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Modern Physics Letters A
Vol. 25, No. 31 (2010) 2655–2666c World Scientific Publishing Company
DOI: 10.1142/S0217732310033876

EFFECTS OF CHERN SIMONS CORRECTIONS ON CONSERVEDQUANTITIES OF RELATIVISTIC FLUID

KOHKICHI KONNO
Department of Natural and Physical Sciences, Tomakomai National College of Technology,
Tomakomai 059-1257, Japankohkichi@gt.tomakomai-ct.ac.jp
TOYOKI MATSUYAMA
Department of Physics, Nara University of Education, Nara 630-8528, JapanSATOSHI TANDA
Center of Education and Research for Topological Science and Technology,
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan

Received 9 April 2010
We consider relativistic fluid flow under Chern–Simons modified Maxwell theory andunder Chern–Simons modified gravity theory. We take account of the effects of Chern–
Simons corrections on the quantities of fluid flow that is conserved without the Chern–
Simons corrections. We find that the conservations of several quantities are generally
broken by the Chern–Simons corrections.
Keywords: Chern–Simons; fluid; Lagrangian description.

1. Introduction
The study of symmetry is requisite in modern physics. A typical example can befound in the standard theory for the interactions between elementary particles.
In contrast, the discoveries of symmetry breaking phenomena have opened newfrontiers in physics.1–4 Further detections of such phenomena might provide a signature of new physics. Chern–Simons (CS) modified theories5,6 are candidates forthe theories that describe the breaking of parity symmetry. The CS modified theories have attracted a lot of attentions in the context of exploring new physics.7–10
The CS modified Maxwell theory5 is constructed from the usual electromagnetic
action with a CS term. Similarly, the CS modified gravity theory6 is constructedfrom the Einstein–Hilbert action with a CS term. The CS terms violate the parity
symmetry. In previous works, electromagnetic and gravitational fields have mainly
been investigated.11–21 In this paper, we discuss the effect of the symmetry breaking
terms on fluid flow under the CS modified theories.
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This paper focuses on conserved quantities of fluid. For its discussion, it is usefulto adopt the Lagrangian description,22,23 in which each fluid particle is assigned alabel, i.e. Lagrangian coordinates at the initial time. In the Lagrangian coordinates,
mass evaluated in a closed region is conserved under time evolution. Similarly, forperfect fluid, circulation is also conserved. These facts are valid even when we takeaccount of general relativity.22,23 Furthermore, we can encounter other conserved
quantities, e.g., fluid helicity,24–27 magnetic helicity and cross helicity.28–30,27 The
last two quantities are considered for fluid interacting with electromagnetic fields.
In this paper, we deal with the above-mentioned conserved quantities within ageneral relativistic framework and take account of the corrections induced by the
CS modified theories.
This paper is organized as follows. In Sec. 2, we consider fluid flow interactingwith electromagnetic fields under the CS modified Maxwell theory. Utilizing the
Lagrangian description, we discuss the conservation of mass (or energy), circulation, and fluid helicity. We also investigate magnetic helicity and cross helicity. In
Sec. 3, we deal with fluid flow under the CS modified gravity theory. We discuss theconserved quantities in the same way as in Sec. 2. Finally we provide a summary
in Sec. 4. Throughout the paper, we use geometrized units with c = G = 1.

2. Fluid Flow in CS Modified Maxwell Theory
2.1. CS modified Maxwell theory
The action of the CS modified Maxwell theory is provided by5
IEM =

Z



1
1
d4 x −
Fµν F µν +vµ ∗F µν Aν ,
16π
8π

(1)

where Aµ is the four-potential, Fµν ≡ ∂µ Aν − ∂ν Aµ is the field-strength tensor,
F≡ 12 εµνλσ Fλσ is the dual field-strength tensor, and vµ is an external fourvector called the embedding vector. Here, εµνλσ denotes the Levi–Cività tensor with
ε0123 = 1. Raising and lowering the indices of tensors are done by the Minkowski
metric ηµν and η µν . The second term in the integrand in Eq. (1) is called the
CS term. The embedding vector vµ is now assumed to satisfy ∂µ vν = 0 to ensure gauge invariance.5 The dual field-strength tensor satisfies the Bianchi identity∂µ ∗F µν = 0. The variation of the action with respect to Aµ gives the left-hand sideof the electromagnetic field equation∗ µν∂ν F µν + vν ∗F µν = 4πJ µ ,(2)where J µ is the electric four-current. The second term on the left-hand side in thisequation stems from the CS term in Eq. (1). Thus the field equation of the Maxwell
theory is modified by the CS correction as in Eq. (2).

