Torsion, Chern-Simons Term and Diffeomorphism Invariance

Abstract

In the $torsion \otimes curvature$ approach of gravity Chern-Simons
modification has been considered here. It has been found that Chern-Simons
contribution to the bianchi identity has become cancelled from that of the
scalar field part. But "homogeneity and isotropy" consideration of present day
cosmology is a consequence of the "strong equivalence principle" and
vice-versa.

Full text

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arXiv:1006.3749v1 [gr-qc] 18 Jun 2010
Torsion, Chern-Simons Term and
Diffeomorphism Invariance
Prasanta Mahato∗
Partha Bhattacharya†
Abstract
In the torsion⊗curvature approach of gravity Chern-Simons mod-
ification has been considered here. It has been found that Chern-
Simons contribution to the bianchi identity has become cancelled from
that of the scalar field part. But “homogeneity and isotropy” con-
sideration of present day cosmology is a consequence of the “strong
equivalence principle” and vice-versa.
PACS: 04.20.Fy, 04.20.Cv
KEY WORDS: Chern-Simons modified gravity, Torsion, Diffeomor-
phism invariance
1Introduction
Chern-Simons (CS) modified gravity is a 4-dimensional deformation of Gen-
eral Relativity (GR), postulated by Jackiw and Pi[1]. This theory of gravity
is an extension of general relativity by adding a parity violating Pontrya-
gin density∗RR coupled to a scalar field θ to Einstein-Hilbert Lagrangian.
This scalar field θ can be viewed as either a prescribed background quantity
or as an evolving dynamical field. The∗RR is defined as the contraction
of the Riemann tensor with its dual and it is odd under a parity transfor-
mation, thus potentially enhancing gravitational parity-breaking. The CS
correction introduces a means to enhance parity-violation through a pure
curvature term, as opposed to through the matter sector, as more commonly
∗Narasinha Dutt College, Howrah, West Bengal, India 711101, e-mail:
hato@dataone.in
†BangabasiMorningCollege,Kolkata,
parthab68@yahoo.co.in
pma-
WestBengal,India 700009,e-mail:
1

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Torsion, Chern-Simons Term and ....
happens in GR. One of the important feature of CS modified gravity is its
emergence within predictive frameworks of more fundamental theories. For
example, the low energy limit of string theory comprises general relativity
with a parity violating correction term, that is nothing but the Pontryagin
density. This term is crucial for cancelling gravitational anomaly in string
theory through Green-Schwartz mechanism[2]. The Pontryagin density, as
an anomaly cancelling term, also arises in particle physics and in the context
of loop quantum gravity[3]. Ref.[3] gives a recent review on Chern-Simons
Modified General Relativity. In a very recent approach the CS modified
gravity reduces to topologically massive gravity in three dimensions[4].
It is now a well established facts that de Sitter group is the correct un-
derlying gauge group of gravity as unlike Poincare group, it is a semisimple
group yielding consistent field equations. Thus de Sitter gauge theory comes
up as the corrected Poincare gauge theory[5]. Recently a gravitational La-
grangian has been proposed[6], where a Lorentz invariant part of the de Sitter
Pontryagin density has been treated as the Einstein-Hilbert Lagrangian. In
this formalism the role of torsion in the underlying manifold is multiplicative
rather than additive and the Lagrangian looks like torsion⊗curvature. This
indicates that torsion is uniformly nonzero everywhere. In the geometrical
sense, this implies that micro local space-time is such that at every point
there is a direction vector (vortex line) attached to it. This effectively cor-
responds to the non commutative geometry having the manifold M4× Z2,
where the discrete space Z2is just not the two point space but appears as an
attached direction vector[7]. In this approach we consider only a particular
class of U4space, where only the axial vector part of the torsion is present ev-
erywhere in space time.1This may be compared with another approach[10]
where the axial vector torsion is given by the derivative of a pseudoscalar
field and then one gets a propagating torsion wave unlike in the standard
Einstein-Cartan theory where torsion ceases to exist outside spinorial mat-
ter. This propagating torsion extends over whole spacetime. Whereas in the
present formalism the additive torsion decouples from the theory but not the
multiplicative one and this implies the non trivial omnipresence of the axial
torsion. In this model of minimum extension of Einstein-Hilbert theory, we
only take axial torsion as an extra degree of freedom. Considering torsion
and torsion-less connection as independent fields[11], it has been found that,
κ of Einstein-Hilbert Lagrangian appears as an integration constant and is
linked with the topological Nieh-Yan density of U4 space. As a result, κ
has got its definite geometrical meaning in U4 space in comparison to its
1It is to be noted here that in the presence of spinorial matter only the axial vector
part of the torsion couples to the spinor field[8, 9].

