PHYSICAL REVIEW D 78, 024037 (2008)

Flat rotation curves in Chern-Simons modified gravity
Kohkichi Konno,1,* Toyoki Matsuyama,2 Yasuhiro Asano,1 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 5 October 2007; published 23 July 2008)

We investigate the spacetime of a slowly rotating black hole in the Chern-Simons modified gravity. Thelong range feature of frame-dragging effect under the Chern-Simon gravity well explains the flat rotation
curves of galaxies which is a central evidence of dark matter. Our solution provides a different scenario ofrotating space from Gödel’s solution.
DOI: 10.1103/PhysRevD.78.024037

PACS numbers: 04.50.h, 04.70.Bw, 95.35.+dI. INTRODUCTION
There are three fundamental unsolved issues in thetheory of gravity: quantization of gravity, dark energy,
and dark matter. The string theory [1] is a promising
candidate for a consistent quantum theory of gravity.
Many attempts in quantizing gravity, however, have notbeen successful. In astronomy/astrophysics, a number ofobservations suggest the existence of dark energy [2,3] anddark matter [3,4]. Although many surveys of astrophysicalobjects have been conducted [5], it has not yet been revealed what is dark matter. For instance, the flat rotation
curves of galaxies [6] have been considered to be a robust
evidence of dark matter. The velocity v of a star orbiting
around the center of a galaxy becomes a constant at acertain distance r far from the center.
pﬃﬃﬃ While theNewtonian gravity yields a relation v / 1= r. At present,
we usually attribute the discrepancy to dark matter.
However it may be still possible to explain the phenomenon based on a theory without dark matter. In this paper, we
propose a model to solve this discrepancy in the frameworkof the Chern-Simons (CS) gravity.
The CS action [see Eq. (1)]is a universal entity obtained
from an effective action in the string theory [1,7]. In fact,
the action also appears in condensed matter physics such as
the quantum Hall effect [8,9]. Deser et al. [10] originallyconstructed a theory of CS gravity in (2 þ 1) dimensions
and recently Jackiw and Pi extended it to (3 þ 1) dimensions [11]. Several important consequences of the CS
action have been discussed in theories of gravity [7,12–14]. In connection with this study, we summarize two
important features of the CS gravity as follows. First, theSchwarzschild solution satisfies the field equation of CS
gravity, which indicates that the CS theory meets a requirement of classical tests for general relativity [15]. Second,
generally speaking, the CS term enhances angular velocity
or angular momentum. In other words, the CS term rathermodifies the gravitomagnetic part of the gravitational field
than the gravitoelectric part [16]. These facts imply that CS*konno@topology.coe.hokudai.ac.jp

1550-7998= 2008=78(2)=024037(5)gravity may behave as if dark matter exists, which motivated this study.
In this paper, we study a solution for a slowly rotating
black hole in the CS gravity based on a previous work [13].
The frame-dragging effect turns out to be significantly
large in a region far from the gravitational source. As aconsequence, the obtained solution explains the flat rotation curves of a galaxy. By analyzing the frame-dragging
effect on precession of a spinning object, we also discuss away to confirm the CS gravity in observations. This paper
indicates a route connecting the quantum theory of gravity
with the dark matter problem. Hereafter, we use the geometrized units with c ¼ G ¼ 1.
II. CS GRAVITY
We briefly review a theory of the CS modified gravity[11]. The action in this paper is given by the EinsteinHilbert action and the CS one,
1 Z 4
1pﬃﬃﬃﬃﬃﬃﬃ
d x  gR þ l#  R   R  ; (1)I¼16
4where g is the determinant of the metric, R  g R  isthe Ricci scalar, R   @      is the Riemanntensor ( is the Christoffel symbols), l is a coupling
constant, and # is an external quantity. The dual Riemanntensor density is defined by  R    12 " R  ,where " is the Levi-Civita tensor density of weight
one. The variation of the action with respect to g gives afield equationG þ lC ¼ 8T  ;(2)

where G is the Einstein tensor, T  is the energymomentum tensor, and C is the Cotton tensor defined as
1
C ¼  pﬃﬃﬃﬃﬃﬃﬃ ½v ð" r R  þ " r R  Þ2 gþ ðr v Þð R þ  R Þ:(3)Here, v  @ # is called an embedding vector. In thistheory, the condition

