Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk

Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk

P.Čížek and O. Semerák

Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;oldrich.semerak@mff.cuni.cz Received 2017 May 13; revised 2017 July 28; accepted 2017 August 17; published 2017 September 13

Abstract

Will, in 1974, treated the perturbation of a Schwarzschild black hole due to a slowly rotating, light, concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to perturbation by a thin disk(which is more relevant astrophysically), but, due to rather bad convergence properties, the resulting expansions are not suitable for specific(numerical)computations. However, we show that Green’s functions, represented by Will’s result, can be expressed in closed form(without multipole expansion), which is more useful. In particular, they can be integrated out over the source(a thin disk in our case)to yield good converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in thefirst perturbation order, on the simplest case of a constant-density disk, including the physical interpretation of the results in terms of a one- component perfectfluid or a two-component dust in a circular orbit about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, as it is just carried along by the dragging effect of the disk.

Key words:accretion, accretion disks –black hole physics– gravitation

1. Introduction

Disk-like structures around very compact bodies are likely to play a key role in the most energetic astrophysical sources like active galactic nuclei, X-ray binaries, supernovae, and gamma-ray bursts. Analytical modeling of such structures relies on various simplifying assumptions, the basic ones being their stationarity, axial symmetry, and test (non-gravitating)nature(see, e.g., Kato et al.2008). The last assumption is justified by two arguments:(i)in the above astrophysical systems, the mass of the disk is typically much smaller than that of the black hole(or neutron star)in their center, so the latter surely dominates the gravitational potential as well as the “radial”field, and(ii)black holes are the strongest possible(extended)gravitational sources(and neutron stars are just slightly less compact), so they would—at least in a certain region—dominate thefield even if their mass were less than that of the surrounding matter. However, such arguments need not hold for a latitudinal component of thefield(namely, perpendicular to the disk),¹and most importantly, the additional matter may in fact dominate the second and higher derivatives of the metric(curvature). These higher derivatives are in turn crucial for the stability of the matter’s motion, and thus the tricky issue of self-gravity enters the problem. Actually, real, massive matter may thus assume a quite different configuration from test matter(Abramowicz et al.1984). One should also add that even if the accreting matter really only had a tiny effect on the geometry, it could still significantly change the observational record of the source; in particular, it may change the long-term dynamics of bodies orbiting in the system(e.g., Suková & Semerák2013and references therein).

Hence, the properties of accretion systems may be sensitive to the precise shape of thefield(Semerák2003,2004). Unfortunately, general relativity is nonlinear, and the fields of multiple sources mostly cannot be obtained by simple superposition. Such more complicatedfields are being successfully treated numerically(this even applies to strongly time-dependent cases including gravitational collapse, collisions, and waves), but, for the present, the compass of explicit analytical solutions terminates at systems with a very high degree of symmetry, practically at static and axially symmetric cases. It would be most desirable to extend this tostationarycases, namely, those admitting rotation. The stationary axisymmetric problem is usually represented in the form of the Ernst equation, but what is actually being tackled is the corresponding linear problem(the Lax pair of equations whose integrability condition is just the Ernst equation). Exact solutions of this problem have been searched for in several ways. Klein & Richter(2005)and Meinel et al.(2008) summarized a“straightforward”but rather involved treatment of the respective boundary-value problems, providing both the black hole and thin-disk solutions, plus prospects of how to also obtain their nonlinear superpositions. Other attempts employed “solution- generating”techniques—mathematical procedures that transform one stationary axisymmetric metric into another and can in principle provide any solution of this type. The practical power of these methods strongly depends on how simple is the“seed”metric, so these methods usually start from a static one. Using the soliton(inverse-scattering)method of Belinsky and Zakharov, Tomimatsu(1984), Krori & Bhattacharjee(1990), Chaudhuri & Das(1997), and Zellerin & Semerák(2000)generated black holes immersed in external fields, but at least the case corresponding to a hole surrounded by a thin disk(Zellerin & Semerák2000)turned out to be unphysical (Semerák 2002). Bretón et al. (1997), who started from a different representation of the static axisymmetric seed and managed to

“install”a rotating black hole in it(see also Bretón et al.1998for a charged generalization)seem to have been more successful.

If the external-matter gravity is weak, the problem may be treated as a small perturbation of the central-source metric, which is determined by the linearized Einstein equations. In doing so, one can restrict the problem to special types of perturbations, for

© 2017. The American Astronomical Society. All rights reserved.

