Ultrahard fluid and scalar field in the Kerr-Newman metric

Ultrahard fluid and scalar field in the Kerr-Newman metric

E. Babichev,1,2 S. Chernov,2,3 V. Dokuchaev,2 and Yu. Eroshenko2

1APC, Universite Paris 7, rue Alice Domon Duquet, 75205 Paris Cedex 13, France2Institute for Nuclear Research of the Russian Academy of Sciences, Prospekt 60-letiya Oktyabrya 7a, Moscow 117312, Russia

3P. N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninsky Prospekt 53, 119991 Moscow, Russia(Received 11 July 2008; published 24 November 2008)

An analytic solution for the accretion of ultrahard perfect fluid onto a moving Kerr-Newman black hole

is found. This solution is a generalization of the previously known solution by Petrich, Shapiro, and

Teukolsky for a Kerr black hole. We show that the found solution is applicable for the case of a

nonextreme black hole, however it cannot describe the accretion onto an extreme black hole due to

violation of the test fluid approximation. We also present a stationary solution for a massless scalar field in

the metric of a Kerr-Newman naked singularity.

DOI: 10.1103/PhysRevD.78.104027 PACS numbers: 04.20.Dw, 04.70.Bw, 04.70.Dy, 95.35.+d

I. INTRODUCTION

The only known three-dimensional exact solution for theaccretion flow onto the Kerr black hole is the analyticalsolution by Petrich, Shapiro, and Teukolsky [1]. This so-lution describes the stationary accretion of a perfect fluidwith the ultrahard equation of state, p ¼ �, with p beingthe pressure and � being the energy density, onto a movingKerr black hole (see also [2–5]). Here we generalize thissolution to the case of a moving Kerr-Newman black hole.For the Kerr-Newman metric with naked singularity, wepresent the stationary solution for a massless scalar field.

The problem of a steady-state accretion onto a (moving)black hole can be formulated as follows. Consider a blackhole moving through a fluid with a given equation of state,p ¼ pð�Þ [6]. Usually it is assumed that the backreactionof fluid to the metric is negligible, in which case theproblem is being solved in the test fluid approximation. Itis also assumed that the black hole mass changes in timeslowly enough, such that the steady-state accretion is es-tablished (the so-called quasistationary process). The goalis to find a stationary solution to the equations of motionfor the flowing fluid in the gravitational field of the blackhole.

Figure 1 illustrates an example of this solution for thecase of fluid passing the Schwarzschild black hole. It isunlikely that the Kerr-Newman black hole (or a nakedsingularity) is found in astrophysical context, as well asthe accreting fluid with the ultrahard equation of state.However, the theoretical study of these questions can beuseful for better understanding both the principal generalrelativity problems, e.g. the approaching to the extremeblack hole state, and the real matter accretion onto theastrophysical compact objects.

The velocity of relativistic perfect fluid, u�, in theabsence of vorticity can be expressed as the gradient ofthe scalar potential c , see, e. g. [7],

hu� ¼ c ;�; (1)

where h ¼ d�=dn ¼ ðpþ �Þ=n is the fluid enthalpy prop-

erly normalized, h ¼ ðc ;�c ;�Þ1=2, n is the particle number

density. When the equation of state for the perfect fluid hasa simple form, p ¼ pð�Þ, then the number density can beexpressed in terms of the energy density and the pressure:

nð�Þn1

¼ exp

�Z �

�1

d�0

�0 þ pð�0Þ�; (2)

where n1 and �1 are correspondingly the number densityand the energy density at the infinity. The above equation

4 2 0 2 44

2

0

2

4

6

r sin θ M

rco

sθ

M

FIG. 1. The three-velocity field of the fluid v for the case ofSchwarzschild black hole, and we take v1 ¼ 0:5.

PHYSICAL REVIEW D 78, 104027 (2008)

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http://dx.doi.org/10.1103/PhysRevD.78.104027

can be seen as a formal definition of the ‘‘number density’’for a perfect fluid with an arbitrary equation of state p ¼pð�Þ.

A specific feature of a fluid with the ultrahard equationof state is that the resulting equation for the scalar potentialis linear [8]. For p ¼ �, from Eq. (2) one easily obtains,� / n2. As a result, the number density conservation,ðnu�Þ;� ¼ 0, gives the Klein-Gordon equation for a mass-

less scalar field, hc ¼ 0. This reveals a way to obtain anexact solution of the accretion problem in the Kerr metricby separation of variables and decomposition in the spheri-cal harmonic series [1]. Our study of the perfect fluids andscalar fields in the Kerr-Newmanmetric closely follows theanalysis of Petrich et al. [1].

