PHYSICAL REVIEW D 76, 024009 (2007)

Does a black hole rotate in Chern-Simons modified gravity?
Kohkichi Konno,1,* Toyoki Matsuyama,2 and Satoshi Tanda1
1

Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan2
Department of Physics, Nara University of Education, Nara 630-8528, Japan(Received 20 April 2007; published 20 July 2007)

Rotating black hole solutions in the 3  1-dimensional Chern-Simons modified gravity theory arediscussed by taking account of perturbation around the Schwarzschild solution. The zenith-angle
dependence of a metric function related to the frame-dragging effect is determined from a constraint
equation independently of a choice of the embedding coordinate. We find that at least within theframework of the first-order perturbation method, the black hole cannot rotate for finite black holemass if the embedding coordinate is taken to be a timelike vector. However, the rotation can be permitted
in the limit of M=r ! 0 (where M is the black hole mass and r is the radius). For a spacelike vector, therotation can also be permitted for any value of the black hole mass.
DOI: 10.1103/PhysRevD.76.024009

PACS numbers: 04.70.Bw, 04.50.+h

I. INTRODUCTION
The latest observational results of the cosmic microwave
background (CMB) anisotropy from the Wilkinson
Microwave Anisotropy Probe (WMAP) [1] are successfully explained by the  cold dark matter standard model.
However, two big issues still remain: what is dark matter,
and what is dark energy? According to the WMAP results,
unfortunately about 96% of the contents of the Universe isgiven by the dark components that we still do not know.
Therefore, properties of dark matter and dark energy haveeagerly been investigated from observations [2,3].
In contrast with the ordinary approaches [2,3], in which
the existence of the dark components is assumed, it is ofgreat interest to investigate alternative gravity theories [4 –8] to solve the dark matter and dark energy problem. In thispaper, we focus our attention on the Chern-Simons modified gravity theory [9]. This gravity theory was constructedby Deser et al. [9] in (2  1) spacetime dimensions for thefirst time by analogy with the topologically massive U(1)and SU(2) gauge theories. The Chern-Simons modified
gravity theory was relatively recently extended by Jackiwand Pi [10] to (3  1) spacetime dimensions. In the extended theory, the Schwarzschild solution holds without
any modification [10]. Therefore, the theory passes theclassical tests of general relativity [11]. In this gravity
theory, however, the Kerr solution does not hold. Thus,
the solution for a rotating black hole should have a different form from the Kerr solution. In 2  1-dimensional
Chern-Simons modified gravity, a family of rotating blackhole solutions was found by Moussa et al. [12]. Thesolutions have a fascinating feature that observers in thisspacetime behave like ones inside the ergosphere of theKerr spacetime. This feature is similar to that of the rotation of galaxies [13]. Therefore, it is very interesting toinvestigate rotating black hole solutions in the 3 1-dimensional Chern-Simons modified gravity theory. In*konno@topology.coe.hokudai.ac.jp

1550-7998= 2007=76(2)=024009(5)this paper, we discuss rotating black hole solutions taking
account of perturbation around the Schwarzschild solution.
This paper is organized as follows. In Sec. II, we briefly
review the 3  1-dimensional Chern-Simons modified
gravity theory. In Sec. III, we consider the perturbation
around the Schwarzschild solution to discuss slow rotationof the black hole. First we investigate a constraint equation
independently of a choice of the embedding coordinate. InSec. III A, from the first-order equations of the field equation, we obtain the metric solution taking the embedding
coordinate to be timelike. In Sec. III B, we investigate themetric solution for the case in which the embedding coordinate is spacelike. Finally, we provide a summary inSec. IV. In this paper, we use a unit in which c  G  1.
II. BRIEF REVIEW OF CHERN-SIMONSMODIFIED GRAVITY THEORY
We briefly review the Chern-Simons modification ofgeneral relativity developed by Jackiw and Pi [10]. TheChern-Simons modified gravity theory is provided by theaction
Z
1 Z 4 p
1dxI  dx4 L gR  #  RR164
Z
11p
dx4 gR  v K  ;
(1)
16
2
where the first term in the integrand is the Einstein-Hilbert
action, and v  @ # is an external 4-vector, which iscalled the embedding coordinate. The Pontryagin density
RR is defined by  RR   R   R  , using the dual
Riemann tensor  R    12 " R  . The ChernSimons topological current K  is given byK   "  @   23   ;(2)

which is related to the Pontryagin density as @ K  
1
2 RR.