024037-1

Ó 2008 The American Physical Society

KONNO, MATSUYAMA, ASANO, AND TANDA

10 ¼ r C ¼ pﬃﬃﬃﬃﬃﬃﬃ v R   R 
8 g

PHYSICAL REVIEW D 78, 024037 (2008)(4)should be imposed to ensure the diffeomorphism invariance. This condition is derived by using the Bianchi identity r G ¼ 0 and the equation of motion r T  ¼ 0.
Equations (2) and (4) are basic equations in the CS gravity.
In the previous work [13], we obtained a solution ofEqs. (2) and (4) for a slowly rotating black hole by using
the perturbation expansion around the Schwarzschild solution. In this paper, we particularly concentrate our attention on the case of a spacelike vectorv ¼ @ # ¼ @ z ¼ @ ðr cos Þ ¼ ð0; cos ; r sin ; 0Þ.
The metric is provided by
2M 22M 1 2
ds2 ¼  1 drdt þ 1 rr
þ r2 ðd 2 þ sin2 d 2 Þ  2r2 !ðrÞdtd ; (5)where M is mass of a black hole, the dependence of the(t ) component is restricted by Eq. (4), and
C2MC! ¼ 21 1 þ 32 ½r2  2Mr  4M2rrr
þ 4Mðr  2MÞ lnðr  2MÞ:(6)Here, C1 and C2 are constants characterizing the rotation ofa black hole and are related to a small parameter ðJ=MrÞ, i.e., C1 , C2  Oð Þ, where J is the angular momentum. This solution satisfies both Eqs. (2) and (4) up to
the first order in . Within the first order of , the solution
does not depend on l. The differential equation for ! canbe obtained from the nonvanishing components of the fieldequation. In the (t ) component, the Einstein tensor gives
the differential equation, whereas the Cotton tensor vanishes. In contrast in other components, the Einstein tensor
vanishes, and the Cotton tensor gives the same differentialequation. Therefore, Eq. (6) is also a solution in generalrelativity. We emphasis that the Kerr solution does notsatisfy Eq. (4), which is an important difference between
the Einstein gravity and the CS gravity.
Here we briefly mention a relation between the original
CS gravity and the string theory. In the original CS gravity,Eq. (4) is imposed to ensure the diffeomorphism invariance. As shown in Ref. [7], # in Eq. (1) is not an external
quantity but a dynamical variable in the framework ofstring theory. Therefore Eq. (4) is replaced by the fieldequation [7]g r r # ¼ 

lpﬃﬃﬃﬃﬃﬃﬃ  R   R 64 g(7)in the string theory. When l is a nonvanishing value and #is a constant, Eqs. (2) and (7), respectively, reduce toG ¼ 0 and  R   R  ¼ 0 in a vacuum. We emphasis that our solution in Eqs. (5) and (6) also satisfies theseequations. Thus Eqs. (5) and (6) describe a classical field

which includes effects of quantum gravity. Furthermore,
we need not to align the rotational axis with the embedding
vector in this case because of v ¼ 0. Hereafter we consider Eq. (7) rather than Eq. (4) because Eq. (7) makes theCS theory self-consistent.
III. FEATURES OF SOLUTION
We look into a solution of the spacetime of a slowly
rotating black hole. For this purpose, we consider a scalar
invariant R R which is useful to evaluate the radial
dependence of the frame-dragging effect. From the metric,
the scalar invariant at large r is obtained in the form
R R

¼ ðRð0Þþ Rð1Þ ÞðRð0Þ þ Rð1Þ  Þ’48M2
4C2
 4 24 ;6r sinr(8)

where ‘‘(0)’’ and ‘‘(1)’’ denote the zeroth and the first orderin , respectively. The first term stems from theSchwarzschild solution, and the second term corresponds
to the rotation of a black hole. For Þ 0, , the scalar
invariant asymptotically reduces to zero at large r. As aconsequence, the spacetime becomes asymptotically flat atinfinity. The singularity of the rotational axis could beavoided by finding nonlinear or exact solutions. We notethat the frame-dragging part proportional to r4 decays
more slowly than the Schwarzschild part proportional tor6 . On the derivation of our metric solution we use theabsence of the Chern-Pontryagin density  R   R 
which is a source of gravitational anomaly [17]. Thus theabsence of gravitational anomaly gives the long range
nature of the frame-dragging effect. For comparison, werecall the Kerr metric whose (t ) component at large r isgiven by gtðKÞ ¼ 2J0 sin2 =r, where J0  Oð Þ is theangular momentum. For this metric, we obtainRðKÞ  RðKÞ’48M2 144J02
 8 ð2 þ cos2 Þ:
rr6(9)

Thus the Kerr metric gives rise to the rapid decay of theframe-dragging part because the second term is proportional to r8 .
IV. TRAJECTORIES OF A TEST PARTICLE
To show more astrophysical consequences of the solution, we investigate trajectories of a test particle with amass of m. Since the metric does not depend on t and ,the t and components of four-momentum p are conserved for the particle [18]. Hence, it can be assumed thatp0 ¼ mE and p ¼ mL, where E and L are the energy
and angular momentum of the particle, respectively. We
focus on trajectories in the equatorial plane ( ¼ =2).
From g p p ¼ m2 , we obtain