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Torsion, Chern-Simons Term and .... 3
standard meaning of being simply an ad hoc constant. If we consider ax-
ial vector torsion together with a scalar field φ connected to a local scale
factor[12], then the Euler-Lagrange equations also link, in laboratory scale,
the mass of the scalar field with the Nieh-Yan density and, in cosmic scale
of FRW-cosmology, they predict only three kinds of the phenomenological
energy density representing mass, radiation and cosmological constant. In a
recent paper[13], it has been shown that this scalar field may also be inter-
preted to be linked with the dark matter and dark radiation. Recently it has
been shown that[14], using field equations of all fields except the frame field,
the starting Lagrangian reduces to a generic f(R) gravity Lagrangian which,
for FRW metric, gives standard FRW cosmology. But for non-FRW metric,
in particular of Ref.[15], with some particular choice of the functions of the
scalar field φ one gets f(R) = f0R1+v2tg, where vtgis the constant tangential
velocity of the stars and gas clouds in circular orbits in the outskirts of spiral
galaxies. In this letter we are going to study the CS modification of this
formalism where the CS scalar θ has been considered to be a function of the
scalar field φ.
2Formulation
The gravitational Lagrangian, with CS modification, may be defined to be[12,
14, 3]
LG+ LCS
= N{R − u(φ)} +∗BB
∓1
2dφ ∧∗dφ − h(φ)η +1
4θ(φ)ˆRab∧ˆRab,
(1)
where * is Hodge duality operator, Rη =1
d¯ ωba+¯ ωbc∧¯ ωca,ˆRba= dˆ ωba+ˆ ωbc∧ˆ ωca, ¯ ωab= ωab−Tab, ˆ ωba= −ebνeaν:µdxµ,
Ta=
N = dT, ηa=
terior covariant differentiation with respect to the connection one form ¯ ωab,
: represents tensorial covariant differentiation, w.r.t. the Christoffel connec-
tion, acting upon external indices and Bais a two form with one internal
index and of dimension (length)−1and θ(φ), h(φ) are unknown functions of
φ whose forms are to be determined subject to the geometric structure of the
manifold.
Now we write the total gravity Lagrangian in the presence of a spinorial
matter field, given by
2¯Rab∧ ηab, B = Ba∧¯∇ea,¯Rba=
1
2!eaµTµναdxν∧ dxα, Tab= eaµebνTµναdxα, T =
1
3!ǫabcdeb∧ ec∧ edand ηab=∗(ea∧ eb). Here¯∇ represents ex-
1
3!Tµναdxµ∧ dxν∧ dxα,
Ltot.
= LG+ LCS+ LD,
(2)

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Torsion, Chern-Simons Term and ....
where[13]
LD
=[i
2{ψ∗γ ∧ Dψ + Dψ ∧∗γψ} −g
+cψ
γµ := γaeaµ,
1
4γµD{}γµ=1
−i
here D{}, or : in tensorial notation, is Riemannian torsion free covariant
differentiation acting on external indices only; σab=i
and g, cψ are both dimensionless coupling constants. Here ψ and ψ have
dimension (length)−1
SL(2,C) transformation on the spinor field and gamma matrices, given by,
4ψγ5γψ ∧ T
√
∗dTψψη]
∗γ := γaηa, D := d + Γ
(3)
(4)
Γ :=
4γµγµ:νdxν
=
4σabeaµebµ:νdxν
(5)
2(γaγb−γbγa), ψ = ψ†γ0
2 and conformal weight −1
2. It can be verified that under
ψ → ψ′
= Sψ, ψ → ψ′= ψS−1
= SγS−1,
and γ → γ′
4θabσab), Γ obeys the transformation property of a SL(2,C)
gauge connection, i.e.
(6)
where S = exp(i
Γ → Γ′
s. t. Dγ := dγ + [Γ,γ] = 0.
=
S(d + Γ)S−1
(7)
(8)
As in Ref.[12, 13], by varying the independent fields in the Lagrangian
Ltot., we obtain the Euler-Lagrange equations and then after some simplifi-
cation we get the following results
¯∇ea= 0,
∗N =1
(9)
(10)
κ,
i.e. ¯∇ is torsion free and κ is an integration constant having dimension of
(length)2.2
√∗dT =
mψ= cψ
cψ
√κ,
(11)
2In (1),¯∇ represents a SO(3,1) covariant derivative, it is only on-shell torsion-free
through the field equation (9).The SL(2,C) covariant derivative represented by the
operator D is torsion-free by definition, i.e. it is torsion-free both on on-shell and off-
shell. Simultaneous and independent use of both¯∇ and D in the Lagrangian density (2)
has been found to be advantageous in the approach of this article. This amounts to the
emergence of the gravitational constant κ to be only an on-shell constant and this justifies
the need for the introduction of the Lagrangian multiplier Bawhich appears twice in the
Lagrangian density (1) such that ¯ ωaband eabecome independent fields.

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Torsion, Chern-Simons Term and .... 5
i∗γ ∧ Dψ −g
iDψ ∧∗γ −g
4γ5γ ∧ Tψ + mψψη = 0,
4ψγ5γ ∧ T + mψψη = 0.
(12)
(Gba+ Cba)η = −κ[i
8{ψ(γbDa+ γaDb)ψ − (Daψγb+
Dbψγa)ψ}η −g
±1
= −κ[Tba(ψ) + Tba(φ)]η say,
0 = [1
(13)
16ψγ5(γa∗Tb+ γb∗Ta)ψη
(14)
2∂aφ∂bφη +1
2hηδba](15)
(16)
2∇νψ{σba
−(Daψγb− Dbψγa)ψ}
−g
= κ[∇cθǫcde(a¯∇eRb)d+ ∇c∇dθ∗¯Rd(ab)c],
Tba(ψ) =
8{ψ(γbDa+ γaDb)ψ − (Daψγb+
Dbψγa)ψ} −g
Tba(φ) = ±1
2,γν}ψ +i
2{ψ(γbDa− γaDb)ψ
(17)
4ψγ5(γa∗Tb− γb∗Ta)ψ]η,
(18)
Cab
(19)
i
(20)
16ψγ5(γa∗Tb+ γb∗Ta)ψ,
2∂aφ∂bφ +1
(21)
2hδba,
(22)
κd[g
4
∗(ψγ5γψ ∧ T) + Σ] = −g
Σ = ∓1
2
4θ′(φ)ˆRab∧ˆRab= ∓d∗dφ.
4ψγ5γψ,
(23)
∗(dφ ∧∗dφ) + 2h −1
κu
(24)
{1
κu′(φ) − h′(φ)}η +1(25)
Now we see that[1, 3]
¯∇bCba = −1
¯∇bTba(φ) =
8κ∂aθ∗RR
¯∇b[±1
(26)
2∂aφ∂bφ +1
2hδba] =1
8∂aθ∗RR +1
2∂aΣ (27)
Therefore Bianchi identity of Gbaimplies
¯∇bTba(ψ) = −1
2∂aΣ (28)

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Torsion, Chern-Simons Term and ....
Earlier[12, 13, 16], we have seen that Σ = constant gives us standard
isotropic and homogeneous FRW-universe at cosmic scale and from the last
equation we see that this is the case of strong equivalence principle. There-
fore we can state that Isotropy & Homogeneity ⇔ Diffeomorphism
Invariance. Moreover, for Σ = constant, we can define conserved axial
current[16] by
J ≡ κ∗(ψγ5γψ ∧ T)T
(29)
The general theory of relativity is a diffeomorphism invariant theory
where energy momentum tensor is covariantly conserved and there exists
a killing field that generates an isometry of the spacetime. Now the vital
question is which solution of Einstein’s equation corresponds to our universe
or at least an idealized model that approximates our universe. We know
that the structure of the universe as predicted by GR based on simple cos-
mological principle of homogeneity and isotropy approximation in the large
scale structure of our universe. So diffeomorphism invariance of Einstein’s
equation and simple cosmological principle are closely related. In our present
formalism of minimal extension of GR we see that diffeomorphism invariance
is still maintained if and only if the universe is isotropic and homogeneous.
The equation (28) is valid with or without CS term of the action. So this
conclusion remains valid even in the presence of CS modified extended GR.
When we consider other form of cosmic energy density, may be in the early
universe, we have to adopt a non-FRW geometry where we may have to
forgo the isotropy and homogeneity of the universe[13], then the above result
shows that conservation of energy-momentum tensor of baryonic matter is
violated but the total energy-momentum tensor of both baryonic and dark
matter/radiation always conserved.
3 Discussion
In the standard formulation of Chern-Simons modified gravity where θ is
taken as external variable, the presence of Cotton tensor Cbain the modified
Einstein’s equation violates the diffeomorphism invariance because the non
vanishing divergence of cotton tensor is proportional to ∗RR, the Pontrya-
gin density. To maintain the diffeomorphism invariance the consistency of
dynamics forces ∗RR to vanish, so the consistency condition suppressed the
symmetry breaking CS term in the action, even though its variation results
in the modified equation of motion[17]. When θ is taken as local dynamical
variable then this constraints is replaced by evolution equation of θ which can