024009-1© 2007 The American Physical Society

KOHKICHI KONNO, TOYOKI MATSUYAMA, AND SATOSHI TANDA

From the variation of the Lagrange density L withrespect to the metric g , it turns out that the field equation
has the formG

C 8T

;

Using this perturbed metric, from the condition (5), weobtain0  r C  v(3)

where G  R  12 g R is the Einstein tensor, T  isthe energy-momentum tensor, and C is the Cotton tensor
defined as

PHYSICAL REVIEW D 76, 024009 (2007)!r;  (4)Here v  r v  @ @ #   @ # is a symmetrictensor. Corresponding to the Bianchi identity r G 
0 and the equation of motion r T   0, the following
condition should be imposed:
10  r C  p v RR:
8 g(5)This constraint equation implies that diffeomorphism symmetry breaking is suppressed [10].
III. PERTURBATIVE APPROACH TO ROTATINGBLACK HOLE SOLUTIONSIn the Chern-Simons modified gravity theory, theSchwarzschild solution as a nonrotating black hole solution holds without any modification as mentioned above.
The Schwarzschild metric is given by
2M 22M 1 2dt  1 ds2   1 drrr
 r2 d 2  sin2 d 2 ;
(6)where M is the black hole mass. This solution gives C p
0 and r C  v RR=8 g  0 trivially.
In order to take account of rotation of the black hole, let
us consider perturbation around the Schwarzschild solution. In the perturbation, the expansion parameter
isrelated to the angular momentum J of the black hole,i.e., J O . Under the assumption of stationary, axisymmetric spacetime, we can write the form of the perturbed
metric as [14 –16]
2M
1  hr; dt2
ds2   1 r

2M 1
 1
1  mr; dr2r
 r2 1  kr; d 2  sin2 d  !r; dt2 :(7)The functions hr; , mr; , kr; , and !r;  are of thefirst order in . Hereafter, we take account of equations up
to the first order in .

(8)

where a subscript comma denotes the partial differentiation
with respect to the coordinates. In this expression, thefunction !r;  only appears. Therefore, we find thatsolutions for !r;  should have the functional form

1C   p v " r R   " r R  2 g
 v  R   R  :3M
sin !;r  2 cot !;r ;r3$r
;sin2(9)where $ is a function of r only. While !r;  is singularon the rotation axis (  0 and ), the metric is regular atleast up to the first order, because gt  r2 $r O 2 . Note that gt does not vanish on the rotation axis
unless $r is identically zero. This means that the shiftvector Ni  gti i  r; ;  defined in the (3  1) formalism [17] is singular on the rotation axis. It should also be
noted that this result is independent of a choice of theembedding coordinate v .
A. Linear perturbation equations and the metricsolution for timelike v
We adopt a timelike vector for v , i.e., v 
1=0 ; 0; 0; 0, which is derived from #  t=0 . In ouruniverse, there exists the frame of reference in which theCMB radiation can be seen as an isothermal distribution
except for small fluctuations. The frame of reference is
specified by a timelike vector. Such a timelike vector is acandidate for the timelike vector v .
From the (tt), (rr), (r ), (r ), ( ), ( ), and ()components of Eq. (3), we can obtain homogeneous differential equations for the functions h, m, and k. Hence, thehomogeneous differential equations have a simple solutionof hr;   mr;   kr;   0. These differential
equations are completely decoupled from the function !.
Since we are now interested in the rotation of the blackhole, i.e., the function !, we do not seek any other solutions for the functions h, m, and k. From the (tr), (t ), andt -components of Eq. (3), we obtain the equations for !
0  !;

 rr  2M!;rr  2rr  2M cot !;rr

 5 cot !; 22r  5M!;r 42r  5M cot !;r  3cot2  1!; ;0  r2 r  2M!;rrr  r!;r(10)