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FLAT ROTATION CURVES IN CHERN-SIMONS . . .

drd22ML2¼ E2  1 1 þ 2  2EL!ðrÞ;rrPHYSICAL REVIEW D 78, 024037 (2008)(10)

where  is a proper time. Differentiating this equation withrespect to , we derive
1 dd2 r
¼
22 drd
2ML21
1 þ 2  EL!0 ðrÞ; (11)rrwith !0  d!=dr. These equations can be solved perturbatively because the last terms on the right-hand side ofEqs. (10) and (11) are small compared with the other terms,i.e., !, !0  Oð Þ.
We solve Eqs. (10) and (11) up to the first order in forcircular orbits (r ¼ const). The solutions are then given byr  2M
E ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
rðr  3MÞsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
M
r2 ðr  2MÞ 0r!ðrÞ þ! ðrÞ ;r  3M
2ðr  3MÞsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Mr4 ðr  2MÞþ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !0 ðrÞ:L ¼ rr  3M 2 rðr  3MÞ3(12)(13)

Using these quantities, we can calculate d =dt ¼
ðdt=dÞ1 d =d for circular orbits. The circular velocity
is then obtained assﬃﬃﬃﬃﬃ dM
r2
¼
þ r!ðrÞ þ !0 ðrÞ :
v¼r
(14)
dtr
2
This is the central result of this paper. The first term, which
is a monotonically decreasing function proportional tor1=2 , is coming from the Schwarzschild metric. It isvery surprising that the second term becomes a constantat large r, i.e.,sﬃﬃﬃﬃﬃ
M C2
þ :v’
(15)
r
2
Therefore, the same feature can be expected in the rotation
curves of galaxies. On the other hand in the case of the Kerrsolution, the frame-dragging part is negligible at large rpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
because vðKÞ ’  M=r  J0 =r2 is derived. In a galaxy,there are a bulge and a disk associated with the central
black hole. For the spherically symmetric part such as abulge, an outside spacetime solution has the same form asthat of our solution. The nonspherical part such as a disk
would deform the solution in the direction of the rotational
axis. Thus the feature of the flat rotation curve is considered to remain unchanged on the equatorial plane.
At first glance, the constant circular velocity v at infinity
seems to contradict the asymptotically flat spacetime. This,
however, is explained as follows. It is impossible to cover
the spatial infinity by a single Minkowski spacetime. To doso, a number of Minkowski spacetimes are necessary. It is
possible to consider that the spatial infinity is covered byseveral finite-size areas and that each area is covered by aMinkowski spacetime. In such a situation, two adjacentareas are smoothly connected with each other by virtue ofthe infinitesimal curvature which is given by the second
term of Eq. (8). This situation is similar to that for a vector
potential field created by a solenoid with magnetic flux .
For a certain gauge, the vector potential has the form Ai ¼
ð=2Þðy=ðx2 þ y2 Þ; x=ðx2 þ y2 Þ; 0Þ, where i represents
spatial indexes. [This is very similar to Eq. (17) below.] If
the exact form Ai dxi  da is considered, any function fora cannot cover the whole space. Because of this fact, theloop integral
H
H of Ai can give the nonzero value , i.e.,
Ai dxi ¼ da ¼ .
The angular momentum of the Kerr black hole does notdependchoices of a two-surface within the definition ofR pon
ﬃﬃﬃﬃﬃﬃﬃ
J¼g" r  d d =16 [19], where  ¼
ð0; 0; 0; 1Þ is the Killing vector. When we evaluate theangular momentum of a black hole in Eqs. (5) and (6) byusing the same definition, the angular momentum depends
not only on C1 and C2 but also on r. This means that thegravitational field also has the angular momentum whose
degree is given by C1 and C2 . It should also be noted thatsuch a situation is a result of # ¼ const in Eq. (7). When #is changed from a constant at large r, the frame-dragging
effect would also be modified. In particular, when R  R
 Þ 0 in Eq. (7), the solution of Eq. (2) may
have the same feature as that of the Kerr solution.
V. PRECESSION OF SPINNING OBJECTSWe discuss a way to confirm the CS gravity through theprecession of spinning objects. To discuss the precession ofspinning objects, we first adopt the isotropic form of themetric derived from the transformation, ðx; y; zÞ ¼
ð~r sin cos ; r~ sin sin ; r~ cos Þ,
wherer ¼ r~ð1 þ
M=2~
rÞ2 . Then we apply the weak field approximation todiscuss more accessible situations. Namely the postNewtonian approximation is applied to the diagonal components of the metric and the leading term at large r isconsidered in the off diagonal components gti . We obtainds2 ’ ð1  2UÞdt2 þ ð1 þ 2UÞðdx2 þ dy2 þ dz2 Þþ 2Nið1Þ ðx; y; zÞdxi dt;
where U ¼ M=r, and Nið1Þ is given by
y~r
x~r;
C;
0:
Nið1Þ ¼ C2 2
2 2x þ y2x þ y2(16)(17)

