1 Einstein Gravity and the Need for Its Modification
3
ds
2
= c
2 dt
2
− a
2
(t)
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θdφ
2
)
,
(1.6)
where a(t) is the scale factor, and k = 1, 0, −1 for positive, zero and negative curvature respectively. The matter in this case is given by the relativistic fluid:
κ
2 T μν = (ρ + p)v μ v ν + pg μν ,
(1.7)
where ρ is a density of the matter, and p is its pressure, in many case one employs the
equation of state p = ωρ, with ω is a constant characterizing the kind of the matter.
Besides of these solutions, an important example is represented also by the Gödel
solution [3]:
ds
2
= a
2
(dt + e
x dy)
2
− dx
2
−
1
2
e
2x dy
2
− dz
2
,
(1.8)
which, just as the FRW metric, arises if the matter is given by the fluid-like form:
κ
2 T μν = κ
2
ρv μ v ν + g μν ,
(1.9)
but in this case one has v
μ
=
1
a
, ρ =
1
a 2 , and = −
1
2a 2 . Namely these solutions and
their direct generalizations will be considered within our course.
Now, let us make some introduction to quantum gravity. Indeed, it is natural to
expect that the gravity, in a whole analogy with electrodynamics and other field
theories, must be quantized. To do it, one can follow the approach developed by ’t
Hooft and Veltman [4]. We start with splitting of the dynamic metric g μν into a sum
of the background part ¯
g μν and the quantum fluctuation h μν :
g μν = ¯
g μν + κh μν ,
(1.10)
where the κ is introduced to change dimension of h μν to 1. As a result, the action
can be expanded in infinite power series in h μν . For the first step, we can choose
¯
g μν = η μν . The lowest, quadratic contribution to the Lagrangian of h μν is
L 0 =
1
4
∂ μ h
α
α ∂
μ h
β
β −
1
2
∂ β h
α
α ∂
μ h
β
μ −
1
4
∂ μ h αβ ∂
μ h
αβ
+
1
2
∂ α h νβ ∂
ν h
αβ
, (1.11)
where the indices of h αβ are raised and lowered with the flat Minkowski metric. The
Lagrangian (1.11) is called the Fierz–Pauli Lagrangian, it is used within constructing
of some generalizations of gravity.
The corresponding (second-order) equations of motion are actually the linearized
Einstein equations:
3
ds
2
= c
2 dt
2
− a
2
(t)
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θdφ
2
)
,
(1.6)
where a(t) is the scale factor, and k = 1, 0, −1 for positive, zero and negative curvature respectively. The matter in this case is given by the relativistic fluid:
κ
2 T μν = (ρ + p)v μ v ν + pg μν ,
(1.7)
where ρ is a density of the matter, and p is its pressure, in many case one employs the
equation of state p = ωρ, with ω is a constant characterizing the kind of the matter.
Besides of these solutions, an important example is represented also by the Gödel
solution [3]:
ds
2
= a
2
(dt + e
x dy)
2
− dx
2
−
1
2
e
2x dy
2
− dz
2
,
(1.8)
which, just as the FRW metric, arises if the matter is given by the fluid-like form:
κ
2 T μν = κ
2
ρv μ v ν + g μν ,
(1.9)
but in this case one has v
μ
=
1
a
, ρ =
1
a 2 , and = −
1
2a 2 . Namely these solutions and
their direct generalizations will be considered within our course.
Now, let us make some introduction to quantum gravity. Indeed, it is natural to
expect that the gravity, in a whole analogy with electrodynamics and other field
theories, must be quantized. To do it, one can follow the approach developed by ’t
Hooft and Veltman [4]. We start with splitting of the dynamic metric g μν into a sum
of the background part ¯
g μν and the quantum fluctuation h μν :
g μν = ¯
g μν + κh μν ,
(1.10)
where the κ is introduced to change dimension of h μν to 1. As a result, the action
can be expanded in infinite power series in h μν . For the first step, we can choose
¯
g μν = η μν . The lowest, quadratic contribution to the Lagrangian of h μν is
L 0 =
1
4
∂ μ h
α
α ∂
μ h
β
β −
1
2
∂ β h
α
α ∂
μ h
β
μ −
1
4
∂ μ h αβ ∂
μ h
αβ
+
1
2
∂ α h νβ ∂
ν h
αβ
, (1.11)
where the indices of h αβ are raised and lowered with the flat Minkowski metric. The
Lagrangian (1.11) is called the Fierz–Pauli Lagrangian, it is used within constructing
of some generalizations of gravity.
The corresponding (second-order) equations of motion are actually the linearized
Einstein equations: