Non-constant volume exponential solutions in higher-dimensional Lovelock cosmologies

0.26 MB·English

by  Dmitry Chirkov

Non-constant volume exponential solutions in higher-dimensional Lovelock cosmologies
Dmitry Chirkov
Sternberg Astronomical Institute, Moscow State University, Moscow 119991 Russia and
Faculty of Physics, Moscow State University, Moscow 119991 Russia
Sergey A. Pavluchenko
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile
Alexey Toporensky
5
1 Sternberg Astronomical Institute, Moscow State University, Moscow 119991 Russia and
0
Kazan Federal University, Kazan 420008, Russia
2

n
a In this paper we propose a scheme which allows one to find all possible exponential solu-
J
tionsofspecialclass–non-constantvolumesolutions–inLovelockgravityinarbitrarynum-

8
berofdimensionsandwitharbitratecombinationsofLovelockterms. We applythis scheme
1
to (6+1)- and(7+1)-dimensionalflat anisotropiccosmologiesin Einstein-Gauss-Bonnetand

]
c third-order Lovelock gravity to demonstrate how our scheme does work. In course of this
q
demonstration we derive all possible solutions in (6+1) and (7+1) dimensions and compare
-
r
g solutions and their abundance between cases with different Lovelock terms present. As a
[ special but more “physical” case we consider spaces which allow three-dimensionalisotropic

1 subspace for they could be viewed as examples of compactification schemes. Our results
v
suggestthatthesamesolutionwiththree-dimensionalisotropicsubspaceismore“probable”
0
6 to occur in the model with most possible Lovelock terms taken into account, which could
3
be used as kind of anthropic argument for considerationof Lovelock and other higher-order
4
0 gravity models in multidimensional cosmologies.
.
1
0 PACS numbers: 04.20.Jb, 04.50.-h, 98.80.-k
5
1
:
v
I. INTRODUCTION
i
X
r
a Multidimensional paradigm being rather popular and motivated by a number of different approaches
(from string theory to anthropic principle based on very specific and useful for life properties of gravitation
in 3 dimensions) have also well known difficulties. We need to hideextra dimensions, so that it is reasonable
to expect that they are contracting (or at least have been contracting during some period of Universe
evolution), fromtheotherhandthreelargedimensionsareexpandingalmostisotropically. Suchcombination
ofexpandingandcontracting dimensionscanbeachieved intheorieswithhigherordercurvaturecorrections,
however, the nature of this qualitative difference between large and small dimensions is often completely
obscure. Many papers even start from decomposition of metrics as a product of “external” isotropic and
“internal” spaces thus postulating this difference without any attempt to shed light on its origin.
That is why any situation in which this division is not postulated a priori, but appeared due to some
underlying principle is of particular interest. One of such situation have been found recently in Gauss-
Bonnet gravity. Initially the solutions in question appeared when generalisation of Kasner solution for a flat
anisotropic Universe have been studied in the regime when the Gauss-Bonnet term is dominated . It was
found that apart from a solution in which scale factor have a power-law behavior [1, 2] (a direct analog of
the Kasner solution) a solution with exponential time dependence of the scale factor exists [3]. It has no
analogue inGeneralRelativity. Later thisclass solution havebeengeneralized tofullEinstein-Gauss-Bonnet
(EGB) theory (where we do not neglect the Einstein term in the action) as well as to the theory with a
cosmological constant [4,5]. Ina moregeneral setup suchsolutions (dubbedanisotropic inflation) have been
shown to exist in theories with Ricci square corrections [6, 7] , though being absent in f(R) gravity, where
only Gauss curvature enters into the action.
An analysis of existence of such solutions in EGB gravity reveals an interesting fact that they exists only
ifthespacehasisotropicsubspaces[8](thoughnotnecessary 3-dimisotropic subspaces). Theonlyexception
is so-called constant-volume solutions [9], but they are special class and we do not consider them in this
paper. Though applicability of these solutions to a realistic description of our Universe still needs more
efforts in order to incorporate matter and exit from early time inflation, the fact that isotropic subspaces
appears as a condition for this solution to exist is rather promising.
One of the first attempts to find an exact static solutions with metric being a cross product of a (3+1)-
dimensional manifold times a constant curvature “inner space”, also known as “spontaneous compactifi-
cation”, were done in [10], but with four dimensional Lorentzian factor being actually Minkowski (the
generalization for a constant curvature Lorentzian manifold was done in [1]). In the cosmological context it
could be useful to consider Friedman-Robertson-Walker as a manifold for (3+1) section; this situation with
constant sized extra dimensions was considered in [11]. There it was explicitly demonstrated that to have
more realistic model one needs to consider the dynamical evolution of the extra dimensional scale factor
as well. In the context of exact solutions such an attempt was done in [12] where both the (3+1) and
the extra dimensional scale factors where exponential functions. Solutions with exponentially increasing
(3+1)-dimensional scale factor and exponentially shrinking extra dimensional scale factor were described.
Of recent attempts to build a successful compactification particularly relevant are [13] where the dynam-
ical compactification of (5+1) EGB model was considered, [14, 15], with different metric ansatz for scale
factors corresponding to (3+1)- and extra dimensional parts, and [16, 17] where general (e.g. without any
ansatz) scale factors and curved manifolds were considered.
In [1] the structure of the equations of motion for Lovelock theories for various types of solutions has
been studied. It was stressed that the Lambda term in the action is actually not a cosmological constant as
it does not give the curvature scale of a maximally symmetric manifold. In the same paper the equations
of motion for compactification with both time dependent scale factors were written for arbitrary Lovelock
2
order in the special case that both factors are flat. The results of [1] were reanalyzed for the special case of
10 space-time dimensions in [18]. In [19] the existence of dynamical compactification solutions was studied
with the use of Hamiltonian formalism.
Usually when dealing with cosmological solutions in EGB or more general Lovelock gravity [20] one
imposes a certain ansatz on the metric. Two most used (and so well-studied) are power-law and exponential
ansatz. The former of them could be linked to Friedman (or Kasner) stage while the latter – to inflation.
Power-law solutions were intensively studied some time ago [1, 2] and recently [3, 4, 21–23] which leads
to almost complete their description (see also [24] for useful comments regarding physical branches of the
solutions). Exponential solutions, on the other hand, for some reason are less studied but due to their
“exponentiality” could compactify extra dimensions much faster and more reliably. Our first study of
exponential solutions[5]demonstratetheirpotential andsowestudiedexponentialsolutions inEGBgravity
full-scale. We described models with both variable [8] and constant [9] volume and developed general
solution-building scheme for EGB. And now we are generalizing this scheme for general Lovelock gravity.
In the present paper we describe a general scheme which allows us to get such solution in an arbitrary
Lovelock gravity with arbitrary number of dimensions. This scheme is illustrated by third order Lovelock
theory in (6+1) and (7+1) dimensions.
As one of the objectives we want to compare solutions in the same dimensionality but with different
Lovelock terms taken into consideration. To be specific, we will compare (6+1)-dimensional solutions in
EGB with (6+1)-dimensional solutions in L + L + L ; then the same will be performed with (7+1)-
1 2 3
dimensionalsolutions. Thereasoningbehindthiscomparisonissimple–originally, itwasEGBgravitywhich
was motivated by the string theory, and third-order curvature correction which comes from string theory
consideration, do not coincide with third Lovelock term and, through that, are not ghost-free (see [25, 26]).
With this in mind general Lovelock theory is worse-motivated then EGB, but in higher dimensions, if we
want to “estimate” the influence of higher-order terms, Lovelock theory could give us some “insight”. Yet, if
we want to formally follow what comes from M/string theory, we need to stay with EGB – with both these
arguments at hand, we will consider them both and compare the solutions one could get from both of them.
Another important and more “physical” task is to explore the abundance of the solutions with three-
dimensional isotropic subspace. Indeed, when dealing with higher-dimensional cosmological models, one
needs to keep in mind that we observe only three spatial dimensions. This way we pay special attention to
spatial splitting which have three-dimensional isotropic subspace – if these three dimensions expand while
the remaining directions contract, this would be successful dynamical compactification scheme in action.
Thestructureofthemanuscriptisasfollows: firstweintroducethemostgeneralequations wearedealing
with. Then we develop the scheme for finding all possible spatial splittings in any number of dimensions
and with any possible Lovelock terms and their combinations. After that in Section IV we apply the
3
scheme for (7+1)-dimensional space-times for L +L +L case to retrieve all possible spatial splittings.
1 2 3
Then in Section V we describe all possible splittings in (6+1) and (7+1) dimensions and the corresponding
solutions. After that we separate solutions which allow three-dimensional isotropic subspaces. Finally we
draw conclusions and discuss obtained results.
II. THE SET-UP.
Let us spell out the conventions that we will use throughout this work. We choose to use units such that
speed of light and gravitational constant are equal to 1; Greek indices runfrom 0 to D, while Latin one from
1 to D unless otherwise stated; we also will use Einstein summation convention.
Let us consider (D+1)-dimensional flat space-time with Lovelock gravity. The gravitational action is
d n
1 D 1
S = dD+1x g + , = c , d= , = ∆β1...β2n Rα2s−1α2s (1)
2κ2 | | L Lm L nRn 2 Rn 2n α1...α2n β2s−1β2s
Z q (cid:8) (cid:9) nX=0 (cid:22) (cid:23) sY=1
where κ2 is the (D + 1)-dimensional gravitational constant, is the Lagrangian of a matter, g is the
m
L
determinant of the metric tensor, Rαβ stands for the components of the Riemann tensor, c are constants,
µν n
D corresponds to the integer part of D; the generalized Kronecker delta is defined as:
2 2
j k
δβ1 ... δβ2n
α1 α1
∆αβ11......βα22nn = det(cid:12)(cid:12)(cid:12) ... ... ... (cid:12)(cid:12)(cid:12) (2)
(cid:12)(cid:12)(cid:12)δαβ12n ... δαβ22nn (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
We choose a reference system in such a way that met(cid:12)ric has the follo(cid:12)wing form:
(cid:12) (cid:12)
ds2 = dt2+ e2ak(t)dx2 (3)
k
−
k
X
Hereafter we will write a ,a˙ ,a¨ instead of a (t),a˙ (t),a¨ (t) for brevity. It is easily shown that
k k k k k k
R0i = a¨ +a˙2, Rj1j2 = a˙ a˙ , j < j , Rαβ = 0, α,β , µ,ν , (4)
0i i i j1j2 j1 j2 1 2 µν
(cid:8) (cid:9) (cid:8) (cid:9)
the dot denotes derivative w.r.t. t. So, arbitrary component of the Riemann tensor takes the form:
Rµν = a¨ +a˙2 δ[µδν]δ0 δk + a˙ a˙ δ[µδν]δi δj , (5)
λσ  k k 0 k [λ σ] i j i j [λ σ]
Xk (cid:16) (cid:17) Xi<j 
square brackets denote the antisymmetric part on the indicated indices. It canbe shown that
p|2gn|Rn = Snj1<X...<j2nrY2=n1a˙jr +nSn−1Xk a˙2k{j1<...<Xj2n−2},k2rYn=−12a˙jr + ddt j1<..X.<j2n−12rYn=−11a˙jrePi ai
  
 
ai 2n d ai 2n−1 n−1
= Sn −2n2Sn−1 ePi a˙jr + dt ePi a˙jr, Sn = C22n−2p (6)
(cid:16) (cid:17) j1<X...<j2nrY=1 j1<..X.<j2n−1 rY=1 pY=0
 
4
The last term in the rhs of Eq. (6) is a total time derivative of some function; this term does not contribute
to an equation of motion and can be omitted. Let us denote
2n d
1
mod = 2n S 2n2S a˙ , mod = g c mod (7)
Rn n − n−1 jr L 16π | | nRn
(cid:16) (cid:17)j1<X...<j2nrY=1 q nX=0
The superscript ’mod’ means ’modified’. Using modified Lagrangian (7) instead of initial one and varying
the action (1) we obtain dynamical equations
d 2n−2 2n
ζ a¨ +a˙2 a˙ +(2n 1) a˙ = κ2Tm (8)
n k k jr − jr m
nX=1 kX,m(cid:0) (cid:1){j1<...<Xj2n−2},k,m rY=1 {j1<...X<j2n},mrY=1 
and constraint  
d 2n
(2n 1)ζ a˙ = κ2T0, (9)
− n jr 0
nX=1 j1<X...<j2nrY=1
where ζ = c 2n S 2n2S . In the following we consider isotropic perfect fluid with the equation of
n n n n−1
−
state p = ωρ as a(cid:0)matter source,(cid:1)so the energy-momentum tensor takes the form
T0 = ρ, T1 = ... = TD = p (10)
0 − 1 D
and seek for exponential solutions such that
ds2 = dt2+ e2Hktdx2, H const (11)
k k
− ≡
k
X
Using the notations of the Sec. II we see that a (t) = H t, H const. Substituting it in (8)–(9) we see
k k k
≡
d 2n−2 2n
ζ H2 H +(2n 1) H = ωκ (12)
n k jr − jr
nX=1 kX,m {j1<...<Xj2n−2},k,m rY=1 {j1<...X<j2n},mrY=1 
 