The introduction of the potential c , Eq. (1), provides away to solve a problem of hydrodynamics dealing with anequation for a scalar function. This is not surprising, sinceany scalar field c is fully equivalent to the hydrodynamicaldescription, provided that the vector c ;� is timelike. Forexample, the canonical massless scalar field c with theenergy-momentum tensor,

T�� ¼ c ;�c ;� � 12g��g

��c ;�c ;� (3)

is equivalent to a perfect fluid with the ultrahard equationof state, provided the following identifications are takeninto account:

u� � c ;�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic ;�c

;�p ; p ¼ � � 1

2c ;�c

;�: (4)

Therefore, in what follows we do not make a distinctionbetween the massless scalar field and ultrahard perfectfluid, as far as the vector c ;� is timelike. For example,the problem of accretion of a perfect fluid can be viewed asaccretion of a scalar field, with a specific form of theboundary condition at the infinity, c ! _c1t. Here _c1 isexpressed in terms of the fluid density at the infinity, �1, asfollows, _c1 ¼ ð2�1Þ1=2. However, if a solution for thescalar field contains spacelike c ;� somewhere then such asolution can be identified with no perfect fluid analogue.

The paper is organized as follows. Accretion of ultrahardperfect fluid (massless scalar field) onto the moving Kerr-Newman black hole is considered in Sec. II. The basicequation of motion for the scalar potential is derived inSec. II A; in Sec. II B we solve this equation for the case ofthe moving nonextreme black hole; in Sec. II C we analyzein detail the case of the extreme black hole. In Sec. III wefind a stationary solution for the scalar field in the Kerr-Newman singularity. In Sec. IV we briefly describe theresults of the work.

II. ACCRETION OF ULTRAHARD FLUID

A. Metric and equation of motion

The Kerr-Newman metric can be written as [9,10],

ds2 ¼ ��

Adt2 �Asin2�

�ðd��!dtÞ2 � �

�dr2 ��d�2;

(5)

where

� ¼ r2 � 2Mrþ a2 þQ2; (6)

� ¼ r2 þ a2cos2�; (7)

A ¼ ðr2 þ a2Þ2 � a2�sin2�; (8)

! ¼ 2Mr�Q2

Aa: (9)

HereM is the mass of a black hole or a naked singularity, ais the specific angular momentum,Q is the electric charge,and ! is the angular dragging velocity. The event horizonof the Kerr-Newman black hole, r ¼ rþ, is the larger rootof the equation � ¼ 0, i. e. r� ¼ M� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2 � a2 �Q2p

.The event horizon exists only if M2 � a2 þQ2. WhenM2 < a2 þQ2 the metric (5) describes a naked singularity.The Klein-Gordon equation for a massless scalar field in

the background gravitational field g�� is given by

c ;�;� ¼ 1ffiffiffiffiffiffiffi�g

p @

@x�

� ffiffiffiffiffiffiffi�gp

g��@c

@x�

�¼ 0: (10)

For the Kerr-Newman metric (5) the above equation can berewritten as�1

�½ðr2 þ a2Þ@t þ a@��2 � 1

sin2�ð@� þ asin2�@tÞ2

� @rð�@rÞ � 1

sin�@�ðsin�@�Þ

�c ¼ 0: (11)

Equation (11) is the main equation we will use to findanalytic solutions.

B. Accretion onto a moving black hole

First we consider a stationary accretion of an ultrahardperfect fluid onto a nonextreme black hole, i.e. we assumeM2 > a2 þQ2. The first boundary condition for Eq. (11) isa relation at the space infinity [1], r ! 1:

c ¼ �u01tþ u1r½cos� cos�0 þ sin� sin�0 cosð��0Þ�;(12)

where u01 is the 0-component of the black hole 4-velocity

BABICHEV, CHERNOV, DOKUCHAEV, AND EROSHENKO PHYSICAL REVIEW D 78, 104027 (2008)

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at infinity

u� � ðu0;u1Þ ¼ ð1� v2Þ�1=2ð1; v1Þ; (13)

and u1 � ju1j; while v1 is the 3-velocity vector of theblack hole at the infinity with the space orientation speci-fied by the two arbitrary angles, �0 and �0.

There are several approaches to formulate the secondboundary condition for Eq. (11) in the case of the sta-tionary accretion onto the black hole.

The classical approach, originated from the Bondi prob-lem [11,12], fixes the value of the energy flux onto a blackhole at the critical sound surface [13–16]. The critical pointis fixed by the requirement that the physical solution isdouble valued neither in the velocity of the fluid nor in theradial coordinate. This requirement picks up one physicalsolution from the infinite number of formal solutions,parametrized by the flux of the energy onto a black hole.In Fig. 2 several solutions with different fluxes are shown,along with the ‘‘correct’’ (critical) one.

An alternative way to fix the flux is to ask that the energydensity is finite at the black hole event horizon r ¼ rþ (seee.g. [1]). In most cases both approaches give the sameresult. In particular, this is true for a nonextreme Kerr-Newman black hole. As we see later, however, in thespecific case of the extreme Kerr-Newman black hole therequirement for the flow to have a trans-sonic point gives a

solution for which the energy density is infinite at thehorizon (see details in the next Sec. II C).Note that at the critical surface the infall velocity of the

accreting fluid reaches the sound speed cs, which for theultrahard fluid is equal to the speed of light, cs ¼ 1. Forthis reason in the considered case a sound surface coincideswith the event horizon r ¼ rþ.Following Petrich et al. [1], we search a solution of

Eq. (11) separating variables and decomposing c in thespherical harmonic series:

c ¼ �u01tþXl;m

Al;mRlðrÞYlmð�;�Þ; (14)

where the radial part RlðrÞ satisfies the equation

d

dr

��dRlðrÞdr

�þ

��lðlþ 1Þ þm2a2

�

�RlðrÞ ¼ 0: (15)

It is convenient to define a new variable by the relation,

r ¼ Mþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2

q: (16)

Note that the definition of , Eq. (16), does not work in thecase of the extreme black hole. Using (16) we can rewriteEq. (15) in the form

ð1� 2ÞR00 � 2R0

þ�lðlþ 1Þ �m2ði�Þ2

1� 2

�R ¼ 0;

(17)

where � ¼ a=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2

p. This is the Legendre equa-

tion with an imaginary second index [17]. The generalsolution of Eq. (14) is

c ¼ �u01tþXl

½AlPlðÞ þ BlQlðÞ�Yl0ð�;�Þ

þX0

lm

½AþlmP

im�l ðÞ þ A�

lmP�im�l ðÞ�Ylmð�;�Þ: (18)

In the above equation the prime ð0Þ denotes that we do notinclude the term with m ¼ 0; Pl and Ql are the Legendrefunctions of the first and second kind correspondingly; andPim�l is the associated Legendre function which can be

expressed in terms of the hypergeometric function [17]:

Pim�l ðÞ / eimFð�l; lþ 1; 1� im�; ð1� Þ=2Þ; (19)

where

¼ �

2lnþ 1

� 1¼ a

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2

p lnr� r�r� rþ

: (20)

Then the components of the 4-velocity are

2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.2

0.4

0.6

0.8

1.0

r M

ur

FIG. 2. Possible types of solutions for the radial component ofthe fluid four-velocity are shown as a function of the radius forthe accretion of fluid with the equation of state p ¼ �=2 onto aSchwarzschild black hole. Different curves correspond to differ-ent values of the influx. The single solution with a critical valueof inflow, corresponding to the physical solution, is shown by thesolid curve with a dot labeling the critical point.

ULTRAHARD FLUID AND SCALAR FIELD IN THE KERR- . . . PHYSICAL REVIEW D 78, 104027 (2008)

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nut ¼ �u01;

nur ¼fPl½AlðPlÞ0 þ BlðQlÞ0�Yl0 þP0

lm½AþlmðPim�

l Þ0 þ A�lmðP�im�

l Þ0�YlmgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2

p ;

nu� ¼�X

l

½AlPl þ BlQl�@Yl0

@�þX0

lm

½AþlmP

im�l þ A�

lmP�im�l � @Ylm

@�

�;

nu� ¼�X0

lm

½AþlmP

im�l þ A�

lmP�im�l � @Ylm

@�

�;

(21)

where subindex denotes a derivative on the variable . Using the normalization condition u�u� ¼ 1, we obtain

n2 ¼ ð��Þ�1½ðr2 þ a2Þu01 � anu��2 � ð�sin2�Þ�1½nu� � asin2�u01�2 � �

�ðnurÞ2 ��1ðnu�Þ2: (22)

In the limit r ! rþ we find

ðn2��Þjr!rþ ¼�ðr2þ þ a2Þu01 �X0

l;m

imðAþlme

im þ A��lme

�imÞYlmð�;�Þ�2

��aX0

l;m

imð�Aþlme

im þ A�lme

�inÞYlmð�;�Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2

q Xl

BlYlmð�;�Þ�2: (23)

From the finiteness of the fluid density at the event horizonit follows that all Bl equal to zero except B0, and B0 > 0.All coefficients Aþ

lm are also zero, since the terms contain-ing Aþ

lm are irregular at the horizon. The correspondingsolution, which is finite at the horizon, reduces to

c ¼ �u01tþ ðr2þ þ a2Þu012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2


þXl;m

AlmFð�l; lþ 1; 1þ im�; ð1� Þ=2Þ

� Ylmð�;�� Þ; (24)

where the second term comes from the Legendre functionQ0ðxÞ:

Q0ðxÞ ¼ 1

2ln1þ x

1� x: (25)

Using the boundary condition at the infinity, Eq. (12), wefinally obtain

c ¼ �u01tþ ðr2þ þ a2Þu012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � a2 �Q2


þ u1ðr�MÞ cos� cos�0þ u1 Re½ðr�Mþ iaÞ sin� sin�0eið��0�Þ�: (26)

In the case of the Kerr black hole the solution (26) reducesto the known solution by Petrich et al [1]. From (26) onecan find the components of the 4-velocity:

nut ¼ �u01;

nur ¼ �u01r2þ þ a2

�þ u1 cos� cos�0

þ u1 Re

��1þ ia

�ðr�Mþ iaÞ

�

� sin� sin�0eið��0�Þ

�;

nu� ¼ �u1ðr�MÞ sin� cos�0þ u1 Re½ðr�Mþ iaÞ cos� sin�0eið��0�Þ�;

nu� ¼ �u1 Im½ðr�Mþ iaÞ sin� sin�0eið��0�Þ�:

(27)

The accretion rate of the fluid onto the black hole is givenby

_N ¼ �ZSnui

ffiffiffiffiffiffiffi�gp

dSI ¼ 4�ðr2þ þ a2Þn1u01; (28)

while the corresponding rate of the energy flux onto theblack hole is

_M ¼ 4�ðr2þ þ a2Þð�1 þ p1Þu01: (29)

Note that as in the Kerr case, the flux (28) is independenton the direction of the black hole motion.From Eq. (26) one can see that in the limit of the extreme

black hole,M2 � a2 �Q2 ! 0, the solution for the accre-tion is problematic, since c contains a divergent term

proportional to ðM2 � a2 �Q2Þ�1=2. Moreover, in thislimit n ! 1 while ur ! 0 at the event horizon, whichclearly signals on the violation of the test fluid approxima-tion. However, it might happen that going to the limitM2 � a2 �Q2 ! 0 we missed a correct regular solution,valid only in the extreme case. To check this, in the next


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Sec. II C, we will redo all the calculations for the extremecase, M2 ¼ a2 þQ2.

Let us study in more detail some particular cases. First,we set a ¼ 0, i.e. we consider a moving nonrotatingcharged black hole. As in the case of a Schwarzschildblack hole (see [1]), there is a stagnation point: the pointwhere the fluid velocity is zero relative to the black hole. Itis not difficult to find the location of the stagnation point:

rstag ¼ M

�1þ

�1þ 2rþM�Q2ð1þ v1Þ

M2v1

�1=2

�:

As an example, in Fig. 3 we show the three-velocity of thefluid in the case of the Reissner-Nordstrom black hole withQ ¼ 0:99M and v1 ¼ 0:5.

Let us turn to another particular case. Substituting u1 ¼0 into Eq. (22) we find the radial density distribution of theaccreting ultrahard fluid onto the Kerr-Newman black holeat rest:

n2 ¼ A� ðr2þ þ a2Þ2��

: (30)

This radial distribution is shown in Fig. 4. At the eventhorizon r ¼ rþ the ratio of densities at the equator (� ¼�=2) and the pole (� ¼ 0) is

�ðrþ; �=2Þ�ðrþ; 0Þ

¼ 4rþðr2þ þ a2Þ � a2ðrþ � r�Þ4r3þ

: (31)

In the appendix we provide an alternative approach to solve

a problem for the stationary accretion, in the case when ablack hole is at rest.

C. Accretion onto the extreme black hole

In this section we will concentrate specifically on theaccretion onto the extreme Kerr-Newman black hole, try-ing to find a good solution. We search a stationary solutionof the wave equation (11) in the form (14), like in thenonextreme case. The radial part RlðrÞ now satisfies Eq.(15) with � ¼ ðr�MÞ2. Using a new variable,

� ¼ r=M� 1;

we transform (15) to a simpler equation

�2R00�� þ 2�R0

� þ�m2a2

M2�2� lðlþ 1Þ

�R ¼ 0: (32)

For m ¼ 0 Eq. (32) reduces to the Euler’s equation with asolution

R ¼ C1

�r

M� 1

�l þ C2

�r

M� 1

��l�1: (33)

The corresponding potential is

c ¼ �u01tþXl;m

�C1lm

�r

M� 1

�l þ C2

lm

�r

M� 1

��l�1�Ylm:

(34)

Using the boundary condition at infinity we find

c ¼ �u01tþ u01M2

r�Mþ u1ðr�MÞ cos�: (35)

4 2 0 2 44

2

0

2

4

6

r sin θ M

rco

sθ

M

FIG. 3. The three-velocity of the fluid in the case of the almostextremal Reissner-Nordstrom black hole, Q ¼ 0:99M. We takev1 ¼ 0:5. The stagnation point is indicated by a star.