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Torsion, Chern-Simons Term and .... 7
be viewed as Klein-Gordon equation in the presence of a potential and Pon-
tryagin density as source term[3]. Then from modified Einstein’s equation it
can be shown that strong equivalence principle is satisfied provided θ satisfies
its evolution equation. So there is no need to impose Pontryagin constraint
to maintain Lorentz symmetry and thereby strong equivalence principle.
In the first order formalism of CS modified gravity[10], curvature tensor
is taken as SO(3,1) field strength such that the gravitational part gives the
standard first order Einstein-Hilbert action but the CS part∗RR, being not
torsion free, serves as the effect of the CS deformation on the space-time
geometry through an effective contribution of torsion which depends on the
external field θ[10]. In the absence of CS term the action reduces to standard
Einstein-Hilbert action. But in our model of minimal extension of Einstein-
Hilbert action a multiplicative torsion is already present in the lagrangian
which looks like torsion ⊗ curvature. The additive torsion may decouples
from the theory but not the multiplicative one. This indicates that torsion is
uniformly nonzero everywhere. This torsion is not present in the curvature
tensor and in the CS term. Here we take the CS term as θ(φ)ˆRab∧ˆRab, where
ˆRabis the curvature 2-form from the metric only.
In our formulation of minimal extension of GR the evolution equation
for φ is given by equation (25). Here we see that strong equivalence is not
automatically satisfied but it depends upon the isotropy and homogeneity of
spacetime (28). This is because here we take CS modification of minimal ex-
tended gravity, φ is taken as dark matter field and θ is taken as a function of
φ. As a consequence we see that energy momentum conservation of ordinary
(baryonic) matter depends upon the constancy of Σ (or isotropy and homo-
geneity of spacetime). And therefore strong equivalence principle of ordinary
baryonic matter depends entirely upon the evolution of dark matter to form
the FRW-geometry of the present universe in cosmic scale.
References
[1] R. Jackiw and S. Y. Pi. Phys. Rev., D68 : 104012, 2003.
[2] M. B. Green and J. H. Schwartz. Phys. Lett., B149 : 117, 1984
[3] S. Alexander and N. Yunes. Phys. Rept., 480 : 1, 2009
[4] H. Ahmedov and A. N. Aliev. Phys. Lett., B 690 : 196, 2010
[5] R. Aldrovandi and J. G. Pereira. J. Math. Phys., 29 : 1472, 1988
[6] P. Mahato. Mod. Phys. Lett., A17 : 1991, 2002

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8
Torsion, Chern-Simons Term and ....
[7] A. Connes. Noncommutative Geometry, Academic Press, New York,
1994
[8] V. Desabatta and M. Gaspereni. Introduction to Gravitation, World
Scientific, Singapore, 1994
[9] E. Mielke and D. Kreimer Gen. Rel. Grav., 31 : 701, 1999
[10] M. B. Cantcheff. Phys. Rev., D78 : 0801.025002, 2008
[11] P. Mahato. Phys. Rev., D70 : 124024, 2004
[12] P. Mahato. Int. J. Theor. Phys., 44 : 79, 2005
[13] P. Mahato. Int. J. Mod. Phys., A 22 : 835, 2007
[14] P. Mahato. Annales de la Fondation Louis de Broglie, 32 : 297, 2007
[15] C. G. Bohmer, L. Hollenstein and F. S. N. Lobo. Phys. Rev., D76 :
084005, 2007
[16] P. Mahato. ICFAI Univ. Jour. of Phys., 2 : 92, 2009
[17] R. Jackiw. Lorentz Violation in a Diffeomorphism-Invariant Theory,
arXiv : 0709.2348v1 [hep-th]