d 2n
(2n 1)ζ H = κ, κ =κ2ρ (13)
− n jr −
nX=1 j1<X...<j2nrY=1
It could be useful to rewrite Eqs. (12)–(13) in terms of elementary symmetric polynomials (see e.g. [27]):
n
e 1, e = H , e 0, n N (14)
0 ≡ n jr −n ≡ ∈
j1<X...<jnrY=1
In what follows we will also use helpful notation for e with parameters H ,...,H excluded:
n k1 kl
n
ek1,...,kl = H , n N (15)
n jr ∈
{j1<...<jnX},{k1,...,kl}rY=1
It is easy to check that
2n 2n−2 2n
e e = H H = H2 H +2n H (16)
1 2n−1 i jr i jr jr
Xi j1<X...<j2nrY=1 Xi {j1<...<Xj2n−2},i rY=1 j1<X...<j2nrY=1
5
Then with (14)–(16) taken into account Eqs. (12)–(13) read as
d d
0= ζ eiei ei ωκ, i= 1,...,D; (2n 1)ζ e = κ (17)
Ei ≡ n 1 2n−1− 2n − − n 2n −
nX=1 (cid:16) (cid:17) nX=1
Since we assume H const, it follows from Eqs. (12)–(13) that ρ const, so that the continuity equation
i
≡ ≡
ρ˙+(ρ+p) H = 0 (18)
i
i
X
reduces to
(ρ+p) H = 0, (19)
i
i
X
which allows several different cases: a) ρ 0 (vacuum case), b) ρ+ p = 0 (Λ-term case), c) H = 0
i i
≡
(constant volume case which we call CVS for brevity) and their combinations: d) vacuum CPVS and e)
Λ-term CVS. So, all possible exponential solutions can be divided into two large groups: solutions with
constant volume and solutions with volume changing in time (non-constant volume solutions); for the latter
case we have only two possibilities: vacuum case and Λ-term case; on the contrary, the first case does not
impose constraints on choice of matter a-priory.
Let N N and let N be the set of the first N natural numbers:
∈
(cid:2) (cid:3)
N l N l 6 N (20)
≡ ∈
(cid:2) (cid:3) (cid:8) (cid:12) (cid:9)
(cid:12)
Initial system of D dynamical equations is equivalent to a system that consist of D 1 difference equations
−
and one dynamical equation:
i [D] = 0 i [D 1] = 0 m [D] = 0 (21)
i i+1 i m
∀ ∈ E ⇐⇒ ∀ ∈ − E −E ∧ ∃ ∈ E
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
It is easy to check that
d
= e (H H ) ζ ei1,i2 = 0 (22)
Ei1 −Ei2 1 i2 − i1 n 2n−2
n=1
X
Let us introduce the following notations:
d
1 H H , 2 ζ ej1,...,jq (23)
Ci1,i2 ≡ i1 − i2 Cj1,...,jq ≡ n 2n−q
n=1
X
Then we have
= e 1 2 = 0 1 = 0 2 = 0 H = 0 (24)
Ei+1−Ei 1Ci,i+1Ci,i+1 ⇐⇒ Ci,i+1 ∨ Ci,i+1 ∨ k
k
X
Equation H = 0 leads to so called constant volume solution (see [8, 9] for details) and requires separate
k k
consideratPion, so in what follows we deal with equations 1 = 0 and 2 = 0 only.
Ci,i+1 Ci,i+1
6
One more thing needs to be explained before continuing with the general scheme, namely, the coupling
constants ζ . In previous papers dedicated to study of the exact solutions in EGB and Lovelock gravity [4,
n
5, 8, 9, 22, 24] we used different couplings constant and for a reader’s convenience in comparing solutions
we write down their relation (remember, ζ = c 2n S 2n2S with S given in (6)):
n n n n−1 n
−
(cid:0) (cid:1)
ζ = 2,ζ = 8α,ζ = 144β. (25)
1 2 3
− − −
In the absence of ζ and ζ natural normalization for dimensional units would be
2 3
ζ = 8πG 4πG 1, (26)
1
− ⇒ ≡
so that when we put some dimensional numerical values, we do it with (26) in mind.
III. THE GENERAL SCHEME.
Inthissectionwedescribethegeneralschemethatallows toobtainawideclassofsolutionswithisotropic
subspaces in the framework of general Lovelock model in (D+1) dimensions.
Let m Z , N N; m 6 N. Let us introduce the following definitions:
≥0
∈ ∈
1. Let m be a collection of all m-element subsets of the set N :
I[N]
(cid:2) (cid:3)
m = Z N Z = m (27)
I[N] ⊆ | |
n (cid:2) (cid:3)(cid:12)(cid:12) o
Notation Z in (27) stands for cardinality of a set Z.(cid:12)
| |
2. Let r,k ,l N, l D, kr ; let us introduce the following notations:
r r−1 ∈ 0 ≡ Xr ∈Ilr−1−1
[D], = , = 1 2 , where
J0 ≡ Jk Jk−1\Jk−1 Jk−1 Jk−1∪Jk−1
Jk1−1 = js js ∈ Jkb−1 ∧ sb∈ Xk ∧bs 6 lk−b21−1 (28)
Jk2−b1 = jsn+k (cid:12)(cid:12)(cid:12)js+k ∈ Jk−1 ∧ s ∈ Xk ∧ sj> lk−21k−o1
n (cid:12) j ko
In what follows we usebspecial notati(cid:12)on yk for elements of , i.e. if we write yk somewhere it means
