Comparing thin accretion disk properties of naked singularities and black holes

The resources for understanding astrophysical phenomena have increased significantly in the wake of recent experimental breakthroughs in astrophysical observations, such as gravitational wave detection by the LIGO & Virgo Collaboration LIGOScientific:2016aoc and observation of the black hole image in Galaxy M87 EventHorizonTelescope:2019dse ; EventHorizonTelescope:2019uob ; EventHorizonTelescope:2019jan ; EventHorizonTelescope:2019ths ; EventHorizonTelescope:2019pgp ; EventHorizonTelescope:2019ggy by the Event Horizon Telescope group. Meanwhile, the next generation Event Horizon Telescope (ngEHT) group is also attempting to capture the sharpest image of the galactic center of our Milky Way. It is widely believed that the compact object at the center of almost every galaxy is a supermassive black hole. Nevertheless, there are numerous astrophysical compact objects that can replicate the observational properties of black holes, such as naked singularities, gravastars, wormholes, and so on. In Vagnozzi:2019apd ; Shaikh:2019hbm ; Gyulchev:2019tvk ; Shaikh:2019fpu ; Dey:2013yga ; Dey+15 ; Shaikh:2018lcc ; Joshi2020 ; Paul2020 ; Dey:2020haf ; Dey:2020bgo ; bambi_2013a ; ohgami_2015 , authors have shown that apart from the black holes, different horizon-less compact objects can also produce shadows.

In general relativity, the dynamical equations describing gravitational collapse do not require that the eventual end state of the collapse of a massive matter cloud must be a black hole. Depending upon the initial conditions with which the collapse is initiated, the gravitational collapse of a massive star may ultimately result in a black hole or a naked singularity. As we know, the cosmic censorship conjecture rules out the possibility of horizon-less compact objects penrose . However, several pieces of literature propose that the continuous gravitational collapse of an inhomogeneous matter cloud can result in a naked singularity eardley ; christodoulou ; joshi ; goswami ; joshi2 ; vaz ; jhingan0 ; deshingkar ; mena ; magli1 ; magli2 ; giambo1 ; harada1 ; harada2 ; joshi7 ; mosani1 ; mosani2 ; mosani3 ; mosani4 ; mosani5 . The theoretical investigations alone are not sufficient to confirm whether the naked singularities exist or not in nature. Therefore, studying their distinct observational properties is necessary to distinguish them from black holes. Moreover, the causal structure of the singularity also remains a mystery. As per general relativity, spacetime singularities can be classified into three categories based on their causal structure: spacelike, nulllike, and timelike. A spacelike singularity is causally detached from other spacetime points, while nulllike and timelike singularities are causally related to the other points of spacetime. The difference in the spacetime geometries around black holes and naked singularities may produce differences in their observable signatures, allowing us to discriminate between these two classes of compact objects. The causal nature of the compact object can also be probed by the orbital dynamics of the stars orbiting that compact object. There are several papers on the detailed study of the orbital precession of timelike geodesics around various compact objects Dey:2020haf ; Bambhaniya:2019pbr ; Dey:2019fpv , where it is demonstrated that the orbital motion around a naked singularity can be substantially different from the orbital motion around a Schwarzschild black hole of the same mass.

The availability of an abundance of data in the electromagnetic realm has piqued the interest of researchers all around the world because such a study might help us better comprehend the nature of compact objects at the centers of active galactic nuclei (AGN) or in the X-ray binaries. In this paper, we attempt to distinguish naked singularities from black holes by investigating the electromagnetic properties of the thin accretion disk around them. The majority of the astrophysical structures grow via accretion. The particles orbiting an astrophysical body lose angular momentum and energy as they flow in, which prevents them from directly falling into the central body and settling into a disk-like structure. The first explicit non-relativistic study of the accretion disks around compact objects was given by Shakura and Sunyaev Shakura:1973 . Later, Novikov and Thorne Novikov , and Page and Thorne Page:1974he investigated the general relativistic model of the accretion disk. There is much literature in which the electromagnetic properties of the accretion disks have been studied for different compact objects Liu:2020vkh ; Liu:2021yev ; Joshi:2013dva ; Bambhaniya:2021ugr ; Rahaman:2021kge ; Harko:2008vy ; Harko:2009xf ; Kovacs:2010xm ; Harko:2009gc . In Guo:2020tgv ; Chowdhury:2011aa , the authors have demonstrated that the thermal properties of the accretion disk around a naked singularity can differ considerably from those of a black hole of the same mass.

The matter density around the central supermassive object in the galaxy is very high. Hence, describing its surrounding geometry using vacuum solutions of Einstein’s field equations may not always yield a good approximation. To account for the effect of the surrounding matter on the galactic geometry, here we consider two non-vacuum solutions, namely, NNS and JMN1 spacetimes. These spacetimes contain the matter distribution around the central singularity. We carry out a comparative study of the accretion properties of the thin accretion disk surrounding a Schwarzschild black hole, an NNS Joshi:2020tlq , and a JMN1 naked singularity Joshi:2011zm . The Schwarzschild and NNS spacetime have, respectively, a spacelike and nulllike singularity at their center. On the other hand, JMN1 spacetime possesses a timelike or nulllike singularity at its centre, depending on the parameter space. Hence, we particularly try to analyse the different astrophysical signatures provided by the radiation emitted by accretion disks around spacelike, nulllike, and timelike singularities. We study the radiated energy flux, spectral luminosity distribution, accretion efficiency, and the temperature distribution on the disk surface for equally massive black holes and naked singularities. From the analysis of the radiating flux and spectral luminosity, we show that the accretion disks surrounding naked singularities are more luminous than that of the black holes. In the Novikov-Thorne accretion disk model, several reasonable assumptions are made. As one of the assumptions, in general, a “zero-torque” condition is considered at the inner boundary of the accretion disk to compute the time-averaged flux Novikov ; Page:1974he , assuming that there is a minimal quantity of material inside the innermost circular orbit (ISCO) that can “torque up” the material immediately outside ISCO. This nearly vanishing torque could be a reasonable assumption for spacetimes possessing ISCO. However, if spacetime has circular orbits all the way up to the zero radial distance, there is a possibility of the existence of non-zero torque owing to the viscosity between the fluid layers near the inner edge induced by the strong gravity near the singularity. This torque might be minimal, but it may have some non-zero value. To address this possibility, we also examine the differences observed in the spectra of the emerging radiation when a non-zero torque value is employed at the inner edge of the disk.

The outline of the paper is as follows. In Sec. II, we introduce the naked singularity and black hole spacetimes. Then we obtain the physical parameters of the circular orbits such as specific energy, specific angular momentum, and angular velocity in static and spherically symmetric spacetimes in section III. In Sec. IV, we review the physical properties of the Novikov-Thorne accretion disks and obtain the equation of flux for the spacetimes with the matching hypersurface. The radiating flux, spectral distribution, accretion efficiency, and temperature distribution for black hole and naked singularity spacetimes are discussed in Sec. V. We conclude our results in Sec. VI. We choose the system of units such that $c=h=G=\sigma=k=1$, where $c$ is the speed of light, $h$ is the Planck’s constant, $G$ is the Newton’s gravitational constant, $\sigma$ is the Stefen-Boltzmann constant, and $k$ is the Boltzmann constant. Here we use $(-,+,+,+)$ sign convention.

The line element for a general spherically symmetric and static spacetime is given by

$$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)d\Omega^{2}\,\,.$$

(5)

Due to the spherical symmetry of spacetime, we can consider $\theta=\pi/2$, ensuring that the test particle moves along the timelike geodesic in the equatorial plane. Since spacetime has temporal and spherical symmetries, the energy per unit rest mass ($E$) and the angular momentum per unit rest mass ($L$), respectively, of the particle orbiting the compact object are conserved along the geodesic. These are defined as

$$E=A(r)\,\dot{t}\,,\,\,\,\,\,L=C(r)\,\dot{\phi}\,.$$

(6)

Here, an overdot represents the derivative with respect to the proper time, $\tau$. Using above equations and the timelike geodesics condition ($g_{\mu\nu}u^{\mu}u^{\nu}=-1$) for equatorial plane, we can write,

$$AB\,\dot{r}^{2}+V_{eff}(r)=E^{2},$$

(7)

and the effective potential $V_{eff}(r)$ is defined as

$$V_{eff}(r)=\left(1+\frac{L^{2}}{C(r)}\right)A(r).$$

(8)

For stable circular orbits in the equatorial plane, the condition $V_{eff}=E^{2}$, $V^{\prime}_{eff}=0$ and $V^{\prime\prime}_{eff}>0$ must hold. Here the $(^{\prime})$ denotes the differentiation with respect to the coordinate $r$. From the first two conditions, it is easy to obtain

	$\displaystyle E$	$\displaystyle=$	$\displaystyle\frac{A}{\sqrt{A-C\mathcal{W}^{2}}},$		(9)
	$\displaystyle L$	$\displaystyle=$	$\displaystyle\frac{C\mathcal{W}}{\sqrt{A-C\mathcal{W}^{2}}},$		(10)
	$\displaystyle\mathcal{W}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{A^{\prime}}{C^{\prime}}}\,,$		(11)

where, $\mathcal{W}=d\phi/dt$ is the orbital angular velocity of the test particle. The radius of the marginally stable circular orbit $r_{ms}$ (or innermost stable circular orbit) of the particle orbiting the compact object can be found out using the condition

$$V^{\prime\prime}_{eff}=0.$$

(12)

The marginally stable orbit is the innermost stable circular orbit in which a particle can stably revolve around a massive object.