1.6 2 2.5

3

4

5

6

r/M

n2 (r)

θ=π/2

1.6 2 2.5

3

4

5

r/M

n2 (r)

θ=0

FIG. 4 (color online). Radial density distribution of the sta-tionary accreting ultrahard fluid around the Kerr-Newman blackhole is shown in the equatorial plane (� ¼ �=2) and along thepolar axis (� ¼ 0). In both plots the solid lines correspond to thecase Q ¼ ð2=3ÞM, a ¼ M=2, dashed lines are for Q ¼ ð2=3ÞM,a ¼ 0, and dotted lines are for Q ¼ 0, a ¼ M=2, respectively.


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For m � 0 the solution of Eq. (15) is

R ¼ffiffiffiffiffiffiffiffima

M

s �C1Jlþ1=2

�ma

M

�þ C2�lþ1=2

�ma

M

��; (36)

where J and � are the Bessel functions of the first andsecond kind correspondingly. The general solution of Eq.(32) is

c ¼ �u01tþXl

�C1l0

�r�M

M

�l þ C2

l0

�r�M

M

��l�1�Yl0

þX0

l;m

ffiffiffiffiffiffiffiffima

M

s �C1þlm Jlþ1=2

�ma

M

�þ C2�

lm �lþ1=2

�ma

M

��

� Ylm; (37)

where it used the Bessel functions [18]:

J�ðxÞ ¼X1k¼0

ð�1Þkðx=2Þ�þ2k

k!�ð�þ kþ 1Þ ; (38)

��ðxÞ ¼ J�ðxÞ cos�� J��ðxÞsin��

; (39)

J�3=2ðxÞ ¼ffiffiffiffiffiffiffi2

�x

s �� cosx

x� sinx

�: (40)

From the first boundary condition at space infinitywe find C1

l0 ¼ C2�1m ¼ 0, C1

10 ¼ Mu1 cos�0, C2�11 ¼

� ffiffiffiffiffiffiffiffiffi�=2

pu1a sin�0, while the second boundary condition

fixes the value of the influx at the sound surface: C1þlm ¼

C2l0 ¼ 0, C2

00 ¼ u01ðM2 þ a2Þ=M. With these boundary

conditions we obtain

c ¼ �u01tþ u1ðr�MÞ cos� cos�0 þ ðM2 þ a2Þu01r�M

þ u1a sin� sin�0 cosð��0Þ�

�r�M

acos

a

r�Mþ sin

a

r�M

�: (41)

The components of the 4-velocity are given by

nut ¼ �u01;

nur ¼ �ðM2 þ a2Þu01ðr�MÞ2 þ u1 cos� cos�0 þ u1 sin�0 sin� cosð��0Þ cos

�a

r�M

�

��1� a2

ðr�MÞ2 þa

r�Mtan

�a

r�M

��;

nu� ¼ �u1ðr�MÞ sin� cos�0 þ u1a cos� sin�0 cosð��0Þ�r�M

acos

a

r�Mþ sin

a

r�M

�;

nu� ¼ �u1a sin�0 sin� sinð��0Þ�r�M

acos

a

r�Mþ sin

a

r�M

�:

(42)

Using the above solution, one can calculate the accretionrate

_N ¼ 4�ðM2 þ a2Þu01n1; (43)

and radial density distribution (for u1 ¼ 0)

n2 ¼ ðr2 þ a2Þ2 � ðM2 þ a2Þ2�ðr�MÞ2 � a2sin2�

�: (44)

From (42) and (44) one can see that at the event horizon ofthe extreme black hole rþ ¼ M both the radial 4-velocityur and the energy density � behave as ur ! 0 and � /n2 / ðr�MÞ�1 ! 1. And the full mass of fluid near theblack hole tends to infinity too. This behavior is an indi-cation of violation of the test fluid approximation. For thisreason the solution (41), (42), and (44) is not self-consistent, and backreaction of the accreting fluid mustbe taken into account to obtain a physically relevantsolution.