(cid:12) s k s
J
there exists a set such that yk .
k s k
J ∈ J
1st step. For a given solution some of equations = 0 are satisfied due to 1 = 0, others
Ei+1 −Ei Ci,i+1
are satisfied due to 2 = 0, thus the initial system becomes equivalent to a system that consist of several
Ci,i+1
(say, k ) equations 1 = 0, several (say, l ) equations 2 = 0 and one dynamical equation:
1 Ci,i+1 1 Ci,i+1
k Z k1 i 1 = 0
∃ 1 ∈ ≥0 ∃X1 ∈ I[D−1] ∀ ∈X1 Ci,i+1
 l Z l1 j (cid:16) 2 = 0(cid:17)
∀i∈ [D] (cid:16)Ei = 0(cid:17) ⇐⇒  X∃11∩∈Y1≥=0 ∃∅Y∧1 ∈XI1[D∪−Y11] ∀=[D∈Y−11(cid:16)]Cj,j+1 (cid:17) (29)
= 0
1
E
 k1+l1 = D−1
7
Note that if k = 0 (l = 0) then = ∅ ( = ∅).
1 1 1 1
X Y
2nd step. The idea is to consider equations 2 = 0, j as a new basic equations, find the
Cj,j+1 ∈ Y1
respective difference equations and obtain result analogous to (29). The difference 2 2 is factorized
Cj1,j2−Cj3,j4
iff one of the elements of the pair (j ,j ) equals to one of the elements of the pair (j ,j ); for example, let
3 4 1 2
us assume that j = j , then
4 2
2 2 = 1 2 (30)
Cj1,j2 −Cj2,j3 Cj1,j3Cj1,j2,j3
According to (23) Ci1′,i′+1 = 0 ⇐⇒ Hi′ = Hi′+1, therefore we can identify indices i′,i′ +1; it is easy to
check that one can always find such i′ , j′ that i′ = j′ +1 (or i′ +1 = j′); using these facts we
1 1
∈ X ∈ Y
replace the set in (29) by . By analogy with (21) we have
1 1
Y J
k [l ] 2 = 0
∀ ∈ 1 (cid:18)Cyk1,yk1+1 (cid:19)
(31)
m
k [l 1] 2 2 = 0 k1 [l ] 2 = 0
∀ ∈ 1− (cid:18)Cyk1,yk1+1 −Cyk1+1,yk1+2 (cid:19) ∧ ∃ 0 ∈ 1 Cyk101,yk101+1 !
It follows from (30) that
2 2 = 0 1 = 0 2 = 0 (32)
Cyk1,yk1+1 −Cyk1+1,yk1+2 ⇐⇒ Cyk1,yk1+2 ∨ Cyk1,yk1+1,yk1+2
Some of equations 2 2 = 0 are satisfied due to 1 = 0, others are satisfied due to
Cyk1,yk1+1 − Cyk1+1,yk1+2 Cyk1,yk1+2
2 = 0, so that
Cyk1,yk1+1,yk1+2
k Z k2 p 1 = 0
∃ 2 ∈ ≥0 ∃X2 ∈ I[l1−1] ∀ ∈ X2 Cyp1,yp1+2
 (cid:18) (cid:19)
∀k ∈ [l1] (cid:16)Cy2k1,yk1+1 = 0(cid:17) ⇐⇒ ∃∃lk201∈∈Z[l≥10] ∃ YCy22k11∈,yIk11[ll+211−1=] ∀0!q ∈Y2 (cid:18)Cy2q1,yq1+1,yq1+2 = 0(cid:19) (33)
0 0
kX22+∩lY22==l1∅−∧1 X2∪Y2 = [l1−1]
Taking into account 1st and 2nd steps we see that
1 = 0 for all i k1
Ci,i+1 ∈X1 ∈ I[D−1]
 1 = 0, for all p k2
 ED1.=.=. 00 ⇐⇒  CCCyyy22kqp1111,,yy,qpy11++k1112+,y1q1+=2 =0 0fo,rfsoormae∈llkXq012∈∈∈Y[lI21[]l∈1−I1[]ll21−1] (34)
E 0 0
  Ek1m+=k02+folr2s=omDe−m2∈ [D]
8
r-th step. Continuing this procedure at r-th step we have
1 = 0 for all p k1
Cp1,p1+1 1 ∈X1 ∈ I[D−1]
 1 = 0 for all p k2
 EED1.=.=. 00 ⇐⇒  C.CCC.yyyy122.ppqk1rr1r2r−−01.,,.11yy,,.p1k1y..201pr.+,+r−y+21qr1r−r=+1=r0=0f0of,orrsfoaomlrlaep2lrkl∈0∈1qXr∈X2∈r[∈lY1∈]rII[∈l1[kl−rrI−1[l1]lrr−−11]−1], lr > 1 (35)
The process terminates when l 61 .CEkfo.1ym2r.kr+−0r.=−s.1o1..,m0.....e,+fyokrr−0rrk−1rs=1o++mrr−l∗er1m==D∈0N;[f−Dotrhr]esoimnietika0rl−s1y∈ste[lmr−1b]ecomes equivalent to the
r
∈
system consisting of k1+...+kr∗ = D−r∗ equalities of Hubble parameters (equations Cy1pii,ypii+2 = 0) and r∗
additional conditions. The maximal value of r∗ is (2d 1). Indeed, when r∗ = r∗ (2d 1) degree of the
max
− ≡ −
last symmetric polynomial in the sum (23) becomes zero and 2 ζ . Equation 2 ζ = 0
Cj1...jrm∗ax ≡ d Cj1...jrm∗ax ≡ d
means vanishing of the highest Lovelock term in the Lagrangian and must be omitted, in other words one
must set lrm∗ax ≡ 0, krm∗ax ≡ lrm∗ax−1−1. Thus, r∗ can vary from 1 to rm∗ax and, respectively, (k1+...+kr∗)
varies from (D 1) to 1 for even D and to 2 for odd D.
−
Each collection of (k1 +...+kr∗) equalities of Hubble parameters gives a number of splittings of space
into isotropic subspaces. Splittings corresponding to various values of (k1 +... +kr∗) are represented in
the Table I. We see that even-dimensional spaces has one more splitting as compared with odd-dimensional
one. Numbers in the column "Splitting" means numbers of equal Hubble parameters; braces stand for
pairing of Hubble parameters that give rise to the next splittings; subscripts after round brackets are used

