3.2 Chern–Simons Modified Gravity
25
This divergence plays the crucial role when the modified Einstein equations are
considered. Their explicit form is
G
μν
+ C
μν
= κ
2 T
μν
,
(3.16)
so, due to the Eq. (3.15), we find that the conservation of the energy-momentum tensor
requires the vanishing of the divergence of the Cotton tensor, which, according to
(3.15), yields an additional consistency condition called the Pontryagin constraint:
∗ R R = 0,
(3.17)
which must be checked for any solution. However, since in many cases, including,
among others, the rotational symmetry, the curvature tensor has the structure R [ab][ab] ,
i.e. its only non-zero components are R 0101 , R 0202 , . . ., this consistency condition will
be automatically satisfied in these cases.
The further extension of the Chern–Simons modified gravity (CSMG) was carried
out through assuming the nontrivial dynamics for the ϑ CS coefficient. The key idea
is as follows [41]: we assume that the action of CSMG includes the kinetic term for
ϑ, looking like
S =
1
2κ 2
d
4 x
|g|
R +
1
2
∇
m
ϑ∇ m ϑ − V (ϑ) −
1
α
ϑ
∗ R R
,
(3.18)
with now
∗ R R ≡
1
2
ε
μναβ R μνγδ R
γδ
αβ , i.e. it is redefined with the Levi–Civita tensor
ε
μνλρ
=
μνλρ
√ |g|
, and instead of the Pontryagin constraint (3.17), one has the equation
of motion for ϑ:
∗ R R = −α
ϑ +
∂V
∂ϑ
.
(3.19)
If we have a metric consistent within the non-dynamical CS framework with a specific
ϑ, it is consistent in the dynamical case if the r.h.s. of this equation is zero. Then, the
ϑ field generates the additional contribution to the energy-momentum tensor T
μν ,
and hence, to the r.h.s. of (3.16).
Now, we present, first, some classical solutions for the CS modified gravity, second, the methodology allowing the gravitational CS term as a quantum correction.
3.2.2 Classical Solutions
So, our task will consist in solving the Eq. (3.16) with the additional condition (3.17).
As a first example, we consider a static spherically symmetric metric [39]:
ds
2
= N
2
(r )dt
2
− A
2
(r )dr
2
− r
2 d
2
.
(3.20)
25
This divergence plays the crucial role when the modified Einstein equations are
considered. Their explicit form is
G
μν
+ C
μν
= κ
2 T
μν
,
(3.16)
so, due to the Eq. (3.15), we find that the conservation of the energy-momentum tensor
requires the vanishing of the divergence of the Cotton tensor, which, according to
(3.15), yields an additional consistency condition called the Pontryagin constraint:
∗ R R = 0,
(3.17)
which must be checked for any solution. However, since in many cases, including,
among others, the rotational symmetry, the curvature tensor has the structure R [ab][ab] ,
i.e. its only non-zero components are R 0101 , R 0202 , . . ., this consistency condition will
be automatically satisfied in these cases.
The further extension of the Chern–Simons modified gravity (CSMG) was carried
out through assuming the nontrivial dynamics for the ϑ CS coefficient. The key idea
is as follows [41]: we assume that the action of CSMG includes the kinetic term for
ϑ, looking like
S =
1
2κ 2
d
4 x
|g|
R +
1
2
∇
m
ϑ∇ m ϑ − V (ϑ) −
1
α
ϑ
∗ R R
,
(3.18)
with now
∗ R R ≡
1
2
ε
μναβ R μνγδ R
γδ
αβ , i.e. it is redefined with the Levi–Civita tensor
ε
μνλρ
=
μνλρ
√ |g|
, and instead of the Pontryagin constraint (3.17), one has the equation
of motion for ϑ:
∗ R R = −α
ϑ +
∂V
∂ϑ
.
(3.19)
If we have a metric consistent within the non-dynamical CS framework with a specific
ϑ, it is consistent in the dynamical case if the r.h.s. of this equation is zero. Then, the
ϑ field generates the additional contribution to the energy-momentum tensor T
μν ,
and hence, to the r.h.s. of (3.16).
Now, we present, first, some classical solutions for the CS modified gravity, second, the methodology allowing the gravitational CS term as a quantum correction.
3.2.2 Classical Solutions
So, our task will consist in solving the Eq. (3.16) with the additional condition (3.17).
As a first example, we consider a static spherically symmetric metric [39]:
ds
2
= N
2
(r )dt
2
− A
2
(r )dr
2
− r
2 d
2
.
(3.20)