1 For a Schwarzschild black hole, which is spherically symmetric, there is of course no latitudinalfield, so any additional source would automatically dominate this component.

example, to stationary and axisymmetric ones. The method can be iterated; in a limit case(many iterations), it goes over to a solution in terms of series. The result then need not represent any longer a tiny variation of any“almost right”metric: it may even be put together on a Minkowski background, with the “strong”part (e.g., a black hole) “dissolved” within the fundamental systems of equations. The main problem with this scheme is the convergence and meaning of the series.

Forty-three years ago, the paper by Will(1974), published in this journal, became a seminal reference on the subject. It provided the gravitationalfield of a light and slowly rotating thin equatorial ring around a(originally Schwarzschild)black hole by mass-and- rotational perturbation of the Schwarzschild metric.(See also the succeeding paper, Will1975, where basic properties of the obtained solution were discussed.)Unfortunately, the perturbation-scheme success depends strongly on how simple the background metric is

—and the Schwarzschild metric is exceptionally simple: Will’s procedure cannot be simply extended to a Kerr background.

Consequently, only partial questions have been answered explicitly in this direction, in particular the one regarding the deformation of the Kerr black hole horizon(Chrzanowski1976; Demianski1976; interestingly, these two results do not agree on certain points, mainly in the limit of an extreme black hole).²

Recently, Will’s black hole–ring problem was revisited by Sano & Tagoshi(2014), this time using the perturbation approach of Chrzanowski, Cohen, and Kegeles, in which the metric is found on the basis of the solution of the Teukolsky equation for the Weyl scalars. Will’s results have also been followed by Hod, who analyzed the behavior of the innermost stable circular orbit in the black hole–ringfield(Hod2014)and the relation between the angular velocity of the horizon, and the black hole and ring angular momenta (Hod2015).

In the present paper, we check whether Will’s scheme can be adapted to the case of a thin equatorial stationary and axisymmetric disk. (Preliminary results were presented inČížek & Semerák2009andČížek2011.)In converging to a positive answer, wefirst observe that the expansions in spherical harmonics that typically come out in this approach(even in computing just the linear terms) converge rather badly, and their numerical processing is problematic. Much more effective is the use of the Green’s functions of the problem, namely, the perturbations generated by an infinitesimal ring. We have been able to express Green’s functions in such a way that the linear perturbation due to a thin disk can be obtained in closed form.

This paper is organized as follows. In Section 2, we introduce the equations describing the gravitational field of a thin disk.

Section3summarizes Will’s approach, and Section4discusses its(un)suitability for a numerical treatment. In Section5, we compute the Green’s functions of the problem in closed form, and in Section6we show, onlinearperturbation by a thin annular concentric disk, that they can be integrated in order to obtain a perturbation generated by a given(stationary and axisymmetric)distribution of mass. The resulting series converge much better and allow specific configurations to be computed explicitly.

For our notations and conventions: our metric signature is(−+++)and geometrized units are used, in whichc= =G 1;Greek indices run from 0 to 3, and the partial derivative is denoted by a comma. Complete elliptic integrals are given in terms of the modulus k, so

ò

^p _- ^a _a

ò

^{p - a a P}

ò

^p _{- a} ^a _{- a}

( ) ≔ ( ) ≔ ( ) ≔

( )

K k d

k

E k k d n k d

n k

1 sin

, 1 sin , ,

1 sin 1 sin

.

0 2 2 0

2 2

0 2 2 2

2 2 2

2. Black Hole and Thin-disk System: Einstein Equations and Boundary Conditions

We will search for the black hole–diskfield by perturbation of the Schwarzschild metric, while restricting the search to the simplest spacetimes that can hostrotatingsources, namely, to those which are stationary and axially symmetric. In addition, we will consider asymptoticallyflat spacetimes, without a cosmological term, and will require their orthogonal transitivity(i.e., the motion of the sources will be limited to stationary circular orbits). In such spacetimes, the time and axial Killing vectorfieldsh^m=^¶_¶^xm

t andx^m=^¶_¶^x_f^m exist and commute, and the planes tangent to the meridional directions(locally orthogonal to both Killing vectors)are integrable. Needless to say, it is assumed that there exists an axis of space-like symmetry, namely, a connected 2D (time-like)set offixed points of space-like isometry. In isotropic-type spheroidal coordinates (t, r, θ, f) (of which t and f have been chosen as parameters of the Killing symmetries), the metric with these properties can, for instance, be written in the“Carter–Thorne–Bardeen”form(e.g., Bardeen1973),

q f w q

= - ⁿ + ^{- n} ( - ) + ^{z- n}( + ) ( )

ds² e dt^{2 2} B r e^{2 2 2} sin² d dt ² e^{2 2} dr² r d^{2 2} , 1 where the unknown functionsν,B,ω, andζdepend only onrandθcovering the meridional surfaces. Besides the above coordinates, we will also occasionally use the Weyl-type cylindrical coordinatesr=rsinq andz=rcosq.