to indicate the number of units in these brackets. For example, record 3 + 2+(1 +1 +...+1)D−5 means
that H ,...,H = H,H,H,h,h,H ,...,H , H , ... , H and at thze n}|ex{t step one can obtain the
1 D 6 D 6 D
| {z } | {z }
follow(cid:8)ing splitting(cid:9)s: (cid:8) (cid:9)
5+(1+...+1) , 3+3+(1+...+1) , 3+2+2+(1+...+1) (36)
D−5 D−6 D−7
9
k1+...+kr∗ Splitting Number of additional conditions

1 2 + ( 1+1+...+1)D−2 D-1

 z}|{
even D  23 23| {++34|||z ++{{{ 22zz}zzz ++ ((}} 11zz }}}|| ((}} ++ 11|{{|| {++ 11{{z11++ }++..........++..++11))11DD))−−DD34−−45 DD--23  odd D
 ....D..-..1.... Tabl2|e {+|Iz.{ 2}zS+e}v2zer+}a||l(is|1{{s..o zp ..{+tl..z}ri ..t o1}..tp+i..nicg.s..o+f D1)-Ddi−m6ensional space.......1...... 
IV. APPLICATION OF THE GENERAL SCHEME.
In this section we consider application of the of the scheme described above to the theory with three
Lovelock terms in (7+1) dimensions. Let
ζ ζ
1 3
ψ = , ψ = (37)
12 32
ζ ζ
2 2
I. r∗ = r∗max = 5; l5 0, k1+...+k5 = 2. It is easy to check that
≡
k +...+k = 2
1 5
m
1.(k = 2,l = 0); (k = 0,l = 3); (k = 0,l = 4); (k = 0,l = 5); (k =0,l = 6), or
5 5 4 4 3 3 2 2 1 1
2.(k = 1,l = 0); (k = 0,l = 2); (k = 0,l = 3); (k = 0,l = 4); (k =1,l = 5), or (38)
5 5 4 4 3 3 2 2 1 1
3.(k = 1,l = 0); (k = 0,l = 2); (k = 0,l = 3); (k = 1,l = 4); (k =0,l = 6), or
5 5 4 4 3 3 2 2 1 1
4.(k = 1,l = 0); (k = 0,l = 2); (k = 1,l = 3); (k = 0,l = 5); (k =0,l = 6), or
5 5 4 4 3 3 2 2 1 1
5. (k = 1,l = 0); (k = 1,l = 2); (k = 0,l = 4); (k = 0,l = 5); (k = 0,l = 6)
5 5 4 4 3 3 2 2 1 1
We consider subcases (1) and (2) in more details; other subcases can be considered analogously.
(1) It is easy to check that
= ∅, = [6], = ∅, = [5], = ∅, = [4], = ∅, = [3], = 1,2 , = ∅
1 1 2 2 3 3 4 4 5 5
X Y X Y X Y X Y X { } Y (39)
= = = = [7]
1 2 3 4
J J J J
10