Black hole shadow of a rotating polytropic black hole by the Newman--Janis algorithm without complexification

TL;DR

This paper derives a rotating polytropic black hole solution using the Newman--Janis algorithm without complexification, analyzing its horizon, shadow, and physical properties like temperature and emission rate.

Contribution

It introduces a novel method to obtain rotating black holes from static solutions without complexification, expanding the understanding of polytropic black hole characteristics.

Findings

The shape of the black hole shadow is characterized.

Horizon and causality conditions are analyzed.

Hawking temperature and emission rate are discussed.

Abstract

In this work, starting from a spherically symmetric polytropic black hole, a rotating solution is obtained by following the Newman--Janis algorithm without complexification. Besides studying the horizon, the static conditions and causality issues of the rotating solution, we obtain and discuss the shape of its shadow. Some other physical features as the Hawking temperature and emission rate of the rotating polytropic black hole solution are also discussed.

Equations

$$d{s}^2=G(r)d{t}^2-\frac{d{r}^2}{F(r)}-H(r)(d{\theta }^2+{\sin }^2\theta d{\varphi }^2).$$

$$du=dt-dr/\sqrt{FG},$$

$$l^{\mu }$$

$$n^{\mu }$$

$$m^{\mu }$$

$$r\to r+ia\cos \theta ,$$

$$u\to u-ia\cos \theta ,$$

$$G(r)\to A(r,\theta ,a),$$

$$F(r)\to B(r,\theta ,a),$$

$$H(r)\to \Psi (r,\theta ,a),$$

$$l^{\mu }$$

$$n^{\mu }$$

$$m^{\mu }$$

$$d{s}^2$$

$$-2a{\sin }^2\theta \sqrt{\frac{A}{B}}drd\varphi -\Psi d{\theta }^2$$

$$-{\sin }^2\theta (\Psi +a^2{\sin }^2\theta (2\sqrt{\frac{A}{B}}-A))d{\varphi }^2).$$

$$du$$

$$d\varphi$$

$$\lambda$$

$$\chi$$

$$K=\sqrt{\frac{F}{G}}H,$$

$$A(r,\theta )$$

$$B(r,\theta )$$

$$d{s}^2$$

$${\rho }^2$$

$$(K+a^2{y}^2{)}^2(3{\Psi }_{,r}{\Psi }_{,y^2}-2\Psi {\Psi }_{r,y^2})=3a^2{K}_{,r}{\Psi }^2$$

$$T^{\mu \nu }=ϵe_t^{\mu }{e}_t^{\nu }+p_r{e}_r^{\mu }{e}_r^{\nu }+p_{\theta }{e}_{\theta }^{\mu }{e}_{\theta }^{\nu }+p_{\varphi }{e}_{\varphi }^{\mu }{e}_{\varphi }^{\nu },$$

$$e_t^{\mu }$$

$$e_r^{\mu }$$

$$e_{\theta }^{\mu }$$

$$e_{\varphi }^{\mu }$$

$$(K_{,r}^2+K(2-K_{,rr})-a^2{y}^2(2+K_{,rr}))\Psi ,$$

$$+(K+a^{2}y)(4y^2{\Psi }_{y^2}-K_{,r}{\Psi }_{,r})=0.$$

$$\Psi =r^2+a^2{\cos }^2\theta ,$$

$$d{s}^2$$

$${\rho }^2$$

$$2f$$

$$\Delta$$

$$\Sigma$$

$$\frac{\partial S}{\partial \tau }=\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S,$$

Full text

I Introduction

After the historical direct detection of gravitational waves from a black hole (BH) merger by the LIGO collaboration in 2015 ligo , and the first image of the supermassive BH located at the centre of the giant elliptical galaxy Messier 87 (M87) by the Event Horizon Telescope (EHT) project project only a few weeks back, there is currently a lot of ongoing research in investigating the properties of BHs in several different contexts. The LIGO direct detection provided us with the strongest evidence so far that BHs do exist in Nature and that they merge, while at the same time it offers us the tools to test strong gravitational fields using radio wave astronomy. However, the LIGO detection has provided us with no information about the horizon of the BH, which after all is its defining property.

The gravitational field produced by a BH is so strong that nothing, not even photons, can escape. Therefore, a BH cannot be seen directly. There is, however, the possibility of seeing a dark shadow of a BH via strong gravitational lensing and photon capture at the horizon, if the BH stands between the observer and nearby light sources. The photons emitted from the source, or of the radiation emitted from an accretion flow around the event horizon of the BH, are expected to create a characteristic shadow-like image, that is a darker region over a brighter background. Indeed, recently the EHT project, a global very long baseline interferometer array observing at a wavelength of $1.3$ mm, announced and showed the first image of the supermassive BH located at the centre of M87 L1 , while the corresponding image from the centre of the Milky Way is yet to come. For instrumentation, data processing and calibration, physical origin of the shadow, etc, see L2 ; L3 ; L4 ; L5 ; L6 .

The observation of the shadow probes the spacetime geometry in the vicinity of the horizon, and therefore it tests the existence and properties of the latter psaltis . It should be noted, however, that other horizonless compact objects that possess light rings also cast shadows horizonless1 ; horizonless2 ; horizonless3 ; horizonless4 ; horizonless5 ; horizonless6 ; shakih2018 ; shakih2019 , and therefore the presence of a shadow does not by itself imply that the object is necessarily a BH. Therefore, strong lensing images and shadows offer us an exciting opportunity not only to detect the nature of a compact object, but also to test whether or not the gravitational field around a compact object is described by a rotating or non-rotating geometry. For a recent brief review on shadows see review .

Within the framework of Einstein’s General Relativity GR the most general BH solution is the Kerr-Newman geometry characterized by its mass, angular momentum and electric charge, see e.g. solutions . Since, however, astrophysical BHs are expected to be electrically neutral, the most interesting cases to be considered are either the Schwarzschild SBH or the Kerr geometry kerr . More rotating BH solutions may be generated starting from non-rotating seed spacetimes applying the Newman-Janis algorithm (NJA), described in algorithm1 ; algorithm2 .

Non–rotating solutions have been obtained in non-standard scenarios, such as polytropic BHs polytropic1 ; polytropic2 or BHs with quintessencial energy quint1 , to mention just a few. Over the years the shadow of the Schwarzschild geometry was considered in synge ; luminet , while the shadow cast by the Kerr solution was studied in bardeen (see also monograph ). Shadows of Kerr BHs with scalar hair and BH shadows in other frameworks have been considered in carlos1 ; carlos2 and Bambi:2008jg ; Bambi:2010hf ; study1 ; study2 ; Moffat ; quint2 ; study3 ; study4 ; study5 ; study6 ; bobir2017 ; Konoplya:2019sns ; shakih2019b ; sudipta2019 , respectively.

To explore the physics behind the so-called BH shadows, an alternative tool is provided by the well-known NJA. As already mentioned, this method allows one to pass from a static spherically symmetric BH solution to a rotating one. To be more precise, in the present work we will take a variation of the usual NJA, the only difference being the omission of one of the steps of the NJA, namely the complexification of coordinates quint1 . Instead of this, we will follow an “alternate” coordinate transformation, which will be explained in the next section.

In the present work we propose to investigate the shadow of the rotating polytropic BH. The non-rotating, static, spherically symmetric geometry was obtained in polytropic1 . The metric tensor is a solution to Einstein’s field equations with negative cosmological constant, and the thermodynamics of the BH precisely matches that of a polytropic gas. One one hand, as stated in polytropic2 , from the point of view of a possible astrophysical tests of the non-rotating polytropic solution, the so-called static radius, which defines the equilibrium region between gravitational attraction and dark energy repulsion, would be of importance Panotopoulos:2017clp . On the other hand, in the rotating case, which will be studied in the present work by using the NJA, the computation of the shadow would constitute a valuable tool in order to confirm or refute theoretical predictions regarding the intimate structure of space and time at the strong field regime.

The plan of our work is the following. After this introduction, we briefly summarize how the NJA works in the next section. In section III we study the conditions leading to unstable null trajectories for a general parametrization of a rotating BH while section IV is devoted to the study of the Hawking temperature and the emission rate of a generic $3+1$ rotating BH. In section V we construct the rotating solution starting from a static and spherically symmetric polytropic BH as the seed geometry and we study some features of the solution as for example horizon and static conditions, causality issues, BH shadow, Hawking temperature and emission rate. Finally we conclude our work in the last section. We adopt the mostly negative metric signature $(+,-,-,-)$, and we choose natural units where $c=1=G$.

II Newmann–Janis algorithm without complexification

In this section we review the main aspects on the NJA to generate rotating solutions introduced by M. Azreg-Aïnou in Ref. azreg2014 . In this work, the author performed a modification in the algorithm with the purpose to avoid the complexification process in the original protocol as follows.

