expression , with .

In the equatorial approximation (), the components of the metric (1) reduce in the equatorial coordinate system to the form

(7)
(8)
(9)
(10)
(11)

where

The scalar field is given, in this approximation, by . In the equatorial plane the frame dragging frequency, given by Eq. (6), has the expression

(12)

where , and

respectively, with . For the Kerr solution (), reduces to the form NoTh73 (); PaTh74 ()

Iii Circular geodesic motion in stationary and axisymmetric spacetimes

Let us consider, in the coordinate system , an arbitrary stationary and axially symmetric geometry, with line element given by

(13)

The metric (13) is adapted to the symmetries of the spacetime, endowed with the time- and space-like Killing vectors and for time translations and spatial rotations, respectively. In the equatorial approximation (), and the metric functions , , , and in Eq. (13) depend only on the radial coordinate . Then the geodesic equations in the equatorial plane take the form

(14)

and

(15)

respectively, where is the affine parameter, and are the specific energy and specific angular momentum of the particles moving along the time-like geodesics, and the potential term is defined by

(16)

For circular orbits in the equatorial plane, the conditions and , respectively, must hold. These conditions determine the specific energy , the specific angular momentum and the angular velocity of particles moving on circular orbits as

(17)
(18)
(19)

Any stationary observer, moving along a world line and with a uniform angular velocity , has a four-velocity vector , which lies inside the surface of the future light cone. Therefore, the condition

(20)

appearing in Eqs. (17) and (18), provides the constraint for the angular velocity of the stationary observers, with

(21)
(22)

where the frame dragging frequency of the spacetime is defined by .

Only circular orbits for which the condition (20) holds do exist. The limiting case of this condition, , gives , the innermost boundary of the circular orbits for particles, called photon orbit. The circular orbits , for which the condition holds, are bound, and the condition gives the radius of the marginally bound circular orbit, i.e., the innermost orbits. The marginally stable circular orbits around the central object are determined by the condition

(23)

where the condition holds for all stable circular orbits. The marginally stable orbit is the innermost boundary of the stable circular orbits of the Keplerian rotation. By inserting Eqs. (17)-(18) into Eq. (23), and solving the resulting equation for , we obtain the marginally stable orbit, once the metric coefficients , and are explicitly given.

Iv Equatorial geodesic motion around rotating naked singularities

Inserting the metric components (7)-(9) into Eqs. (17)-(19), we obtain the specific energy, the specific angular momentum, and the angular velocity of the particles orbiting along circular geodesics in the equatorial plane of the rotating naked singularity. Hence , and can be written as

(24)
(25)
(26)

where

with . For we have and , and the latter formulae reduce to the familiar expressions

obtained for the equatorial approximation of the Kerr solution NoTh73 (); PaTh74 ().

If we compare the locations of the marginally stable, of the marginally bound, and of the photon orbits, plotted in Fig. 1 as functions of the spin parameter , we see that they have a strong dependence on the parameter .

Figure 1: Radii of circular equatorial orbits around a black hole (), and a naked singularity (, 0.7 and 0.5), with a total mass , as functions of the spin parameter . The dashed curves show the direct orbits, and the dotted curves represent the retrograde ones. The event horizon/singularity is denoted by a solid curve, while the horizontal dotted-dashed line at represents the ergosphere in the equatorial plane.

For static black holes ( and on the top left hand plot), the orbits with radii , and , are located at , and , respectively. As the spin increases from zero they pass through the ergosphere, and approach the even horizon at the maximal spin value . For (see the other three plots in Fig. 1), there is a critical value of the spin, where for each type of direct orbit the curves have a cut-off. The behavior of the retrograde orbits exhibits only a slight dependence on . The value of is minimal for the radius , and maximal for , e.g., it is and for (see the top right hand plot). After reaching this critical value of the rotation, the marginally stable, marginally bound and photon orbits ”jump” to the singularity, located at . If the rotation exceeds the value , corresponding to the marginally stable orbit, then all the radii , , and become degenerated. With increasing values of , these critical values are decreasing and even in the case of slow rotation we obtain at the singularity fully degenerated orbits.

Figure 2: The function versus for a rotating black hole, with (), and a naked singularity (, 0.7, and 0.5), with a total mass . The spin parameter is set to 0.99, 0.9, 0.7 and 0, respectively.

