561
Prog. Theor. Phys. Vol. 122, No. 2, August 2009, Letters

Rotating Black Hole in Extended Chern-Simons Modified Gravity
Kohkichi Konno,1,2,∗) Toyoki Matsuyama3 and Satoshi Tanda1,2
1 Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
2 Center of Education and Research for Topological Science and Technology,

Hokkaido University, Sapporo 060-8628, Japan

3 Department of Physics, Nara University of Education, Nara 630-8528, Japan(Received February 27, 2009; Revised April 20, 2009)
We investigate a slowly rotating black hole in four-dimensional extended Chern-Simons
modiﬁed gravity. We obtain an approximate solution that reduces to the Kerr solution
when a coupling constant vanishes. The Chern-Simons correction eﬀectively reduces theframe-dragging eﬀect around a black hole in comparison with that of the Kerr solution.
Subject Index: 420, 453

Introduction
There is growing interest in modiﬁcations of the theory of generalrelativity. The theory of quantum gravity mostly motivates such modiﬁcations.1), 2)
Quantum gravity is requisite for dealing with extreme situations in the Universe,such as the vicinity of a black hole and the initial state of the Universe. For thetheory of quantum gravity, several candidates have been proposed.1), 2) Those candidates certainly require modiﬁcations of general relativity. In recent years, serious
astrophysical problems of dark matter and dark energy also motivate modiﬁed gravity theories. According to recent observational results,3) the dark components are
considered to comprise over 90% of the energy constituents in the Universe. Modiﬁcations of general relativity would provide an alternative way to resolve the unsolved
issues of the dark components. Among modiﬁed gravity theories, we focus on fourdimensional Chern-Simons (CS) modiﬁed gravity4), 5) in this paper.
In CS modiﬁed gravity, the action is modiﬁed by adding a CS term to the
Einstein-Hilbert action. Deser et al.6) originally developed the CS modiﬁed gravity in (2+1) dimensions and Jackiw and Pi4) extended it to (3+1) dimensions byintroducing an external scalar function. Furthermore, Smith et al.5) have recently
considered an extended CS modiﬁed gravity in which the scalar function is regarded
as a dynamical scalar ﬁeld. The most interesting points of the four-dimensional
CS modiﬁed gravity4), 5), 7)–13) are summarized as follows. First, the CS modiﬁed
gravity can be obtained explicitly from the superstring theory.5), 7) In the superstring theory, a CS term in the Lagrangian density is essential to cancel anomaly.
Second, as shown by Weinberg,14) the parity-violating higher-order correction in aneﬀective ﬁeld theory for inﬂation also provides the CS modiﬁed gravity. Third, theSchwarzschild solution holds in the CS modiﬁed gravity.4) Thus, the theory passes
the classical tests of general relativity. Finally, in the CS modiﬁed gravity, the axial
part of gravitational ﬁelds is mainly modiﬁed in comparison with the cases of gen∗)E-mail: kohkichi@eng.hokudai.ac.jp

562

Letters

Vol. 122, No. 2eral relativity. In fact, the Kerr solution does not hold in the CS modiﬁed gravity.
This is because the Chern-Pontryagin density, which can be derived from the partial
integral of the CS term, is closely coupled to the axial part of gravitational ﬁelds.
In relation to the last point, rotating black hole solutions have been discussed in theframework of the Jackiw-Pi model.8)–10) In Ref. 8), an approximate solution for arotating black hole was obtained by using a slow rotation approximation. The approximate solution interestingly shows the same feature as that of ﬂat rotation curves
seen in spiral galaxies.9) In Ref. 10), an exact solution for a rotating mathematical
black hole was obtained. Somewhat exotic features of those solutions are considered
to arise from the constraint of vanishing Chern-Pontryagin density in the Jackiw-Pi
model. In this paper, we investigate an approximate solution for a rotating blackhole in the extended CS modiﬁed gravity, in which the constraint is replaced with
a ﬁeld equation for a scalar ﬁeld. To obtain a black hole solution, we assume bothslow rotation of a black hole and a weak CS coupling in this study. Throughout the
paper, we use geometrized units with c = G = 1.
Extended CS modified gravity
The extended CS modiﬁed gravity is provided by
the action4), 5)




1
1 μν
4 √
∗ τ μν σR+ϑ R σ R τ μν − g (∂μ ϑ) (∂ν ϑ) + Lm , (1)I = d x −g −
16π
64π
2
where g is the determinant of the metric gμν , R ≡ g αβ Rαβ (Rαβ ≡ Rλαλβ ) is the Ricciτ − · · · is the Riemann tensor (Γ α is the Christoﬀel symbols),
scalar, Rτσαβ ≡ ∂β Γσαβγ is a coupling constant, ϑ is a dynamical scalar ﬁeld, and Lm is the Lagrangian
for matter. The dual Riemann tensor is deﬁned by ∗Rτσμν ≡ 12 εμναβ Rτσαβ , where√
ε0123 ≡ 1/ −g is the Levi-Civita tensor. In this paper, we neglect the surface
integral term (see Ref. 11) for a detailed discussion) and the potential term for ϑ inthe action. When we neglect the kinematic term of ϑ in Eq. (1), the action reduces
to the Jackiw-Pi model developed in Ref. 4). From the variations in the action
with respect to the metric gμν and the scalar ﬁeld ϑ, we obtain the ﬁeld equations,
respectively,(2)Gμν + C μν = −8π Tm μν + Tϑμν , ∗ τ μν σ
R σ R τ μν ,(3)g μν ∇μ ∇ν ϑ = −64πwhere Gμν ≡ Rμν − 12 g μν R is the Einstein tensor, C μν is the Cotton tensor deﬁned
by

1
μν
C ≡ − (∇σ ϑ) εσμαβ ∇α Rνβ + εσναβ ∇α Rμβ + (∇σ ∇τ ϑ) ( ∗Rτ μσν + ∗Rτ νσμ ) ,2(4)
Tm μν is the energy-momentum tensor for matter, and Tϑμν is the energy-momentum
tensor of the scalar ﬁeld ϑ,1Tϑμν = (∇μ ϑ) (∇ν ϑ) − g μν (∇λ ϑ)(∇λ ϑ).
2(5)Letters

August 2009

563

Thus, Eqs. (2) and (3) are basic equations for gravitational ﬁelds under the extended
CS modiﬁed gravity.
Approximate solution for a rotating black hole
We discuss a slowly rotating blackhole by considering the perturbation of a spherically symmetric static spacetime. For(0)2dxμ dxν = −e2Φ(r) dt2 + e2Λ(r) dr2 +such a2 background, we have the metric ds = gμν
2
2
2
(0)
r dθ + sin θdφ and the scalar ﬁeld ϑ = ϑ (r), where Φ and Λ are functionsof r, and the superscript ‘(0)’ denotes the zeroth order in perturbation. The ﬁeld
equations for the background reduce to12)μνGμν(0) = −8πTϑ(0) ,μν (0) (0) (0)∇μ ∇ν ϑ = 0.
g(0)

(6)
(7)

Hereafter, we assume Tm μν = 0. We now adopt the Schwarzschild spacetime with
ϑ(0) = 0 as a background solution for Eqs. (6) and (7). The metric in polar coordinates is given by(0)dxμ dxν = − 1 −
gμν

2M
r

dt2 + 1 −

2M
r

−1



dr2 + r2 dθ2 + sin2 θdφ2 , (8)where M is the mass of a black hole. Let us consider the perturbation of the Schwarzschild spacetime,
(0)
(1)
+ gμν
,
gμν = gμν

(9)

(1)

(10)

ϑ=ϑ
(1)

,

where gμν and ϑ(1) are at least ﬁrst-order quantities of a small parameter that(1)characterizes the slow rotation of a black hole, i.e., gμν , ϑ(1) ∼ O ( ). Thus, we takeaccount of a scalar ﬁeld that is purely excited by the spacetime curvature throughEq. (3). As discussed in detail below, we also adopt the approximation of a weak CS
coupling, that is, we also regard  as an expansion parameter. The scalar ﬁeld obeys
Eq. (3). While the right-hand side in Eq. (3), i.e., ∗ Rτσμν Rστ μν , exactly vanishes forspherically symmetric spacetimes, ∗ Rτσμν Rστ μν becomes nonzero at ﬁrst order in .
Thus, the scalar ﬁeld should be of order  . Then the CS correction in Eq. (2) has
order 2 . Therefore, when we take the Schwarzschild black hole as a background,
the CS correction to the rotation of the black hole is of order 2 .
We discuss the basic equations for such a slowly rotating black hole in the extended CS modiﬁed gravity. Using the Regge-Wheeler gauge,15) we have the ﬁrstorder axisymmetric metric as⎞⎛ − 1 − 2Mh(r, θ)H(r, θ)0−r2 sin2 θω (r, θ)
r