As usual, the starting point is a static spherically symmetric metric parametrized as

[TABLE]

The next step consists in introducing the advanced null coordinates $(u,r,\theta,\phi)$ defined by

[TABLE]

from where, the non–zero components of the inverse metric can be written as $g^{\mu\nu}=l^{\mu}n^{\nu}+l^{\nu}n^{\mu}-m^{\mu}\bar{m}^{\nu}-m^{\nu}\bar{m}^{\mu}$ with

[TABLE]

and $l_{\mu}l^{\mu}=m_{\mu}m^{\mu}=n_{\mu}n^{\mu}=l_{\mu}m^{\mu}=n_{\mu}m^{\mu}=0$ and $l_{\mu}n^{\mu}=-m_{\mu}\bar{m}^{\mu}=1$. Now, introducing the complex transformation

[TABLE]

and assuming that

[TABLE]

we obtain

[TABLE]

Using the above transformations, the line element, in the so–called rotating Eddington-–Finkelstein coordinates reads

[TABLE]

In order to write the metric (II) in the Boyer–-Lindquist coordinates, we proceed to perform the global coordinate transformation

[TABLE]

where $\lambda$ and $\chi$ must depend on $r$ only to ensure the integrability of Eq. (15). As it is well known, in the original NJA the next step in the construction of the rotating metric, consists in complexifying $r$. However, in order to circumvent the complexification, Azreg-Aïnou (see Ref. azreg2014 ) proposed an ansatz for the unknown functions involved. Namely, taking

[TABLE]

where

[TABLE]

and

[TABLE]

the metric (II) takes the Kerr–like form

[TABLE]

with

[TABLE]

where $a=J/M$, with $M,J$ being the mass and the rotation speed, respectively, of the black hole.

At this point some comments are in order. First, note that the function $\Psi(r,\theta,a)$ remains unknown but it must satisfy the following differential equation

[TABLE]

which corresponds to imposing that the Einstein tensor satisfies $G_{r\theta}=0$. Second, it can be shown (see appendix A in Ref.azreg2014 ) that the metric (22) satisfies Einstein’s field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$ with the source given by

[TABLE]

with

[TABLE]

Even more, in order to ensure the consistency of Einstein’s field equations, the unknown $\Psi$ must satisfy another constraint, namely

[TABLE]

Finally, the above expressions can be simplified in the particular case $G=F$ and $H=r^{2}$. Indeed, in this case it can be shown that one solution of Eq. (24) is given by

[TABLE]

and the metric (22) takes the form

[TABLE]

with

[TABLE]

It is easy to verify that when $a=0$ we recover the non rotating black hole solution.

III Null geodesics around the rotating black hole

In this section we implement the standard Hamilton–Jacobi formalism to separate the null geodesic equations in the rotating space–time. Our main goal here is to obtain the celestial coordinates parametrized with the radius of the unstable null orbits and to study the shadow of a generic rotating solution.

Let us start with the Hamilton–Jacobi equations carter

[TABLE]

where $\tau$ is the proper time and $S$ is the Jacobi action. As usual, if we assume that Eq. (37) has separable solutions, the action takes the form

[TABLE]

where $E$ and $\Phi$ are the conserved energy and angular momentum respectively. Now, replacing (38) in (37) we obtain that

[TABLE]

where

[TABLE]

where $Q$ is the so–called Carter constant. As it is well-known, the unstable photon orbits in the rotating space–time must satisfy the constraints $R=0$ and $R^{\prime}=0$ which lead to

[TABLE]

where $\xi=\Phi/E$ and $\eta=Q/E^{2}$ correspond to the impact parameters. Hence,

[TABLE]

It is worth mentioning that, in the above expressions, $r$ corresponds to the radius of the unstable null orbits.

Now, the apparent shape of the shadow is obtained by using the celestial coordinates which are defined as vazquez

[TABLE]

where $(r_{0},\theta_{0})$ correspond to the coordinates of the observer. Finally, the after calculating the limit in the above expressions, the celestial coordinates read

[TABLE]

and the shadow corresponds to the parametric curve of $\alpha$ and $\beta$ with $r$ as a parameter.

IV Hawking temperature and emission rate of a rotating black hole

In Ref. ma , the author derived the Hawking temperature of a general four–dimensional rotating BH using the null–geodesic tunneling method developed by Parikh and Wilczek parikh1 ; parikh2 ; parikh3 . In this section we shall summarize the main result obtained in ma .