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2.2. Basic equations for fluid flow
The basic equations for fluid flow under the CS modified Maxwell theory can beobtained from the conservation law of energy–momentum tensor T µν . The energy–
momentum tensor is composed of the fluid part Tmµν and the electromagnetic partTem µν , i.e.
T µν = Tmµν + Tem µν .

(3)

For the fluid part, we assume perfect fluid
Tmµν = (ρ + p)uµ uν + pη µν ,(4)

where ρ is the energy density, p is the pressure, and uµ denotes the four-velocity
field of fluid particles (uµ uµ = −1). In the CS modified Maxwell theory, the electromagnetic part Tem µν is given by5


1
1
1
(5)
Tem µν =F µλ F ν λ − η µν Fλσ F λσ + v ν ∗F µλ Aλ .
4π
42
From the conservation equation ∂µ T µν = 0, using Eqs. (2), (3) and (5), we obtain∂µ Tmµν = F νλ Jλ .

(6)This equation can be divided into two parts, i.e. the component parallel to uµ andthe components orthogonal to uµ . The former gives the continuity equation(ρuµ ),µ + puµµ = 0 ,(7)where a comma denotes the partial differentiation with respect to coordinates. The
latter gives the equation of motion(ρ + p)uν uµν + P µν p,ν = F µν Jν ,(8)

where P µν ≡ η µν + uµ uν is the projection tensor. In deriving Eqs. (7) and (8), we
used the ideal magnetohydrodynamics approximation uµ F µν = 0. We now assume
barotropic fluid, for which new variables s and h can be introduced as31,23
Z ρ

dρ
s ≡ exp
,
(9)
ρ+p
Z p

dp
h ≡ exp.
(10)
ρ+p
The functions s and h correspond to the specific entropy and the specific enthalpy,
respectively.31,23 When the pressure vanishes, we have s = ρ and h = 1 by taking
appropriate constants of integration in the integrals in Eqs. (9) and (10). By usings and h, Eqs. (7) and (8) are rewritten, respectively, as(suµ ),µ = 0 ,(11)September 15, 2010 9:23 WSPC/146-MPLA

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K. Konno, T. Matsuyama & S. Tandauν uµ ,ν + P µν (ln h),ν = N µ − V µ ,

(12)

where
Nµ ≡

1
F µ F λσ,σ ,4π(ρ + p) λ

(13)

Vµ ≡

1
∗ λσ
F Fλσ v µ .
16π(ρ + p)

(14)

To derive Eq. (12), we used the identity
1 µ λσδ F Fλσ ,(15)
4 νwhere δ µν denotes the Kronecker delta. In Eq. (12), V µ comes from the CS correction. Thus while the continuity equation (11) is unchanged, the equation of motion(12) is changed due to the CS correction.
We also discuss vorticity of fluid. For this purpose, let us define the four-vorticity
ω µ as1ω µ ≡ εµνλσ uν uλ,σ .
(16)
2The spatial components of the four-vorticity give the usual three-vorticity ∇ × vf
when the motion is nonrelativistic, i.e. uµ ≃ (1, vf i ) and |vf | ≪ 1, where the Latin
index i runs over the spatial coordinates. We emphasize that the temporal component of ω µ gives the density of fluid helicity vf · (∇ × vf ) in the nonrelativistic
case (see also Refs. 24, 25 and 27). Hence, ω µ may be regarded as the four-currentof fluid helicity. This fact gives us a new insight that we can treat both helicity
and circulation of fluid in a unified way by adopting the four-vorticity ω µ definedin Eq. (16). Furthermore, we can easily recognize the transformation law of helicity density under a coordinate transformation. Under a coordinate transformation
xµ → x′µ , the helicity density ω 0 is transformed according to the transformation′law ω µ → ω µ = (∂x′µ /∂xν )ω ν . From Eq. (12), we obtain the differential equationfor ω µ ,
 µ ν hωuhω ν uµ−s ,ν h
s
h ,ν
∗ µλ

F

=

Fνλ =

1
[(Nν ω ν uµ + ∗M µν uν ) − (Vν ω ν uµ + ∗ W µν uν )] ,s(17)

1 µνλσε
Nλ,σ ,
2

(18)

where
M µν ≡∗

1 µνλσεVλ,σ .
(19)
2
Here ∗M µν and ∗ W µν are interpreted as rotational parts of the derivatives of Nµ
and Vµ , respectively. Thus when an electromagnetic field or the CS correction exists,
the vorticity equation becomes an inhomogeneous equation as seen in Eq. (17).
∗W µν ≡

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Consequently, the basic equations for fluid flow under CS modified Maxwell
theory are given by the field equation (2), the continuity equation (11), the equationof motion (12) and the vorticity equation (17).
2.3. Lagrangian description of fluid flow
We deal with the fluid motion from the viewpoint of the Lagrangian description.
We adopt the Lagrangian coordinates22,23
xµ = (τ, xi ) = (τ, x) ,(20)
iwhere τ is the proper time of a fluid particle and x is constant along a line of fluid
flow. In this coordinates, the four-velocity becomesdxµ= δ 0µ = (1, 0, 0, 0) .
(21)dτAdopting the Lagrangian coordinates is equivalent to taking the four-velocity inthe form of Eq. (21). Equation (21) is called the Lagrangian condition.22,23 When
we use the Lagrangian condition, the metric ηµν and the partial derivative ∂µ in
Eqs. (2), (11), (12) and (17) must be replaced with the general metric gµν andthe covariant derivative ∇µ , respectively, because the coordinates are no longer√
Cartesian. Furthermore, the Levi–Cività tensor is redefined as ε0123 = 1/ −g,where g is the determinant of gµν . In the Lagrangian description, the metric ischanged with fluid motion, while the coordinates of fluid particles are fixed.
Let us discuss the continuity equation (11) using the Lagrangian condition (21).
Equation (11) gives√−gs ,0 = 0 .
(22)
uµ =

Thus we obtains(τ, x) =sg(τ0 , x)s(τ0 , x) .
g(τ, x)When the pressure is negligible, Eq. (23) reduces tosg(τ0 , x)ρ(τ, x) =ρ(τ0 , x) .
g(τ, x)

(23)

(24)

√
Therefore, Eqs. (23) and (24) lead to the conservations of entropy density −gs√
and energy density −gρ, respectively, in the Lagrangian coordinates.
Next we discuss the vorticity equation (17) using the Lagrangian condition (21).
Here it should be noted that the temporal component of ω µ is not independent of
the spatial components because we have ω 0 = g0i ω i from the equality uµ ω µ = 0.
Hence we focus on the spatial components of ω µ . The spatial components of Eq. (17)
give
 i
hω
h
= ( ∗M iν − ∗W iν )uν .
(25)
s ,0
s

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Thus we obtainih(τ, x)ω (τ, x) =s"g(τ0 , x)h(τ0 , x)ω i (τ0 , x) + s(τ0 , x)g(τ, x)

×

Z τ

dτ

τ0

′h ∗

s

( M

iν

#

iν

∗

− W )uν ,(26)

where we have used Eq. (23). When the pressure is negligible, Eq. (26) reduces to
s
"
#
Z τ
g(τ0 , x) ii
′ 1 ∗ iν∗ iνω (τ, x) =ω (τ0 , x) + ρ(τ0 , x)dτ ( M − W )uν . (27)g(τ, x)ρτ0√√Equation (26) (or (27)) means that the spatial components −ghω i (or −gω i )is conserved in the absence of both the rotational part of electromagnetic force
and the CS correction. In particular, when the pressure is negligible, we can obtain√the conservation of four-vector density −gω µ directly from Eq. (17). In such acase, therefore, the conservations of helicity and circulation can be obtained in a
unified way within a relativistic framework. In the presence of the rotational part ofelectromagnetic force or the CS correction, the conservation law concerning vorticity
is broken. The integral terms in Eqs. (26) and (27), which are independent of ω i ,become sources of three-vorticity. It means that even if ω i vanishes everywhere at
the initial time, the spatial components of vorticity may be created at some time.
The temporal component of the four-vorticity can be derived from ω 0 = g0i ω i as
mentioned above. When the pressure vanishes, ω 0 is obtained directly from Eq. (17),s"Z τg(τ0 , x) 010
ω (τ, x) =ω (τ0 , x) + ρ(τ0 , x)dτ ′g(τ, x)
ρ
τ0
ν

∗

0i

ν

∗

0i

#

× {(Nν ω + M ui ) − (Vν ω + W ui )} .