2M −1
⎟
⎜
m(r, θ)0−r2 u (r, θ)
H(r,
θ)
1
−
(1)
⎟,
⎜
r
gμν = ⎝⎠
00
0
r2 k (r, θ)
2
2
2
2
2
−r u (r, θ)0r k (r, θ) sin θ−r sin θω (r, θ)(11)where (h, H, m, k, ω, u) ∼ O ( ). The scalar ﬁeld is also assumed to depend on r andθ, i.e., ϑ = ϑ(1) (r, θ) ∼ O ( ). We do not pay attention to the order in  for a while.

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Vol. 122, No. 2

We can obtain diﬀerential equations for the ﬁrst-order functions (h, H, m, k, ω, u, ϑ)from Eqs. (2) and (3). Diﬀerential equations of ﬁrst order in are completely divided
into four groups as {h, m, k}, {H}, {u}, and {ω, ϑ}. The (tt), (rr), (rθ), (θθ), and(φφ) components of Eq. (2) give homogeneous diﬀerential equations for h, m, and k.
The (tr) and (tθ) components of Eq. (2) give homogeneous diﬀerential equations forH, and the (rφ) and (θφ) components give homogeneous  diﬀerential equations for u.
For those components of Eq. (2), we have C μν ∼ O 2 . Of course, a simple solutionof h = m = k = H = u = 0 satisﬁes the diﬀerential equations for h, m, k, H, andu. Finally, we obtain diﬀerential equations for ω and ϑ from the (tφ) component of
Eq. (2) and the ﬁeld equation (3) for the scalar ﬁeld,r(r − 2M )∂r2 ω + 4(r − 2M )∂r ω + ∂θ2 ω + 3 cot θ∂θ ω
1
1
6M (r − 2M )∂r ∂θ ϑ −∂θ ϑ ,=4rsin θr sin θ1
∂θ (sin θ∂θ ϑ)r(r − 2M )∂r2 ϑ + 2(r − M )∂r ϑ +sin θ
3M(sin θ∂r ∂θ ω + 2 cos θ∂r ω) .
= −
8πr

(12)

(13)

Equations (12) and (13) govern the rotation of a black hole under the extended CS
modiﬁed gravity.
We discuss solutions for Eqs. (12) and (13) to clarify how the black hole rotates.
For this purpose, let us expand ω and ϑ asω(r, θ) =ϑ(r, θ) =

∞

n=1
∞


ω̃n (r)1∂θ Pn (cos θ),
sin θ

ϑ̃n (r)Pn (cos θ),

(14)
(15)

n=1

where Pn denotes the Legendre function of the ﬁrst kind. From Eqs. (12) and (13),
we obtain for each n


4
2
−
n(n
+
1)
6M
ϑ̃
n
ω̃n =  5 ϑ̃n −,(16)ω̃n + ω̃n +rr(r − 2M )rr
ϑ̃n +2(r − M ) n(n + 1)3n(n + 1)M ϑ̃n −ϑ̃n = ω̃ ,r(r − 2M )r(r − 2M )8πr2 (r − 2M ) n(17)

where the prime denotes the diﬀerentiation with respect to r. To solve Eqs. (16) and(17), let us assume   1. We solve Eqs. (16) and (17) by the method of iteration byconsidering  to be an expansion parameter. The homogeneous diﬀerential equationsof Eqs. (16) and (17) give the equations of order 0 . We obtain the homogeneous
solutions asrr  −n − 1 , n+ B0 G2,0, (18)
ω̃n = A0 2 F1 1 − n, 2 + n, 4;