III. SCALAR FIELD AROUND NAKEDSINGULARITY

It is worthwhile to note that the space-time of the Kerr-Newman naked singularity is pathological when both Qand a are nonzero [19]. In general, this space-time is notglobally hyperbolic, which means, in particular, that theCauchy problem for the scalar field is not well posed. Theorigin of the pathology is inside the region A< 0, wherefunctionA is given by Eq. (8). This region contains closedcausal curves, and it is possible to start from a point at theasymptotically flat region far from the singularity, to gointo the ‘‘pathological’’ region, and to travel back in timethere and then come back to the starting point. Thus a timemachine can be constructed. If, however, Q ¼ 0 or a ¼ 0the space-time is not pathological anywhere, except for thepoint of the physical singularity at r ¼ 0.Below we describe an analytic solution for the stationary

distribution of the scalar field in the Kerr-Newman geome-try. For simplicity we consider the case when the naked


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singularity is at rest, with respect to the scalar field at theinfinity. Thus we impose the boundary condition at theinfinity in the following form:

c ! _c1tþ c1 at r ! 1; (45)

where _c1 and c1 are constants.The corresponding solution of the wave equation (11) is

given by Eq. (18), provided that the following replace-ments are made:

! �i; � ! i�; �u01 ! _c1:

The boundary condition at the infinity gives Aþlm ¼ A�

lm ¼0 for all l andm, since the solution should not depend on�as r ! 1. To satisfy Eq. (45) it is also necessary to putAl ¼ 0 and Bl ¼ 0 for all l � 0, since the terms corre-sponding to l � 0 contain solutions which diverge at r !1. The term containing A0 is a constant which can beidentified with c1. The full solution which satisfies theboundary condition (45) can be written as

c ¼ _c1tþ c1 þ B

�arctan

�r�Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þQ2 �M2p

��

2

�;

(46)

where the last term comes from Q0, and B is an arbitraryconstant. The components of the energy-momentum tensorare


Trt ¼ � _c1B sin�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þQ2 �M2

q; (47)

Ttt ¼ �Tr

r ¼ B2ða2 þQ2 �M2Þ þA _c 212��

; (48)

T�� ¼ T�

� ¼ B2ða2 þQ2 �M2Þ �A _c 212��

: (49)

The energy flux onto the singularity is nonzero whenB _c1 � 0. The corresponding radial 4-velocity of the in-flowing fluid is

ur ¼ � Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þQ2 �M2

p�

_c 21A� B2ða2 þQ2 �M2Þ : (50)

Note that the influx (47) is not fixed when both constants Band _c1 are nonzero, because in the case of the nakedsingularity one physical boundary condition is missing,which for the black hole was the boundary condition atthe event horizon or at the critical point. The energy fluxonto the central singularity is zero, in particular, cases B ¼0 or _c1 ¼ 0.

A particular solution (46) with B ¼ 0 and _c1 � 0describes the stationary distribution of the massless scalar

field (or, equivalently, the ultrahard fluid with p ¼ �)without the influx around the naked singularity. This solu-tion has a very specific feature: the energy density of thescalar field �ðr; �Þ ¼ Tt

t from Eq. (48) becomes zero at thesurface A ¼ 0, inside of which the causality is violated.Since the space-time with both a and Q nonzero is

pathological, in what follows we consider only the Kerrand Reissner-Nordstrom cases, correspondingly Q ¼ 0and a ¼ 0, where the noncausal region A< 0 is absent.For example, in Fig. 5 the distribution of the energy densityof ultrahard fluid with zero flux around the Kerr nakedsingularity is shown.Another particular solution (46) with _c1 ¼ 0 and B �

0 describes a stationary distribution of the massless scalarfield with zero energy density at infinity. This solution doesnot have a perfect fluid analogue. In Fig. 6 the energydensity distribution for the scalar field is shown. The totalmass of the scalar field for this solution is finite, Mf <1.

To calculate this mass we use the quasistatic coordinate

frame ðt; r; �; ~�Þ with azimuthal coordinate ~�,

d ~� ¼ d��!dt: (51)

This coordinate frame is ‘‘rotating with the geometry’’[20,21] with the angular velocity ! given by (9). Anobserver at rest in this frame rotates in the azimuthaldirection with an angular velocity !. The total mass ofthe scalar field stationary distributed around the nakedsingularity is

-4 -2 0 2 4r cosθ M

-4

-2

0

2

4

rsi

nθM

FIG. 5 (color online). Density plot in the poloidal plane of theenergy density distribution of ultrahard perfect fluid (masslessscalar field) around the Kerr naked singularity with a ¼ 2M. Theparticular solution (46) with _c1 � 0 and B ¼ 0 is taken, so thatthe influx onto the singularity is zero.