Using the conditions for circular orbits and Eq. (12), we find that the marginally stable circular orbit is located at $r_{ms}=6M_{T}$ for the Schwarzschild black hole. Similarly, we confirm that NNS and JMN1 spacetimes do not have an innermost stable circular orbit; therefore, circular orbits are possible from large radii up to $r\to 0$.

The comparative study of the properties of stable circular orbits $E$, $L$, and $\mathcal{W}$ for the NSS, JMN1, and Schwarzschild spacetimes is shown in Fig. (1). The matching boundary of JMN1 is set to $r_{b}=6.667$, where it matches with the exterior Schwarzschild spacetime. The total mass $M_{T}$ is set to $1$ throughout the paper. Figs. (1) and (1) represent the energy per unit rest mass $E$ and angular momentum per unit rest mass $L$ as a function of radius $r$. In NNS and JMN1, both $E$ and $L$ go to zero as $r\to 0$. The behaviour of $E$ and $L$ in NNS and JMN1 at a large distance from the centre resembles that of the Schwarzschild black hole. For the Schwarzschild black hole, the lowest values of $E$ and $L$ are at $r_{ms}=6M_{T}$, and below $r=6M_{T}$, $E$ and $L$ increase as $r$ decreases, which is the result of the unavailability of stable circular orbits.

The corresponding results for angular velocity $\mathcal{W}$ are shown in Fig. (1). The distinguished behaviour of circular orbit properties in naked singularities is also expected to be reflected in the spectral properties of their accretion disks, which are discussed later in this paper.

We consider the geometrically thin and optically thick accretion disk model described by Novikov-Thorne Novikov ; Page:1974he . In this section, we first briefly review some of the physical properties of the Novikov-Thorne accretion disk obtained in Page:1974he . Then we obtain the equation for energy flux for the case when the interior spacetime is matched with the exterior spacetime at a matching radius.

The massive particles move in nearly circular orbits in the accretion disk. For naked singularities considered here, we assume that the test particle orbiting inside the matter cloud is influenced only by the cloud’s gravity and has no interaction with the matter cloud that constitutes naked singularity spacetime. The accretion disk is optically thick if the free mean path of the scattering photons $l$ is much smaller than the height of the disk $H$, described by the maximum half-thickness of the disk, $l<<H$ 2017bhlt.book…..B . Here, we consider a thin accretion disk whose vertical size is negligible in comparison to its horizontal extension, i.e., the disk height $H$, is considerably less than its characteristic radius $R$, $H<<R$ Novikov ; Page:1974he . This means that most of the matter lies in the equatorial plane. In this case, the vertical size specified along the z-axis in cylindrical coordinates $(r,\phi,z)$ is insignificant. The efficient cooling provided by radiation across the disk surface keeps the disk from accumulating heat produced by stresses and dynamical friction. Due to this cooling process, the disk remains in hydrodynamic equilibrium. In a steady-state accretion disk model, the mass accretion rate, $\dot{m}$, is considered to be constant and does not vary with time. The inner edge of the accretion disk is at a marginally stable circular orbit $r_{ms}$, and the accreting matter at the higher radii follows Keplerian motion. The physical characteristics defining the accreting matter are averaged over a specific time scale, such as $\Delta t$, the entire period of the orbits, the azimuthal angle $\Delta\phi=2\pi$, and the disk height $H$. The particles in the accretion disk move with a rotational velocity $\mathcal{W}$, specific energy $E$, and specific angular momentum $L$ that depends entirely on the radial coordinate $r$. The accreting particles orbit with the four-velocity $u^{\mu}$ and constitute the disk of average surface density $\Sigma$, which is described as

$$\Sigma=\int_{-H}^{H}<\rho_{0}>dz\,.$$

(13)

