Self-gravitating razor-thin disks around black holes via multi-hole seeds

TL;DR

This paper constructs stable, self-gravitating, razor-thin disks of counterrotating dust around Schwarzschild black holes using seed solutions with multiple holes, satisfying energy conditions and being asymptotically flat.

Contribution

It introduces a method to generate self-gravitating disk solutions around black holes from multi-hole seed solutions, expanding the set of known BH+disk configurations.

Findings

Disks are infinite, annular, and linearly stable up to the innermost stable circular orbit.

All energy conditions are satisfied along the disk.

The spacetime is asymptotically flat with finite ADM mass.

Abstract

We construct self-gravitating razor-thin disks of counterrotating dust around Schwarzschild black holes (BHs) by applying the "displace, cut, and reflect" method to known seed solutions representing multi-holes. All but one of the sources of the seed solution generate the surrounding annular disk, whereas the remaining BH is kept unaltered and lies at the disk center after the transformation. The disks are infinite in extent, have annular character, and are linearly stable up to the innermost stable circular orbit of the system. Moreover, all energy conditions are satisfied along the disk and the spacetime is asymptotically flat, having finite ADM mass. We also comment on charged disks around extremal Reissner-Nordstr\"om BHs constructed from a Majumdar-Papapetrou $N$-BH seed solution. These simple examples can be extended to more general "BH + disk" solutions, obtained by the same method from seeds of the type "BH + arbitrary axisymmetric source". A natural follow-up of this work would be to construct disks around Reissner-Nordstr\"om BHs with arbitrary charge-to-mass ratio and around Kerr BHs.

Equations

$$d{s}^2=-e^{2\psi }d{t}^2+e^{-2\psi }[e^{2\gamma }(d{\rho }^2+d{z}^2)+{\rho }^{2}d{\phi }^2],$$

$$z\to |z|+\lambda$$

$$\sigma=\frac{e^{\psi-\gamma}}{4\pi}\left(2\frac{\partial\psi}{\partial|z|}-\frac{\partial\gamma}{\partial|z|}\right)\Bigg{|}_{z=0}\,,$$

$$P_{\varphi}=\frac{e^{\psi-\gamma}}{4\pi}\frac{\partial\gamma}{\partial|z|}\Bigg{|}_{z=0}\,.$$

$$\psi (\rho ,z)=\sum _{i=1}^N{\psi }_i(\rho ,z),$$

$${\psi }_i(\rho ,z)=\frac{m_i}{L_i}\log \left[\frac{R_i^{+}+R_i^{-}-L_i}{R_i^{+}+R_i^{-}+L_i}\right],$$

$$\gamma (\rho ,z)=\sum _{i=1}^N\sum _{j=1}^N\frac{m_i{m}_j}{L_i{L}_j}\log \left[\frac{E_{ij}^{+-}{E}_{ij}^{-+}}{E_{ij}^{++}{E}_{ij}^{--}}\right],$$

$$E_{ij}^{\pm \pm }=R_i^{\pm }{R}_j^{\pm }+z_i^{\pm }{z}_j^{\pm }+{\rho }^2.$$

$$R_i^{\pm }=\sqrt{{\rho }^2+(z_i^{\pm }{)}^2}$$

$$z_{i}^{\pm}=z-\Big{(}Z_{i}\mp\frac{1}{2}L_{i}\Big{)}\,.$$

$$R_1^{\pm }=\sqrt{{\rho }^2+(z_1^{\pm }{)}^2}$$

$$z_1^{\pm }=|z|\mp \frac{1}{2}{L}_1$$

$$d{s}^2=-U^{-2}d{t}^2+U^2(d{\rho }^2+{\rho }^{2}d{\phi }^2+d{z}^2),$$

$$U=1+\frac{M_1}{\sqrt{{\rho }^2+z^2}}+\sum _{i=2}^N\left(\frac{M_i}{\sqrt{{\rho }^2+{\zeta }_i^2}}\right)$$

$$\sigma =\frac{{\sum }_{i=2}^N\frac{M_i(Z_1-Z_i)}{{\left((Z_1-Z_i{)}^2+{\rho }^2\right)}^{3/2}}}{2\pi {\left(1+\frac{M_1}{\rho }+{\sum }_{i=2}^N\frac{M_i}{\sqrt{(Z_1-Z_i{)}^2+{\rho }^2}}\right)}^2}.$$