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Mf ¼Z

�ðtÞT

��


d�� ¼Z

~Ttt

ffiffiffiffiffiffiffi�~gp

d3~x; (52)

where d3~x ¼ drd�d ~� andffiffiffiffiffiffiffi�~g

p ¼ �sin�. Using the ex-pression for the energy-momentum tensor (3), we calculatethe scalar energy density ~Tt

t for the solution (46) in the

quasistatic frame ðt; r; �; ~�Þ:~T tt ¼ � 1

2~grrðc ;rÞ2 ¼ 1

2B2 2

��M2; (53)

where we define a superextreme parameter

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þQ2 �M2

M2

s; a2 þQ2 �m2 > 0: (54)

Substituting ~Ttt from (53) into (52), we obtain

Mf ¼ 2�B2 2M2Z 1

0

dr

�¼ �B2ð�þ 2arccot Þ M:

(55)

This equation relates the constant B in (46) with a totalmass of the scalar field.

IV. CONCLUSION

We found an analytic solution for the stationary accre-tion of ultrahard perfect fluid onto the moving Kerr-Newman black hole. The presented solution is a general-ization of the Petrich, Teukolsky, and Shapiro solution for

the accretion onto the moving Kerr black hole. Our solu-tion describes the velocity and the density distribution ofthe perfect fluid with the equation of state pð�Þ ¼ � inspace, when the Kerr-Newman black hole moves throughthe fluid. The solution is unequally determined in terms ofthe scalar potential, the exact solution for which is given byEq. (26). One can think of the found solution as of one tothe problem for a canonical massless scalar field in themetric of a moving Kerr-Newman black hole, since thesetwo descriptions are equivalent for the considered problem.In this case the solution for the scalar field coincides withthe scalar potential for the fluid, Eq. (26).Our results are only valid in the test fluid approximation,

put differently, the backreaction of the accreting fluid isignored. In the case of a nonextreme Kerr-Newman blackhole one can always find such a range of parameters thatthe backreaction can indeed be safely ignored, while this isnot the case for the extreme black hole. We showed that forthe extreme Kerr-Newman black hole the test fluid ap-proximation is always violated, since the energy densityof ultrahard fluid diverges at the event horizon. A similarbehavior of the energy density for the perfect fluids withsound velocity cs � 1 occurs for the extreme Reissner-Nordstrom black hole [22]. Apparently, if the black hole isalmost extreme, M2 �Q2 � a2 ! þ0, the value of theenergy density at the infinity must be tuned considerably,to avoid the violation of the test fluid approximation. Thusto solve correctly the problem of accretion in the case of an(almost) extreme black hole the backreaction of the fluidmust be taken into account. This is in accord with [23,24],where it was suggested that the backreaction is importantfor a description of a scalar particle absorption with a largeangular momentum by a near extreme black hole. We willinvestigate this problem in a separate work [25].We also presented an analytic solution for the massless

scalar field in the metric of the Kerr-Newman naked sin-gularity, i.e. when M2 <Q2 þ a2. This solution describesthe scalar field distribution around the naked singularity, ingeneral with nonzero influx. The found solution containssome pathologies which are the consequences of the cau-sality violation in the vicinity of the Kerr-Newman space-time with a naked singularity. For the Kerr or the Reissner-Nordstrom naked singularity the presented solution is wellbehaved in the whole space-time, except for the point ofthe physical singularity.

ACKNOWLEDGMENTS

We would like to thank V. Beskin, Ya. Istomin,V. Lukash, and K. Zybin for useful discussions. Thework of E. B. was supported by the EU FP6 Marie CurieResearch and Training Network ‘‘UniverseNet’’ (MRTN-CT-2006-035863). The work of other coauthors was sup-ported in part by the Russian Foundation for BasicResearch Grant No. 06-02-16342 and the RussianMinistry of Science Grant No. LSS 959.2008.2.

-4 -2 0 2 4r cosθ M

-4

-2

0

2

4r

sin

θM

Q 0a 3M

FIG. 6 (color online). Density plot in the poloidal plane of theenergy density distribution of a scalar field around the Kerrnaked singularity with a ¼ 3M. A solution (46) with _c1 ¼ 0and B � 0 is taken, and the influx onto the singularity is zero.This solution does not have a perfect fluid analogue, and the totalmass of the scalar field for this solution is finite.


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APPENDIX: ALTERNATIVE FORMALISM FORACCRETION

Here we present an alternative approach to solve theproblem of accretion in a specific case when a black hole isat rest, u01 ¼ 1, u1 ¼ 0. The way we solve the problemhere is closer to the original ‘‘hydrodynamic’’ approach,described in [11–13], rather that the ‘‘scalar field ap-proach’’ by Pertich et al [1].

