16
2 Modifications of the Pure Gravitational Sector
The equations of motion are found to look like [25, 26]
f R R μν −
f
2
g μν + 2 f Q R
β
(μ R ν)β + g μν f R − ∇ (μ ∇ ν) f R +
+
f Q R μν
− 2∇ λ
∇ (μ
f Q R
λ
ν)
+ g μν ∇ α ∇ σ
f Q R
ασ
= κ
2 T
m
μν , (2.29)
where f Q =
∂ f
∂ Q
, f R =
∂ f
∂ R
, and T
m
μν is the energy-momentum tensor of the matter.
As an example, we consider the Gödel-type metric (2.26). One can show, that,
unlike general relativity, such solutions are possible not only for dust but also for
the vacuum (with non-zero cosmological constant), in particular, completely causal
vacuum solutions are present [26]. Clearly, the solutions of this form are possible also
for the presence of the matter given by the relativistic fluid and a scalar field. Again, as
in [22], all Einstein equations will take the form of purely algebraic relations between
density, pressure, field amplitude and constants from the gravity Lagrangian. As for
the cosmological metric, the possibility of accelerating solutions can be shown just
in the same manner as in the previous sections. Among other possible solutions in
f (R, Q) gravity, it is worth to mention Reissner–Nordström black holes [27] and
wormholes [28]. Further generalization of this theory would consist in consideration
of function not only of R and Q, but also of P = R μναβ R
μναβ , with study of the
corresponding theory called f (R, Q, P) gravity is in principle not more difficult,
see f.e. [29].
Now, let us make the next step—suggest that the dimension of the space-time is
not restricted to be four but can be arbitrary. This step allows us to introduce the
Lovelock gravity. Its key idea is as follows.
Let us consider the gravity model defined in the space-time of an arbitrary dimension [30], called the Lovelock gravity:
S =
d
D x
|g|(c 0 + c 1 R + c 2 G + · · · ).
(2.30)
Here c 0 , c 1 , c 2 , . . . are some constants possessing nontrivial dimensions. It is natural
to suggest that they, up to some dimensionless numbers, are given by various degrees
of the gravitational constant. Each term with 2n derivatives is topological, i.e. it represents itself as a total derivative at D = 2n, and identical zero in minor dimensions.
We note that there is no higher derivatives of the metric in the action. This action is
characterized the following properties displayed by the Einstein–Hilbert action: (i)
the tensor A αβ , the l.h.s. of the corresponding equations of motion, is symmetric; (ii)
the covariant divergence of A αβ vanishes; (iii) the A αβ is linear in second derivatives
of the metric.
The general form of the term with 2n derivatives in the Lagrangian contributing
to (2.30) can be presented as [31]:
L n =
1
2 n δ
i 1 ...i 2n
j 1 ... j 2n
R
j 1 j 2
i 1 i 2
. . . R
j 2n−1 j 2n
i 2n−1 i 2n
,
(2.31)
2 Modifications of the Pure Gravitational Sector
The equations of motion are found to look like [25, 26]
f R R μν −
f
2
g μν + 2 f Q R
β
(μ R ν)β + g μν f R − ∇ (μ ∇ ν) f R +
+
f Q R μν
− 2∇ λ
∇ (μ
f Q R
λ
ν)
+ g μν ∇ α ∇ σ
f Q R
ασ
= κ
2 T
m
μν , (2.29)
where f Q =
∂ f
∂ Q
, f R =
∂ f
∂ R
, and T
m
μν is the energy-momentum tensor of the matter.
As an example, we consider the Gödel-type metric (2.26). One can show, that,
unlike general relativity, such solutions are possible not only for dust but also for
the vacuum (with non-zero cosmological constant), in particular, completely causal
vacuum solutions are present [26]. Clearly, the solutions of this form are possible also
for the presence of the matter given by the relativistic fluid and a scalar field. Again, as
in [22], all Einstein equations will take the form of purely algebraic relations between
density, pressure, field amplitude and constants from the gravity Lagrangian. As for
the cosmological metric, the possibility of accelerating solutions can be shown just
in the same manner as in the previous sections. Among other possible solutions in
f (R, Q) gravity, it is worth to mention Reissner–Nordström black holes [27] and
wormholes [28]. Further generalization of this theory would consist in consideration
of function not only of R and Q, but also of P = R μναβ R
μναβ , with study of the
corresponding theory called f (R, Q, P) gravity is in principle not more difficult,
see f.e. [29].
Now, let us make the next step—suggest that the dimension of the space-time is
not restricted to be four but can be arbitrary. This step allows us to introduce the
Lovelock gravity. Its key idea is as follows.
Let us consider the gravity model defined in the space-time of an arbitrary dimension [30], called the Lovelock gravity:
S =
d
D x
|g|(c 0 + c 1 R + c 2 G + · · · ).
(2.30)
Here c 0 , c 1 , c 2 , . . . are some constants possessing nontrivial dimensions. It is natural
to suggest that they, up to some dimensionless numbers, are given by various degrees
of the gravitational constant. Each term with 2n derivatives is topological, i.e. it represents itself as a total derivative at D = 2n, and identical zero in minor dimensions.
We note that there is no higher derivatives of the metric in the action. This action is
characterized the following properties displayed by the Einstein–Hilbert action: (i)
the tensor A αβ , the l.h.s. of the corresponding equations of motion, is symmetric; (ii)
the covariant divergence of A αβ vanishes; (iii) the A αβ is linear in second derivatives
of the metric.
The general form of the term with 2n derivatives in the Lagrangian contributing
to (2.30) can be presented as [31]:
L n =
1
2 n δ
i 1 ...i 2n
j 1 ... j 2n
R
j 1 j 2
i 1 i 2
. . . R
j 2n−1 j 2n
i 2n−1 i 2n
,
(2.31)