Full text

I Introduction

It is now widely accepted that black holes (BHs) are ubiquitous in nature; moreover, recent gravitational-wave detections of binary BH mergers LIGO2016PhRvL ; LIGO2016ApJL and observations of the shadow of the central BH of M87 EHT2019paper1 confirmed with great precision that they behave as predicted by general relativity. It is also a paradigm that BHs are generally surrounded by accretion disks. However, these disks are usually treated as test fluids abramowiczEtal1988ApJ ; frank2002accretion ; abramowiczFragileLRR ; sadowskiEtal2015MNRAS ; lasotaVieiraEtal2016AA ; it is clear that, in order to take into account the disk’s self gravity, we must deal with distorted BH fields gerochHartle1982JMP . Many efforts have been made in this direction, particularly in the quest for exact solutions of Einstein’s field equations representing razor-thin disks morganMorgan1969PR ; morganMorgan1970PRD ; letelierOliveira1987JMP ; bicakLyndenbellKatz1993PRD ; gonzalezLetelier1999CQGra ; gonzalezLetelier2000PRD ; ujevicLetelier2004PRD ; gonzalezGutierrez2012CQGra ; gonzalezPimentel2016PRD ; gutierrezpineres2015GRG ; vogtLetelier2003PRD ; vieiraLetelier2014GRG ; vieiraRamoscaroSaa2016PRD ; semerak2016PRD ; freitasSaa2017PRD and “BH + razor-thin disk” structures lemosletelier1993CQG ; lemosLetelier1994PRD ; lemosLetelier1996IJMPD ; saaVenegeroles1999PhLA ; semerakZacek2000PASJ ; semerakZacek2000CQGra ; zacekSemerak2002CzJP ; semerak2002review ; karasHureSemerak2004CQGra ; vogtLetelier2005PRD ; gutierrezpineresEtal2014IJMPD . Charged disks around extremal Reissner-Nordström (R-N) BHs are also recently getting attention loraclavijo-ospinahenao-pedraza2010PRD ; semerak2016PRD ; polcarSukovaSemerak2019ApJ . Also, the dynamics of test particles in these systems was widely studied saaVenegeroles1999PhLA ; semerakZacekZellerin1999MNRAS ; semerakSukova2010MNRAS ; semerakSukova2012MNRAS ; sukovaSemerak2013MNRAS ; witzanySemerakSukova2015MNRAS ; polcarSukovaSemerak2019ApJ ; vieiraRamoscaroSaa2016PRD . Therefore, in addition to their important theoretical interest, exact “BH + disk” solutions of Einstein’s field equations became a crucial tool to understand the effects of the accretion-disk self gravity around astrophysical BHs.

We present in this work a new family of exact solutions of Einstein’s field equations representing annular razor-thin disks around Schwarzschild BHs. The method presented here provides a simple procedure to construct these composite systems once we have an adequate seed solution representing a Schwarzschild BH plus an arbitrary external axisymmetric source. It also allows us to construct charged disks around extremal R-N BHs.

Section II gives a brief overview of the formalism used to construct the “BH + disk” systems. Section III presents the central result of the paper: the construction of annular razor-thin disks around Schwarzschild BHs via multihole seed solutions. Section IV briefly comments on the extreme R-N case. We present our conclusions in Section V.

II Razor-thin disks from seed solutions

Let us assume we have spacetime composed of a BH plus an external axisymmetric source, represented by the Weyl metric

[TABLE]

which we will denominate a “seed” solution. Interpreting $z$ as a “vertical” coordinate, we may apply a Kuzmin-like transformation binneytremaineGD to the metric functions of the seed solution in order to obtain a razor-thin disk source, meaning a delta-like distributional source on the symmetry plane of the new system. This procedure is often called the “displace, cut, and reflect” (DCR) method vogtLetelier2003PRD ; vieiraLetelier2014GRG ; gonzalezPimentel2016PRD ; freitasSaa2017PRD ; navarronogueraEtal2018GRG and consists of making the transformation

[TABLE]

to the metric functions $\psi$ and $\gamma$, with $\lambda>0$. As we will see, we can construct self-gravitating disks around a central Schwarzschild BH by a suitable choice of the seed solution and of the DCR parameter $\lambda$.

