2.4 Functions of Other Curvature Invariants
17
where the 2n-order Kronecker-like delta symbol is
δ
i 1 ...i 2n
j 1 ... j 2n
=
δ
i 1
j 1
. . . δ
i 1
j 2n
. . . . . . . . .
δ
i 2n
j 1
. . . δ
i 2n
j 2n
.
(2.32)
It is easy to check that at n = 1, we have the scalar curvature, and at n = 2, the
Gauss–Bonnet term. The term with n = 0 is naturally treated as the cosmological
constant. As a result, we can write down the action;
S =
1
κ 2
d
D x
|g|
0≤n
2(n−1)
L n .
(2.33)
Here, zero order is for , first—for R, second—for G. The α n are some numbers,
and λ is a length scale, f.e. Planck length, it is given by κ in D = 4 where the κ
−1
has a dimension of inverse length.
The l.h.s. of the modified Einstein equations looks like [31]
G αβ =
0≤n≤D/2
α n λ
2(n−1) G (n)αβ ;
G
A
(n) β = −
1
2 n+1 δ
αi 1 ...i 2n
β j 1 ... j 2n
R
j 1 j 2
i 1 i 2
. . . R
j 2n−1 j 2n
i 2n−1 i 2n
.
(2.34)
It is clear that G (0)αβ = −
1
2
g αβ , G (1)αβ = G
E H
αβ ≡ R αβ −
1
2
Rg αβ is the usual Einstein tensor. The r.h.s. of the modified Einstein equations is not modified within this
approach, so we have G αβ = κ
2 T αβ .
It turns out to be that although the l.h.s. (2.34) of the modified Einstein equations is
very complicated, these equations admit some exact solutions for an arbitrary spacetime dimension, i.e. for the presence of terms with very high orders in curvatures.
The most interesting cases are the maximally symmetric (anti) de Sitter space and
the FRW cosmological metric.
In the (a)dS space, the Riemann curvature tensor is given by
R αβγδ =
σ
λ 2 (g αγ g βδ − g αδ g βγ ),
(2.35)
with σ is a some number. In this case the vacuum equation yields
0≤n
n
= 0,
with β n =
(D−1)!
(D−2n−1)!
α n , and this equation possesses some roots for σ (in general
complex ones). Each value of σ allows to find the corresponding scalar curvature.
We can solve the modified Einstein equations also for the FRW metric (1.6):
R i0 j0 = −g i j
¨
a
a
; R i jkl =
˙
a
2
+ k
a 2 (g ik g jl − g il g jk ),
(2.36)
17
where the 2n-order Kronecker-like delta symbol is
δ
i 1 ...i 2n
j 1 ... j 2n
=
δ
i 1
j 1
. . . δ
i 1
j 2n
. . . . . . . . .
δ
i 2n
j 1
. . . δ
i 2n
j 2n
.
(2.32)
It is easy to check that at n = 1, we have the scalar curvature, and at n = 2, the
Gauss–Bonnet term. The term with n = 0 is naturally treated as the cosmological
constant. As a result, we can write down the action;
S =
1
κ 2
d
D x
|g|
0≤n
2(n−1)
L n .
(2.33)
Here, zero order is for , first—for R, second—for G. The α n are some numbers,
and λ is a length scale, f.e. Planck length, it is given by κ in D = 4 where the κ
−1
has a dimension of inverse length.
The l.h.s. of the modified Einstein equations looks like [31]
G αβ =
0≤n≤D/2
α n λ
2(n−1) G (n)αβ ;
G
A
(n) β = −
1
2 n+1 δ
αi 1 ...i 2n
β j 1 ... j 2n
R
j 1 j 2
i 1 i 2
. . . R
j 2n−1 j 2n
i 2n−1 i 2n
.
(2.34)
It is clear that G (0)αβ = −
1
2
g αβ , G (1)αβ = G
E H
αβ ≡ R αβ −
1
2
Rg αβ is the usual Einstein tensor. The r.h.s. of the modified Einstein equations is not modified within this
approach, so we have G αβ = κ
2 T αβ .
It turns out to be that although the l.h.s. (2.34) of the modified Einstein equations is
very complicated, these equations admit some exact solutions for an arbitrary spacetime dimension, i.e. for the presence of terms with very high orders in curvatures.
The most interesting cases are the maximally symmetric (anti) de Sitter space and
the FRW cosmological metric.
In the (a)dS space, the Riemann curvature tensor is given by
R αβγδ =
σ
λ 2 (g αγ g βδ − g αδ g βγ ),
(2.35)
with σ is a some number. In this case the vacuum equation yields
0≤n
n
= 0,
with β n =
(D−1)!
(D−2n−1)!
α n , and this equation possesses some roots for σ (in general
complex ones). Each value of σ allows to find the corresponding scalar curvature.
We can solve the modified Einstein equations also for the FRW metric (1.6):
R i0 j0 = −g i j
¨
a
a
; R i jkl =
˙
a
2
+ k
a 2 (g ik g jl − g il g jk ),
(2.36)