Schwarzschild black hole encircled by a rotating thin disc: Properties of perturbative solution
aa r X i v : . [ g r- q c ] A p r Schwarzschild black hole encircled by a rotating thin disc:Properties of perturbative solution
P. Kotlaˇr´ık, ∗ O. Semer´ak, † and P. ˇC´ıˇzek ‡ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic (Dated: April 9, 2018)Will [1] solved the perturbation of a Schwarzschild black hole due to a slowly rotating light con-centric thin ring, using Green’s functions expressed as infinite-sum expansions in multipoles and inthe small mass and rotational parameters. In a previous paper [2], we expressed the Green functionsin closed form containing elliptic integrals, leaving just summation over the mass expansion. Such aform is more practical for numerical evaluation, but mainly for generalizing the problem to extendedsources where the Green functions have to be integrated over the source. We exemplified the methodby computing explicitly the first-order perturbation due to a slowly rotating thin disc lying betweentwo finite radii. After finding basic parameters of the system – mass and angular momentum ofthe black hole and of the disc – we now add further properties, namely those which reveal how thedisc gravity influences geometry of the black-hole horizon and those of circular equatorial geodesics(specifically, radii of the photon, marginally bound and marginally stable orbits). We also realizethat, in the linear order, no ergosphere occurs and the central singularity remains point-like, andcheck the implications of natural physical requirements (energy conditions and subluminal restric-tion on orbital speed) for the single-stream as well as counter-rotating double-stream interpretationsof the disc.
I. INTRODUCTION
Disc accretion onto a black hole is a likely engine ofsome of the most energetic astrophysical sources, like ac-tive galactic nuclei, X-ray binaries or gamma-ray bursts.Modelling of such sources naturally starts from determi-nation of the gravitational field of the black hole encir-cled by a disc. Although the disc typically has just atiny fraction of the black-hole mass, its gravitational ef-fect may be important on the level of higher derivativesof the potential (space-time curvature), which in turn arecrucial for stability of motion of its own matter. To takethe disc’s gravity into account is however only easy ina static (non-rotating) and axially symmetric case whenexact “superposition” of the two sources can at least par-tially be obtained analytically. The simple static settinghas been studied many times in the literature; it can approximate some properties of the actual accretion sys-tems, but it lacks a significant feature suggested by everydepiction of accretion onto compact objects: rotation.If the rotation of the central black hole is slow, one ofthe analytical options is to perform a small perturbationof a Schwarzschild solution and adjust it to the boundaryconditions corresponding to a chosen source – usually aring, a disc or a toroid. This means to expand the rele-vant quantities in the Einstein equations in correspond-ing small parameters (typically related to mass and an-gular momentum of the additional matter) and then tryto solve the equations to a desired order of expansion.In the previous paper [2] we considered a linear (first-order) perturbation of the Schwarzschild black hole due ∗ [email protected] † oldrich.semerak@mff.cuni.cz ‡ [email protected] to a slowly rotating concentric finite thin disc. Inspiredby [1] who calculated a perturbation due to an infinites-imally thin ring, we expressed in closed form the Greenfunctions (for metric functions representing gravitationalpotential and rotational dragging), given in his paper asseries in orthogonal polynomials. The closed form useselliptic integrals and is more practical for numerical eval-uation, but mainly for studying extended sources whenthe Green functions have to be integrated over the sourcevolume. We illustrated the method on linear perturba-tion due to a simple disc existing between two finite radiiand having constant Newtonian density.In the cited paper we provided a longer introductionand described there thoroughly the perturbation methodas well as interpretation of the disc both in terms of a one-component ideal fluid and in terms of a two-component(counter-rotating) geodesic dust. Let us thus only remindthat the general metric considered is that of circular, i.e.stationary and axisymmetric plus orthogonally transitivespace-times,d s = − e ν d t + B r e − ν sin θ (d φ − ω d t ) ++ e ζ − ν (d r + r d θ ) , (1)where the unknown functions ν , B , ω and ζ depend onlyon r and θ covering the meridional surfaces. Note thatthe above coordinates (mostly called isotropic) are re-lated to the Weyl-type cylindrical coordinates ρ and z by ρ = r sin θ, z = r cos θ. The unknown functions are given by Einstein equationswith appropriate boundary conditions. The function B can in our case be chosen to read B = 1 − k r = (2 r − k )(2 r + k )4 r , (2)which corresponds to a horizon lying on r = k/ B = 0); this also provides the meaning of the pa-rameter k . We start from a Schwarzschild background,d s = − (cid:18) r − M r + M (cid:19) d t ++ (2 r + M ) r (cid:2) d r + r (d θ + sin θ d φ ) (cid:3) , so in our case k coincides with the black-hole mass M .Note that the Schwarzschild background is described by ν ≡ ν Schw = ln 2 r − M r + M (3)and that the radial derivative of this potential (often oc-curring in calculations) reads, with our choice (2) for B , ν ,r = M Br . The main task of any stationary and axisymmetricproblem is to find the functions ν and ω which have themeaning of a gravitational potential and of a draggingangular velocity, respectively. We derived the perturba-tive (linear-order) solution for these functions in the firstpaper [2], in particular, we expressed in closed form theGreen functions for both the functions and illustrated, ona simple example of a disc with constant Newtonian den-sity stretching between two finite radii r out > r in > M/ ν (denoted by ν and representing entirely the effect of the disc) and equa-tions (84)–(86) for the perturbation of ω (denoted by ω and equal to the total ω ). The solution depends, besidesthe black-hole mass M , on four parameters: the innerand outer radius of the disc ( r in and r out in terms of theisotropic radius) and two densities, one (denoted by S )having the meaning of Newtonian surface mass densityand scaling the disc mass and (thus) potential, and theother (denoted by W ) having similar role for the drag-ging function ω . Examples of the disc-potential profilesare shown in Figure I, as plotted in the equatorial plane(given by the disc) and along the symmetry axis.In the previous paper we found parameters of the one-component as well as two-component interpretation ofthe resulting disc (density and pressure in the formercase, while two densities in the latter one, plus the cor-responding velocities). We also computed the mass andangular momentum of the black hole and of the disc (seeequations (113) and (114) of Paper I), the main point be-ing that we adjusted the black-hole rotation (which canbe chosen rather freely within the linear order) in sucha manner that the hole keeps, in perturbation, its massas well as zero angular momentum, while the angular ve-locity of its horizon becomes non-zero (positive). This means that the hole is just being “dragged along” by thedisc. Asking about total mass and angular momentum,we found the asymptotic behaviour of ν and ω at radialinfinity. We also checked how the solution looks at im-portant locations, namely on the axis, in the equatorialplane and on the horizon.The present paper is devoted to further propertiesof the perturbative black-hole–disc solution. Section IIdeals with the influence of the disc on geometry of theblack-hole horizon, as revealed by isometric embedding ofits meridional outline into E , by the latter’s Gauss cur-vature, by its proper area and surface gravity, and by itsequatorial and meridional proper circumferences. Then,in Section III, we briefly realize that the first-order per-turbation does not give rise to an ergosphere and that thecentral singularity remains given by that of the original,Schwarzschild space-time. Section IV focuses on the in-fluence of the disc on the properties of circular equatorialgeodesics, in particular on the positions of the photon,marginally bound and marginally stable orbits, as wellas on that of a “marginally possible” (or “Lagrangian”)orbit where the gravitational influence from the hole andfrom the external source just compensate. Finally, in Sec-tion V, we check implications of several natural physicalrequirements (energy conditions and subluminal charac-ter of the disc-matter motion) for interpretations of thedisc in terms of a one-component fluid as well as in termsof a two-component dust.We use geometrized units in which c = 1, G = 1, met-ric tensor g µν has the signature ( − +++), Greek indicesrun from 0 to 3 and index-posed comma denotes partialderivative. A. Validity of the linear perturbation order
Restricting to the linear perturbation order means thatduring derivation of the solution one neglects all termsquadratic and higher-order in the perturbation quantities ν and ω ≡ ω as well as in any of its derivatives. Validityof such a result is, roughly speaking, restricted to regionswhere ν and ω as well as their derivatives are small withrespect to the unperturbed potential ν . Practically thisdepends on where the disc is placed – if its radius is large,it lies where the black-hole influence is already weak, soeven a very low values of the densities S and/or W canmake its effect dominant, mainly in its vicinity. How-ever, one is clearly more interested in the case when thedisc lies on radii where astrophysical accretion discs aresupposed to have their inner radii, i.e. around or some-what below 10 M . It is of course simple to evaluate andcompare ν , ω (and their derivatives) with ν for any The freedom in the choice of the black-hole angular momentumarises because dragging ( ω ) enters the potential ( ν ) only in thesecond order, so in the first order it has no back effect on thepotential. r rν ν equatorial planeaxis FIG. 1. Radial profiles of the constant-density-disc potential, plotted in the disc ( ≡ equatorial) plane (top row) and along thesymmetry axis (bottom row). In the left column, the disc stretches between the radii r in = 5 M and r in = 8 M and its densityis (from top to bottom curve) S = 0 . /M , 0 . /M , 0 . /M , . . . , 0 . /M . In the right column, the density is set to S = 0 . /M and the disc has radial width 3 M , with inner radius (from top to bottom curve again) r in = 2 M , 4 M , 6 M , . . . , 20 M . Thepotential clearly behaves according to expectation. (There is no black hole included in these plots, so the parameter M can beunderstood just as a certain mass scale.) given set of parameters, but let us only restrict here toproviding some idea by saying that for a disc lying – interms of the isotropic radius r – between r in = 5 M and r out = 8 M (which is just above the pure-Schwarzschildradius of the innermost stable circular orbit), the linearapproximation is valid up to some S ≃ . /M and upto some W ≃ M : for such values, the disc potential | ν | and the dragging function | ω | are at worst (close to thedisc) about 3 × smaller than the black-hole potential | ν | ,so the neglected quadratic terms are about 10 × smaller.Let us add that S ≃ . /M and W = 20 /M imply, forthe 5 M < r < M disc, that its mass is about M . = 0 . M and its angular momentum is about J = 7 . M . Wechecked that a similar bound also holds for gradient of ω (this is important, because it is the gradient squaredthrough which ω enters the equation for ν – see equation(6) or (12) in Paper I); however, dragging generally fallsoff much faster than potential when receding from the source, and one has to be careful if using higher (thanthe first) derivatives of ω – these are typically larger, atleast close to the disc (especially close to its edges).A minor note concerning the term e ν which frequentlyappears in the formulas below: the perturbation of thepotential ν is one of the quantities which should be leftonly in linear order, yet we often do not expand e ν to(1+ ν ) (though it can of course be done easily). Namely,since the dragging ( ω ) only “back-affects” the potentialin the second perturbation order, the properties which donot contain ω (in the linear order) behave like in the exact static axisymmetric case. If – for example for the illustra-tions (of such properties) to better show some tendency– one also admits larger values of ν than those strictlycomplying with the linear regime, it is thus more safe touse the full formula than its linear-in- ν part. (The resultis anyway relevant only where the linear approximationis acceptable.) II. GEOMETRY OF THE HORIZON
With the choice B = 1 − k r , the black-hole horizon liesat r = k/ k = M in our case, since we start fromthe Schwarzschild metric), so its coordinate picture is ex-actly spherical irrespectively of the perturbation. How-ever, the intrinsic shape of the horizon (given by properdistances in the two angular directions) does change dueto the presence of the additional source. In order to re-veal this, let us take the horizon as the two-dimensionalsurface { t = const , r = M/ } . From (1), its metric readsd s = ( g θθ ) H d θ + ( g φφ ) H d φ == M e ζ − ν ) H d θ + M B e − ν ) H sin θ d φ == M (cid:20) B e ν (0) (cid:21) H (cid:20) e ν ( θ ) e ν (0) d θ + e ν (0) e ν ( θ ) sin θ d φ (cid:21) H , where the relation valid on stationary and axisymmetrichorizon ζ H ( θ ) = 2 ν H ( θ ) − ν H (0) + ln B (4)has been employed (see e.g. [1], equation (24)). Substi-tuting now ν = ν + ν , where ν ≡ ν Schw and ν is theperturbation brought by the external source, one has( B e − ν ) H = 16 e − ν and the Schwarzschild part of the exponent 4 ν H ( θ ) − ν H (0) cancels out, so the horizon metric reduces tod s = 4 M e ν (0) (cid:20) e ν ( θ ) e ν (0) d θ + e ν (0) e ν ( θ ) sin θ d φ (cid:21) (5)evaluated at r = M/
2. Hence, the horizon geometry de-pends only on the metric function ν and thus in the firstperturbation order it behaves like in the exact static-casesuperposition. The latter has been solved at many places,see e.g. [3]. A. Isometric embedding into E In that paper (actually an erratum), the isometric em-bedding was summarized (as taken from [4]) of the hori-zon two-surface in a three-dimensional Euclidean space(revealing the actual horizon shape), and also the pre-scription for its Gauss curvature was given. Let us justbriefly recall that the isometric imbedding starts fromwriting the metric in the formd s = η (cid:2) f − ( µ ) d µ + f ( µ ) d φ (cid:3) , where µ := cos θ and, in our case, η = 2 M e − ν ( µ =1) ,f ( µ ) = (1 − µ ) e ν ( µ =1) − ν ( µ ) , with ν already understood to be evaluated at the horizon( r = M/ x, y, z ) is then givenby xη = p f cos φ, yη = p f sin φ, zη = µ Z s − ( f ,µ ) f d µ . (6) c azim / (2 π ) FIG. 2. Meridional section ( φ = const) of the horizon’s isomet-ric embedding into E , with the symmetry axis going in thevertical direction and the equatorial plane perpendicular toit. The horizontal axis thus represents azimuthal circumfer-ential radius (proper azimuthal circumference of the horizondivided by 2 π ), while in the vertical direction the contour goesin such a way that its length represents proper distance mea-sured along the horizon in the meridional (polar) direction.For a disc lying between r in = 0 . M and r out = 1 M andhaving density S = 0 .