The necessary formalism to deal with relativistic razor-thin disks is the theory of distributional sources with support on timelike spacetime hypersurfaces taub1980JMP ; lemosLetelier1994PRD ; vieiraLetelier2014GRG ; vieiraRamoscaroSaa2016PRD ; freitasSaa2017PRD . The stress-energy tensor is written as $T^{\mu}_{\ \nu}=Q^{\mu}_{\ \nu}\,\hat{\delta}(z)+D^{\mu}_{\ \nu}$, where $\hat{\delta}(z)=\delta(z)/\sqrt{g_{zz}}$ is the covariant delta distribution in curved spacetimes vieiraLetelier2014GRG ; vieiraRamoscaroSaa2016PRD , $Q^{\mu}_{\ \nu}$ is the disk’s stress-energy tensor, and $D^{\mu}_{\ \nu}$ represents the smooth matter-energy content of the spacetime (such as a halo or a thickened disk, for instance). The form of the metric allows us to write $Q^{\mu}_{\ \nu}=diag(-\sigma,P_{\rho},P_{\varphi},0)$, the proper surface energy density $\sigma$ and the principal pressures $P_{\rho}$ (radial) and $P_{\varphi}$ (azimuthal) of the disk. It is worth noting that, when we consider the covariant delta distribution $\hat{\delta}$, the formalism gives us, in a self-consistent manner, the “physical” (or “true”) stress-energy tensor $Q^{\mu}_{\ \nu}$ of the disk vieiraLetelier2014GRG ; vieiraRamoscaroSaa2016PRD without the need to differentiate between the “formal” and the “true” stress-energy tensors mentioned for instance in lemosLetelier1994PRD ; gonzalezLetelier1999CQGra ; gonzalezLetelier2000PRD ; vogtLetelier2003PRD ; vieiraLetelier2014GRG ; gutierrezpineresEtal2014IJMPD ; gonzalezPimentel2016PRD ; navarronogueraEtal2018GRG .

In terms of the metric derivatives, the surface density $\sigma$ and azimuthal pressure $P_{\varphi}$ of the disk are written as vieiraRamoscaroSaa2016PRD

[TABLE]

[TABLE]

Static razor-thin disks described by a Weyl metric have radial pressure $P_{\rho}=0$, and therefore are made of counterrotating dust morganMorgan1969PR ; lemosLetelier1994PRD ; klein1997CQG ; gonzalezEspitia2003PRD ; freitasSaa2017PRD . Moreover, the DCR procedure keeps unaltered the “upper” part of the seed spacetime, above the disk; the stress-energy tensor of the upper part is preserved, as well as any singularities/black holes. These structures are then reflected to the “lower” part of the new spacetime, below the disk. The procedure is illustrated in Fig. 1 for the $N$-rod solution letelierOliveira1998CQG mentioned in Section III.

III Annular disks around Schwarzschild BHs

We consider as seed the “$N$ collinear Schwarzschild BHs” solution israelKhan1964NCim or, more generally, the “$N$ collinear Weyl rods” solution letelierOliveira1998CQG . The metric has the Weyl form (1) with functions $\psi$ and $\gamma$ given by letelierOliveira1998CQG

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

Here,

[TABLE]

and

[TABLE]

The function $\psi$ represents the gravitational potential of $N$ one-dimensional rods of length $L_{i}$ and mass $m_{i}$ along the $z$-axis, with center at at $z=Z_{i}$ letelierOliveira1998CQG . We order the $Z_{i}$ such that $Z_{i}-L_{i}/2>Z_{j}+L_{j}/2$ if $i<j$, meaning that there will be no intersection between the rods. They will have $Z_{1}>Z_{2}>...>Z_{N}$, in such a way that the rod labeled by the index 1 will be at highest $z$ among all sources. When $L_{k}=2m_{k}$, the $k$-th Weyl rod reduces to a Schwarzschild BH of mass $m_{k}$ israelKhan1964NCim ; letelierOliveira1998CQG .