(28)

The terms ∗M 0i ui and ∗W 0i ui in the integrand are independent of ω µ . This factmeans that helicity density may also be created irrespective of the values of ω µ .
Therefore, not only the conservation of circulation but also the conservation ofhelicity is generally broken in the CS modified Maxwell theory.
2.4. Magnetic helicity and cross helicity
Let us discuss magnetic helicity and cross helicity in the CS modified Maxwell
theory. These quantities are conserved under certain conditions in ordinary magnetohydrodynamics. In this subsection, we use the Cartesian coordinates, in which
the metric reduces to the flat Minkowski metric.
We define the magnetic helicity four-current Hm µ as
Hm µ ≡ ∗F µν Aν .

(29)

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Using the identity (15), we derive for ideal magnetohydrodynamics∗ µνFFµν = −4uµ ∗F µλ uν Fνλ = 0 .

(30)

Thus we obtain
Hm µ,µ =1 ∗ µν
F Fµν = 0 .
2(31)Therefore, the magnetic helicity four-current Hm µ is conserved even in the CS
modified Maxwell theory.
Next we discuss the cross helicity four-current Hc µ defined by
Hc µ ≡ ∗ω µν Aν ,(32)

where ∗ ω µν ≡ 21 εµνλσ uλ,σ . For ideal magnetohydrodynamics, we obtain
Hc µ,µ =1 ∗ µν
1
ω Fµν = B µ [(ln h),µ + Vµ ] ,
2
2

(33)

where B µ ≡ ∗F µν uν denotes the magnetic field. The last term on the right-handside, which is proportional to B µ Vµ , arises from the CS correction. When the righthand side in Eq. (33) does not vanish, the conservation of cross helicity is broken.
Therefore, the cross helicity four-current is not generally conserved in the CS modified Maxwell theory.
3. Fluid Flow in CS Modified Gravity
3.1. CS modified gravity
The action of the dynamical CS modified gravity is provided by6,18ZRℓ ∗ τ µν σ
1 µν
4 √IG = d x −g −+
ϑ R σ R τ µν − g (∂µ ϑ)(∂ν ϑ) − V (ϑ) + Lm ,16π 64π2(34)where R ≡ g αβ Rαβ (Rαβ ≡ Rλαλβ ) is the Ricci scalar, Rτσαβ ≡ ∂β Γτσα − · · · isthe Riemann tensor (Γαβγ is the Christoffel symbols), ℓ is a coupling constant, ϑ
is a dynamical scalar field, V is a potential, and Lm is the Lagrangian density
for matter. The dual Riemann tensor is defined by ∗Rτσµν ≡ 21 εµναβ Rτσαβ , where√
ε0123 ≡ 1/ −g. The second term in Eq. (34) is called a Chern–Pontryagin term
and connected to the CS term, via partial integration,
Zℓ
2 σ τ η
µαβγ
σ
τ
4 √
ICS = −
d x −g(∂µ ϑ)εΓατ ∂β Γγσ + Γατ Γβη Γγσ .
(35)
32π
3
Thus nontrivial ∂µ ϑ leads to the CS modification. In this paper, we neglect the
problem concerning the surface integral term (see Ref. 32 for the serious treatment).

