4
1 Einstein Gravity and the Need for Its Modification
G
(0)
μν ≡ −
1
2
(∂
λ
∂ μ h λν + ∂
λ
∂ ν h λμ ) +
1
2
h μν +
1
2
η μν ∂ α ∂ β h
αβ
−
−
1
2
η μν h
λ
λ +
1
2
∂ μ ∂ ν h
λ
λ = 0.
(1.12)
We have linearized gauge symmetry δh μν = ∂ μ ξ ν + ∂ ν ξ μ in the l.h.s., and linearized
Bianchi identities ∂
μ G
(0)
μν = 0. As a consequence, afterwards one must fix the gauge,
which can be done by adding the term
L G F = −
1
2
C μ C
μ
,
(1.13)
where C μ = ∂
α h αμ −
1
2
∂ μ h
α
α , so one has a new Lagrangian
L = L 0 −
1
2
C μ C
μ
= −
1
4
∂ μ h αβ ∂
μ h
αβ
+
1
8
∂ μ h
α
α ∂
μ h
β
β ,
(1.14)
which can be rewritten as
L = −
1
2
∂
λ h αβ V
αβμν
∂ λ h μν ,
(1.15)
where V
αβμν
=
1
2
η
αμ
η
βν
−
1
4
η
αβ
η
μν , which implies the following propagator in the
momentum space:
< h αβ (−k)h μν (k) >= i
η μα η νβ + η να η μβ −
2
D−2
η μν η αβ
k 2 − i
,
(1.16)
where D is the space-time dimension (the singularity at D = 2 is related with the
fact that the D = 2 Einstein–Hilbert action is a pure surface term).
Now, let us expand the Einstein–Hilbert action (1.3) in series in h μν by making
again the substitution (1.10) but with the arbitrary background ¯
g μν . In this case we
see that the metric determinant and curvature scalar are expanded up to the second
order in h as
|g| →
¯
|g|
1 +
1
2
h
α
α −
1
4
h
β
α h
β
α +
1
8
(h
α
α )
2 + · · ·
;
(1.17)
R → R + h
β
β − ∇
α ∇
β h αβ − R
αβ h αβ −
1
2
∇ α (h
β
μ h
μ,α
β ) +
1
2
∇ β [h
β
ν (2h
να
,α − h
α,ν
α )] +
+
1
4
(h
ν
β,α + h
ν
α,β − h
,ν
αβ )(h
β,α
ν + h
βα
,ν − h
α,β
ν ) −
−
1
4
(2h
να
,α − h
α,ν
α )h
β
β,ν −
1
2
h
να h
β
β,να +
1
2
h
ν
α ∇ β (h
β,α
ν + h
βα
,ν − h
α,β
ν ) + h
ν
β h
β
α R
α
ν .
where h
μ,α
β ≡ ∇
α h
μ
β , etc., and the covariant derivative is constructed on the base of
the background metric. This expression is sufficient for the one-loop calculations
1 Einstein Gravity and the Need for Its Modification
G
(0)
μν ≡ −
1
2
(∂
λ
∂ μ h λν + ∂
λ
∂ ν h λμ ) +
1
2
h μν +
1
2
η μν ∂ α ∂ β h
αβ
−
−
1
2
η μν h
λ
λ +
1
2
∂ μ ∂ ν h
λ
λ = 0.
(1.12)
We have linearized gauge symmetry δh μν = ∂ μ ξ ν + ∂ ν ξ μ in the l.h.s., and linearized
Bianchi identities ∂
μ G
(0)
μν = 0. As a consequence, afterwards one must fix the gauge,
which can be done by adding the term
L G F = −
1
2
C μ C
μ
,
(1.13)
where C μ = ∂
α h αμ −
1
2
∂ μ h
α
α , so one has a new Lagrangian
L = L 0 −
1
2
C μ C
μ
= −
1
4
∂ μ h αβ ∂
μ h
αβ
+
1
8
∂ μ h
α
α ∂
μ h
β
β ,
(1.14)
which can be rewritten as
L = −
1
2
∂
λ h αβ V
αβμν
∂ λ h μν ,
(1.15)
where V
αβμν
=
1
2
η
αμ
η
βν
−
1
4
η
αβ
η
μν , which implies the following propagator in the
momentum space:
< h αβ (−k)h μν (k) >= i
η μα η νβ + η να η μβ −
2
D−2
η μν η αβ
k 2 − i
,
(1.16)
where D is the space-time dimension (the singularity at D = 2 is related with the
fact that the D = 2 Einstein–Hilbert action is a pure surface term).
Now, let us expand the Einstein–Hilbert action (1.3) in series in h μν by making
again the substitution (1.10) but with the arbitrary background ¯
g μν . In this case we
see that the metric determinant and curvature scalar are expanded up to the second
order in h as
|g| →
¯
|g|
1 +
1
2
h
α
α −
1
4
h
β
α h
β
α +
1
8
(h
α
α )
2 + · · ·
;
(1.17)
R → R + h
β
β − ∇
α ∇
β h αβ − R
αβ h αβ −
1
2
∇ α (h
β
μ h
μ,α
β ) +
1
2
∇ β [h
β
ν (2h
να
,α − h
α,ν
α )] +
+
1
4
(h
ν
β,α + h
ν
α,β − h
,ν
αβ )(h
β,α
ν + h
βα
,ν − h
α,β
ν ) −
−
1
4
(2h
να
,α − h
α,ν
α )h
β
β,ν −
1
2
h
να h
β
β,να +
1
2
h
ν
α ∇ β (h
β,α
ν + h
βα
,ν − h
α,β
ν ) + h
ν
β h
β
α R
α
ν .
where h
μ,α
β ≡ ∇
α h
μ
β , etc., and the covariant derivative is constructed on the base of
the background metric. This expression is sufficient for the one-loop calculations