The case $\lambda>Z_{1}$ generates a disk-only solution and was treated in bicakLyndenbellKatz1993PRD . The case $Z_{2}<\lambda<Z_{1}$ generates two equal-mass BHs with a disk in their mid-plane; however, the DCR method will, in general, preserve a “strut” israelKhan1964NCim ; synge1960relativity connecting the two BHs (and therefore crossing the disk). We regard this solution as unphysical.

In order to construct annular disks around Schwarzschild BHs, let us consider the $N$-rod seed solution presented above, with $L_{1}=2m_{1}$ in such a way that the first rod is a Schwarzschild BH with mass $m_{1}$ (we assume $Z_{1}>0$). If we choose the DCR parameter as $\lambda=Z_{1}$, the resulting spacetime will be composed of a Schwarzschild BH of mass $m_{1}$ surrounded by a razor-thin disk generated by the other ($N-1$) sources of the seed solution, as shown below.

Intuitively, the DCR method will preserve the upper part of the rod, source of the seed potential $\psi_{1}$, and reflect it with respect to the horizontal plane, generating in the DCR-transformed spacetime an equal rod but now encircled by a razor-thin disk (see Fig.1). This rod will then be a Schwarzschild BH of masss $m_{1}$. Indeed, putting $\lambda=Z_{1}$, the term $\psi_{1}$ will be a function of the sum $R_{1}^{+}+R_{1}^{-}$, where

[TABLE]

and

[TABLE]

after the DCR transformation. If $z\geq 0$ we have the usual Schwarzschild field. Now, if $z<0$, we have $z_{1}^{\pm}=-z\mp\frac{1}{2}L_{1}$, in such a way that $(z_{1}^{\pm}(-z))^{2}=(z_{1}^{\mp}(z))^{2}$ and thus $R_{1}^{\pm}(\rho,-z)=R_{1}^{\mp}(\rho,z)$. Therefore $R_{1}^{+}(\rho,-z)+R_{1}^{-}(\rho,-z)=R_{1}^{-}(\rho,z)+R_{1}^{+}(\rho,z)$ and $\psi_{1}(\rho,-z)=\psi_{1}(\rho,z)$. It follows that $\psi_{1}$ is smooth in the whole region outside the resulting rod and is correctly identified with the field of a Schwarzschild BH of mass $m_{1}$. Moreover, the other functions $\psi_{i}$, $i>1$, will contain linear terms in $|z|$ when expanded around $z=0$. They give, in this way, a direct contribution to the disk’s stress-energy tensor, according to Eqs. (3)–(4).

Although the explicit expressions for the surface density and azimuthal pressure of the disk are too lengthy to be presented here, they can be obtained straightforwardly from the DCR procedure, Eq. (2), applied to the seed metric functions $\psi$ and $\gamma$ (5)–(10), by means of expressions (3)–(4). The disk’s surface density $\sigma$ always vanishes at the event horizon ($\sigma(0)=0$ in Weyl coordinates) and the azimuthal pressure $P_{\varphi}$ satisfies, in general, $P_{\varphi}(0)>0$. The disks have infinite extent; their density profile goes asymptotically as $\sigma\sim 1/\rho^{3}$ (as in the case of the classical Kuzmin disk) and their pressure profile as $P_{\varphi}\sim 1/\rho^{4}$. Accordingly, the circular speed of the counterrotating streams [$V^{2}=P_{\varphi}/\sigma$, see bicakLyndenbellKatz1993PRD ; lemosLetelier1994PRD ; gonzalezEspitia2003PRD ] has a Keplerian asymptotic profile $V\sim\rho^{-1/2}$. We always have $V^{2}<1$ outside the marginally stable orbit (see below), in such a way that the streams are physically plausible (although we may have $V^{2}>1$ in regions very close to the BH, but these regions are not considered as part of the disk, as explained below). We also note that the annular character of this “BH + disk” solution ($\sigma(0)=0$) is a purely relativistic effect, being related to the existence of an event horizon; the corresponding superposition “monopole + $N$ Kuzmin potentials” in Newtonian gravity always gives us a positive surface density at the disk center.