As an illustration of the dependence of on the spin and on the scalar charge parameter , we present in Fig. 2 the second order derivative of the effective potential with respect to the radial coordinate, given by Eq.  (23), as a function of the radius . Here the values of are the same as in Fig. (1), and the spin parameter takes different values. The top left hand panel shows the case of the black hole . For a static black hole vanishes at , and its zero shifts to lower radii with increasing spin parameter. For , is already approaching the static limit at , and for , the marginally stable orbit enters the ergosphere, and it approaches the event horizon. The other panels in Fig. 2, presenting for the naked singularity (), show that Eq. (23) has no longer solutions for higher values of the spin parameter, and remains negative everywhere. As a result, the particles rotating around the singularity have stable circular orbits in the whole spacetime, up to its boundary, i.e., to the singularity itself. With decreasing , the critical value of , above which has no zeros any more, is also decreasing. For , only very slowly rotating naked singularities have a marginally stable orbit, which does not coincide with the singularity.

V The frame dragging effect

An interesting question related to the geodesic motion in the gravitational potential of the rotating naked singularity is the frame dragging effect, i.e., the relation between the rotational velocity of the spacetime itself, and the azimuthal component of the four-velocity of the freely falling (or rotating) massive test particles. The frame dragging frequency at the singularity can be calculated from Eq. (12) as

(27)

where . From a physical point of view can be interpreted as the angular velocity of the singularity, . We note that the limit vanishes for , but it becomes unity for . The latter result is not trivial, but if it holds, we reobtain for the Kerr solution, since for . For the rotating naked singularity metric given by Eq. (1), the expression (27) reduces to

which is clearly valid only for a rotating singularity, since would diverge in the static case. Even if holds, the explicit expression for might be puzzling, since it does not depend on , and shows that the angular velocity of a rotating naked singularity is inversely proportional to its spin parameter. We would expect that the angular frequency of any rotating object to be proportional to the angular momentum of the body - and this is indeed true almost over the whole spacetime, except in a very small domain, close to the naked singularity. In a very close vicinity of the naked singularity, at , becomes bigger for lower values of , as compared to the values corresponding to higher spins. We can determine the radius at which the proportionality relation for is inverted by inserting Eq. (12) into the equation , with . The result is , which can be further simplified to the equation

(28)

For this equation reduces to the cubic equation , which has the triple real root , showing that, as expected, there is no inversion in the relation for Kerr black holes. For , the biggest real root of Eq. (28) provides the outer boundary of the region between and , where the frame dragging effect of the spacetime with the spin parameter is weaker than the frame dragging effect measured for a lower spin value . Outside this region, i.e., for all radii greater than , the relation holds. Fig. 3, where we plotted as a function of the radial distance for naked singularities with same mass and same scalar charge (), but different spin parameters, shows this behavior. In Fig. 3, the solid black line, representing the frame dragging frequency obtained for the naked singularity with the spin value of 0.99, has the highest values at radii greater than . For lower radii, a radius , where the curves for lower spin values intersect the black solid curve, always exists. The lower the spin value is, the closer shifts to . For radii less than , the frame dragging frequency for becomes higher than the value obtained for . This result holds for any pair of spin values, i.e., there exist radii , where for any value of , the curve , and the curve for , intersect each other. In all cases we obtain the relation for .

Figure 3: The radial profile of near the rotating naked singularity with , for different spin parameters.

A similar pathological behavior can be found for the angular velocity of massive test particles. For the first term of appearing in Eq. (24), as we approach the singularity we obtain the limit

(29)

For the angular velocity we obtain

(30)

By supposing that the naked singularity rotates fast enough, so that , then, if , for the orbits located close to the singularity decreases to zero. For , the angular velocity of the rotating particles has a non-vanishing value at the singularity, and it remains finite even in the static case, ( is supposed again). For , is proportional to , and for it diverges to infinity, as also does. Since at the singularity coincides with , the difference also diverges at for a vanishing spin parameter. However, this divergence does not involve any violation of causality, since the velocity of the orbiting particles, measured in the locally non rotating frame, does not exceed the speed of light, i.e., still holds, where

for circular equatorial orbits. At we have , and , a relation that provides the upper limit of the velocity for time-like particles that satisfy the condition (20) when moving along circular geodesics in the equatorial plane.