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When we neglect the kinematic term and the potential term of ϑ in Eq. (34), theaction reduces to that of the Jackiw–Pi model.6 From the variations in the action
with respect to the metric gµν and the scalar field ϑ, we obtain the field equations,
respectively,Gµν + ℓC µν = −8π(Tm µν + Tϑµν ) ,dV (ϑ)ℓ ∗ τ µν σ−
R σ R τ µν ,dϑ64πwhere Gµν is the Einstein tensor, C µν is the Cotton tensor defined byg µν ∇µ ∇ν ϑ =

(36)
(37)

1
C µν ≡ − [(∇σ ϑ)(εσµαβ ∇α Rνβ + εσναβ ∇α Rµβ ) + (∇σ ∇τ ϑ)( ∗Rτ µσν + ∗Rτ νσµ )] ,2
(38)
Tm µν is the energy–momentum tensor for matter, and Tϑµν is the energy–
momentum tensor of the scalar field,


1
(39)
Tϑµν = (∇µ ϑ)(∇ν ϑ) − g µν (∇λ ϑ)(∇λ ϑ) + V (ϑ) .
2Thus Eqs. (36) and (37) are basic equations in the CS modified gravity.
3.2. Basic equations for fluid flow
We obtain basic equations for fluid motion under the CS modified gravity from thecovariant divergence of Eq. (36),ℓ∇ν C µν − ∇ν Tϑµν ≡ Θµ ,(40)
8πwhere Θµ is defined as force exerted on the ordinary matter. We can regard Θµ as
a CS correction. Using Eq. (39) and the equality6∇ν Tm µν = −∇ν C µν =1 µ ∗ σ νλ τ(∇ ϑ) R τ R σνλ ,8(41)

we obtainΘµ (ϑ) = (∇ν ϑ)(∇µ ∇ν − ∇ν ∇µ )ϑ .

(42)

Thus we find that Θµ vanishes when the scalar field is regular everywhere. Then
we derive the usual equation of motion for matter, ∇ν Tm µν = 0. In this case, themotion of fluid is affected only through the change of the metric. When we consider
a test particle, we derive the usual geodesic equation from ∇ν Tm µν = 0, which is
independent of the mass. This fact means that the equivalence principle is valid forregular ϑ under the CS modified gravity. On the other hand, if there is a singularityin ϑ, Θµ would not vanish at the singularity. For example, if ϑ = arctan(y/x) ≡ φin rectangular Cartesian coordinates, we derive


2πx
2πy
δ(x)δ(y),δ(x)δ(y),0,(43)Θµ = 0, 2
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where we have used the formula33(∂x ∂y − ∂y ∂x )φ = 2πδ(x)δ(y) .

(44)Here, δ(x) denotes the Dirac’s delta function. Therefore, when such a topologicalsingularity exists, the force term Θµ plays an important role at the singularity.
Hereafter, we take account of the effect of the CS correction Θµ on fluid motion.
We now assume perfect fluid again. Equation (40) can be divided into two parts,
i.e. the component parallel to uµ and the components orthogonal to uµ . The former
gives the balance equation, i.e. the continuity equation with a source term,(ρuµ );µ + puµ;µ = −uµ Θµ ,(45)

where a semicolon denotes the covariant derivative. The latter gives the equationof motion(ρ + p)uν uµ;ν + P µν p,ν = P µ ν Θν ,(46)

where P µν is the projection tensor onto the hypersurface orthogonal to uµ . Using
the variables s and h, we can write Eqs. (45) and (46), respectively, as(suµ );µ = −S ,(47)uν uµ ;ν + P µν (ln h),ν = V µ ,(48)

where the CS corrections S and V µ are defined byS≡suµ Θµ ,
ρ+p

(49)

Vµ ≡

1
P µ Θν .
ρ+p ν

(50)

The balance equation (47) and the equation of motion (48) govern fluid dynamics
under the CS modified gravity.
From Eq. (48), we also obtain the differential equation for vorticity ω µ as


hω µ
s

uν
hω ν
−
s
;ν h



 µ
u
1
1
= 2 Sω µ + [Vν ω ν uµ + ∗ W µν uν ] ,
h ;ν
s
s

(51)

where
∗

W µν ≡

1 µνλσεVλ;σ .
2(52)

Here ∗ W µν is regarded as the rotational component of the force V µ . Thus, Eq. (51)governs the time evolution of vorticity in fluid motion under the CS modified gravity.