0, 0 .
1, 0 .
2, 0 . . /M ), the horizon becomes more and more flattened. Bothaxes are in the units of M . B. Gauss curvature
The Gauss curvature is itself a good indicator of howthe horizon behaves when subjected to a tidal effect of theexternal source. In particular, it is a common experiencethat, when the source is about the equatorial plane, thehorizon’s axial parts may be so “strained” that the Gausscurvature decreases below zero there (a similar distortionalso arises as a consequence of rotation, as in the Kerrcase). The Gauss curvature equals half of the Ricci cur-vature scalar, computed for the chosen two-dimensionalsurface. For the metric (1), the r = const surface hasGauss curvature K ( r = const) = 1 + ν ,θθ + ( ν ,θ + ζ ,θ ) cot θ − ν ,θ ζ ,θ r e ζ − ν , which specifically for the horizon ( r = M/ ζ ,θ = 2 ν ,θ )gives K H = 1 + ν ,θθ + 3 ν ,θ cot θ − ν ,θ ) M e ν ( θ ) − ν (0) , (7)where ν again represents just the external source. (Thelast, quadratic term should therefore be omitted in orderto comply with the first-order approximation, and the ex-ponential downstairs can also be expanded accordingly.)Like for the isometric embedding, the explicit expressionvalid for our constant-density disc is rather cumbersome,but on the axis it reduces to K H ( θ = 0) = 1 − πM S (cid:16) x in − x out (cid:17) M e πMS ( x out − x in ) , (8)where we have used the horizon value (equation (90) inthe previous paper) ν ( x = 1) = − πM S (cid:18)q x − sin θ − q x − sin θ (cid:19) , (9)as written in the radial variable x := rM + M r (in terms of the latter, the horizon is on x = 1). Clearly(8) can become negative for sufficiently large Newtoniandensity of the disc S (and for the outer radius of the disc x out sufficiently larger than the inner radius x in ). For S = 0, the expression reduces to the Schwarzschild value1 / (4 M ).The effect of the constant-density disc on the horizon’sintrinsic geometry is illustrated in Figure 2. As clear fromabove, the effect is solely determined by the external-source potential (the parameter W is thus irrelevant inthe linear perturbation order), so in the case of our discequations (83) and (81) of [2] are important. The plotmay look almost like a repetition of the figure presentedin [3], but it is not so – the sources considered there weredifferent from the present disc (there, it was a Bach-Weylthin ring and an infinite disc obtained by inversion ofthe first member of the Morgan-Morgan counter-rotatingfamily). Anyway, the plot does not need much comment-ing – the horizon inflates towards to external source asexpected. What could however be mentioned here is thepaper by [5] who solved the black-hole–thin-disc problemnumerically and obtained, in some slowly-rotating cases,a prolate horizon (see discussion at the very end of ourprevious paper [2]). Such an observation has not been re-peated in any other study, and there also does not seem tobe any chance for it in our case. Actually, this is alreadyclear from equation (5): the prolate-horizon eventualitywould require ν ( θ ) > ν (0) (the potential well generatedby the external-source would have to be deeper at thehorizon’s poles than elsewhere), which, for an attractiveequatorial source, is never true.Figure 3 shows how the Gauss curvature of the hori-zon at the axis pole behaves in dependence on the disc For our choice B = 1 − M / (4 r ), it holds x ,r = B/M . It is alsouseful to note that Be − ν = (2 r + M ) / (4 r ). density S . Several cases with different inner disc radii(while the same width r out − r in ) are plotted. Generally,the curvature decreases from the Schwarzschild value of1 / (4 M ) and eventually falls below zero when the discdensity (and thus mass) is increased. Again, the plot re-sembles figure 2 of [6] where the same effect was studiedfor an exact superposition of a Schwarzschild black holewith various members of the inverted Morgan-Morgancounter-rotating disc family. SK ( θ = 0) FIG. 3. Gauss curvature of the horizon at the axis of symme-try ( θ = 0) in dependence on the disc density S , plotted fordiscs of radial width r out − r in = 0 . M and having – from thebottom to the top curve – inner radius at r in = 0 . M , 0 . M ,1 . M , . . . , 1 . M . For all of these, the Gauss curvature falls tonegative values if the disc is sufficiently dense/massive. Axesare in the units of 1 /M and 1 /M , respectively. C. Proper area and surface gravity
Another horizon properties on which the external-source effect can be studied are the horizon’s proper areaand surface gravity. The area is given by A H = I H √ g θθ g φφ d θ d φ = 2 π Z π ( Br e ζ − ν ) H sin θ d θ and in our case – with (4) and (9) – amounts to A H = 16 πM e − ν ( x =1 ,θ =0) = 16 πM e πMS ( x out − x in ) . (10)The surface gravity on a stationary and axisymmetrichorizon is given by κ := lim N → + ( g µν N ,µ N ,ν ) = (cid:26) e ν − ζ (cid:20) ( ν ,r ) + ( ν ,θ ) r (cid:21)(cid:27) H , where the lapse function N is simply N ≡ e ν . Substitut-ing again our case, we have κ H = e ν ( x =1 ,θ =0) M = 14 M e πMS ( x out − x in ) = 4 πMA H . (11)Note that this result is independent of θ ( κ H is uniformall over the horizon) as it should be, according to thezeroth law of black-hole thermodynamics, for any sta-tionary horizon. Clearly the horizon area grows rapidly(exponentially) while the surface gravity falls down whenthe disc density S increases. See Figure 4 for illustration. D. Equatorial and meridional circumferences
Deformation of the horizon due to the external sourceis naturally accompanied by a change of the ratio betweenits equatorial and meridional proper circumferences. Theequatorial (actually any azimuthal) one is very simpledue to the axial symmetry, c equa = 2 π q g φφ ( r = M/ , θ = π/
