2.4 Functions of Other Curvature Invariants
19
simplified situation G = const = ±120b
2 . Explicitly, for positive B(= G/120) =
b
2 , the solution is
y + C =
4
5
d A
(A
)
2
b 2 − (A ) 4 ,
(2.41)
and for the negative B = −b
2 —in the form
y + C = −
4
5
d A
(A
)
2
b 2 + (A ) 4 .
(2.42)
In principle, there are more situations when the modified Einstein equations in the
Gauss–Bonnet gravity can be solved. We note that the braneworld solutions could
be found not only for Gauss–Bonnet gravity but for other gravity models including
the already discussed f (R) gravity (see f.e. [35] and references therein), however,
we do not discuss the details of braneworld solutions here because of the restricted
volume of this review.
2.5 Conclusions
We discussed various extensions of Einstein gravity characterized by modifications in
the purely gravitational sector. These modifications are based on adding new scalars
representing themselves not only as various degrees of the scalar curvature but also
as functions of higher order curvature invariants. We explicitly demonstrated that
the R
2 gravity is all-loop renormalizable, and that the most important solutions of
general relativity, such as cosmological FRW metric and Gödel metric continue to
be solutions within modified gravity. Moreover, we showed that modifications of the
pure gravitational sector allow for accelerated cosmological expansion being thus
examples of reasonable solutions for the dark energy problem, so that the problem
of choosing a better modification of the gravity apparently can be solved in principle
while the problem of choice for the most adequate modification of the gravity is
actually more observational and experimental than theoretical.
Within this section we presented several other interesting results. First, we
described the argumentation allowing for establishing the equivalence between modifications in the pure gravitational sector and adjusting the action of the extra scalar
field coupled to gravity, which implies that the f (R) gravity is equivalent to the
scalar-tensor gravity with an appropriate potential. Second, we discussed the 1/R
terms whose form seems to be highly controversial since the observed curvature of
the space-time is very small hence these terms are very large. Third, we considered possible generalizations of the gravity consistent within the extra dimensions
concept.
Further development of a general gravity model consists in the idea that for the
complete description of the gravity it is not sufficient to study only the metric, so that
19
simplified situation G = const = ±120b
2 . Explicitly, for positive B(= G/120) =
b
2 , the solution is
y + C =
4
5
d A
(A
)
2
b 2 − (A ) 4 ,
(2.41)
and for the negative B = −b
2 —in the form
y + C = −
4
5
d A
(A
)
2
b 2 + (A ) 4 .
(2.42)
In principle, there are more situations when the modified Einstein equations in the
Gauss–Bonnet gravity can be solved. We note that the braneworld solutions could
be found not only for Gauss–Bonnet gravity but for other gravity models including
the already discussed f (R) gravity (see f.e. [35] and references therein), however,
we do not discuss the details of braneworld solutions here because of the restricted
volume of this review.
2.5 Conclusions
We discussed various extensions of Einstein gravity characterized by modifications in
the purely gravitational sector. These modifications are based on adding new scalars
representing themselves not only as various degrees of the scalar curvature but also
as functions of higher order curvature invariants. We explicitly demonstrated that
the R
2 gravity is all-loop renormalizable, and that the most important solutions of
general relativity, such as cosmological FRW metric and Gödel metric continue to
be solutions within modified gravity. Moreover, we showed that modifications of the
pure gravitational sector allow for accelerated cosmological expansion being thus
examples of reasonable solutions for the dark energy problem, so that the problem
of choosing a better modification of the gravity apparently can be solved in principle
while the problem of choice for the most adequate modification of the gravity is
actually more observational and experimental than theoretical.
Within this section we presented several other interesting results. First, we
described the argumentation allowing for establishing the equivalence between modifications in the pure gravitational sector and adjusting the action of the extra scalar
field coupled to gravity, which implies that the f (R) gravity is equivalent to the
scalar-tensor gravity with an appropriate potential. Second, we discussed the 1/R
terms whose form seems to be highly controversial since the observed curvature of
the space-time is very small hence these terms are very large. Third, we considered possible generalizations of the gravity consistent within the extra dimensions
concept.
Further development of a general gravity model consists in the idea that for the
complete description of the gravity it is not sufficient to study only the metric, so that