Chapter 2
Modifications of the Pure Gravitational
Sector
2.1 Motivations
As we already noted in the Introduction, one of the ways to modify gravity consists
in introducing additional terms to the gravitational sector. Such terms are given by
scalars constructed on the base of the metric tensor, i.e. these scalars are functions
of the Riemann tensor, the Ricci tensor, possibly, their covariant derivatives, and the
scalar curvature. In the simplest case the Lagrangian is the function of the scalar
curvature only, so, the action is
S =
1
16πG
d
4 x
|g| f (R),
(2.1)
where, f (R) is a some function of the scalar curvature. Since the Einstein gravity is
very well observationally confirmed, and the curvature of the Universe is known to be
small, it is natural to suggest that f (R) = R + γ R
n , with n ≥ 2, so, Einstein–Hilbert
term dominates. The case n = 2 is very interesting by various reasons, from renormalizability to possibility of cosmic acceleration, so, it will be discussed in details.
However, other values of n, including even negative ones which called attention
recently, are also interesting. Another generalization of this action is the suggestion
that the Lagrangian depends also on invariants Q = R μν R
μν and P = R μναβ R
μναβ ,
such class of theories is called f (R, Q, P) gravity, the paradigmatic example is the
Weyl gravity (see f.e. [9] and references therein), where the Lagrangian is given by
the square of the Weyl tensor. Besides of these situations, it is interesting also to
abandon the restriction for the space-time to be four-dimensional. In this context we
will consider also higher-dimensional space-times and discuss Lovelock gravities
whose action involves higher curvature invariants.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_2
7
Modifications of the Pure Gravitational
Sector
2.1 Motivations
As we already noted in the Introduction, one of the ways to modify gravity consists
in introducing additional terms to the gravitational sector. Such terms are given by
scalars constructed on the base of the metric tensor, i.e. these scalars are functions
of the Riemann tensor, the Ricci tensor, possibly, their covariant derivatives, and the
scalar curvature. In the simplest case the Lagrangian is the function of the scalar
curvature only, so, the action is
S =
1
16πG
d
4 x
|g| f (R),
(2.1)
where, f (R) is a some function of the scalar curvature. Since the Einstein gravity is
very well observationally confirmed, and the curvature of the Universe is known to be
small, it is natural to suggest that f (R) = R + γ R
n , with n ≥ 2, so, Einstein–Hilbert
term dominates. The case n = 2 is very interesting by various reasons, from renormalizability to possibility of cosmic acceleration, so, it will be discussed in details.
However, other values of n, including even negative ones which called attention
recently, are also interesting. Another generalization of this action is the suggestion
that the Lagrangian depends also on invariants Q = R μν R
μν and P = R μναβ R
μναβ ,
such class of theories is called f (R, Q, P) gravity, the paradigmatic example is the
Weyl gravity (see f.e. [9] and references therein), where the Lagrangian is given by
the square of the Weyl tensor. Besides of these situations, it is interesting also to
abandon the restriction for the space-time to be four-dimensional. In this context we
will consider also higher-dimensional space-times and discuss Lovelock gravities
whose action involves higher curvature invariants.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_2
7