2) == πM h Be − ν ( θ = π/ i H , and in our black-hole–disc case it comes out c equa = 4 πM e − ν ( x =1 ,θ = π/ == 4 πM exp (cid:20) πM S (cid:18)q x − − q x − (cid:19)(cid:21) . (12)The meridional (in the given coordinates, it means lati-tudinal) circumference c meri = 2 Z π p g θθ ( r = M/
2) d θ = M Z π ( e ζ − ν ) H d θ == M h Be − ν (0) i H Z π e ν H ( θ ) − ν H (0) d θ is more difficult to compute explicitly. For our specificsituation, it reads c meri = 4 M e − ν ( r = M/ , Z π e ν ( r = M/ ,θ ) d θ , (13)with (9) substituted for ν ( r = M/ , θ ) again.Ratio of the horizon’s meridional to equatorial circum-ferences is plotted (together with the proper area andsurface gravity) in Figure 4 for several sequences of black-hole–disc configurations, in order to illustrate its depen-dence on the disc density (mass) and location. The plotscorrespond to flattening of the horizon due to the disc. Sometimes the behaviour of the c meri /c equa ratio is suggestedas a sufficient indicator of the horizon shape. However, this isnot always reliable, namely c meri exceeds c equa when the horizongets stretched along the axis as well as if it gets concave in theaxial regions (while c equa kept constant). III. STATIC LIMIT AND SINGULARITY
A rotating horizon is usually surrounded by a staticlimit – a surface which limits the possibility to stay at restrelative to an asymptotic rest frame (namely to “resist”rotational dragging caused by the source). In a non-rotating case, the static limit coincides with the horizon.Our first-order perturbation does not separate a staticlimit from the horizon. Actually, if “staying at rest withrespect to infinity” means to stay in spatial coordinates,i.e. to have four-velocity u µ = ( u t , , ,
0) with u t = 1 √− g tt , the static limit is given by g tt = 0, which for the metric(1) means − e ν + B r ω e − ν sin θ = 0 , so in the first order just e ν = e ν e ν = 0 . The disc potential is nowhere infinitely deep, e ν > e ν = 2 r − M r + M = 0 ⇐⇒ r = M . Another possible question is whether the physical sin-gularity of the solution is perturbed off its original, point-like character. A detailed answer is difficult and we willnot provide it here. First, one should admit that inisotropic coordinates the black-hole interior is not cov-ered. Second, the answer should be reached by iden-tifying possible divergence(s) of (e.g.) the Kretschmanncurvature invariant, which for the perturbed metric leadsto quite a long expression. Nevertheless, one can judgethe answer from the structure of the Kretschmann scalar.Computing this scalar for the metric (1) written in termsof the functions N ≡ e ν , g φφ ≡ B r e − ν sin θ , g rr ≡ e ζ − ν and ω and omitting all the terms quadratic or higher-order in ω , one is left with an expression which does not contain ω at all . All the other terms, however, are determinedpurely by ν = ν + ν , of which ν is nowhere singular, soall the singularities of the resulting space-time must begiven by singularities of the (Schwarzschild) background. IV. PROPERTIES OF EQUATORIALCIRCULAR GEODESICS
Although there is no self-gravity (non-linear effect ofthe matter on itself) involved in the first perturbation or-der, one can still estimate some features on this level al-ready – simply from how the gravitational field is changed
S r in A H κ H c meri /c equa FIG. 4. Basic properties of the horizon – its proper area A H (first row), surface gravity κ H (middle row), and ratio ofits meridional to equatorial circumferences c meri /c equa (bottom row), plotted for discs of radial width r out − r in = 0 . M , independence on their density S (left column) and inner radius r in (right column). Specifically, in the left plots, the abovequantities are plotted, against S , for ten different inner radii r in = 0 . M , 0 . M , 0 . M , . . . , 1 . M , while in the right plots theyare plotted against r in for ten different densities S = 0 . /M , 0 . /M , 0 . /M , . . . , 0 . /M . Identification of curves: in the leftcolumn, with S growing from zero, A H increases from 16 πM , κ H decreases from 1 / (4 M ) and c meri /c equa decreases from 1, themore steeply the smaller is r in ; in the right plot, with r in growing (from M/
2, which is the radius of the horizon), A H increases, κ H decreases and c meri /c equa increases, the more steeply the larger is S . Units: [ M ] for A H , [1 /M ] for κ H , dimensionless for c meri /c equa , [1 /M ] for S and M for r in . due to the perturbation. In particular, the modified fieldimplies a modified geodesic structure, ergo a differentworld-lines of free particles. Indeed, these are being fol-lowed by free test particles, but, in the given approxi- mation, also by the matter which is generating the per-turbation. We will focus on important equatorial circulargeodesics and check how they are shifted due to the grav-ity of the additional disc source. A. Light-like limits of circular motion
The first important property are the light-like limitsof circular motion, namely the light-cone boundaries ex-pressed in terms of the angular velocity Ω := d φ/ d t ,which are given byd s ( ρ = const , z = const) = ( g tt + 2 g tφ Ω + g φφ Ω ) d t = 0 . Substituting g tt = − e ν − g φφ ω , g φφ = B ρ e − ν , g tφ = − g φφ ω , one obtains Ω min , max = ω ∓ e ν ρB (14)(where for the equatorial motion everything is to beevaluated at z = 0). In our linear perturbation ofSchwarzschild, we have, in the equatorial plane,Ω min , max = ω ∓ r (2 r − M )(2 r + M ) e ν . (15) B. Zero-speed limit of free circular motion