First of all, the radiated particles are considered as $s$-waves because for an observer at infinity the radiation is spherically symmetric whether the BH is rotating or not. Now, the tunnelling rate of a $s$-wave from the inside to the outside of the BH is given by

[TABLE]

where $\mathcal{I}$ is the action of the tunnelling particle and $\Gamma_{0}$ the normalization factor. Moreover, as it is well-known, the emission rate satisfies

[TABLE]

where $E$ is the energy of the emitted particle, $\beta=2\pi/\kappa$ and $\kappa$ is the surface gravity of the horizon. Now, from (51) and (52) we arrive to

[TABLE]

from where the Hawking temperature can be derived form the standard relation

[TABLE]

In order to obtain an expression of the Hawking temperature in terms of the components of the metric we proceed to consider a generic rotating line element of the form

[TABLE]

It can be demonstrated that (see section II of Ref. ma for details)

[TABLE]

where the prime stands for derivative respect to the radial coordinate, $r_{+}$ is the horizon radius and

[TABLE]

with $\Omega_{+}$ the angular velocity of the event horizon, which is a constant defined by

[TABLE]

Finally, replacing (56) in (53) an using (54), the Hawing temperature takes the form

[TABLE]

which can be written alternatively as

[TABLE]

Please, notice that the Eq. (60) is a generalization of the standard formula for the Hawking temperature. In particular, when the angular velocity is taken to be zero, we recover the well–known formula, namely:

[TABLE]

In what follows, we shall study the emission rate of a rotating BH. As it is claimed in references amir2016 ; papnoi2014 , the BH shadow corresponds to its high energy absorption cross section for the observer located at infinity. On the other hand, it is well known that for a BH endowed with a photon sphere, the cross section has a limiting value, namely, $\sigma_{lim}$, which coincides with the geometrical cross section of this photon sphere misner . Now, for a spherically symmetric BH it can be demonstrated that shao2013 , $\sigma_{lim}\approx\pi R_{s}^{2}$ with $R_{s}$ the radius of the photon sphere defined by (see amir2016 )

[TABLE]

In the previous expression, the quantities $\alpha_{t},\alpha_{r},\beta_{t}$ correspond to particular values of the celestial coordinates (see Eqs. (49) and (50)). In particular, $\alpha_{r}$ corresponds to the most right value of $\alpha$ and the pair $(\alpha_{t},\beta_{t})$ stands for coordinates of the top of the shadow hioki2009 . With all the quantities defined in this section, the emission rate spectrum ; wei2013 ; amir2016 ; Panotopoulos:2016wuu ; Panotopoulos:2017yoe ; Destounis:2018utr ; Panotopoulos:2018pvu ; Rincon:2018ktz can be calculated by

[TABLE]

where $\omega$ represents the frequency of photons. In the next section we obtain the Hawing temperature and the emission rate for a rotating polytropic BH.

V Rotating polytropic black hole solution

In this section we construct the rotating BH solution from a static and spherically symmetric polytropic BH. Then we shall study some of its physical properties as well as the BH shadow and its emission.

As a starting point, we consider the polytropic BH solution obtained in polytropic1

[TABLE]

where $\mathcal{F}=\left(\frac{r^{2}}{L^{2}}-\frac{2M}{r}\right)$, $L^{2}=-3/\Lambda$, with $\Lambda$ being the cosmological constant, and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the line element of the unit two-dimensional sphere. It is worth mentioning that $\mathcal{F}$ in (64) has the same form of the black string solution found in lemos ; cai in the context of plane symmetric solutions of Einstein’s equations solutions .

Now, using $F=G=\frac{r^{2}}{L^{2}}-\frac{2M}{r}$, the line element (32) reads

[TABLE]

with

[TABLE]

At this point some comments are in order. First, the horizons are solutions of $\Delta(r_{\pm})=0$ which, in this particular case, corresponds to

[TABLE]

It is worth mentioning that from the horizon condition we can obtain a bound on the spin parameter, namely $a/M$. In this case, the allowed values are constrained by

[TABLE]

Note that above result differs from the obtained in the case of the Kerr solution, where the constraint is given by $a/M<1$. Second, the static limit, namely, the surface from where observers can remain static, corresponds to $g_{tt}=0$. More precisely,

[TABLE]

Note that, the event horizon, $r_{+}$, coincides with the static radius $r_{st}$ at the poles $\theta=0$ and $\theta=\pi$ as in the Kerr solution. Third, note that causality violation and closed time–like curves are possible if $g_{\phi\phi}>0$. To be more precise, the condition is