In Fig. 4 we have plotted , , and for the same values of and as we have used in Fig. 2. The top left hand panels show the frame dragging effect in the Kerr spacetime. When (the static case), for the orbiting particles can have a Keplerian rotation, with the frequency . At lower radii there are no circular orbits, and the particles fall freely onto the event horizon, with a velocity with azimuthal component restricted to values between and . For the static black hole at the even horizon these values tend to , i.e., there is no frame dragging. With increasing , the frame dragging effect becomes more and more important: as the orbital radius is approaching , the particle is dragged with the rotating spacetime. No static observer exists in the ergosphere, since any particle passing through the static limit is forced to rotate with a positive angular velocity. For rapid rotations , is also approaching the event horizon, where the particles move along circular orbits with the angular velocity of the horizon, namely, . At we obtain , as .

Figure 4: The radial profiles of , , and for black holes with and for the rotating naked singularity with , 0.9, 0.7, 0.5, 0.4, and 0.2, respectively. The spin parameter is set to 0.99, 0.7 and 0, respectively.

These effects are considerably different for rotating naked singularities, presented in the rest of the panels in Fig. 3. At , the lapse function for the metric vanishes, , as it does in the static limit for the black holes, and Eqs. (21) and (22) give and , without fixing the value of at the singularity. The result that at the singularity, depending on the values of , is either zero, or takes some non-zero values between 0 and , follows from Eq. (30). For relatively small scalar charges, , the static case is still similar to the static case of the black holes: the marginally stable orbit is located at around (the top right hand panel in Fig. 2), and for the orbits with radii greater than , the Keplerian rotation can still be maintained. At lower radii is restricted to values between and , and vanishes at the singularity (as it does at the event horizon of a Schwarzschild black hole). For rotating singularities with and , is still greater than , but the photon radius has jumped to the singularity (the top right hand panel in Fig. 1). Particles can therefore move along circular orbits in the whole spacetime, although not all these orbits are stable. However, at the singularity, located at , vanishes, even if has a finite, non-zero value. For , has already a non-zero value at the singularity, which is inversely proportional to , but it is still less than , as shown in the middle right hand panel in Fig  4. The panels in the bottom of Fig. 4 show that both and increase very rapidly close to the singularity - in fact exceeds the frame dragging frequency at as , and the relation holds for . The relation for is shown in Fig. 5, where we have plotted the ratio as a function of the radius. As mentioned before, for this ratio tends to at the singularity, the particles orbiting at have finite velocities, with the speed of the light like particles tending towards the speed of light.

Figure 5: The radial profiles of the ratio for (left hand panel) and (right hand panel), for different values of the spin parameter.

The angular velocity of particles moving along stable circular geodesics around black holes is maximal at the marginally stable orbit, and decreases monotonically, with the increasing coordinate radius . Eq. (30), and Fig. 4, show that this is not the case for the particles rotating in the gravitational potential of rotating naked singularities with . At the singularity drops to zero from its maximal value at , which is still close to the singularity (). By inserting Eq. (24) into the condition we obtain the equation

(31)

with the solution

(32)

For the limiting case of the Kerr black holes (), Eq. (32) has only one root, , or , which is due to multiplication of by in the computation of Eq. (31). At , the derivative of is not zero, indicating a monotonic decrease in the angular velocity with increasing radial coordinate. For Eq. (32) has two roots, where is the value of the radial coordinate where is maximal. Since Eq. (32) does not depend on the spin parameter , the dimensionless radius is only determined by the parameter of the singularity. The location of the marginally stable orbit still depends on , as in the case of Kerr black holes, and it decreases as the singularity rotates faster and faster. At orbits with radii higher than , the angular velocity is increasing with decreasing , but if is greater than , then there is an annulus between the inner edge of the accretion disk and , where for particles moving along circular orbits is decreasing for small distances from the singularity.

Vi Standard accretion disks

Accretion discs are flattened astronomical objects made of rapidly rotating gas which slowly spirals onto a central gravitating body, with its gravitational energy degraded to heat. A fraction of the heat converts into radiation, which partially escapes, and cools down the accretion disc. The only information that we have about accretion disk physics comes from this radiation, when it reaches radio, optical and -ray telescopes, allowing astronomers to analyze its electromagnetic spectrum, and its time variability. The efficient cooling via the radiation over the disk surface prevents the disk from cumulating the heat generated by stresses and dynamical friction. In turn, this equilibrium causes the disk to stabilize its thin vertical size. The thin disk has an inner edge at the marginally stable orbit of the compact object potential, and the accreting plasma has a Keplerian motion in higher orbits.