The opposite limit of free circular motion is the caseof zero speed. Actually, when speaking of a very com-pact centre, one immediately imagines high orbital speed,necessary to produce sufficient centrifugal effect to bal-ance the centre’s gravity. However, if there is (also) someheavy enough source external to the orbit (the disc in ourcase), it may attract the test body so strongly that “noangular velocity is small enough” (even a body at restis pulled outwards). Put simply, the test particle has toorbit below the Lagrangian point (in fact a whole circle)of the system.The limit, Lagrangian orbit, has to lie in the equatorialplane and is given by a very simple condition: the radialacceleration must vanish, i.e., in the linear-perturbationorder (when dragging does not enter), ν ,r ( θ = π/
2) = 0.Substituting ν Schw + ν for the total potential, one ob-tains quite a long result (due to ν ) which is not worthpresenting. However, we will evaluate its behaviour nu-merically and include it in a summarizing figure below. C. Condition for free circular motion
The condition that the circular motion be free(geodesic; in astrophysics usually called Keplerian) iseasily obtained by demanding that the acceleration cor-responding to the four-velocity u µ = u t (1 , , , Ω) van-ishes. In the Killing-type coordinates, the accelerationhas just meridional components, of which the latitudinal(or “vertical”) one generally vanishes only in the equa-torial plane, and vanishing of the radial component has two solutionsΩ ± = − g tφ,ρ g φφ,ρ ± s(cid:18) g tφ,ρ g φφ,ρ (cid:19) − g tt,ρ g φφ,ρ == ω + g φφ ω ,ρ g φφ,ρ ± s(cid:18) ω + g φφ ω ,ρ g φφ,ρ (cid:19) − g tt,ρ g φφ,ρ . (16)In the linear approximation of our problem, clearly theparenthesis under the square root is to be omitted, sinceit is quadratic in ω . Also, of g tt = − e ν + g φφ ω one keepsonly the first term, and in the term before the squareroot one substitutes ν = ν into g φφ = B ρ e − ν . Moreprecisely, the linear approximation here means to take g φφ g φφ,ρ = Br M + 2 Br (1 − r ν ,r ) = r r + M r − M = r e ν , (17) s(cid:18) ω + g φφ ω ,ρ g φφ,ρ (cid:19) − g tt,ρ g φφ,ρ . = s − g tt,ρ g φφ,ρ . = . = 8 √ M r e ν (2 r + M ) (cid:18) r + M r − M r + M M ν ,r (cid:19) , (18)where we have already fixed to the equatorial plane(where ρ ≡ r ) and indicated by . = the restriction to thelinear order in ν (or its derivative). In the Schwarzschildlimit, the Keplerian frequencies reduce toΩ ± = ± s − g tt,ρ g φφ,ρ = ± √ M r (2 r + M ) . D. Marginally stable circular geodesics
For any kind of theoretical behaviour to be realistic,the principal realizability is a necessary condition, butphysically not a sufficient one: the behaviour should alsobe stable. In connection with thin accretion discs, thestability of their orbits with respect to perturbations act-ing within the disc plane is usually emphasized; it isknown to lead to the condition that the angular momen-tum of circular motion has to increase in the outwardradial direction, i.e. that u φ,r >
0. Actually, when sub-ject to such an equatorial perturbation, a circular orbitoscillates, with respect to an asymptotic inertial frame,with the so-called radial epicyclic frequency (see e.g. [7]for derivation in the same notation) whose square reads κ = e ν − λ ( u t ) u φ g αφ,ρ u α g tβ u β,ρ == e ν − λ u φ,ρ ( u t ) ρ B (cid:2) u φ,ρ − ( u t ) ρ B Ω ,ρ (cid:3) . (19)For a geodesic orbit, the geodesic value(s) of Ω (16)should be employed. Let us restrict to the equatorialplane and notice that the result depends on the signsof u φ,r and Ω ,r . Since the perturbation values shouldbe much smaller than the “background”, Schwarzschildones, we assume the usual argumentation (valid for iso-lated stationary black holes) can be applied: Ω + de-creases and Ω − increases with r (namely, their magnitudefalls of with distance). Hence, for “prograde” orbits, forwhich u φ > ,r <
0, stability is ensured by u φ,r > κ comes out positive). For “retrograde” orbits,on the other hand, u φ < ,r >
0, and stabilityis ensured by u φ,r < magnitude increases).In order to calculate u φ,ρ , one uses the relations (validfor any motion in the given type of space-times) u φ = g φφ u t (Ω − ω ) ,u t = [ − g tt − g φφ Ω (Ω − ω )] − / . For an unperturbed Schwarzschild, the geodesic-orbitvalue of the latter is given by( u t ± ) − = − g tt − g φφ (Ω ± ) = (2 r − M ) − M r (2 r + M ) . Calculation of u φ,r yields quite a cumbersome result, evenafter the restriction to linear perturbation, the more soafter substituting the geodesic values Ω ± , so we only il-lustrate it numerically below. E. Marginally bound circular geodesics
It is also useful to check down to where the circularorbits (geodesics in our case) are energetically bound,i.e. having − u t <
1. In terms of the contravariant four-velocity components, the marginal case thus reads1 = u t (cid:2) e ν + B ρ e − ν ω (Ω − ω ) (cid:3) . = u t (cid:0) e ν e ν + B r e − ν ω Ω (cid:1) , (20)where the second row restricts to the linear perturba-tion. In the second term, one can use B r e − ν =(2 r + M ) / (16 r ) and substitute just the unperturbed,pure-Schwarzschild value for the geodesic values of Ω,thus finding B r e − ν ω Ω ± . = ± r Mr (2 r + M ) ω . F. Photon geodesics
The “innermost” possible tracks for free circular mo-tion are found by equating the light-like limits of station-ary circular motion (15) with the geodesic values (16)–(18). One thus obtains the conditions(2 r + M ) ω ,r e ν = ∓ r − M )(2 r − M − √ M r ) ± r rM (2 r + M )(4 r + M ) ν ,r . (21) For a pure-Schwarzschild limit, they reduce to(2 r − M ) = 4 M r which gives the correct Schwarzschild value.