Apart from the asymptoticflatness, the boundary conditions have to befixed on the symmetry axis, on the black hole horizon, and on the external-source surface. Regularity of the axis(localflatness of the orthogonal surfacesz=constatr=0)requires thate^zB atr0⁺. The invariantsg_tt=g_abh h^{a b},g_tf=g_abh x^{a b}, andg_ff=g_abx x^{a b} have to be even functions ofρ(in order not to induce a conical singularity on the axis). Should the circumferential radius g_ff grow linearly with proper cylindrical radiusr[e^{z n-} ]_r=0, thus with ρ, there must beg_ff»O(r²), and demanding thefiniteness ofω, also-g_tf=g_ffw»O(r²).

The stationary horizon is characterized bye²ⁿ=0andw=const≕wH(in our coordinates, it specifically means thatw,_q=0there). In order for the azimuthal and latitudinal circumferences of the horizon to be positive andfinite, the functionsBre^-2nande^2z-2nhave

2 Note that another approximation possibility is the post-Newtonian expansion. The composition of a rotating gravitational center with a massive ring in Keplerian rotation was tackled, using the gravito-electromagnetic analogy, by Ruggiero(2016).

to be positive andfinite: the latter ensures the regularity ofg_rras well. Hence, Br=0 and e^2z=0 on the horizon. (Let us add in advance that thefield equations also imply thatw_,r Bandn w_{,r ,r}have to befinite on the horizon, sow_,r has to vanish there as well.) Now for boundary conditions on the external source. We assume that the latter has the form of an infinitesimally thin disk in the equatorial plane z=0, stretching over some interval of radii lying above the central black hole horizon. We assume that the disk bears neither charge nor current(there are no EMfields)and that the spacetime is reflection-symmetric with respect to its plane. The metric is then continuous everywhere, but hasfinite jumps in thefirst normal derivativesg_ab,zacross the disk. The functionsν,B,ω, and ζmust be even inz, theirz-derivatives odd in z, and even powers and multiples of derivatives(for example,B_,zn_,z)even inz (therefore, they do not jump acrossz=0).

In order for the spacetime to be stationary, axially symmetric, and orthogonally transitive, the disk elements must only move along surfaces spanned by the Killingfields, namely, they must follow spatially circular orbits with steady angular velocityW =^df

dt. This corresponds to the four-velocity

h x

h x d r w

= + W

+ W = W = - + -

a a a

a a a n

∣ ∣ ( ) a ( ) ( )

u u^t 1, 0, 0, , u u e^{t 2 t} u B v^t , 0, 0, 1 2

with

r w r w w

= - W - =

- W - = W -

n n

n n

ff n -

-

- - -

( )u e ( ) ≔ ( ) ( )

B e

e

v v B e g e

1 1 , where

t 2

2

2 2 4 2

2 2

2

represents the linear velocity with respect to the local zero-angular-momentum observer. For the thin disks(Tzz=0,Tz^r=0)without radial pressure(T_r^r=0), the surface energy-momentum tensor

ò

_-¥^{¥ ba =}

ò

₌⁼- ^{ba z- n ba r ba z- n ba r d} +

≕ ( ) ⟺ ≕ ( ) ( ) ( )

T g dz_zz T e dz S T e S z 3

z z

0

0 2 2 2 2

has only three non-zero components (S_t^t,S_f^t,S_f^f), representing energy density, orbital-momentum density, and azimuthal pressure, respectively. If(S_f^f-Stt) + 4S S_ft ^f0

2 t , it can be diagonalized toS^ab=su u^{a b} +Pw w^{a b}, whereσandP(more precisely,se^{n z-} and Pe^{n z-} ) stand for the surface density and azimuthal pressure in a co-moving frame ,and w^a is the “azimuthal” vector perpendicular tou^a, with components ^a= -

r ( f )

w _B u , 0, 0, ut

1 ,w_a=rB(-u^f, 0, 0,u^t). Hence, the surface-tensor components read

s s s s

= - - + ^f f f= + f ff= + + f

( ) ( ) ( ) f ( )

Stt P u u , St P u ut , S P P u u . 4

Orthogonally transitive stationary and axisymmetric spacetimes are described byfive independent Einstein equations. In our case of a thin disk, the energy-momentum tensor only has the T_t^t,T_f^t, andT_f^f components, and the equations read³

r

 · ( B)=0, ( )5

n r w p w p s d

  -  = - - = + +

n z n -

ff - f

· (B ) B ( ) ( ) ( ) ( ) ( )

e Be T T T B P v

v z

2 4 2 4 1

1 , 6

t t 3 2 t

4

2 2 2 2

2

r w p p r s d

  = - = - +

-

n z n

f n

- - -

· (B e ) Be T B e ( P) v ( ) ( )

v z

16 16

1 , 7

3 2 4 2 2 t 2 2

2

z +z + n + n - r w + w = p -w = p s + d

rr r n r z n -

ff - f

( ) ( ) B [( ) ( ) ] ( ) ( ) ( )

e e T T v P

v z

3

4 8 8

1 , 8

zz z z t

, , , 2

, 2 2 2

4 , 2

, 2 2 2 2

2

z_r r r -z r = - r nr - n - r rr - r + r n w - w

- r

(B ) (B ) B [( ) ( ) ] 1[(B ) (B ) ] B e [( ) ( ) ] ( )

2

1

4 , 9

z z z zz z

, , , , , 2

, 2

, , 3 3 4

, 2

, 2

z_r r +z r r = - rn nr - r r + r nw w

- r

(B ) (B ) 2B (B ) 1 B e ( )

2 , 10

z z z z z

, , , , , , , 3 3 4

, ,

where ∇and · denote the gradient and divergence in a (auxiliary) Euclidean three-space. The last two equations (for ζ) are integrable, provided the first three vacuum equations hold. The axis boundary condition e^z=B implies that the ζfunction can elsewhere be obtained according to^z(r,q)=

ò

₀^{qz z,qd +lnB, wherez},qfollows from the last twofield equations.

The treatment of the stationary axisymmetric problem, Equations (5)–(10), usually starts from a suitable solution of the first Equation(5). In the parametrization (coordinates)we use here, it is convenient to choose⁴

= - ( )

B k

1 r

4 . 11

2 2

3 Equation(8)isnotindependent, but we include it here since it provides the jump ofz_,zacross the equatorial plane given later in Section2.2.

4 When working in Weyl-type coordinates(t,ρ,z,f)and with the corresponding Weyl–Lewis–Papapetrou form of the metric, theB-equation is usually satisfied by

=

B 1. This choice is advantageous in most respects, but it makes the horizon“degenerate”to a central segment of the symmetry axisr=0, which is not suitable for a discussion of its properties. Different solutions forBare also possible for the“Carter–Thorne–Bardeen”form of the metric we use here; however, changingB generally does not imply a real physical difference—it effectively corresponds to a certain redefinition of coordinates, cf. Section6.1(only theB=1 rchoice leads to different, plane-wave solutions).

With such a choice, the horizon lies whereB=0, hence onr=k 2. This reveals the meaning of the constantk(which is supposed to be positive); in particular, for a Schwarzschild metric, one hask=M;for Kerr, it would bek=M+ M²-a² , withathe center’s specific angular momentum(k=0 would correspond to an extreme black hole, or to a Minkowski spacetime).

The main task is to solve the coupled Equations(6)and(7), and then to integrate Equations(9)and(10)forζ. With the choice

= -

B 1 ₄^k_r²₂ (thus withB,_q=0)and written out explicitly in the (t,ρ,z,f)coordinates, these equations read

n + n +n +n q= q w + w + p s+ + d

qq q n q -

(r ) r ( B) B r [ ( ) ( ) ] ( ) ( ) ( )

e r r P v

v z

ln cot

2 sin 4 1

1 , 12

r r r r r

2 , , 2

, , , ,

2 2 4

2 2

, 2

, 2 2 2

2

w w n w w w q w n p

q s d

+ - + + + - = - +

qq q q q -

n

( ) ( ) ( ) ( ) ( )

r r r r B re

B P v

v z

4 1 3 ln 3 cot 4 16

sin 1 , 13

rr r r r r

2 , , , 2

, , , , , ,

2

2

z z q n n q w w

- - _q = - _q + - - ^-n - _q

(2 B r) B cot B r[ ( ) ( ) ] 2B 2 1 B r e [r ( ) ( ) ] ( )