Moreover, the structure of the metric functions guarantees us that the “BH + disk” spacetime is asymptotically flat. Therefore its ADM mass poisson2004toolkit is finite. It is given by $M_{\rm ADM}=\sum_{i=1}^{N}m_{i}$ and coincides with the ADM mass of the seed spacetime. Other definitions for the disk mass vogtLetelier2003PRD ; vieiraLetelier2014GRG give us a finite total mass for the disk itself.

Since the strong energy condition $\sigma+P_{\varphi}>0$ is satisfied along the whole disk, timelike circular geodesics are always vertically stable according to the definite vertical stability criterion developed in vieiraRamoscaroSaa2016PRD , extension of the Newtonian result vieiraRamosCaro2016CeMDA . On the other hand, Rayleigh’s radial stability criterion letelier2003PRD ; abramowiczKluzniak2005ApSS ; vieiraMarekEtal2014PRD shows us that the disk has an innermost stable circular (geodesic) orbit (ISCO) in the presence of the BH. In this “particle” approximation, the disk may be regarded as linearly stable for radii larger than the ISCO radius $\rho_{ISCO}$. We argue below that, by an adequate choice of parameters, the disk density, azimuthal pressure, and the disk’s influence on the system’s field lines are negligible for radii smaller than $\rho_{ISCO}$. We note that the disk satisfies all energy conditions for $\rho\geq\rho_{ISCO}$.

The simplest solution is the $N=2$ case, with $L_{2},\,m_{2}>0$, in which the disk is generated by only one seed source $m_{2}$. The general picture in this case is of an annular disk ($\sigma(0)=0$) around a BH of mass $m_{1}$ with a regular density profile peaked at $\rho>0$. The pressure profile always has $P_{\varphi}(0)>0$ and $P_{\varphi}$ is at least one or two orders of magnitude smaller than $\sigma$.

In order to have a physically plausible disk, we must guarantee that $\sigma$ is negligible (in some sense) for $\rho<\rho_{ISCO}$ and that $P_{\varphi}(0)\approx 0$. However, the $N=2$ disk has, in general, a non-negligible surface density for $\rho<\rho_{ISCO}$. We overcome this issue by considering at least a third rod in the seed solution, and this rod must have negative mass $m_{3}<0$ in order to diminish the disk density near its center (the ordering of the rods will depend on the situation; the only requisite is that at least one of the masses must be negative). By choosing adequate parameters for the negative-mass rod, we can make the disk’s surface density negligible for $\rho<\rho_{ISCO}$ and, at the same time, reduce the azimuthal pressure $P_{\varphi}(0)$ at the disk center to arbitrarily low values. We show examples of this construction in Figs. 2 and 3 (we consider $m_{2}<0$ and $m_{3}>0$ for practical purposes).

We also analyze the system’s field lines semerakZellerinZacek1999MNRAS (the stream lines of the 4-acceleration $a_{\mu}=u^{\nu}\nabla_{\nu}u_{\mu}$ of a static observer) in the Schwarzschild coordinates ($r,\theta$) given by $\rho=\sqrt{r(r-2m_{1})\,}\cos\theta$, $z=(r-m_{1})\sin\theta$. It can be shown that, for a static observer, $a_{\mu}=\partial_{\mu}\psi$ semerakZellerinZacek1999MNRAS . The event horizon is located at the hypersurface $r=2m_{1}$. The distortion of the field lines from spherical symmetry shows the disk influence on the test-particle dynamics; field lines extending up to the equatorial plane represent the dominant contribution of the disk to the 4-acceleration of static observers, as we exemplify in Figs. 2 and 3. The disk has influence on the 4-acceleration of static observers only for radii comparable to (and greater than) the radius of maximum surface density; near the BH ($r<10\,m_{1}$ for the disks of Figs. 2 and 3, larger than the ISCO radius) the distortion from the spherically symmetric field lines is negligible.

All the above arguments lead us to the conclusion that it is a good approximation to neglect the disk contribution for $\rho<\rho_{ISCO}$, considering then an “effective” inner rim for the disk close to $\rho_{ISCO}$. We then justify its annular character.