G. Coordinate versus geometrical measures
The statements about space-times are often being ex-pressed in coordinate terms, and this is especially thecase when speaking of location of the important orbitswe have just focused on. Although the isotropic, Weyl(or Schwarzschild) coordinates do represent some of thespace-time features quite adequately, such statementsshould be made with caution. Specifically in the caseof orbital radii, one should check them by employingmore invariant measures, like proper radial distance orcircumferential radius (given as proper circumference ofa given circle divided by 2 π ). This is mainly desirable inspace-times where there is another source present like insituation we consider here, because this additional sourcealso contributes to the metric and thus to the way howsuch measures are computed. Let us briefly check how itgoes for our black-hole–disc system.The proper radial distance from a horizon ( r = M/
2) toa given r > M/
2, calculated along all the three remainingcoordinates ( t , θ , φ ) constant, is given by r Z M/ √ g rr d r = r Z M/ e ζ − ν d r , while the proper circumference corresponding to a certain r (computed along constant t , r and, in the equatorialplane, θ = π/
2) reads π Z √ g φφ d φ = 2 π √ g φφ = 2 πBre − ν . The proper distance would thus require to know ζ which is however quite a difficult task. Actually, thisfunction is given, in the first perturbation order, by equa-tions(2 − B ) rζ ,r − Bζ ,θ cot θ + 2 − B = B (cid:2) r ( ν ,r ) − ( ν ,θ ) (cid:3) , (2 − B ) ζ ,θ + Brζ ,r cot θ − (2 − B ) cot θ = 2 Br ν ,r ν ,θ (in the first order, the function ω does not enter at all),which could be only solved numerically (with our given ν ) and we have not done it in the first paper nor here. Onthe other hand, the circumferential radius of the r = constrings is much easier to find, r cf = Bre − ν = (2 r + M ) r e ν , (22)where, for the equatorial case, one substitutes the equa-torial form of the disc potential for ν . Results obtainedfor the important circular orbits are shown in Figure 5,together with their coordinate values.0 H. Illustrations
In Figure 5, we exemplify the conditions derived abovefor a disc lying between r in = 7 M and r out = 10 M ,evaluating them numerically in terms of the coordinate(isotropic) as well as circumferential radii. The figureshows how the locations of the important circular equa-torial geodesics depend on the disc mass (actually on itsNewtonian density S ) for several values of W . With in-creasing S , the photon and marginally bound orbits godown (from their Schwarzschild values) in coordinate ra-dius, but their circumferential radii increase, because theexponential in (22) “beats” the decrease of r . The inter-val of stable circular motion, existing between the discand the pure-Schwarzschild location (in isotropic radius,it is r . = 4 . M ), shrinks and finally disappears for acertain S ; for example, in the W = 0 case this happensjust above 0 . /M (which corresponds to the total discmass M . = 0 . M ). Note that with increasing S thecircumferential radii of the disc inner and outer edges(the disc between them is grey shaded) first slightly re-cede from each other, but than come closer and finally“intersect” for S ≃ . /M . Well understandable fromexpression (22), it indicates rather strong spatial curva-ture due to the potential valley generated by the disc.(Keep in mind that S ≃ . /M is far beyond validity ofthe linear approximation – see Section I A.)In Figure 6, the properties of free circular motion (bothbelow, within and above the disc) are shown for discslying at three different radii relatively close to the blackhole. They are indicated by shades of grey: dark greyis where the geodesics are possible, time-like, bound andstable, lighter grey are possible, time-like and bound butnot stable, still lighter grey shows where they are onlypossible and time-like (but neither stable nor bound),and pure white indicates where they are only possible orwhere even none of the conditions is satisfied. We remindthe reader that “possible” means that there exists a valueof angular velocity for which a circular track at a givenradius is realizable as a geodesic. The boundary of aregion where this is fulfilled (given by the Lagrangiancircle ν ,r = 0) is indicated by the dot-dashed curve.The plots shown in Figure 6 are quite complicated,but one should bear in mind that i) their most compli-cated, right-hand parts are typically far beyond validityof the linear approximation (something like left quarterof the plots may be relevant, see Section I A), and thatii) probably only the leftmost of the plots is reasonableastrophysically, because in the others the disc is too closeto the black hole. Anyway, the best understandable andsimply behaving are the orbits which exist above the disc:these represent photon, marginally bound and marginallystable orbits of the whole system; for low disc mass, thephoton and the marginally bound orbits are seen to prac-tically coincide with the edge of the disc. Between theblack hole and the disc lie the orbits which “belong to theblack hole” but shift from their Schwarzschild positionsdue to the disc presence. As expected, this shift is slow and gradual for the photon and marginally bound orbits,whereas the marginally stable orbit(s) are much moresensitive to the details of the field; focusing already onthe disc entirely lying above the Schwarzschild radii of allthe important orbits (leftmost plot), it is seen that thereoriginally (for a negligible-mass disc) exists a region ofstable circular geodesics between the Schwarzschild valueof r ISCO and the inner disc edge, but with increased discmass this region shrinks and quite soon disappears, leav-ing the whole region below the disc unstable. In themiddle plot, the inner edge of the disc lies below thepure-Schwarzschild radius of the ISCO, and the situa-tion is seen to be just opposite to the previous case:there first exists an unstable region between the innerdisc edge and the Schwarzschild value of r ISCO , whichquite quickly shrinks and disappears with increased discmass. In any case, the disc orbits quite soon (in the senseof increasing the disc mass) become unstable at the outer disc edge. And just a final note: interesting is the regionaround the mass value of 0 . M in the middle plot,because there all the disc orbits are stable (and boundand time-like), although the inner disc edge is below thepure-Schwarzschild value of r ISCO .The dependence of the important-orbit radii on theradius of the disc is plotted in Figure 7, specifically forthe orbits lying between the horizon and the disc (the co-rotating as well as counter-rotating ones) and for smallvalues of the disc mass density. With increasing discradius, the orbits approach their Schwarzschild locations,except for the marginally bound orbits (determined byenergy with respect to infinity and thus affected evenwhen the disc lies on large radii).