For the general relativistic case the theory of mass accretion around rotating black holes was developed by Novikov and Thorne NoTh73 (). They extended the steady-state thin disk models introduced in ShSu73 () to the case of the curved space-times, by adopting the equatorial approximation for the stationary and axisymmetric geometry. For a steady-state thin accretion disk described in the cylindrical coordinate system most of the matter lies close to the radial plane. Hence its vertical size (defined along the -axis) is negligible, as compared to its horizontal extension (defined along the radial direction ), i.e, the disk height , equal to the maximum half thickness of the disk, is always much smaller than the characteristic radius of the disk, . The thin disk is in hydrodynamical equilibrium, and the pressure gradient and a vertical entropy gradient in the accreting matter are negligible. In the steady-state accretion disk models, the mass accretion rate is supposed to be constant in time, and the physical quantities of the accreting matter are averaged over a characteristic time scale, e.g. , and over the azimuthal angle , for a total period of the orbits and for the height . The plasma moves in Keplerian orbits around the compact object, with a rotational velocity , and the plasma particles have a specific energy , and specific angular momentum , which depend only on the radii of the orbits. The particles are orbiting with the four-velocity in a disk having an averaged surface density . The accreting matter is modeled by an anisotropic fluid source, where the rest mass density (the specific internal energy is neglected), the energy flow vector and the stress tensor are measured in the averaged rest-frame. The energy-momentum tensor describing this source takes the form

(33)

where , . The four-vectors of the energy and of the angular momentum flux are defined by and , respectively. The four dimensional conservation laws of the rest mass, of the energy and of the angular momentum of the plasma provide the structure equations of the thin disk. From the structure equations the flux of the radiant energy over the disk can be expressed as (PaTh74, ; Th74, )

(34)

where the no-torque inner boundary conditions were prescribed PaTh74 (). This means that the torque vanishes at the inner edge of the disk, since the matter at the marginally stable orbit falls freely into the black hole, and cannot exert considerable torque on the disk. The latter assumption is valid as long as strong magnetic fields do not exist in the plunging region, where matter falls into the hole.

The geometry of the space-time near to the equator, or the metric potential determines the radial dependence of , and for the particles moving on circular orbits around the central object. We can therefore calculate the averaged radial distribution of photon emission for accretion disks around the rotating singularity in the equatorial approximation, by applying the flux integral Eq. (34). Evaluating of the specific energy at the inner edge of the disk, we can also determine the efficiency of conversion of the rest mass into outgoing radiation.

The accreting matter in the steady-state thin disk model is supposed to be in thermodynamical equilibrium. Then the radiation emitted by the disk surface can be considered as a perfect black body radiation, where the energy flux is given by ( is the Stefan-Boltzmann constant) and the observed luminosity has a redshifted black body spectrum To02 ():

(35)

Here is the distance to the source, is the Planck distribution function, is the disk inclination angle (we set it to zero), and and indicate the position of the inner and outer edge of the disk, respectively. We take and , since we expect the flux over the disk surface vanishes at for any kind of general relativistic compact object geometry. The emitted frequency is given by , where the redshift factor can be written as

(36)

where we have neglected the light bending Lu79 (); BMT01 ().

The efficiency with which the central object converts rest mass into outgoing radiation is the other important physical parameter characterizing the properties of the accretion disks. The efficiency is defined by the ratio of two rates measured at infinity: the rate of the radiation of the energy of the photons escaping from the disk surface to infinity, and the rate at which mass-energy is transported to the compact object. If all the emitted photons can escape to infinity, the efficiency depends only on the specific energy measured at the marginally stable orbit ,

(37)

For Schwarzschild black holes the efficiency is about 6%, no matter if we consider the photon capture by the black hole, or not. Ignoring the capture of radiation by the black hole, is found to be 42% for rapidly rotating black holes, whereas the efficiency is 40% with photon capture in the Kerr potential.