 3r cot !;r

 6rr  2M!;rr  4r  3M!;r ;0  rr  2M!;rr  4r  2M!;r  !;(11)
 3 cot !; :(12)Here, Eqs. (10) and (11) are obtained, respectively, only
from the nonvanishing (tr) and (t )-components of the024009-2

DOES A BLACK HOLE ROTATE IN CHERN-SIMONS . . .

PHYSICAL REVIEW D 76, 024009 (2007)

Cotton tensor, and Eq. (12) is obtained only from the (t )component of the Einstein tensor. Note that these equations
do not include 0 . This is due to a shortcoming of the firstorder perturbation method.
From the result of Eq. (9), Eq. (10) is automaticallysatisfied. From Eqs. (11) and (12), we obtain the differential equations for $r, respectively,
r2 $000  6r$00  6$0  0;(13)

the black hole cannot rotate in the Chern-Simons modified
gravity for the timelike vector.
However, in the limit of M=r ! 0, the derivative ofEq. (14) coincides with Eq. (13). Hence, the solution
$

C1 C2
 2;
rr

(17)

i.e.,
gt  C1 r  C2 ;rr  2M$00  4r  2M$0  2$  0;

(14)

where a prime denotes the differentiation with respect tothe coordinate r. The solution of Eq. (13) is given by$  C0 C1 C2
 2;
rris permitted in this limit. Since the metric component gt is
proportional to r at infinity, the frame-dragging effect of
this solution works in the whole space.
B. Linear perturbation equations and the metricsolution for spacelike v

(15)where C0 , C1 , and C2 are constants of integration. On theother hand, the solution of Eq. (14) is given byr  2M D2 2 3 r  2Mr  4M2  4Mr  2M
r3rlnr  2M ;(16)$  D1

where D1 and D2 are constants of integration. Thus, thesolution that satisfies both differential equations (13) and(14) is given only by $r  0. Therefore, we conclude
that within the framework of the first-order perturbation,

Next we take another choice of #  r cos = 0 .
This provides a spacelike vector v  0; cos = 0 ;r sin = 0 ; 0, which becomes a unit vector parallel tothe rotation axis at infinity. The discrepancy between theobservational result and the theoretical prediction in thequadrupole moment of the CMB anisotropy may imply theexistence of such a spacelike vector [18].
From the (tt), (t ), (rr), (r ) ( ), and ()components of the field equation, we obtain the nonvanishing equations, respectively,

2rr  2Mk;rr  2r  2Mm;r  23r  5Mk;r  k;(18)

 cot k;  m;

 cot m;  2k  2m

1
r2 r  2M2 $000  rr  2M6r  11M$00  2r  2M3r  2M$0  2M$ ;0 r  2M

1rr  2M$00  4r  2M$0  2$sin2
1
rr  2M2 rh;rrr  cot h;rr   rr  2M2 rk;rrr  cot k;rr   rr  2Mh;r
2 0 r3 k; r  2M cot h; k;r   rr  2Mf2r  Mh;rr  r  3Mm;rr  3r  2Mk;rr g r  2M cot f3r  Mh;r  3r  2Mk;r  3Mm;r g  2r  5Mh; r  2Mcot2 h;(19)

 r  3Mm; r  2Mk; k;   2r2  9Mr  10M2 h;r  9Mr  2Mm;r

 2r2  9Mr  14M2 k;r  3r  3M cot h;  m;   r  2Mcot3 h;  k;   2r  2Mm  k ;(20)

r  2Mh;r  3M

 cot h;  k;

 cot k;  2r  2Mh;r  k;r   2k  mrr  2M$00  4r  2M$0  2$ ;(21)

0rr  2Mh;r  k;r   r  3Mh;  r  Mm;  r cot
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rr  2M$00  4r  2M$0  2$ ;(22)