V. PHYSICAL REQUIREMENTS ON THE DISC
In a previous paper [2], we considered two interpreta-tions of the disc – the interpretation in terms of a singleideal fluid in stationary circular motion, characterizedby surface density σ , azimuthal pressure P and orbitalvelocity v , and the interpretation in terms of two counter-rotating but non-interacting (dust) streams orbiting theblack hole on prograde and retrograde circular geodesics,characterized by surface densities σ ± and orbital veloc-ities v ± (all the velocities are taken with respect to thelocal zero-angular-momentum observer, ZAMO). Let ussupplement this part by listing basic physical require-ments imposed on the disc and checking what they implyfor the parameters of the above two pictures.Let us remind, from the first paper, that the two in-terpretations correspond to writing the surface energy-momentum tensor S αβ ( ρ ) := z =0 + Z z =0 − T αβ e ζ − ν d z r cf (photon) r cf (mb) r cf (ms) r (photon) r (mb) r (ms) S S S
FIG. 5. Dependence of the location of the photon (left), marginally bound (middle) and marginally stable (right) circular orbitson the disc mass (actually its density S ), drawn for a disc existing between r in = 7 M and r out = 10 M , for several values of the“dragging density” W . The bottom halves of the plots are drawn in terms of the isotropic radius r , while the top halves aregiven in terms of the corresponding circumferential radius r cf ≡ √ g φφ . In all the plots, the middle (solid) line corresponds to W = 0 and the two side curves correspond to W = 1 /M and W = 5 /M ; the orbits co-rotating with the disc are represented bydashed lines lying above the middle W = 0 curve (or, to the right of it in the right plot), while the orbits counter-rotating withrespect to the disc are represented by dot-dashed lines lying below the middle curve (to the left of it in the right plot; just oneof these exists there). In the top plots, the region filled with the disc is shaded in grey. The radii are given in the units of M ,the density S is in the units of 1 /M . in the forms S αβ = σu α u β + P w α w β (1 stream)= σ + u α + u β + + σ − u α − u β − (2 streams) . Here u α is the “bulk” four-velocity, u α = u t (1 , , , Ω) ,u α = ρBu t (cid:18) − e ν ρB − ωv, , , v (cid:19) , where ( u t ) = e − ν − B ρ e − ν (Ω − ω ) = e − ν − v and v := ρBe − ν (Ω − ω ) = √ g φφ e − ν (Ω − ω )represents linear velocity with respect to the localZAMO, and w α is the “azimuthal” vector perpendicu-lar to u α , with components w α = u t (cid:18) v, , , e ν ρB + ωv (cid:19) ,w α = ρBu t ( − Ω , , , . The four-velocities u α ± of the counter-rotating picture areof the u α ± = u α ± (1 , , , Ω ± ) form again, now with Ω ± (andthe corresponding v ± ) given by free circular motion (seeSection IV C). The parameters of the two interpretationsare related in a number of ways, which we mentioned inthe previous paper. A. Energy conditions
In terms of the surface energy-momentum tensor S µν ,the energetic conditions (slightly different requirementsfor the attractive character of gravity) readweak condition : S µν ˆ u µ ˆ u ν ≥ , dominant condition : g αβ S αµ S βν ˆ u µ ˆ u ν ≤ , strong condition : S µν ˆ u µ ˆ u ν + S ≥ , where ˆ u µ represents any future-pointing time-like four-velocity ( ≡ a physical observer). Substituting the one-stream expression for S µν , one finds easily S µν ˆ u µ ˆ u ν = ˆ γ ( σ + ˆ v P ) ,g αβ S αµ S βν ˆ u µ ˆ u ν = ˆ γ ( − σ + ˆ v P ) ,S µν ˆ u µ ˆ u ν + S γ σ + P )(1 + ˆ v ) , where ˆ γ := − u ν ˆ u ν = 1 √ − ˆ v is the Lorentz factor corresponding to the relative speedˆ v of the fluid ( u µ ) with respect to the observer (ˆ u µ ).Combining the two extreme cases ˆ v = 0 and ˆ v = 1 (plus,for the dominant condition, the requirement that the en-ergy flow − S αµ ˆ u µ should be oriented towards future), we2 r ph r mb r ms ν ,r = 0 ν ,r = 0 r ms r mb r ph r ms r ph r mb r ms r ms r mb r ph r ms r ph r mb r ms r ms r mb r ph r ms S S S r r
FIG. 6. Dependence on the disc density S of the isotropic radii r of the photon ( r ph ), marginally bound ( r mb ) and marginallystable ( r ms ) circular geodesics, drawn, together with the (dot-dashed) boundaries of the region(s) where such orbits are possibleas geodesics ( ν ,r = 0), for the W = 0 disc lying between r in = 7 M and r out = 10 M (left plot), between r in = 4 M and r out = 7 M (middle plot), and between r in = 2 M and r out = 5 M (right plot); the inner and outer radii of the discs are indicated by dashedlines. Meaning of the lines is clear from shading: dark shaded are regions where the geodesics are possible, time-like, bound andstable (in the horizontal direction), lighter grey are possible, time-like and bound but not stable, still lighter grey shows wherethey are only possible and time-like (but neither stable nor bound), and pure white indicates where they are only possible orwhere even none of the conditions is satisfied. It is helpful to keep in mind that for S = 0, i.e. when there is no disc, all thegraphs reduce to the pure-Schwarzschild values r ms . =4 . M , r mb . =2 . M and r ph . =1 . M (remember r is the isotropic radius,not the Schwarzschild one). It is also seen that with increasing mass density the counter-rotating dust interpretation of the discbecomes problematic, because – even for the disc on the largest radius (left plot) – free circular orbits within the disc graduallycease to be stable/bound/possible/time-like. The radius is in the units of M and the density is in the units of 1 /M . have, from the respective energy conditions,weak : ( σ ≥ ∧ ( σ + P ≥ , dominant : σ ≥ | P | , strong : σ + P ≥ . (Hence, the dominant condition implies the weak one,which even holds in general.) For the two-stream inter-pretation, the energy conditions imply that the densities σ ± be non-negative.Physical intuition tells that the counter-rotating in-terpretation is only possible for discs with non-negative azimuthal pressure, P ≥
0. Mathematically, this isseen from the trace of the energy-momentum tensor, σ + + σ − = − σ + P . Together with the P ≥ σ ≥ P ≥ , σ ± ≥ . (23)Finally, let us recall how σ ± , σ and P came out for ourexample of constant-density disc, as given in equations(100) and (102), (103) in [2]: σ ± = 4 r − M r + M r (cid:20) S ± W ( M r ) / π (2 r + M ) (cid:21) , (24) σ = +Σ + s Σ + 16 M r (2 r − M ) σ + σ − (4 r − M r + M ) , (25) P = − Σ + s Σ + 16 M r (2 r − M ) σ + σ − (4 r − M r + M ) , (26)where we have denoted Σ := ( σ + + σ − ) / S is the (con-stant) Newtonian surface density and W is its counter-part (also constant) which appeared in integration of thedragging differential equation, both assumed to be pos-itive and having the dimension of 1 /M . Now the en-ergy conditions (23) can be checked easily. Most notably, σ ≥ P ≥ σ ± ≥
0. The value of σ + only comes out negative at very low radii, namely at r < (1 + √ / M . = 1 . M (this is still above horizonwhich lies on r = M/ σ + might also turnnegative elsewhere if W were too large relative to S . B. Subluminal motion of the disc fluid
Another obvious requirement is that the disc fluidshould be moving with subluminal speed, v <
1. In[2], we got, for our constant-density disc, v = 4 M rσ − (2 r − M ) P (2 r − M ) σ − M rP , (27)3 r in r in r in δ r ph δ r ph δ r mb δ r mb δ r ISCO δ r ISCO