Vii Electromagnetic signatures of accretion disks around rotating naked singularities

After analyzing, in Sections IV and V, the circular geodesic motion around a rotating naked singularity, we are now ready to discuss the properties of the disk radiation for standard accretion disk models in the spacetime of the naked singularity. In Fig. 6 we present the flux profile, calculated from Eq. (34), for the physical parameters of the configurations already shown in Fig. 2. In the following we set the total mass to , and the accretion rate to /yr. The top left hand panel shows for the static and the rotating black holes. The dependence of the flux distribution on the spin has distinct features: the inner edge of the disk is located at for the static black hole, and shifts to lower radii, approaching , as the black hole spins up to . The radii of the marginally stable orbits are determined by the zeros of in Fig. 2. With increasing spin, the maximal flux is also increasing with at least three orders of magnitude, as compared to the cases with and . For higher values of the spin, the locations of the maxima of the spectra also shift to lower radii, located closer to the inner edge of the disk.

Figure 6: The energy flux radiated by the disk around a rotating black hole (), and a naked singularity (, 0.7, 0.5, 0.4 and 0.2), with the total mass , for different values of the spin parameter . The mass accretion rate is set to /yr.

The rest of the panels in Fig.  6 shows the flux distribution of the thermal radiation of the accretion disks around naked singularities. Depending on the values of the scalar charge parameter , and of the spin parameter , naked singularities and black holes could exhibit either similar, or rather different properties. For , we obtain flux profiles similar to the corresponding cases of the black holes, with the same spin, provided that the equation has a real solution. The top right hand panel in Fig. 2 shows that for , 0.4 and 0.7 there exists a radius where the second order derivative of the effective potential vanishes, whereas for and 0.99 in the whole spacetime. Hence for , has similar characteristics to the flux profiles derived in the case of the black holes for , 0.4 and 0.7., but the flux distribution around the black hole and the naked singularity is rather different for higher spin values.

In the slowly rotating case, when the inner edge of the disk is located at the marginally stable orbit, the radial profiles of have the same shape in both the top left and the right hand side plots, and the flux maxima are somewhat smaller for the black holes as compared to flux maxima for the naked singularity. This difference is enhanced by the fast rotation of the central naked singularity. Since stable circular orbits around the naked singularity with and exist in the whole equatorial plane, the inner edge of the accretion disk reaches the singularity. The radius is bigger than for the rotating black holes with the spin of 0.9 or 0.99, and therefore the surface of the accretion disk around the rotating naked singularity is smaller than the one for its Kerr black hole counterpart. Even this smaller disk surface is not the whole area that can radiate thermal photons in the thin accretion disk model, since the left end of the flux profile is pushed to a radius somewhat higher than . This is due to the fact that changes its sign at , as seen in Fig. 4, and this also changes the sign of the cofactor of the integral in the flux formulae, given by Eq.  (34). As a result, we obtain negative flux values for the radii between and , (since the sign of the integral remains the same at ).

The negative flux values involve non-physical states in the framework of the stationary thin disk model. It indicates that the thermodynamical equilibrium cannot be maintained in this region and other forms of the energy and momentum transport become dominant over the radiative cooling such as advection or convection. Then the approximation of the steady-state and geometrically thin disk model breaks down here. Only outside this region, which forms a very thin annulus between and though, one can apply the standard accretion disk scheme. We assume the thin disk exists only in the region for , where the radiation gives the main contribution for energy and momentum transport, and other forms of the transport processes becomes negligible.

Here we consider the radiation properties only for the geometrically thin, relatively cold disk, which is truncated at , and we discuss the contribution of the hot matter in the region to the total radiation of the disk-naked singularity system in the next Section.

Thus we assume that the accretion disk emits thermal photons only at radii higher than , and that for the photon flux can be computed in the thin disk approximation. Since does depend only on , but not of , the left edge of is located always at the same radii, irrespectively of the rotational speed of the naked singularity. In spite of this reduction in the effective radiant area of the disk surface, as compared with the Kerr black hole case, with the same spin parameters (0.9 and 0.99), the flux maxima is much higher for the rotating naked singularity as compared to the black hole case. The two plots on the top of Fig. 6 show that the rotating naked singularity with has a disk ten times more luminous than the disk of a Kerr black holes with the same spin. Comparing the rapidly rotating naked singularity with its static case we also see a rise of 3 orders of magnitude in the flux maximum, whereas this rise is only 2 orders of magnitude for the black hole. Thus, is much more sensitive for the variation in in the case of the naked singularity. The maximal flux is somewhat higher for than for , which indicates that the maximum of for fast rotation is inversely proportional to the spin parameter.