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3.3. Lagrangian description of fluid flow
We discuss fluid motion from the viewpoint of the Lagrangian description.
We deal with the balance equation (47). Applying the Lagrangian condition (21)to Eq. (47), we derive√√−gs ,0 = − −gS .
(53)
Thus we obtain

s

Z τ

√
dτ ′ −gS .

(54)

When the pressure vanishes, Eq. (54) reduces tosZ τ√g(τ0 , x)1dτ ′ −gS .
ρ(τ0 , x) − pρ(τ, x) =g(τ, x)−g(τ, x) τ0(55)

s(τ, x) =g(τ0 , x)1s(τ0 , x) − pg(τ, x)−g(τ, x)τ0

From Eqs. (54) and (55), we see that if the CS correction S vanishes, the entropy√√
density −gs or the energy density −gρ is conserved in the Lagrangian coordi√√
nates. However, if the CS correction S does not vanish, −gs and −gρ are nolonger conserved. The function S provides a source of entropy or energy as seenin Eqs. (54) and (55). Thus in general, the creation of entropy or energy may occur when there is a topological singularity in the scalar field appearing in the CS
modified gravity.
Next we discuss the vorticity equation (51) using the Lagrangian condition (21).
As mentioned above, the temporal component of ω µ is not independent of the spatial
components because ω 0 = g0i ω i . Hence we focus on the spatial components of ω µagain. The spatial components of Eq. (51) give
 i
hω
h
h
= 2 Sω i + ∗ W iν uν .
(56)
s ,0 ss
Then we obtains(τ, x)h(τ, x)ω (τ, x) =h(τ0 , x)ω i (τ0 , x) + s(τ, x)s(τ0 , x)
i

Z τ
τ0

dτ

′




h
h ∗ iνiSω +W uν .
s2
s(57)

When the pressure vanishes, Eq. (57) reduces toZ τρ(τ, x) i
11
ω i (τ, x) =ω (τ0 , x) + ρ(τ, x)dτ ′ 2 Sω i + ∗ W iν uν .
ρ(τ0 , x)
ρ
ρ
τ0

(58)

From Eqs. (57) and (58), we see that if the CS corrections S and ∗ W i0 vanish,hω i /s or ω i /ρ is conserved in the Lagrangian coordinates. However, if S 6= 0 or∗
W iν 6= 0, hω i /s or ω i /ρ is no longer conserved. In particular, the second termsof the integrands in Eqs. (57) and (58) may produce vorticity, regardless of theSeptember 15, 2010 9:23 WSPC/146-MPLA

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values of vorticity vector. Therefore, even if the vorticity vanishes everywhere at
the initial time, circulation may be created when there is a topological singularity
in the scalar field of the CS modified gravity.
4. Summary
We have considered fluid flow under the CS modified Maxwell theory and under the
CS modified gravity theory. We investigated the effects of the CS corrections on conserved quantities of relativistic fluid. First of all, we obtained the CS corrections to
the equations for fluid, i.e. the continuity equation, the equation of motion and thevorticity equation. For the discussion of vorticity, we introduced the four-vorticity
in Eq. (16). We also pointed out that the four-vorticity provides the expressionsof both three-vorticity and fluid helicity density in the nonrelativistic limit. This
fact means that the four-vorticity unifies the physical pictures of circulation andfluid helicity in a relativistic framework. To discuss conserved quantities of fluid,
we utilized the Lagrangian description of fluid flow. In the CS modified Maxwell
theory, we found that while the entropy or energy is conserved, the circulation andfluid helicity are not generally conserved. We also found that the conservation of
the magnetic helicity holds even in the CS modified Maxwell theory, while the conservation of the cross helicity does not hold in general. In the CS modified gravity,
when the CS scalar field is regular everywhere, the CS corrections do not appear
in the equations for fluid flow. In this case, we have the validity of the equivalenceprinciple. This fact means that for a regular CS scalar field, we cannot find the CS
correction directly from the test of equivalence principle. Furthermore, we foundthat when there is a topological singularity in the CS scalar field, not only thecirculation and fluid helicity but also the entropy and energy are not conserved ingeneral at the singularity.
Acknowledgments
We thank Prof. Y. Fukumoto for useful conversations. One author (K.K.) thanksProf. H. Asada for fruitful discussions. Analytical calculations were performed inpart on computers at YITP in Kyoto University.
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