4 sin , 14

r r r

, , 2

, 2

, 2 3 2 4 2 2

, 2

, 2

z q+( - )z_q= n n_q+ ( - ) q- ^-nw w_q q ( )

Br cot 2 B 2Br 2 1 B cot 1 B r e

2 sin . 15

r r r

, , , , 3 3 4

, , 2

2.1. Counter-rotating Interpretation of Thin Disks

When asking about the counterbalance to its(and the black hole’s)gravity, the disk may either be considered as a solid structure(a set of circular hoops)or, on the contrary, as a non-coherent mix of azimuthal streams. In the astrophysical context, one usually adheres to the latter extreme possibility: that the disk is composed of two non-interacting streams of particles that follow stationary circular orbits in opposite azimuthal directions(Morgan & Morgan1969; Lynden-Bell & Pineault1978; Lamberti & Hamity1989;

Bičák et al.1993; Bičák & Ledvinka1993; Klein & Richter1999; González & Espitia2003; García-Reyes & González2004). These orbits are geodesic if there is no radial stress acting within the disk (T_r^r=0). The surface energy-momentum tensor is thus decomposed as

s s

= +

ab a b a b

+ + + - - - ( )

S u u u u , 16

where the+ -signs indicate the stream orbiting in a positive/negative sense off, taken with respect to the“average”fluid four- velocityu^a. The four-velocities are of theu_^a=u_^a(1, 0, 0,W)form again, so

r s s

w s s

w s s

= - +

- - =

- +

- - - =

- +

f n -

ff

f f

- + +

+

- - -

+ + +

- - -

+ +

- -

⎛

⎝⎜ ⎞

⎠⎟ ( )

S B e v

v

v

v S S v

v

v

v S S

v v

1 1 ,

1 1 ,

1 1 , 17

t t

t

t t

2

2 2

2 2

2

2 2 2

now with W_ (and the corresponding v) given by free circular motion, i.e., by the roots of the equation

+ W + W =

a f a ff a

g_tt, 2g_{t ,} g _, ² 0. Such a motion is only possible in the equatorial plane (z=0) in general, where just the radial component of acceleration remains relevant, vanishing if

w w

w w

W = W = + ^{ff r}  + -

ff r

ff r ff r

r ff r



⎛

⎝⎜⎜ ⎞

⎠⎟⎟ ( )

g g

g g

g

g^tt ; 18

, ,

, ,

2 ,

,

in particular, with theB=1choice, this expression reduces to

w r w r w rn rn

r rn

W = +  + -

-

r r n

r r

r



( ) ( )

( 4e ) 1 ( )

2 1 . 19

2 , 2

, 2 4

, ,

,

The interpretation is only possible when the expression under the square root is non-negative; physically, this is not satisfied for disks with“too much matter on larger radii”: in such a case, the total gravitational pull at a given location pointsoutwards, so“no angular velocity is low enough”to admit Keplerian orbiting there. Parameters of the counter-rotating picture (s,W)and the“total,”one- stream parameters (σ,P,Ω) are related by comparing the two respective forms of the energy-momentum tensor, su u^{a b} +Pw w^{a b}=s_{+ + +}u u^{a b}+s_{- - -}u u^{a b} . From the trace of this equation and from its projections onto u u_{a b} and w w_{a b}, while using a suitable expression of the scalar products

= -

- - = -

- - = -

- -

a a a

a ab a b

 



 



+ - + -

+ -

( )

u u vv

v v

u w v v

v v

g u u v v

v v

1

1 1

,

1 1

, 1

1 1

, 20

2 2 2 2 2 2

one finds

s- =s +s s= s - s s s

- - + -

- - = -

- - + -

- -

+ - + +

+

- -

-

+ + +

- -

-

( )

( )( )

( )

( )( )

( )

( )( )

( )

( )( ) ( )

P vv

v v

vv

v v P v v

v v

v v

v v

, 1

1 1

1

1 1 ,

1 1 1 1 , 21

2

2 2

2

2 2

2

2 2

2

2 2

s

s

s s

= = - -

- -

+

+ = + - =

a a b b

l l k k

s s +

+ - -

+ -

+ - + -

( )( )

( )( ) ⟹ ( )

P u w

u u u w u u

v v v v

vv vv

P

1 1

1

1 ..., 22

P P

s =  - W s s

W - W - =  - - -

- -

f

  

+ -





 

+ -

( ) ⟹ ( ) ( )

( )( ) ( )

u S S

v

v v Pv vv

v v v

1

1

1 . 23

t t tt

2

2 2

We may also, for example, compare expressions forS_abS^ab,finding the relation

s + =s +s + s s =s +s + s s -

- -

ab a b

+ - + - + - + - + - + -

+ -

( ) ( )

( )( ) ( )

P g u u v v

v v

2 2 1

1 1 , 24

2 2 2 2 2 2 2 2

2 2

which in combination withs- =P s₊+s_-also leads to

s =s s - s s s s

- - - = + - -

+ - + - -

+ -

+ - + -

+ -

( )

( )( ) ( ) ( ) ( )

( ) ( )

P v v

v v P P v v

v v

1 1 , 4 1

. 25

2

2 2

2 2 2

2

2.2. One-stream and Two-stream Interpretations: Integrating Jumps in the Field Equations

Relation between the jumps of g_{ab m,} across the disk and S_b^a are obtained by integrating the field Equations (5)–(8) over the infinitesimal intervalá =z 0 ,^- z=0⁺ñ. Only the terms proportional tod( )z (i.e., the source terms on the right-hand sides and the terms linear in B_,zz, n_,zz, w_,zz, and z_,zz on the left-hand side) contribute according to

ò

_z=^z=₀^{-0+n,zzdz=2n,z(z=0+) (etc.)}, so we have

= ⁺ =

( )

B,z z 0 0 and

n = = p - w - = p s+ + p s s

- = +

- + +

ff -

+ f

+ +

+

- -

-

⎛

⎝⎜ ⎞

⎠⎟

(z ) (S S S) ( P) v ( )

v

v v

v

0 2 2 2 1 v

1 2 1

1

1

1 , 26

z t

tt ,

2 2

2 2

2 2

w p

r p s w p s w s w

= = - = - + W -

- = - W -

- + W -

-

f n +

- + +

+

- -

-

⎛

⎝⎜ ⎞

⎠⎟

(z ) S ( ) ( )

B e P

v v v

0 8

8 1 8

1 1 , 27

z

t

, 2 2 4 2 2 2

z = = p -w = p s + p s s

- =

- +

ff -

+ f + +

+

- - -

⎛

⎝⎜ ⎞

⎠⎟

(z ) (S S) v P ( )

v

v v

v

0 4 4 v

1 4

1 1 , 28

z

t ,

2 2

2 2

2 2

where we have expressed the results in terms of the one-stream as well as the two-stream form ofS_b^a. 3. Perturbation Scheme

We will look for a solution of Equations(6)and (7)in the form of series, expanding

å å å

n= n l w= w l z= z l

=

¥

=

¥

=

¥

( )

, , , 29

j j j

j j j

j j j

0 0 0

where the coefficientsn_j,w_j, andz_jdepend onrandθ(orρandz), and the dimensionless parameterλis proportional to the ratio of the disk mass to the black hole massM, and more specifically, they are related by

s+ d º Sl r d = Sl d q = - Sl d q p-

( P) ( )z ( ) ( )z ( )r ( ) ( ) ( ) ( )

r r

r

1 cos 1

2 , 30

whereδdenotes theδ-distribution andlS º +s Pis an“effective”surface density. The functionsn₀,w₀, andz₀represent the black hole background, i.e., the Schwarzschild metric which in isotropic coordinates(recall thatr=rsinqand z=rcosq)reads

q q f

= - -

+ +^⎜ + ^⎟ + +

⎛

⎝⎜ ⎞

⎠⎟ ⎛

⎝ ⎞

⎠ ( ) ( )

ds r M

r M dt M

r dr r d r d

2

2 1

2 sin , 31

2

2 2

4

2 2 2 2 2 2

hence

n = - w

+ = = ^z = - ( )

r M

r M B e M

ln2 r

2 , 0, 1

4 , 32

0 0

2

0 2

and the corresponding orbital velocity is

W = + =

-

( Mr ) ( )

r M v Mr

r M

8

2 , 2

2 . 33

0

3

3 0

Substituting Equations(29)and(30)into Equations(6)and (7), and subtracting the pure Schwarzschild terms, one obtains

å å å å å

å

l n l q l n w w

p d q l

  = +

- -  

+ S +

-

=

¥

=

¥

= -

=

¥

= - -

- -

=

¥ +

⎧

⎨⎪

⎩⎪

⎡

⎣

⎢⎢

⎛

⎝⎜⎜ ⎞

⎠⎟⎟⎤

⎦

⎥⎥

⎫

⎬⎪

⎭⎪

⎡

⎣⎢

⎤

⎦⎥

· ( ) ( )

( ) ·

( ) ( )

B r M

r r M

B r

v v

2 sin

2 2 exp 4

4 cos 1

1 , 34

k

k k

k k

l k

j j j

l m

k l

m k l m

k k

k

1 2

7 2

7 4

0 2

1 1

1

0 1

2 2

å å å å

å

l q w l q l n w

p d q l

 +

-  = -  +

- - 

- + S

-

=

¥

=

¥

= -

=

¥

-

=

¥ +

⎜ ⎟

⎧⎨

⎩

⎫⎬

⎭

⎧

⎨⎪

⎩⎪

⎡

⎣

⎢⎢

⎛

⎝⎜⎜ ⎞

⎠⎟⎟⎤

⎦

⎥⎥

⎫

⎬⎪

⎭⎪

⎛

⎝ ⎞

⎠

⎡

⎣⎢

⎤

⎦⎥

· ( )

( ) · ( )

( )

( ) ( )

r M

r r M

r M

r r M

M r

v v

2 sin

2 2

2 sin

2 2 exp 4

16 1

2 cos

1 , 35

k

k k

k k

l k

j j j

k l l

k k

k 1

7 2

6 4

2 1

1 7 2

6 4

1 4

0 1

2

where[ ]f _k means the coefficient standing atl^k in a Taylor expansion off. Since the background is static, thefirst-order equations only contain the mass–energy terms multiplied by the background metric on the right-hand sides, because thefirst-line sums do not contribute. In higher orders, the right-hand sides contain only lower-order terms.(For non-static backgrounds, the equations do not decouple as easily.)

Now, the eigenfunctions with respect to θ of the operator on the left-hand sides of Equations(34) and (35)are the Legendre polynomials Pl(cosq) and the Gegenbauer polynomials C_l^{(3 2)}(cosq), respectively.⁵ Introducing a dimensionless radius

(

+

)

≔

x _M^r 1 ^M_4r²₂ , we may thus write

å å

n = n q w = w q

=

¥

=

¥

( )x P(cos ), ( )x C^{( )}(cos ). (36)

l j

lj j l

j

lj j

0 0

3 2

Substituting this into Equations (34)and (35)leads to

å

^{- n - + n q = q}

=

¥ ⎧

⎨⎩

⎡

⎣⎢

⎤

⎦⎥

⎫⎬

( ) ( ) ⎭ ( ) ( ) ( )

d

dx x d

dx j j P R x

1 1 cos , , 37

j

lj

lj j l

0

2

å

^{- + w - + + w q = q}

=

¥ ⎧

⎨⎩

⎡

⎣⎢

⎤

⎦⎥

⎫⎬

(x ) d ( ) ( ) ( ) ⎭ ^{( )}( ) ( ) ( )

dx x d

dx x j j C S x

1 1 1 3 cos , , 38

j

lj

lj j l

0

2 4 4 3 2

whereR x,l( q)andS x,l( q)stand, up to anl-independent multiplication factor, for the coefficients of expansion(with respect toλ)of the right-hand sides of Equations(34)and(35); specifically,

å

^{q l =}

å

^{q l =} _q

=

¥

=

¥

( ) [ ( )] ( ) [ ( )] ( )

R x r

B S x Br

, r.h.s. of 34 , , M

sin r.h.s. of 35 . 39

l

l l

l

l l

0

2

0

4

4 2

Provided thatR_land S_ldo not diverge on the axisq=0,p, these coefficients can also be decomposed as

å å

q = q q = q

=

¥

=

¥

( ) ( ) ( ) ( ) ( ) ^{( )}( ) ( )

R xl , R x P cos , S x, S x C cos , 40

j

lj j l

j

lj j

0 0

3 2

where

ò

^{q q q}

= +

( ) - ( ) ( ) ( ) ( )

R x j

R x P d

2 1

2 , cos cos , 41

lj l j

1 1

ò

^{q q q q}

= +

+ + -

( ) ( )( ) ( ) ^{( )}( ) ( ) ( )

S x j

j 2 j3 S x C d

2 1 2 , cos sin cos . 42

lj l j

1

1 3 2 2

5 In Will’s article, the latter are denoted byT_l^{3 2}(cosq).

Demanding that Equations(37)and(38)hold for each order(multipole moment)separately, one obtains a system of independent ordinary differential equations:

n n

- - + =

⎡

⎣⎢

⎤

( ) ⎦⎥ ( ) ( )

d

dx x d

dx j j R

1 ^lj 1 lj lj, 43

2

w w

- ⎡ + - + + =

⎣⎢

⎤

(x ) d ( ) ⎦⎥ ( ) ( ) ( )

dx x d

dx x j j S

1 1 ^lj 1 3 lj lj, 44

2 4 4

wherelÎ,j Î⁺₀. They only contain lower-order source terms(assumed to be given)for everyland are to be supplemented by boundary conditions on the horizonx=1(nljandwljare supposed to be regular there)and at spatial infinity(nljandwljshould vanish there). Of the many techniques available for such equations, we shall focus onfinding their Green’s functions.

Let us start from the fundamental systems of Equations(43)and(44). Thefirst one is the Legendre differential equation whose fundamental system can be expressed as a linear combination of Legendre functions of thefirst and second kinds:

ò

_x ^x _x

= -

¥

( ) ( ) ( )

( )[ ( )] ( )

P x Q x P x d

and P

1 . 45

j j j

x 2 j 2

For the second equation, Will (1974) used the substitution t=(x+1) 2 which transforms it to the hypergeometric differential equation, having two generators of the fundamental system,

ò

_x ^x _x

= - + +

= +

¥

⎜ ⎟

⎛

⎝

⎞

( ) ⎠ ( ) ( )

( ) [ ( )] ( )

F x F j j x

G x F x d

, 3; 4; 1 F

2 and

1 , 46

j j j

x j

2 1 4 2

where 2 1F a b c( , ; ;x)denotes the Gauss hypergeometric function. Note that since jÎ⁺₀,F(x)is in fact a polynomial of degreej.

Asymptotically(asx ¥),P xj( )~xj,Q x_j( )~x^{- -l 1},F x( )~x^j, andG x( )~x^{- -j 3}. At the horizon(x=1),Q_j(x)andG_j(x)diverge (exceptG x₀( ), which will be discussed later).

Given the above boundary conditions, the Green’s functions of Equations(37)and(38)can be found in the form

 

¢ = - ¢  ¢

- ¢ ¢

n ⎧

⎨⎩

( ) ( ) ( )

( ) ( ) ( )

x x Q x P x x x

P x Q x x x

, for

for , 47

j

j j

j j

 

¢ = - ¢  ¢

- ¢ ¢

w ⎧

⎨⎩

( ) ( ) ( )

( ) ( ) ( )

x x G x F x x x

F x G x x x

, for

for , 48

j

j j

j j

and their inhomogeneous solutions as

ò



nlj( )x = ^¥ R xlj( )¢ ⁿj(x x dx, ¢) ¢, (49)

1

ò



w d

= ¢ ¢ ¢ +

+

¥ w

( ) ( ) ( )

( ) ( )

x S x x x dx J

, x

1 , 50

lj lj j

l j

1

0 3

where J_l are arbitrary constants, representing a choice of the black hole spin (see Section 3.1). Such a solution is unique up to a coordinate transformation. Note that one could add to the Green functions terms proportional to P x0( )=F x0( )=1 which is everywherefinite, but such an addition only corresponds to a rescaling of time and(thus)of the coordinate angular velocity, so it has no invariant physical effect.

Using Equation(36), we may write





ò ò ò

å å

n q n q q

q q q q

= = ¢ ¢ ¢

= ¢ ¢ ¢ ¢ ¢ ¢

n

n

=

¥ ¥

=

¥

¥ -

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x x P R x x x P dx

R x x x dx d

, cos , cos

, , , , cos , 51

l

j

lj j

j

lj j j

l

0 1 0

1 1

1





ò ò ò

å å

w q w q q

q q q q

= = ¢ ¢ ¢

= ¢ ¢ ¢ ¢ ¢ ¢

w

w

=

¥ ¥

=

¥

¥ -

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

x x C S x x x C dx

S x x x dx d

, cos , cos

, , , , cos , 52

l

j

lj j

j

lj j j

l 0

3 2

1 0

3 2

1 1

1