2.3 f (R)-Gravity
15
Therefore, the f (R) gravity turns out to be equivalent to the general relativity with
the extra scalar, i.e. to the scalar-tensor gravity. The form of the potential is therefore
related with the form of the function f (R).
Clearly, the natural question is about possibility to obtain other important gravitational solutions within the f (R) gravity context. First, for the Gödel metric (1.8),
as well as for its straightforward generalization defined in [22] as Gödel-type metric:
ds
2
= (dt + H (r )dφ)
2
− D
2
(r )dφ
2
− dr
2
− dz
2
,
(2.26)
where
H
D
= 2ω,
D
D
= m
2
,
(2.27)
with ω, m are constants, the scalar curvature is constant, hence the Eq. (2.15) are simplified drastically since the term involving covariant derivatives of f (R) goes away,
and the l.h.s. of these equations turns out to be a mere combination of constants. It
was shown in [22] that both causal and non-causal solutions are possible, with f (R)
is an arbitrary function of the scalar curvature, while to achieve causality, it is not
sufficient to have only a relativistic fluid as in [3], and one must add as well a scalar
matter—one should remind that since the Einstein equations are nonlinear, the solution generated by a sum of two sources is not equal to the sum of solutions generated
by each source. As for the black holes, we strongly recommend the excellent book
[23] where Schwarzschild-type BH solutions in f (R) gravity are considered, see
also [24] and references therein.
In [23], a wide spectrum of possible generalizations of f (R) gravity was discussed, such as f (R, L m ) and f (R, T ) models, where L m is the matter Lagrangian,
and T is the trace of the energy-momentum tensor. However, within our study we
will pursue another aim—we will suggest that the matter is coupled to the gravity
in the usual form while the free gravity action depends on other scalars constructed
on the base of the Riemann tensor and metric. This will be the subject of the next
section.
2.4 Functions of Other Curvature Invariants
Let us suggest that instead of the function of the scalar curvature only, we have also
functions of other scalars. There are many examples of studies of such models, so we
discuss only some most interesting ones, the f (R, Q) gravity, the Lovelock gravity
and the Gauss–Bonnet gravity.
We start our discussion from the f (R, Q) gravity. In this theory, the Lagrangian
is a function not only of the scalar curvature, but also of Q = R μν R
μν , so,
S =
d
4 x
|g| f (R, Q) + S m .
(2.28)
15
Therefore, the f (R) gravity turns out to be equivalent to the general relativity with
the extra scalar, i.e. to the scalar-tensor gravity. The form of the potential is therefore
related with the form of the function f (R).
Clearly, the natural question is about possibility to obtain other important gravitational solutions within the f (R) gravity context. First, for the Gödel metric (1.8),
as well as for its straightforward generalization defined in [22] as Gödel-type metric:
ds
2
= (dt + H (r )dφ)
2
− D
2
(r )dφ
2
− dr
2
− dz
2
,
(2.26)
where
H
D
= 2ω,
D
D
= m
2
,
(2.27)
with ω, m are constants, the scalar curvature is constant, hence the Eq. (2.15) are simplified drastically since the term involving covariant derivatives of f (R) goes away,
and the l.h.s. of these equations turns out to be a mere combination of constants. It
was shown in [22] that both causal and non-causal solutions are possible, with f (R)
is an arbitrary function of the scalar curvature, while to achieve causality, it is not
sufficient to have only a relativistic fluid as in [3], and one must add as well a scalar
matter—one should remind that since the Einstein equations are nonlinear, the solution generated by a sum of two sources is not equal to the sum of solutions generated
by each source. As for the black holes, we strongly recommend the excellent book
[23] where Schwarzschild-type BH solutions in f (R) gravity are considered, see
also [24] and references therein.
In [23], a wide spectrum of possible generalizations of f (R) gravity was discussed, such as f (R, L m ) and f (R, T ) models, where L m is the matter Lagrangian,
and T is the trace of the energy-momentum tensor. However, within our study we
will pursue another aim—we will suggest that the matter is coupled to the gravity
in the usual form while the free gravity action depends on other scalars constructed
on the base of the Riemann tensor and metric. This will be the subject of the next
section.
2.4 Functions of Other Curvature Invariants
Let us suggest that instead of the function of the scalar curvature only, we have also
functions of other scalars. There are many examples of studies of such models, so we
discuss only some most interesting ones, the f (R, Q) gravity, the Lovelock gravity
and the Gauss–Bonnet gravity.
We start our discussion from the f (R, Q) gravity. In this theory, the Lagrangian
is a function not only of the scalar curvature, but also of Q = R μν R
μν , so,
S =
d
4 x
|g| f (R, Q) + S m .
(2.28)