[TABLE]

from where, given that $\sin^{2}(\theta)/\rho^{2}$ is positive, the sign of $\Sigma$ plays a crucial role in the analysis. What is more, the condition to avoid causality violation and closed time–like curves is to impose $\Sigma>0$. In the particular case of the rotating polytropic BH, the condition on $\Sigma$ reads

[TABLE]

It is worth mentioning that, in contrast to Kerr solution, the causality issues can be avoided for particular choices of the parameters. For example, taking $a=L$, the above condition reduces to

[TABLE]

which is trivially satisfied given that all the coefficients involved in the left hand side are positive.

Now, we shall focus our attention in the construction of the BH shadow. Replacing the metric function $F(r)=-\frac{2M}{r}+\frac{r^{2}}{L^{2}}$ in Eqs. (45) and (46), the impact parameters $(\xi,\eta)$ associated to the unstable null geodesics around the rotating BH are given by

[TABLE]

from where, assuming $\theta_{0}=\pi/2$, the celestial coordinates read

[TABLE]

In figure 1 the shadow of the rotating BH is shown for different values of $a/M$ and $L/M$. Note that the shadow undergoes a deformation as $a/M$ increases. Indeed, for $a/M=1.5$ the silhouette losses its symmetry and looks like an oval instead of an ellipse.

The hawking temperature for this rotating BH is given by

[TABLE]

where $r_{+}$ is the event horizon radius obtained from the condition (67), which reads

[TABLE]

with

[TABLE]

Now, for particular values of the parameters $(a,L,M)$, the Hawking temperature can be computed and $R_{s}$ can be obtained from Eq. (62). Finally, replacing (77) and (62) in (63), we can study the behaviour of the emission rate as a function of the frequency $\omega$. The emission rate profile is depicted in figure 2, where it can be shown that it decreases with an increasing value of $L/M$.

VI Conclusions

In this work we have reviewed the main aspects related to the Newman–Janis algorithm without complexification, the construction of unstable null orbits and the computation of the Hawking temperature and emission rate for general rotating black holes and we have used these tools to construct a rotating polytropic black hole solution. It is worth mentioning that, to the best of our knowledge, the construction of such a rotating solution has not been considered before. Besides, we have studied some physical properties as the position of the horizons and the static limit which defines the ergosphere as well as the causality condition. We have obtained that, in contrast to the Kerr solution, the causality issues can be avoided for certain choice of the black hole parameters, namely $a$, $M$ and $L$. Additionally, we have demonstrated that, in contrast to the Kerr black hole, the bound on the spin parameter $a/M$ can be grater that one because the condition of the appearance of horizons entails that this bound is proportional to $L/M$. We have also analysed the shadow of the rotating polytropic solution finding that its shape can change while increasing $L/M$. As a final result, we have studied the emission rate, showing that it increases for large values of $L/M$.

Finally, it would be interesting to investigate how the properties of the solution here obtained are modified in light of the so called scale–dependent scenario. In such cases, the coupling constants acquire a dependence on the scale with a free parameter which encodes the quantum features (which is called scale–dependent parameter). Thus, given that the black hole shadows give observational evidence for black holes, we could establish bounds on the aforementioned parameter (see Koch:2016uso ; Rincon:2017ypd ; Rincon:2017goj ; Rincon:2017ayr ; Contreras:2017eza ; Rincon:2018sgd ; Hernandez-Arboleda:2018qdo ; Contreras:2018dhs ; Rincon:2018dsq ; Contreras:2018gct ; Canales:2018tbn ; Rincon:2019cix ; Rincon:2019zxk ; Contreras:2019fwu ; Contreras:2018gpl ; Contreras:2018swc and references therein). We hope to be able to address this issue in a future work.

Acknowlegements

The author Á. R. acknowledges DI-VRIEA for financial support through Proyecto Postdoctorado 2019 VRIEA-PUCV. The author G. P. thanks the Fundação para a Ciência e Tecnologia (FCT), Portugal, for the financial support to the Center for Astrophysics and Gravitation-CENTRA, Instituto Superior Técnico, Universidade de Lisboa, through the Grant No. UID/FIS/00099/2013. The author P. B. was supported by the Faculty of Science and Vicerrectoría de Investigaciones of Universidad de los Andes, Bogotá, Colombia, under the grant number INV-2018-50-1378.