The equations for stationary distribution of ultrahardfluid with the equation of state p ¼ � in the Kerr-Newman metric is integrated directly, similar to the analo-gous problem in the Schwarzschild [13,26,27] andReissner-Norestrom metrics [22]. From (42) it followsthat the specific azimuthal and longitudinal angular mo-mentum are both zero:

L� ¼ u� ¼ 0; L� ¼ u� ¼ u� ¼ 0: (A1)

This means, in particular, that u� ¼ 0, and so � ¼ constalong the lines of flow. Using this property of the stationaryultrahard fluid, we find the first integrals of the energy-momentum conservation

T��;� ¼ 1ffiffiffiffiffiffiffi�g

p @

@x�ðT�

�

ffiffiffiffiffiffiffi�gp Þ ¼ 1

2g��;�T

�� ¼ 0: (A2)

Integration of this equation for � ¼ 0 gives the first inte-gral of motion (the relativistic Bernoulli energy conserva-tion equation):

ðpþ �Þu0u ffiffiffiffiffiffiffi�gp ¼ C1ð�ÞM2; (A3)

where u ¼ ur is a radial 4-velocity component andC1ð�Þ isa function of �. To find the second integral of motion wewrite a projection equation

u�T��;� ¼ �;�u

� þ ðpþ �Þ 1ffiffiffiffiffiffiffi�gp @

@x�ð ffiffiffiffiffiffiffi�gp

u�Þ ¼ 0;

(A4)

which can be expressed as

�;r

pþ �uþ 1ffiffiffiffiffiffiffi�g

p @

@rð ffiffiffiffiffiffiffi�gp

uÞ ¼ 0: (A5)

Integration of (A5) gives the second integral of motion (theenergy flux conservation):


u exp

�Z �

�1

d�0

�0 þ pð�0Þ�¼ �Að�ÞM2; (A6)

where Að�Þ is a function of �. Using the normalizationcondition u�u� ¼ 1, from integrals of motion (A3) and(A6) we find

ðpþ �Þ exp��Z �

�1

d�0

�0 þ pð�0Þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ð�þ�u2ÞA

s

¼ �C1ð�ÞAð�Þ ¼ p1 þ �1; (A7)

where function A is given by (8). From Eqs. (A6) and(A7) for a ! 0 and Q ! 0, we can fix the functions C1ð�Þand Að�Þ,

C1ð�Þ ¼ �Að�Þðp1 þ �1Þ; Að�Þ ¼ A0 sin�;

where A0 ¼ const is a dimensionless constant. FollowingMichel [13], we find relations at the critical sound pointr ¼ r�:

�ðr�Þ ¼ 0; u2� ¼ ðr2� þ a2Þ2ðr� �MÞ�½2r�ðr2� þ a2Þ � a2sin2�ðr� �MÞ� ;

(A8)

where u� ¼ uðr�Þ. From the first equation in (A8) it fol-lows that for M2 � a2 þQ2, there are two critical pointsr1 ¼ r� and r2 ¼ rþ. The critical point at smaller radius,r1 ¼ r�, is inside the event horizon. The two points coin-cide only for the extreme black hole, M2 ¼ a2 þQ2. Thismeans that for the extreme black hole the boundary con-ditions must be different from the ones in the nonextremecase. Using the general relation b ¼ 2þ i� s, derived in[14–16], between the number of boundary conditions b, thenumber of invariants i, and the number of critical surfacess, we can see that in the extreme case we must put onlythree boundary conditions, very similar to the case of astationary atmosphere. Thus, a stationary atmosphereseems to be a more adequate description in the case ofthe extreme black hole, rather than a stationary accretion.Using the relations (A8) for the critical point r ¼ rþ, we

calculate the constant A0, which determines the value ofthe accretion flow:

A0 ¼ r2þ þ a2

M2: (A9)

Now, from (A6) and (A7) one can find the space distribu-tion of the energy density, � ¼ �ðr; �Þ, and the radial

component of the 4-velocity, u ¼ ðA0M2=�Þð�=�1Þ�1=2,

�

�1¼ ðrþ rþÞðr2 þ r2þ þ 2a2Þ � a2ðr� r�Þsin2�

�ðr� r�Þ ;

u2 ¼ ðr2þ þ a2Þ2ðr� r�Þ�½ðrþ rþÞðr2 þ r2þ þ 2a2Þ � a2ðr� r�Þsin2��

:

(A10)

The solution (A10) coincides with the one found inSec. II B in the case when u1 ¼ 0.


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Ultrahard fluid and scalar field in the Kerr-Newman metric

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