The fact that the flux maximum is higher for the rotating naked singularity than for the black hole, even if it is integrated over a smaller surface area, is the consequence of the considerable difference in the metric determinant, characterizing the four-volume element in which the radiant flux is measured in the vicinity of the equatorial plane. For Kerr black holes holds in the equatorial approximation, but from the metric (7)-(11) of the rotating naked singularity we obtain . Since the shape factor vanishes as , the function has a small value at , which is close to . Then the four-volume element is much smaller for the naked singularity than for the Kerr black hole, and it produces much higher values in the flux integral (34), even if the disk properties determining , and are similar in the two cases, as far as we integrate from in the case of the naked singularity (e.g., in the two plots in the top in Fig. 4, the values of have only a moderate difference for the naked singularity and the black hole).

We obtain similar trends in the characteristics of the radiated flux if we further decrease the values of . By considering the plots for or 0.5 in Fig. 2, one can see that the equation has a real solution only for and (), or only for the static case (). For higher values of , the second order derivative of the effective potential remains negative in the whole equatorial plane, and the inner edge of the disk always jumps to . This effect is shown by the two middle plots in Fig. 6, where the flux profiles for the naked singularity can be separated into the two groups (similarly to the case of ), formed by the curves similar to those obtained for black holes ( has a real solution), and the curves with higher maxima, located almost at their left edge ( holds everywhere). The high flux values obtained for the second group are again the consequence of the rapidly shrinking volume element in the vicinity of the singularity. We also see that the flux maxima for the latter group is inversely proportional to the spin parameter, but the curves fall more rapidly for lower spin.

Figure 7: The temperature distribution on a disk around a rotating black hole (), and a naked singularity (, 0.7, 0.5, 0.4 and 0.2), with the total mass of , for different values of the spin parameter . The mass accretion rate is set to /yr.

Since approaches zero at the singularity for , the inner boundary of the radiant disk area is located at , where we obtain finite flux values for . For , has a non zero value at , and it is a monotonous function in the entire equatorial plane. Therefore, we cannot use as a truncation radius for the thin disk, and the inner edge of the disk would reach the singularity, where the volume element shrinks to zero, and the cofactor of in the flux formula (34) goes to infinity. Their multiplication for causes to remain finite at the singularity, as seen in Fig.  6. For all non-static cases, plotted in the middle right hand plot in Fig. 6, the peaks at the left edge of the flux profiles have infinite amplitudes, which cannot represent physical states. The thin diks model is not a good approximation at the inner edge of the disk, as the steady state of the disk cannot be maintained close to the singularity. We can still state that the accretion disk must be extremely bright in its innermost area, but there must be some other physical mechanisms besides the radiative cooling, such as advection, which have an important role in the energy and angular momentum transport.

For , holds at , and will diverge as approaches zero. Its derivative with respect to also tends to infinity, also giving, together with the vanishing volume element at , an infinite flux value in Eq. (34). This effect is demonstrated by the bottom plots in Fig. 6. Since and its derivative is inversely proportional to , the rate of the rise of the flux profiles as approaching is also inversely proportional to . Thus, the slowly rotating naked singularities have brighter accretion disks than their fast rotating counterparts do have, but in both cases tends to infinity at the inner edge of the disk. Without any upper boundary in the flux maxima at the singularity, the physical state of the matter in motion close to the singularity will recede from the thermodynamical equilibrium. The standard accretion disk model cannot be applied at such small distances from the singularity. At higher radii there must exists a region, where the disk matter can attain thermodynamical equilibrium. If this radius is still close enough to , the flux profiles, shown in the bottom plots of Fig. 6, still resemble the real physical situation, with some uncertainties in the position of their left edges. Obviously, in a physical situation, the amplitudes of the peaks must have some finite values, but these maxima are still much higher for rotating naked singularities as compared with those calculated for disks rotating around Kerr black holes.

In Fig. 7 we present the temperature distribution of the accretion disk for the same configurations which we have used to study the flux profiles. The disk temperature exhibits a similar dependence on the parameters and as does. With decreasing , and increasing , we obtain temperature profiles with much higher and sharper maxima than those for the Kerr black hole. For , there still exist configurations with lower spin, which give radial temperature profiles similar to those obtained for the Kerr black holes. Although these temperature profiles become uncertain in the innermost region of the accretion disk, the disk must still be extremely hot in this region, as compared to the typical disk temperatures obtained for Kerr black holes, with the same spin values.

Figure 8: The accretion disk spectra for a rotating black hole (), and a naked singularity (, 0.7, 0.5, 0.4 and 0.2), with the total mass of , for different values of the spin parameter . The mass accretion rate is set to /yr.

In Fig. 8 we have also plotted, for the same set of values of the parameters and , the disk spectra, which were calculated from the luminosity formula Eq. (35). These plots show the same trends found in the behavior of the radiated flux distribution , and of the disk temperature in the black hole and naked singularity geometries, respectively. For the Kerr black hole, with , the cut-off frequency of the spectra shifts towards higher values, and the maximal amplitudes increase with the increasing spin parameter, i.e, the accretion disks of Kerr black holes become hotter by rotating faster, and they produce a blueshifted surface radiation with higher intensity. In the case of the rotating naked singularity, the two classes already identified in the previous discussion of the radial distribution of and of preserve their different natures. We have seen that there is only a moderate decrease in the disk temperature for the static solution, and for rotating naked singularities with and with spin of and 0.7, as compared to the Kerr black hole with the same spin values. As a result, the disk spectra exhibit only negligible differences for black holes and naked singularities. For fast rotation ( and 0.99), the accretion disk is much hotter in the area close to the singularity. Hence, the cut-off frequencies shift blueward, and the maximal amplitudes of the spectra are much higher then for the black hole disk spectra. The spectral features for the second group are not sensitive to the variations in the spin, and the relative shifts in the cut-off frequencies and the spectral maxima are very small, as shown in the bottom right hand plot in Fig. (8).

In Table 1 we present the conversion efficiency of the accreted mass into radiation for both rotating naked singularities and black holes, for different values of the parameters and . This Table demonstrates that with increasing spin parameter, and decreasing (increasing scalar charge), the efficiency is increasing to a maximal value, and then it starts to decrease again. The rate of the increase and of the decrease depends on the location of the inner edge of the disk, as can be seen in Eq. (37). The behavior of the two groups, with the inner disk edge at and , respectively, is the same in this respect, but varies in different ranges for the two groups. For the configurations with inner disk edge located at , the increase in causes only a moderate variation in . While the efficiency for the slowly rotating case is somewhat greater for the naked singularity, the rapidly rotating black holes have a considerable higher efficiency than the naked singularities, with the same spin parameter, do have. Comparing the first two lines of the table we see that the static and the slowly rotating configurations have the same efficiencies, whereas the accreted mass-to-radiation conversion mechanism is about four times more efficient for extremely rapidly rotating black holes then for the fast spinning naked singularities, with . We find relatively small values in the first two columns of the third line, and in the first column of the fourth line as well. All the values in the last four lines belong to the second group of naked singularities, for which the inner disk edge is located at the singularity (indicated with the same values in the parenthesis in each line as , and which does not depend on ). In these cases the accretion disk radiates a great amount of thermal energy, and the values of the efficiency are considerably higher than those found for the first group. For the static cases the efficiency can reach 50% for , but it drops to 40%, as we decrease to 0.2. With increasing spin, has a very mild increase, reaching the values 70% () and 61% () at , and then it starts to decrease for faster rotating singularities. For the naked singularity with the spin of and , the efficiency falls to 34%, which is still higher than the efficiency for extremely fast rotating Kerr black holes. We can conclude that there is a range of the spin parameter , and of the scalar charge , where Kerr black holes represent more effective engines in the conversion of accreted mass to radiation than the naked singularities do. Nevertheless, we have also found another range of these parameters, where the conversion efficiency for rotating naked singularities is much higher than for Kerr black holes.

1.00 5.72% (6.00) 7.51% (4.62) 10.4% (3.40) 15.6% (2.32) 26.4% (1.46)
0.99 5.72% (6.00) 7.51% (4.62) 10.4% (3.37) 6.03% (2.02) 5.92% (2.02)
0.70 6.03% (6.15) 8.92% (4.14) 73.4% (2.86) 70.8% (2.86) 69.7% (2.86)
0.50 6.94% (6.00) 70.6% (4.00) 65.1% (4.00) 61.8% (4.00) 60.4% (4.00)
0.40 50.7% (5.01) 61.3% (5.01) 56.4% (5.01) 53.2% (5.01) 51.8% (5.01)
0.20 40.8% (10.0) 41.6% (10.0) 38.0% (10.0) 35.3% (10.0) 34.3% (10.0)
Table 1: The efficiency, measured in percents, and (values in parenthesis) at the inner edge of the accretion disk for Kerr black holes, and rotating naked singularities, respectively. The value of corresponds to the Kerr black hole.

