3.2 Chern–Simons Modified Gravity
23
G
μν
+
1
μ
C
μν
= 0.
(3.5)
It is useful also to write the linearized form of the gravitational Chern–Simons action
obtained from (3.2) under the replacement g μν = η μν + κh μν :
S
(0)
= −
1
2μ
d
3 xh
μν
αμρ ∂
ρ
(η γν − ∂ γ ∂ ν )h
γα
.
(3.6)
We see that this action is, first, explicitly gauge invariant under usual linearized gauge
transformations δh μν = ∂ μ ξ ν + ∂ ν ξ μ , second, involves higher derivatives. However,
after obtaining the equations of motion for the full linearized action formed by the
sum of the terms (1.11) and (3.6), one finds that the physical degrees of freedom
satisfy the second-order equation [38], with their propagator behaves as ( + μ
2
)
−1 ,
thus, in the 3D CS modified gravity there is no problems with negative-energy states
discussed in the previous chapter. The similar situation occurs in the four-dimensional
case as well.
The generalization of this theory to the four-dimensional case turns out to be
straightforward, however, in this case, similarly to the electrodynamics, this generalization essentially involves the CPT (and in certain cases Lorentz) symmetry
breaking. From the formal viewpoint such a generalization for the linearized theory is performed through replacement
μνλ
→ b ρ
ρμνλ , with b ρ is a constant vector,
which allows to convert the CS term to the Carroll-Field-Jackiw (CFJ) term which
in the Abelian case looks like
L C F J =
ρμνλ b ρ A μ ∂ ν A λ .
(3.7)
In principle, such a replacement of the three-dimensional Levi–Civita symbol by
the four-dimensional one contracted with a vector already allows to write down the
four-dimensional gravitational CS term:
L C S,grav =
d
4 x
ρμνλ b ρ
b
μa ∂ ν
a
λb +
2
3
b
μa
c
νb
a
λc
,
(3.8)
with its linearized form is
S
(0)
= −
1
2
d
4 xh
μν
αμρλ b
λ
∂
ρ
(η γν − ∂ γ ∂ ν )h
γα
.
(3.9)
We note that this action is invariant under the same linearized gauge transformations
δh μν = ∂ μ ξ ν + ∂ ν ξ μ . Now, it is very interesting to discuss some motivations for this
term.
First of all, already in 1984, much time before the interest to Lorentz-CPT breaking
strongly increased, the gravitational anomalies have been discussed in [40], where
the topological current K
μ was introduced, with its explicit form is
23
G
μν
+
1
μ
C
μν
= 0.
(3.5)
It is useful also to write the linearized form of the gravitational Chern–Simons action
obtained from (3.2) under the replacement g μν = η μν + κh μν :
S
(0)
= −
1
2μ
d
3 xh
μν
αμρ ∂
ρ
(η γν − ∂ γ ∂ ν )h
γα
.
(3.6)
We see that this action is, first, explicitly gauge invariant under usual linearized gauge
transformations δh μν = ∂ μ ξ ν + ∂ ν ξ μ , second, involves higher derivatives. However,
after obtaining the equations of motion for the full linearized action formed by the
sum of the terms (1.11) and (3.6), one finds that the physical degrees of freedom
satisfy the second-order equation [38], with their propagator behaves as ( + μ
2
)
−1 ,
thus, in the 3D CS modified gravity there is no problems with negative-energy states
discussed in the previous chapter. The similar situation occurs in the four-dimensional
case as well.
The generalization of this theory to the four-dimensional case turns out to be
straightforward, however, in this case, similarly to the electrodynamics, this generalization essentially involves the CPT (and in certain cases Lorentz) symmetry
breaking. From the formal viewpoint such a generalization for the linearized theory is performed through replacement
μνλ
→ b ρ
ρμνλ , with b ρ is a constant vector,
which allows to convert the CS term to the Carroll-Field-Jackiw (CFJ) term which
in the Abelian case looks like
L C F J =
ρμνλ b ρ A μ ∂ ν A λ .
(3.7)
In principle, such a replacement of the three-dimensional Levi–Civita symbol by
the four-dimensional one contracted with a vector already allows to write down the
four-dimensional gravitational CS term:
L C S,grav =
d
4 x
ρμνλ b ρ
b
μa ∂ ν
a
λb +
2
3
b
μa
c
νb
a
λc
,
(3.8)
with its linearized form is
S
(0)
= −
1
2
d
4 xh
μν
αμρλ b
λ
∂
ρ
(η γν − ∂ γ ∂ ν )h
γα
.
(3.9)
We note that this action is invariant under the same linearized gauge transformations
δh μν = ∂ μ ξ ν + ∂ ν ξ μ . Now, it is very interesting to discuss some motivations for this
term.
First of all, already in 1984, much time before the interest to Lorentz-CPT breaking
strongly increased, the gravitational anomalies have been discussed in [40], where
the topological current K
μ was introduced, with its explicit form is