FIG. 7. Dependence on the disc inner radius r in of the isotropic radii of those photon ( r ph ), marginally bound ( r mb ) andmarginally stable ( r ms = r ISCO ) circular geodesics which lie between the black hole and the disc, drawn for light discs (density S = 10 − /M ) of radial width r out − r in = 3 M and dragging densities W = 1 /M (upper row) and W = 10 /M (lower row). Actuallyplotted are differences between the radii of the respective co-rotating and counter-rotating orbits (drawn in solid/dashed lines)and the corresponding Schwarzschild values (marked as zeros on the axes) – hence the notation by δ . With increasing radiusof the ring r in , the photon and marginally stable orbits approach their Schwarzschild positions, whereas the marginally boundorbit remains shifted due to the presence of the disc (orbital energy with respect to infinity is naturally affected by the discfor any r in ; in addition, increasing the disc radius while keeping its density generally means increasing the disc mass, so itsgravitational effect does not fall off as quickly as one might expect). All axes are scaled by M . where, when taking the square root (after substitutingfor σ and P from above), the + / − sign should be cho-sen in case that σ + > σ − / σ + < σ − . Combining theabove requirement with v ≥ r − M ) > M r ), one obtains either σ > | P | , or σ < −| P | . If consid-ering the interpretation in terms of two counter-rotatinggeodesic streams, the whole disc should lie above bothcorresponding photon geodesics (Section IV F).Note that in astro physical settings the requirement ofsubluminal motion is not sufficient, namely, if adheringto the two-stream interpretation, one would also add thatthe whole disc should lie where the free circular motionis stable (Section IV D). VI. CONCLUDING REMARKS
Several important properties have been derived ofspace-times generated by a black hole surrounded, in asymmetric manner, by a rotating light finite thin disc.We have thus continued our study of the correspondinglinear perturbation of Schwarzschild solution presentedrecently in [2]. Due to the disc, the black-hole horizongrows bigger and oblate, inflating towards the externalsource as usual. No ergosphere occurs in the first per-turbation order and the central singularity remains likein the original, Schwarzschild space-time. Free circularequatorial motion is affected by the presence of the disc,as best illustrated by how the radii of important orbitschange with the disc mass and radius. Since these or-bits (mainly the innermost stable one, ISCO) are crucialfor disc-accretion scenarios, their shift due to the disc’sown gravity should indicate how a real accretion config-uration might differ from its test-matter approximation.4For some parameter ranges this implies just shift of theinner edge of the accretion disc, while for other it mightindicate a tendency for radial fragmentation. Finally, tocalculation of basic physical parameters of the disc (madein previous paper), we have added a check of the naturalphysical requirements like energy conditions or time-like(subluminal) character of disc-matter motion; they leadto simple conditions for parameters in terms of which thedisc is interpreted.At several places, we compared the present re-sults/figures with those obtained e.g. in [3, 6]. The wholeseries to which these older papers belong (as well as oth-ers’ work cited therein) studied the influence of disc orring matter configurations on space-time of a black holewithin static and axisymmetric class of metrics. In thestatic case, the problem is much easier than in the moregeneral, stationary case considered here, because then ω = 0 and the potential ν superposes linearly. In thepresent paper, stationary rotation of the sources is ad-mitted, though it is only taken into account in the linearperturbation order. We have seen that in this order theproperties which are given solely by the gravitational po-tential ( ν ) keep their static (Schwarzschild-like) form, be-cause rotation (dragging angular velocity ω ) only “back-affects” the potential in the second order.The main literature on gravitating stationary (rotat-ing) discs or toroids around black holes was already men-tioned in the first paper. In particular, we summarizedthe paper by [5] who considered a very similar task (a thindisc around a black hole, assuming stationarity and axialsymmetry), but solved it “exactly” (numerically), witha different kind of disc and under different assumptions.We thus concluded there that it is difficult to compareLanza’s results directly with our linear-perturbation ap-proximation, although in the limit of very light disc theresults should be similar in some sense; we plan to returnto this point in future. However, to mention at least oneclear point, in contrast to the above paper, we have neverobtained a prolate deformation of the black-hole horizon. Let us, in addition, refer here to the paper by [8] whichtreated, numerically, a thick toroid rather than a thin disc(around a rotating black hole), but whose point is veryclose to that of ours: to provide a space-time describing a reasonable deviation from Kerr and test its implicationsfor astrophysical black-hole sources. In particular, theychecked there whether the presence of the massive toroidsomehow changes the gravitational waveforms producedby an equatorial inspiral of a small body onto such a sys-tem (and found that the effect in general is very small).Finally, possible future plans include trying to com-pute the first-order perturbation due to a different typeof disc. Actually, the disc with a constant density isprobably not very realistic astrophysically, although onecan hardly expect to be able to integrate the Green func-tions over a more generic function of radius, specificallyfor some which would really follow from some model ofhigh-angular-momentum accretion. Another challenge isof course to extend the treatment to higher perturba-tion orders. Namely, in the linear order there is no backreaction of the disc to its own gravity (no self -gravity),so the most inherent feature of general relativity is notpresent (specifically, the potential superposes like in thestatic case, because dragging only enters the equation forpotential quadratically). Unfortunately, the differentialequations for the k -th perturbation terms have ( k − ACKNOWLEDGMENTS
We are grateful to N. G¨urlebeck and T. Ledvinka forinterest and useful comments. O.S. thanks for supportfrom the grant GACR-17/13525S of the Czech ScienceFoundation. [1] C. M. Will, “Perturbation of a Slowly Rotating Black Holeby a Stationary Axisymmetric Ring of Matter. I. Equilib-rium Configurations,” Astrophys. J. , 521 (1974).[2] P. ˇC´ıˇzek and O. Semer´ak, “Perturbation of aSchwarzschild black hole due to a rotating thin disc,”Astrophys. J. Suppl. Series , 14 (Paper I) (2017).[3] O. Semer´ak, T. Zellerin, and M. ˇZ´aˇcek, “Erratum: Thestructure of superposed Weyl fields,” Mon. Not. R. Astron.Soc. , 207 (2001).[4] L. Smarr, “Surface Geometry of Charged Rotating BlackHoles,” Phys. Rev. D , 289 (1973). [5] A. Lanza, “Self-gravitating thin disks around rapidly ro-tating black holes,” Astrophys. J. , 141 (1992).[6] O. Semer´ak, “Gravitating discs around a Schwarzschildblack hole: III,” Class. Quantum Grav. , 1613 (2003).[7] O. Semer´ak and M. ˇZ´aˇcek, “Oscillations of staticdiscs around Schwarzschild black holes: Effect of self-gravitation,” Publ. Astron. Soc. Japan , 1067 (2000).[8] E. Barausse, L. Rezzolla, D. Petroff, and M. Ansorg,“Gravitational waves from extreme mass ratio inspiralsin nonpure Kerr spacetimes,” Phys. Rev. D75