KOHKICHI KONNO, TOYOKI MATSUYAMA, AND SATOSHI TANDAPHYSICAL REVIEW D 76, 024009 (2007)rr  2Mh;rr  k;rr   r  Mh;r  2r  Mk;r  r  Mm;r  cot h;  k; 

cot2

rr  2M$00  4r  2M$0  2$ ;(23)

0rr  2Mh;rr  k;rr   h; m;

 r  Mh;r  2r  Mk;r  r  Mm;r1
rr  2M$00  4r  2M$0  2$  rfrr  2M$000  23r  5M$00  6$0 gsin2 :2sin0

While the right-hand side of Eq. (22) has the zenith-angle
dependence of cot , the left-hand side is composed of thefirst-order derivatives of h, m, and k with respect to . Thus
the solution has the form of h; m; k / lnsin . However,these functions become singular along the rotation axis.
Hence, the zenith-angle dependence of the functions h, m,and k should vanish, and therefore these functions depend
on r only. On the other hand, this result, i.e., h  hr, m mr, and k  kr, conflicts with Eq. (23), since the lefthand side becomes a function of r only, and the right-handside has the dependence of cot2 . Therefore, the functions
h, m, and k should vanish. Then, we derive the differentialequations
0  r2 r  2M2 $000  rr  2M6r  11M$00
 2r  2M3r  2M$0  2M$;

(25)0  rr  2M$00  4r  2M$0  2$;(26)

0  rr  2M$000  23r  5M$00  6$0 :(27)Equations (25) and (27) can be derived consistently fromEq. (26). Thus the equation that we have to solve isEq. (26). In the same way as the case for the timelikev , the differential equation does not include the parameter 0 . The solution of Eq. (26) is given by the same
expression as Eq. (16), which leads to~r  2M D
 2 r2  2Mr  4M2rr
 4Mr  2M lnr  2M ;~1gt  D(28)~ 2 are constants. Therefore, for the space~ 1 and D
where D
like vector v , the spacetime rotation is permitted for any~ 2  0, then thevalue of the black hole mass. However, if D
frame-dragging effect extends to infinity, because the second term in Eq. (28) diverges as r increases. Furthermore,
the result of Eq. (28) means that the above-mentioned
string singularity of the shift vector Ni extends to infinity~ 2  0.
even if D(24)

IV. SUMMARY
We have investigated the rotation of a black hole in theChern-Simons modified gravity theory. In particular, wehave considered slow rotation of a black hole using theperturbation method, in which the Schwarzschild solution
was taken to be the background. From the constraint
equation, we obtained the zenith-angle dependence of themetric function !r;  related to the frame-dragging effect, independently of a choice of the embedding coordinate v . Furthermore, by solving the field equation, we
found that the black hole cannot rotate for the timelike
vector v at least within the framework of the first-order
perturbation method. However, in the limit of M=r ! 0,
the spacetime rotation is permitted, whose frame-dragging
effect extends to infinity. In contrast, for the spacelike
vector v , the spacetime rotation is permitted for any valueof the black hole mass. Its frame-dragging effect also
extends to infinity. Therefore, it is still an open question
which form of the metric corresponds to the Kerr solution,
which reduces to the Minkowski metric at infinity.
Derivation of exact solutions for stationary, axisymmetricspacetimes in the Chern-Simons modified gravity theory
may solve this problem. Then, we could also understandeffects of the parameter 0 or 0 , which appears in theChern-Simons term, on the black hole physics. The derivation of exact solutions will be a future work.
Furthermore, it should be noted that the above-mentioned
results might be modified by the extension of the theory inwhich # in the Chern-Simons term is taken to be a dynamical variable. This will also be discussed elsewhere.
ACKNOWLEDGMENTS
This work was supported in part by a Grant-in-Aid for
Scientific Research from The 21st Century COE Program‘‘Topological Science and Technology.’’ Analytical calculations were performed in part by Mathematica (WolframResearch, Inc.) on computers at YITP in Kyoto University.

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