Here $\rho_{0}$ represents the rest mass density, and $<\rho_{0}>$ denotes the average of rest mass density over time $\Delta t$ and angle $2\pi$. The energy-momentum tensor describing the source of orbiting particles is an anisotropic fluid source described as

$$T^{\mu\nu}=\rho_{0}u^{\mu}u^{\nu}+2u^{(\mu}q^{\nu)}+t^{\mu\nu},$$

(14)

where $u_{\mu}q^{\mu}=0$ and $u_{\mu}t^{\mu\nu}=0$. The quantities $q^{\mu}$ and $t^{\mu\nu}$ are the energy flow vector and the stress tensor, respectively, which are measured in the averaged rest-frame of the particle orbiting in the disk. Specific heat is not taken into account in this case. One of the important properties of the time-averaged disk structure is the energy flux $\mathcal{F}(r)$ across the disc surface, which can be calculated by averaging the energy flow vector over time and the azimuthal angle given by,

$$\mathcal{F}(r)=<q^{z}>.$$

(15)

Another significant quantity of the radial structure of the disk is the time-averaged torque due to the stresses in the disk, and it is defined as

$$W_{\phi}^{r}=\int_{-H}^{H}<t_{\phi}^{r}>dz\,.$$

(16)

Here $<t_{\phi}^{r}>$ denotes the time and orbital averaged $\phi$-$r$ component of the stress tensor. The energy and angular momentum flux 4-vectors in terms of the stress-energy tensor $T^{\mu\nu}$ of the disk are respectively given by

$$-E^{\mu}=T^{\mu}_{\nu}\left(\frac{\partial}{\partial t}\right)^{\nu}~{}\text{and}~{}J^{\mu}=T^{\mu}_{\nu}\left(\frac{\partial}{\partial\phi}\right)^{\nu}\,.$$

(17)

Using the rest mass conservation law,

$$\nabla_{\mu}(\rho_{0}u^{\mu})=0,$$

(18)

one can obtain the time-averaged rest mass accretion rate that does not depend on the disk radius,

$$\dot{m}\equiv-2\pi\sqrt{-g}~{}\Sigma~{}u^{r}=~{}\text{constant},$$

(19)

where an overdot denotes the derivative with respect to the time coordinate ($t$), $g$ is the determinant of a metric tensor. From the energy conservation, $\nabla_{\mu}E^{\mu}=0$, one can find,

$$[\dot{m}E-2\pi\sqrt{-g}\,\,\mathcal{W}~{}\,W_{\phi}^{r}]_{,r}=4\pi\sqrt{-g}\mathcal{F}E,$$

(20)

where $(_{,r})$ means the differentiation with respect to the radial coordinate $r$. The Eq. (20) implies that the energy carried by the rest mass of the disk, $\dot{m}E$, and the energy carried by the torques in the disk, $2\pi\sqrt{-g}\,\mathcal{W}\,W_{\phi}^{r}$, are well-balanced by the energy radiated away from the disk’s surface, $4\pi\sqrt{-g}\,\mathcal{F}E$. Using the law of angular momentum conservation, $\nabla_{\mu}J^{\mu}=0$, one can write,

$$[\dot{m}L-2\pi\sqrt{-g}~{}W_{\phi}^{r}]_{,r}=4\pi\sqrt{-g}\mathcal{F}L\,.$$

(21)

The first term on the left-hand side of Eq. (21) represents the angular momentum transported by the rest mass of the disk; the second term denotes the angular momentum carried by the torques in the disk; and the third term on the right-hand side represents the angular momentum transferred from the surface of the accretion disk via outgoing radiation. For the ease of calculation, change variables to Page:1974he

	$\displaystyle f$	$\displaystyle=$	$\displaystyle 4\pi\sqrt{-g}~{}\mathcal{F}/\dot{m},$		(22)
	$\displaystyle w$	$\displaystyle=$	$\displaystyle 2\pi\sqrt{-g}~{}\,W_{\phi}^{r}/\dot{m}\,.$		(23)

In terms of $f$ and $w$, the conservation laws (20) and (21) become

	$\displaystyle(L-w)_{,r}$	$\displaystyle=$	$\displaystyle fL\,,$		(24)
	$\displaystyle(E-\mathcal{W}\,w)_{,r}$	$\displaystyle=$	$\displaystyle fE\,.$		(25)

Using above equations along with universal energy-angular-momentum relation $dE=\mathcal{W}\,dL$, we can write

$$\textit{w}=[(E-\mathcal{W}L)/(-\mathcal{W}_{,r})]f\,.$$

(26)

Eliminating $w$ from Eq. (24) using above expression, and integrating the resulting first-order differential equation for $f$, we get

$$\frac{(E-\mathcal{W}L)^{2}}{-\mathcal{W}_{,r}}f=\int(E-\mathcal{W}L)L_{,r}dr+\text{const}.$$

(27)

Here, the constant of integration was determined considering the physical fact that accreting matter falls off the disk when reaches the innermost stable circular orbit, $r=r_{ms}$, and it spontaneously drops into the centre of the compact object. As a result, there is insignificant matter just within $r=r_{ms}$ to induce torque on the matter just outside $r=r_{ms}$. This implies that the torque $W^{r}_{\phi}$, and hence $f(r)$, must disappear at $r=r_{ms}$. To make $f(r_{ms})$ vanish, the integration constant must be selected in such a way that

$$\frac{(E-\mathcal{W}L)^{2}}{-\mathcal{W}_{,r}}f=\int_{r_{ms}}^{r}(E-\mathcal{W}L)\,L_{,r}\,dr\,.$$

(28)

Therefore the expression of time-averaged flux radiating from the disk surface can be written as Novikov ; Page:1974he ,

$$\mathcal{F}(r)=-\frac{\dot{m}}{4\pi\sqrt{-g}}\frac{\mathcal{W}_{,r}}{(E-\mathcal{W}L)^{2}}\int^{r}_{r_{\rm ms}}(E-\mathcal{W}L)L_{,r}dr.$$

(29)

It should be noted that this expression of flux (Eq. (29)) can only be employed for asymptotically flat spacetimes. For non-continuous spacetimes, where there is a matching between the interior and exterior spacetimes at a particular hypersurface, the boundary conditions in Eq. (27) change at the matching hypersurface. In such cases, Eq. (29) needs to be modified depending on whether the emitting point is in the interior or exterior spacetime.

In this section, we investigate the electromagnetic and thermal properties of accretion disk such as energy flux, spectral luminosity distribution, efficiency, and temperature profile for black hole and naked singularity spacetimes.

In this paper, we study the properties of the thin accretion disks surrounding black holes and naked singularities using the Novikov-Thorne accretion disk model. We carry out an analysis of the thermal properties of the radiation emerging from the disk surface. As for the naked singularities, we considered null and JMN1 naked singularity spacetimes, for which no ISCO exists, and hence accretion disks can reach up to the singularity. We also analysed the physical properties of the particles in the disk, such as specific energy ($E$), specific angular momentum ($L$), and angular velocity ($\mathcal{W}$). The conclusions of the paper are summarised as:

• •

  We obtain the equation for calculating the emitting flux for the spacetimes joined at the junction through the matching hypersurface by employing the appropriate boundary conditions in the derivation of the flux equation. The comparative study of the energy flux emitted by the accretion disks of black holes and naked singularities reveals that the amount of flux emerging from the disk surface of naked singularities is higher compared to that of a black hole of the same mass and accretion rate.

• •

  From the electromagnetic spectra of the radiation, we find that, compared to the Schwarzschild solution, the spectral distribution from the inner regions of the disks in JMN1 spacetime is more luminous in the high-frequency range. In contrast, it is more luminous in the low-frequency range in null naked singularity spacetime. These unique imprints in the luminosity spectrum provide a tool for distinguishing spacelike, timelike, and nulllike singularities by using astrophysical measurements of the radiation spectra emerging from accretion disks.

• •

  In addition to that, we obtain the slopes of the luminosity distribution curve for black holes and naked singularity models. The results indicate a considerable difference in the slopes of black holes and naked singularities at high frequencies. Such distinct characteristics may, in principle, help to discriminate between naked singularities and black holes.

• •

  In general, the “zero-torque” condition is assumed at the inner edge of the accretion disk. We show how the luminosity spectra of naked singularities vary when a non-zero torque is introduced close to the singularity. The inclusion of the non-zero torque at the inner edge of the disk results in a distinguishable change in the luminosity profile of the null and JMN1 naked singularities.

• •

  Furthermore, we also compute the accretion efficiency $\epsilon$ with which the accreting matter is converted into radiation and find that the naked singularities are far more efficient than black holes in transforming mass-energy into electromagnetic radiation. While investigating the temperature distribution on the disk surface, we note that the accretion disks around naked singularities are much hotter than black holes. For null and JMN1 naked singularities, the maximum value of the temperature profile lies close to the singularity, indicating that the inner regions of the accretion disks are significantly hotter and more luminous than the outer ones.

The radiation emerging from the inner regions of accretion disks of compact objects is impacted by strong-gravity effects. Because of this, the emitting signal retains traces of the geometrical structure of the central compact object to a faraway observer. Therefore, investigating the accretion processes of compact objects can reveal a lot about their physical properties. In this context, the typical features emerging in the luminosity spectrum of the accretion disks surrounding the naked singularities analysed in this work suggest the possibility of observing and differentiating naked singularities utilising astrophysical observations of the electromagnetic spectra from accretion disks.