5.2 Basic Definitions
51
These equations are very involved. However, already now we can indicate some
situations where the equations are essentially simplified.
First of all, it is clear that the equations of motion are simplified when the variable
to be found depends only on one argument (examples of such variables are scale
factor and radial function). Second, the case of a diagonal metric simplifies the
system immediately since one has N i = 0 and N =
√
|g 00 |. We note that the static
spherically symmetric metric (f.e. non-rotating black hole) and FRW metric are
diagonal. As for the Gödel metric, it has been considered in a tetrad base which
strongly simplifies calculations (see [97] for details). Now, let us consider examples
of solutions for gravitational field equations.
5.3 Exact Solutions
So, let us consider the exact solutions. Again, as earlier, we consider three examples—
cosmological FRW metric, black hole and Gödel-type metric.
We follow [98]. So, for the cosmological case, one suggests N = N (t), N i = 0
(since the FRW metric is diagonal), and g i j = a
2
(t)γ i j , where γ i j is the maximally symmetric spatial metric yielding constant scalar curvature: R = 6k, and
R i j = 2kγ i j , therefore ∇ i R = 0, and the Cotton tensor is also zero, C i j = 0. The
matter is suggested to be the function of time only, = (t). We can introduce the
new Hubble parameter H =
˙
a
Na
, where a = a(t) is the usual scale factor in (1.6).
It is natural to suggest that the matter is given by a scalar field which, as usual in
cosmology, depends only on time. As a result, the equation of motion for N looks
like:
3α(3λ − 1)H
2
+ σ +
6kξ
a 2 +
12k
2
(ζ + 3η)
a 4
=
˙
2
N 2 + V (().
(5.13)
For g i j , one finds
2α(3λ − 1)
˙
H +
3
2
H
2
+ σ +
2kξ
a 2 −
4k
2
(ζ + 3η)
a 4
=
= −
˙
2
N 2 + V (().
(5.14)
Finally, for a matter the equation is
1
N
∂ t
˙
N
+ 3H
˙
N
+
1
2
V = 0.
(5.15)
One can verify that cyclic or bouncing solutions are possible [99]. In the vacuum
case one can prove directly the possibility of static solutions, while in the presence
of the matter, the solutions can be obtained only numerically [100].
51
These equations are very involved. However, already now we can indicate some
situations where the equations are essentially simplified.
First of all, it is clear that the equations of motion are simplified when the variable
to be found depends only on one argument (examples of such variables are scale
factor and radial function). Second, the case of a diagonal metric simplifies the
system immediately since one has N i = 0 and N =
√
|g 00 |. We note that the static
spherically symmetric metric (f.e. non-rotating black hole) and FRW metric are
diagonal. As for the Gödel metric, it has been considered in a tetrad base which
strongly simplifies calculations (see [97] for details). Now, let us consider examples
of solutions for gravitational field equations.
5.3 Exact Solutions
So, let us consider the exact solutions. Again, as earlier, we consider three examples—
cosmological FRW metric, black hole and Gödel-type metric.
We follow [98]. So, for the cosmological case, one suggests N = N (t), N i = 0
(since the FRW metric is diagonal), and g i j = a
2
(t)γ i j , where γ i j is the maximally symmetric spatial metric yielding constant scalar curvature: R = 6k, and
R i j = 2kγ i j , therefore ∇ i R = 0, and the Cotton tensor is also zero, C i j = 0. The
matter is suggested to be the function of time only, = (t). We can introduce the
new Hubble parameter H =
˙
a
Na
, where a = a(t) is the usual scale factor in (1.6).
It is natural to suggest that the matter is given by a scalar field which, as usual in
cosmology, depends only on time. As a result, the equation of motion for N looks
like:
3α(3λ − 1)H
2
+ σ +
6kξ
a 2 +
12k
2
(ζ + 3η)
a 4
=
˙
2
N 2 + V (().
(5.13)
For g i j , one finds
2α(3λ − 1)
˙
H +
3
2
H
2
+ σ +
2kξ
a 2 −
4k
2
(ζ + 3η)
a 4
=
= −
˙
2
N 2 + V (().
(5.14)
Finally, for a matter the equation is
1
N
∂ t
˙
N
+ 3H
˙
N
+
1
2
V = 0.
(5.15)
One can verify that cyclic or bouncing solutions are possible [99]. In the vacuum
case one can prove directly the possibility of static solutions, while in the presence
of the matter, the solutions can be obtained only numerically [100].