By using the Minkowski metric  , we define h g   . It is shown that h satisfies the Lorentz gauge
condition @ h   @ h  =2 ¼ 0 under Eq. (16). In a
local Lorentz frame momentarily comoving with a spin-024037-3

KONNO, MATSUYAMA, ASANO, AND TANDAPHYSICAL REVIEW D 78, 024037 (2008)

ning object, the spin obeys an equation dSðiÞ =d ¼S u r eðiÞ , where eðÞ denotes the tetrad and SðiÞ ¼eðiÞ S is the spin vector [15]. The four velocity of thespinning object u ¼ eð0Þ is approximated as u ’k
ð1; v Þ. In the weak field approximation in Eq, (16), theequation becomes dS=d ¼  S, where¼ ðv aÞ=2 þ 3ðv  rUÞ=2  ðr  Nð1Þ Þ=2 and a denotes theacceleration vector. This equation seems to have the sameform as that obtained under the Kerr solution. The last termof , however, has a different form. We obtain

C2xz
C2yz
C2(18)¼;;N2 ðx2 þ y2 Þ~
2 ðx2 þ y2 Þ~rrr 2~for our solution and

3J0 xz 3J0 yz J0 ½2z2  ðx2 þ y2 ÞðKÞ; 5 ;;N ’r~5r~r~5dispersion in interacting galaxies may be expected to belarger than that in the Newtonian picture. Future investigation for cosmological density perturbation in this direction is necessary.
We remark the difference between our theory and theother modified gravity theories. The modified Newtonian
dynamics [21] certainly describe the flat rotation curves.
However, The modified Newtonian dynamics are entirely
phenomenological and not supported by a recent experiment [22]. Similarly, the f(R) gravity is phenomenologicalin practice [23]. On the other hand, the CS gravity is
directly related to an essential part in the string theory,and our results are basically parameter free. Therefore, ourtheory should be distinguished from the other theories.
VII. CONCLUSION(19)

for the Kerr solution. For large r, Eq. (18) is proportional tor1 , whereas Eq. (19) decays as r3 . This qualitative
difference can be measured from the radial dependenceof the spin precession. For such verification, observations
should be done for spinning objects far from a black hole.
Thus C2 can be estimated from observed data according toEq. (18).
VI. DISCUSSION
The whole universe given by Eqs. (5) and (6) slightly
rotates. However, it is difficult to observe the rotation at
infinity because the angular displacement of objects decreases proportional to 1=r with increasing r. The behaviorof our solution at infinity is in contrast to Gödel’s solution[20] in the Einstein theory. Equations (5) and (6) also meanthat a galaxy is affected by the frame-dragging pull ofanother galaxy. Thus we should observe the correlation
between the rotational velocity in a galaxy and the peculiar
velocities of other galaxies to confirm the frame-dragging
effect. It would be possible to estimate C2 from analysis of
the correlation. By now, unfortunately, a few studies havebeen made on this correlation. The frame-dragging effectmight explain another evidences of dark matter, i.e., large
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We have investigated features of the spacetime endowedwith a slowly rotating black hole within the framework ofthe Chern-Simons (CS) gravity. We found that the CS
gravity enhances the gravitomagnetic part of the gravitational field far from a black hole. As a consequence, thevelocity of a test particle in a circular orbit surprisingly
becomes a constant far from a black hole as it was found ina galaxy’s rotation curve. Thus the result explains a robust
evidence of dark matter without introducing realistic matter. Our finding indicates a possibility to solve the dark
matter problem by a new theoretical framework of the CS
gravity. To confirm the validity of this approach, we need toexplain other evidences of dark matter such as large velocity dispersion of galaxies in cluster and structure formation in the Universe. Further investigation in thisdirection could bridge the gap between purely theoreticalquantum gravity and more realistic astrophysicalphenomena.
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.’’ We thank Dr. K.
Hayasaki for useful discussions. One author (K. K.) thanks
Professor T. Futamase for stimulating comments.
Analytical calculations were performed in part on computers at YITP at Kyoto University.

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