Some composite accretion disk models consider a geometrically thick and optically thin hot corona, positioned between the marginally stable orbit and the inner edge of the geometrically thin, and optically thick, accretion disk, where an inner edge is lying at a few gravitational radii (ThPr75 (). In other type of composite models, the corona lies above and under the accretion disk, and the soft photons, arriving from the disk, produce a hard emission via their inverse comptonization by the thermally hot electrons in the corona LiPr77 (). In both configurations, the disk is truncated at several gravitational radii - reducing the soft photon flux of the disk - and the soft and hard components of the broad band X-ray spectra of galactic black holes were attributed to the thermal radiation of the accretion disk and the emission mechanism in the corona, respectively.

Although the metric potential gives stable circular geodesic orbits in the entire equatorial plane of the spacetime, it cannot guarantee that the matter in motion can maintain configurations with thermodynamical equilibrium in the whole region, which is the basic assumption of the geometrically thin steady state accretion disk model. A natural limit of the region where the standard accretion disk model applies seems to be the radius , where the Keplerian rotational frequency becomes inversely proportional to the radial distance from the naked singularity. This inversion in the versus relation indicates that the structure equations of the Novikov-Thorne disk model have non-physical solutions in the region below . Since this limiting radius depends only on the parameter and the geometrical mass via Eq. (32), the lowest radial limit of the validity of the thin disk approximation is not a free parameter, but it is determined by the charge parameter of the massless scalar field .

Viii Observational implications

An interesting effect, involving the Eddington luminosity for the case of a boson star, was pointed out in To02 (). The Eddington luminosity, a limiting luminosity that can be obtained from the equality of the gravitational force inwards and of the radiation force outwards, is given by erg/s, where is the proton’s mass, and is the Thompson cross section ste (). Since a boson star- as well as any other transparent object - has a non-constant mass distribution, with , and therefore the Eddington luminosity becomes a coordinate dependent quantity, . A similar effect occurs for the case of the naked singularity solution considered in the present paper. For simplicity in the following we will consider only the case of the static naked-singularity. One can associate to the scalar field described by this energy-momentum tensor a mass distribution along the equatorial plane of the disk, given by . By using the explicit form of the scalar field we obtain

(38)

The corresponding coordinate-dependent Eddington luminosity can be obtained as

(39)

The variation of the Eddington luminosity with respect to the coordinate is represented, for a naked singularity with , in Fig. 9.

Figure 9: Coordinate dependent Eddington luminosity as a function of the radial coordinate for the case of a static naked singularity with total mass , and for different values of .

It was argued in YuNaRe04 () that any neutron star, composed by matter described by a more or less general equation of state, should experience thermonuclear type I bursts at appropriate mass accretion rates. The question asked in YuNaRe04 () is whether an ”abnormal” surface may allow such a behavior. The naked singularities may also have such a zero velocity, particle trapping, abnormal surface. The presence of a material surface located at the singularity implies that energy can be radiated, once matter collides with that surface. One can also consider composite accretion disk models as an alternative solution to this problem, and set the truncation radius of the thin disk at . The hot corona of the composite model can lie in the region between and , producing hard X-ray spectra via the inverse Compton scattering of photons radiated from the disk and the electron gas in the corona. This hot corona could represent the ”surface” of the naked singularity. Thus, at least in principle, naked singularity models, characterized by high mass, normal matter crusts/surfaces and type I thermonuclear bursts can be theoretically constructed.

Since naked singularities could be surrounded by a thin shell of matter, the presence of a turning point for matter (the point where the motion of the infalling matter suddenly stops) at the surface of the naked singularity may have important astrophysical and observational implications. Since the velocity of the matter at the naked singularity surface is zero, matter can be captured and deposited on the surface of the naked singularity. Moreover, because matter is accreted continuously, the increase in the size and density of the surface will ignite some thermonuclear reactions YuNaRe04 (). The ignited reactions are usually unstable, causing the accreted layer of gas to burn explosively within a very short period of time. After the nuclear fuel is consumed, the naked singularity also reverts to its accretion phase, until the next thermonuclear instability is trigge