2
1 Einstein Gravity and the Need for Its Modification
dynamical variable in our theory. As usual, the action must be (Riemannian) scalar,
and for the first step, it is assumed to involve no more than second derivatives of the
metric tensor, in a whole analogy with other field theory models where the action
involves just up to second derivatives. The unique scalar involving only second derivatives is a scalar curvature R (we follow the definitions from the book [2] except of
special cases):
R = g
μν R μν ; R μν = R
α
μαν ;
R
κ
λμν = ∂ μ
κ
λν − ∂ μ
κ
λν +
κ
ρμ
ρ
λν −
κ
ρν
ρ
λμ ,
(1.1)
where
μ
νλ are the Christoffel symbols, that is, affine connections expressed in terms
of the metric tensor as
μ
νλ =
1
2
g
μρ
(∂ ν g ρλ + ∂ λ g ρν − ∂ ρ g νλ ).
(1.2)
The Einstein–Hilbert action is obtained as an integral from the scalar curvature over
the D-dimensional space-time:
S =
d
D x
|g|
1
2κ 2 R + L m
,
(1.3)
where g is the determinant of the metric. We assume the signature to be (+ − −−).
The κ
2
= 8πG is the gravitational constant (it is important to note that its mass
dimension in D-dimensional space-time is equal to 2 − D); nevertheless, in some
cases we will define it to be equal to 1. The L m is the matter Lagrangian.
Varying the action with respect to the metric tensor, we obtain the Einstein equations:
G μν ≡ R μν −
1
2
Rg μν = κ
2 T μν ,
(1.4)
where T μν is the energy-momentum tensor of the matter. The conservation of the
energy-momentum tensor presented as ∇ μ T
μν
= 0 is clearly consistent with the
Bianchi identities ∇ μ G
μν
= 0.
Among the most important solutions of these equations, one should emphasize the
Schwarzschild metric (arising for the vacuum, T μν = 0) which describes the simplest
black hole with mass m, looking like
ds
2
=
1 −
2m
r
c
2 dt
2
−
1 −
2m
r
−1
dr
2
− r
2
(dθ
2
+ sin
2
θdφ
2
) (1.5)
(actually, in many cases we will consider a more generic spherically symmetric static
metric (3.20)), and the Friedmann–Robertson–Walker (FRW) metric corresponding
to the simplest (homogeneous and isotropic) cosmological solution:
1 Einstein Gravity and the Need for Its Modification
dynamical variable in our theory. As usual, the action must be (Riemannian) scalar,
and for the first step, it is assumed to involve no more than second derivatives of the
metric tensor, in a whole analogy with other field theory models where the action
involves just up to second derivatives. The unique scalar involving only second derivatives is a scalar curvature R (we follow the definitions from the book [2] except of
special cases):
R = g
μν R μν ; R μν = R
α
μαν ;
R
κ
λμν = ∂ μ
κ
λν − ∂ μ
κ
λν +
κ
ρμ
ρ
λν −
κ
ρν
ρ
λμ ,
(1.1)
where
μ
νλ are the Christoffel symbols, that is, affine connections expressed in terms
of the metric tensor as
μ
νλ =
1
2
g
μρ
(∂ ν g ρλ + ∂ λ g ρν − ∂ ρ g νλ ).
(1.2)
The Einstein–Hilbert action is obtained as an integral from the scalar curvature over
the D-dimensional space-time:
S =
d
D x
|g|
1
2κ 2 R + L m
,
(1.3)
where g is the determinant of the metric. We assume the signature to be (+ − −−).
The κ
2
= 8πG is the gravitational constant (it is important to note that its mass
dimension in D-dimensional space-time is equal to 2 − D); nevertheless, in some
cases we will define it to be equal to 1. The L m is the matter Lagrangian.
Varying the action with respect to the metric tensor, we obtain the Einstein equations:
G μν ≡ R μν −
1
2
Rg μν = κ
2 T μν ,
(1.4)
where T μν is the energy-momentum tensor of the matter. The conservation of the
energy-momentum tensor presented as ∇ μ T
μν
= 0 is clearly consistent with the
Bianchi identities ∇ μ G
μν
= 0.
Among the most important solutions of these equations, one should emphasize the
Schwarzschild metric (arising for the vacuum, T μν = 0) which describes the simplest
black hole with mass m, looking like
ds
2
=
1 −
2m
r
c
2 dt
2
−
1 −
2m
r
−1
dr
2
− r
2
(dθ
2
+ sin
2
θdφ
2
) (1.5)
(actually, in many cases we will consider a more generic spherically symmetric static
metric (3.20)), and the Friedmann–Robertson–Walker (FRW) metric corresponding
to the simplest (homogeneous and isotropic) cosmological solution: