24
3 Scalar-Tensor Gravities
K
ρ
= 2
ρμνλ
b
μa ∂ ν
a
λb +
2
3
b
μa
c
νb
a
λc
,
(3.10)
with its divergence is
∂ ρ K
ρ
=
1
2
μναβ R μνγδ R
γδ
αβ
≡
∗ R R.
(3.11)
We note that the 3D gravitational Chern–Simons term, up to overall multiplier, is
equal to the K
3 component, i.e. the component of this current directed along “extra”,
z axis.
It is clear that the integral from (3.11) over the space-time is a surface term. To
include it into the action in a consistent form, one should introduce a new field ϑ
called the CS coefficient. As a result, we can add to the usual Einstein–Hilbert action
the new term proportional to ϑ which we call the CS action S C S :
S C S =
1
2κ 2
d
4 x
−
1
2
v μ K
μ
=
1
2κ 2 I C S ;
S E H+C S =
1
2κ 2
d
4 x
√
−g R +
1
4
ϑ
∗ R R
.
(3.12)
Here, v μ = ∂ μ ϑ is a vector. We note that in principle this vector is rather a function
of space-time coordinates than the constant, hence, in general the gravitational CS
term breaks the CPT symmetry. However, the ϑ can be treated as an external, but
not dynamical, field, therefore one can choose v μ to be the constant vector. This
immediately implies the Lorentz symmetry breaking, therefore in this case the 4D
CS modified gravity whose action is given by the second equation in (3.12) turns out
to be the first example of the gravity model with the Lorentz symmetry breaking.
The equations of motion for the CS modified gravity can be easily obtained.
Varying the CS term I C S defined by the first equation in (3.12), we get
δ I C S =
d
4 x
√ −gC
μν
δg μν ,
(3.13)
with ε
αβγδ
=
αβγδ
√ |g|
is a Levi–Civita tensor (not a simple symbol!), and
C μν = −
1
2
[v σ (ε σμαβ ∇ α R ν
β + ε σναβ ∇ α R
μ
β ) + v στ ( ∗ R τμσν + ∗ R τνσμ )], (3.14)
is the Cotton tensor, and v στ = ∇ σ v τ . One can check that the covariant divergence
of the Cotton tensor is proportional to the invariant
∗ R R:
∇ μ C
μν
=
1
8
v
ν ∗ R R.
(3.15)
3 Scalar-Tensor Gravities
K
ρ
= 2
ρμνλ
b
μa ∂ ν
a
λb +
2
3
b
μa
c
νb
a
λc
,
(3.10)
with its divergence is
∂ ρ K
ρ
=
1
2
μναβ R μνγδ R
γδ
αβ
≡
∗ R R.
(3.11)
We note that the 3D gravitational Chern–Simons term, up to overall multiplier, is
equal to the K
3 component, i.e. the component of this current directed along “extra”,
z axis.
It is clear that the integral from (3.11) over the space-time is a surface term. To
include it into the action in a consistent form, one should introduce a new field ϑ
called the CS coefficient. As a result, we can add to the usual Einstein–Hilbert action
the new term proportional to ϑ which we call the CS action S C S :
S C S =
1
2κ 2
d
4 x
−
1
2
v μ K
μ
=
1
2κ 2 I C S ;
S E H+C S =
1
2κ 2
d
4 x
√
−g R +
1
4
ϑ
∗ R R
.
(3.12)
Here, v μ = ∂ μ ϑ is a vector. We note that in principle this vector is rather a function
of space-time coordinates than the constant, hence, in general the gravitational CS
term breaks the CPT symmetry. However, the ϑ can be treated as an external, but
not dynamical, field, therefore one can choose v μ to be the constant vector. This
immediately implies the Lorentz symmetry breaking, therefore in this case the 4D
CS modified gravity whose action is given by the second equation in (3.12) turns out
to be the first example of the gravity model with the Lorentz symmetry breaking.
The equations of motion for the CS modified gravity can be easily obtained.
Varying the CS term I C S defined by the first equation in (3.12), we get
δ I C S =
d
4 x
√ −gC
μν
δg μν ,
(3.13)
with ε
αβγδ
=
αβγδ
√ |g|
is a Levi–Civita tensor (not a simple symbol!), and
C μν = −
1
2
[v σ (ε σμαβ ∇ α R ν
β + ε σναβ ∇ α R
μ
β ) + v στ ( ∗ R τμσν + ∗ R τνσμ )], (3.14)
is the Cotton tensor, and v στ = ∇ σ v τ . One can check that the covariant divergence
of the Cotton tensor is proportional to the invariant
∗ R R:
∇ μ C
μν
=
1
8
v
ν ∗ R R.
(3.15)