2,2
−3
, 0
2M
2M
 r
 r
− 1 + D 0 Qn
−1 ,
(19)
ϑ̃n = C0 Pn
MMLetters

August 2009565

where A0 , B0 , C0 , and D0 are constants, 2 F1 is a hypergeometric function, G2,0
2,2 is16)
and Qn is the Legendre function of the second kind. Notea Meijer G-function,that while Pn (r/M − 1) diverges at r → ∞, Qn (r/M − 1) diverges at r = 2M .
Thus, we should have C0 = D0 = 0 at order 0 . Otherwise, ω̃n would also diverge
at r = 2M or r → ∞ at higher orders. The metric solution in Eq. (18) for n = 1
gives the linear approximation of the Kerr solution. The solution for n = 1 allows us
to compare a rotating black hole in the extended CS modiﬁed gravity with that ingeneral relativity. Hence, we focus on the solution for n = 1 hereafter. The solutionof order 0 is summarized as
2J(20)ω̃1 = − 3 , ϑ̃1 = 0,rwhere J is the angular momentum of a black hole. Substituting Eq. (20) into theright-hand side of Eq. (17), we obtain a solution of order  , r
 r− 1 + D 1 Q1
−1ϑ̃1 = C1 P1
M
M



J
2M 15r4 + 5M 2 r2 + 10M 3 r + 18M 4−54
256πM r




r
4
 ,

(21)
−15r (r − M ) ln r − 2M 
where C1 and D1 are constants. When we impose the regularity of ϑ̃1 both at r = 2M
and at r → ∞, we derive
15J
.
(22)
256πM 5Thus, we obtain the regular solution of order  as
 2
J
5r + 10M r + 18M 2 .
(23)ϑ̃1 = −24128πM rIn a similar way, substituting this solution into the right-hand side of Eq. (16), we
obtain the solution of order 2 as 2
J
70r + 120M r + 189M 2 .
(24)ω̃1 = 281792πM r
C1 = 0,D1 = −Consequently, we obtain the approximate solution up to order 2 for a slowly rotating black hole,


 2

1
2J
2
2
70r + 120M r + 189M,
(25)
ω = 3 1−r3584πM r5 2

J
5r + 10M r + 18M 2 cos θ.
(26)
ϑ = −
2
4128πM rFrom Eq. (25), we obtain the metric solution for a slowly rotating black hole underthe extended CS modiﬁed gravity
ds2 = − 1 −2Mr

dt2 + 1 −

2M
r

−1

dr2 + r2 dθ2 + r2 sin2 θdφ2

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Vol. 122, No. 2



 2

4J
1
2
2
1−
−
70r + 120M r + 189M
sin2 θdtdφ.
r3584πM r5(27)

At inﬁnity, the (tφ) component of the metric and scalar ﬁeld reduce to

At r
gtφ

2J
r

gtφ

−

ϑ

−

1 − 25256πM r3(28)

5J
cos θ.
128πM 2 r2

2M , we have
709
J1 − 2−M
114688πM 4

J
−
2M 2

(29)

1727
1−
57344πM 42

43(r − 2M )29
cos θ.
−−J
1024πM 41024πM 5ϑ

sin2 θ,(r − 2M ) sin2 θ,
(30)
(31)

As shown in Eqs. (28)–(31), both the metric and scalar ﬁeld are regular at r → ∞and at r = 2M . From Eqs. (27), (28), and (30), we also ﬁnd that the higherorder correction eﬀectively reduces the frame-dragging eﬀect around the black holecompared with the ﬁrst-order approximation of the Kerr black hole. This result
provides contrast to the result obtained in Refs. 8) and 9) within the Jackiw-Pi
model. The reduction of the frame-dragging eﬀect is attributed to the existence of
the kinematic term of the scalar ﬁeld.
Observational implications
We discuss observational implications for the blackhole solution given in Eq. (27). The CS correction in gtφ has the same polar angle
dependence as that of the ﬁrst-order rotational approximation of the Kerr solution.
Thus, the CS correction can be found only from the radial dependence of gtφ . The CS
correction is composed of higher-order terms of 1/r, i.e., at least terms of O 1/r4 .
This fact denotes that the CS correction eﬀectively modiﬁes the frame-dragging eﬀect
in the vicinity of the black hole. Therefore, to provide an upper limit for the CS
coupling, it is useful to observe an object passing close by the black hole. Let us
consider a massive particle with mass m and four-momentum pμ on the equatorial
plane of the black hole spacetime. For a particle with energy E = −pt /m and angularmomentum L = pφ /m, from pμ pμ = −m2 , we obtain
dr
dτ

2

= E 2 − V 2 (r),(32)

where τ is the proper time of the particle, and



L2
2M
1  2
4
2
1 + 2 + LEJ 3 − ˜2
+
120M
r
+
189M
70r
.
V2 = 1−
r
r
r
M r8
(33)
Here, we deﬁne ˜2 ≡ 2 /896π. Particle trajectories are permitted for a region where
E 2 > V 2 . Figure 1(a) shows the potential under a certain condition with the CS

August 2009

Letters567

Fig. 1. (a) Potential V 2 (r). (b) Particle trajectories in the equatorial plane. The starting point isindicated by a small ﬁlled circle. The results in the presence of the CS coupling are denoted bysolid curves, while the results in the absence of the CS coupling are denoted by dashed curves.
In both ﬁgures, we assume E = 2.35, L/M = 10.0, J/M 2 = 0.5, and ˜2 = 0.1.

coupling (solid curve) and that without the CS coupling (dashed curve). In thisﬁgure, we assume corotating trajectories. The CS correction reduces the potential
for a region close to the horizon r = 2M . At the peaks of the potentials, unstable
circular orbits are realized if E 2 = V 2 (rpeak ). As seen in Fig. 1(a), the peak isshifted outward by the CS correction. However, when a particle that has energy ofE 2 < V 2 (rpeak ) falls freely from the outside (r > rpeak ), the periastron is shifted
inward by the CS reduction of the potential. Multiplying Eq. (32) by (dφ/dτ )−2
and diﬀerentiating that with respect to φ, we obtain the diﬀerential equation for
trajectories


3
 2J
1  2 3
d2 r
3
2
2
2
2
3 (3r − 8M )r= 2 2E r − 2r + 3M r − L r + M L + 3 2Er − E
dφ2
LL(r − 2M )2


2
2
J
3(20r + 63M )3 130r + 189M r − 189M
−
E
.
(34)
+˜2 3 E
Lr3
r2 (r − 2M )2We can obtain particle trajectories by solving this equation. Figure 1(b) shows
trajectories for the potentials given in Fig. 1(a). We used the Runge-Kutta method
to integrate Eq. (34) numerically. As expected, we ﬁnd that the periastron is closer
to the black hole in the presence of the CS coupling than its absence. As seen from
Fig. 1, the CS correction eﬀectively modiﬁes the trajectory around the periastron.
The eﬀect would also be magniﬁed at the apastron. Therefore, observations of asatellite in an extended elliptical orbit, for example, a system such as OJ287,17)
would provide useful information about the CS coupling.
Concluding remarks
We have investigated a rotating black hole in the extended
Chern-Simons (CS) modiﬁed gravity. We considered the perturbation of the
Schwarzschild spacetime with a vanishing scalar ﬁeld to discuss the rotation of a
black hole. The assumption of the vanishing scalar ﬁeld at the zeroth order allows us

568

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Vol. 122, No. 2

to compare a result with that of the Kerr solution. By taking account of the scalarﬁeld that is purely excited by a gravitational ﬁeld through the Chern-Pontryagin
density, we obtained an approximate solution for a slowly rotating black hole. In
obtaining the solution, we adopt both the slow rotation approximation and the approximation of a weak CS coupling. Consequently, we found that the CS correction
gives an eﬀective reduction of the frame-dragging around a black hole in comparison
with that of the Kerr solution. We also showed that observations of systems thathave a satellite in an extended elliptical orbit would be useful to provide an upper
limit for the CS coupling. As a future work, it would be interesting to seek exact
solutions for a rotating black hole in the extended CS modiﬁed gravity. From suchinvestigations, the nonlinear characteristics of the CS modiﬁed gravity would also be
revealed.
Acknowledgements
One author (K.K.) thanks Prof. R. Jackiw for his hospitality at Massachusetts
Institute of Technology. This work was supported in part by a Grant-in-Aid for
Scientiﬁc Research from the 21st Century COE Program “Topological Science andTechnology”. Analytical calculations were performed in part on computers at YITP
of Kyoto University.

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