IV Charged disks around extremal BHs

The formalism presented here can be readily applied to the Majumdar-Papapetrou-type seed solution of $N$ collinear extremal R-N BHs majumdar1947PR ; griffithsPodolsky2009exact , each one at position $Z_{i}$ along the $z$-axis with $Z_{1}>...>Z_{N}$. After the DCR transformation (2) with $\lambda=Z_{1}>0$, the metric becomes

[TABLE]

where

[TABLE]

and $\zeta_{i}=|z|+Z_{1}-Z_{i}$. The corresponding 4-potential is $A_{\mu}=(1/U,0,0,0)$ griffithsPodolsky2009exact ; ryznerJofka2015CQG . The system represents an extremally charged dust disk [$\sigma_{e}=\sigma$, see loraclavijo-ospinahenao-pedraza2010PRD ] around an extremal R-N BH of mass $M_{1}$. The disk’s surface density profile (3) has a simple expression given by

[TABLE]

It vanishes at the origin of Weyl coordinates (corresponding to the event horizon), $\sigma(0)=0$, and therefore the disk also has annular character; it also has a $\sigma\sim 1/\rho^{3}$ power-law tail.

We mention that the DCR method can generate a strut-free equilibrium configuration of two equal-mass extremal R-N BHs with a charged razor-thin disk in its mid-plane, by taking $Z_{2}<\lambda<Z_{1}$ in the DCR method. The BHs will both have mass $M_{1}$ and will be located at $z=\pm(Z_{1}-\lambda)$. If we take instead $\lambda>Z_{1}$ we will have a disk-only solution. The corresponding expressions for the DCR-transformed metric and for the disk surface density follow directly from the formalism. In both cases the disks are infinite in extent and satisfy $\sigma>0$ for $\rho>0$.

V Discussion

We presented in this paper exact solutions of Einstein’s field equations corresponding to Schwarzschild BHs surrounded by annular, counterrotating razor-thin disks of infinite extent. The superposed “Schwarzschild BH + disk” solutions were obtained via the application of the DCR method (Section II) to seed solutions representing $N$ collinear Weyl rods letelierOliveira1998CQG , the extension of the corresponding $N$-BH solution israelKhan1964NCim . We remark that, by considering $L_{1}\neq 2m_{1}$ in the DCR procedure with $\lambda=Z_{1}$ (Section III), we obtain annular disks around an arbitrary Weyl rod.

Although we considered only the $N=3$ case for presentation purposes, the procedure of constructing razor-thin disks around Schwarzschild BHs via multihole seeds allows for an arbitrary (but finite) number of Weyl rods in the seed spacetime, generating an infinite family of razor-thin disks around Schwarzschild BHs. The formalism can also be extended to more general seed spacetimes constituted of a BH plus an arbitrary axially symmetric source, such as for instance an additional Schwarzschild BH supported by an Appell ring semerakEtal2019PRD or a line source of variable density bicakLyndenbellKatz1993PRD below the Schwarzschild BH. The crucial step of the method, which gives us a central Schwarzschild BH, is to choose the DCR parameter $\lambda$ in such a way that the corresponding hypersurface ‘cuts in half’ the highest Schwazrschild rod (in Weyl coordinates), as depicted in Fig. 1.

The stability analysis of the disk presented here takes into account only the linear stability of the corresponding circular geodesics, since we interpret these disks as being composed of counterrotating dust lemosLetelier1994PRD . The disks are vertically vieiraRamoscaroSaa2016PRD and radially letelier2003PRD stable according to this approximation. However, the disk fluid elements may not follow geodesics, since the disk has azimuthal pressure; this pressure must be taken into account in the stability analysis of the fluid freitasSaa2017PRD . A complete treatment would involve the linearised conservation laws for small perturbations to the disk fluid quantities (density and pressure) such as in ujevicLetelier2004PRD ; ujevicLetelier2007GRG ; ujevicLetelier2007MNRAS . This kind of analysis is beyond the scope of the present work.

We also obtained extremally charged dust disks around extremal R-N BHs as another application of the method. We may then extend our results to (neutral or charged) disks around R-N BHs with arbitrary charge-to-mass ratio, following for instance the seed solution of azumaKoikawa1994PThPh . Another possibility is to construct stationary disks around Kerr BHs via the multihole DCR method presented here, for instance from the double-Kerr solution kramerNegembauer1980PLA . These issues are left for future work.

Acknowledgements.

R.S.S.V. thanks Alberto Saa for stimulating discussions. This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.