arXiv:0704.1234v2  [hep-th]  19 Oct 2007
IFT–07–1
Generalization of Einstein–Lovelock theory
to higher order dilaton gravity
D. Konikowska, M. Olechowski
Institute of Theoretical Physics, University of Warsaw
ul. Hoża 69, PL-00 681 Warsaw, Poland
Abstract
A higher order theory of dilaton gravity is constructed as a generalization of the
Einstein–Lovelock theory of pure gravity. Its Lagrangian contains terms with higher
powers of the Riemann tensor and of the first two derivatives of the dilaton. Neverthe-
less, the resulting equations of motion are quasi–linear in the second derivatives of the
metric and of the dilaton. This property is crucial for the existence of brane solutions
in the thin wall limit. At each order in derivatives the contribution to the Lagrangian
is unique up to an overall normalization. Relations between symmetries of this theory
and the O(d, d) symmetry of the string–inspired models are discussed.
http://arxiv.org/abs/0704.1234v2
1 Introduction
The equations of motion in the Einstein theory of gravity in 4 space–time dimensions
are the most general divergence–free tensor (rank 2) equations bilinear in the first
derivatives and linear in the second derivatives of the metric. They can be obtained
from the Hilbert–Einstein action which is linear in the Riemann tensor. In more than
4 space–time dimensions, this theory can be generalized to contain higher powers of
the Riemann tensor in the action. The corresponding equations of motion involve
higher powers of the first derivatives of the metric and are quasi–linear in the second
derivatives (all terms are at most linear in the second derivatives, while multiplied by
powers of the first derivatives). It has been shown that the contribution to the action
of a given order in the Riemann tensor is unique up to an overall normalization. The
quadratic contribution is called the Gauss–Bonnet action or the Lanczos action [1]. It
has been generalized to higher orders by Lovelock [2]. The quasi–linearity is a very
important feature of the Einstein–Lovelock equations of motion. It guarantees that
they can be formulated as a Cauchy problem with some constraints on the initial data
[3]. On the other hand, it is crucial for the existence of non–singular domain wall
solutions in the thin wall limit. This problem for arbitrary order in derivatives was
discussed in [4]. Many aspects of the Einstein–Lovelock gravity were discussed in the
literature1.
Higher derivative corrections to the gravity interactions are present in effective
Lagrangians obtained from string theories. The first correction has exactly the form
of the Gauss–Bonnet term [9], [10]. The lowest order dilaton interactions were added
to the Gauss–Bonnet theory in [11]. However, the α′ expansion in string theories
predicts higher derivative corrections not only for the gravitational interactions, such
corrections appear also for the dilaton. The effective action for the dilaton gravity
with terms up to four derivatives was given in [12], [13]. The effective action with six
derivatives was presented in [14], but its gravitational part has a form different from
that of the corresponding Einstein–Lovelock action.
The dilaton gravity at the field theory level has been investigated by many authors.
Some of them included also certain higher order corrections. Yet in most cases such
corrections were considered only for gravitational interactions and not for the dilaton.
Some higher derivative corrections for both the dilaton and the gravitational interac-
tions were considered in [15]–[21] (certain Riemann tensor combinations with dilaton
dependent coefficients were analyzed in [22]–[25]). The terms predicted by superstrings
up to four derivatives have also been considered in [26]–[28].
The purpose of the present work is to find a generalization of the lowest order
dilaton gravity theory to an arbitrary order in derivatives. We start with the Einstein–
Lovelock higher order gravity and couple it to the dilaton. There are many ways to do
this but we are only interested in the theories where dilaton and gravity interactions
1Quasi–linearity of the Einstein–Gauss–Bonnet theory was reviewed in [5]. A discussion of general
quasi–linear differential equations can be found in [6]. For a review on brane–world gravity see eg. [7].
For a discussion of the Lovelock gravity in the context of the equivalence of the Palatini and metric
formulations see eg. [8].
are as similar to each other as possible. Equations of motion in such a theory are
presented in Section 2. We begin with formulating the conditions which should be
fulfilled by such equations. Most of them are simple generalizations of the conditions
fulfilled by the Einstein–Lovelock equations of motion. One condition is added in order
to eliminate at least some of the possible theories in which the dilaton interactions are
not related to the gravitational ones. The equations of motion satisfying all those
conditions are constructed in Subsection 2.3. It turns out that at each order those
equations are unique up to a numerical normalization. Moreover, they can be obtained
by the standard Euler–Lagrange procedure from the Lagrangian presented in Section 3.
Section 4 contains the proof that our equations of motion are quasi–linear in the second
derivatives of both the metric and the dilaton. The relation between the gravity and
the dilaton interactions is discussed in Section 5. We point out that the Lagrangian of
our higher order dilaton gravity can be obtained in a simple way from the pure gravity
Einstein–Lovelock Lagrangian. We also discuss the relation of the resulting theory to
the O(d, d) symmetric theories. We conclude in Section 6. The Appendix contains the
explicit formulae for the Lagrangian and the equations of motion up to terms of the
sixth order in derivatives.
2 Equations of motion
2.1 Notation
Let us start with introducing certain generalizations of the Kronecker delta and the
trace operator which will be used later to make the formulae more compact. The
generalized Kronecker delta is defined by
j1j2···jn
i1i2···in
= det
· · · δj1in
· · · · · ·
· · · · · ·
· · · δjnin
, (1)
and should be only employed when the spacetime dimensionality D is sufficient: D ≥ n.
Using this definition it is easy to prove some relations among Kronecker deltas of
different order. For example:
ν j1j2...jn
µ i1i2...in
= δνµδ
j1j2...jn
i1i2...in
− δνi1δ
j1j2...jn
µ i2...in
− δνi2δ
j1j2...jn
i1µ ...in
− . . .− δνinδ
j1j2...jn
i1i2...µ
. (2)
The generalized Kronecker delta can be used to define the following trace–like linear
mapping from tenors of rank (n, n) into numbers
T (M) = δj1j2···jni1i2···in M
i1i2···in
j1j2···jn
, (3)
which reduces to the ordinary trace for n = 1. We will also employ an extension of
this operation which maps tensors of rank (n, n) into tensors of rank (1, 1):
T νµ (M) = δ
ν j1j2···jn
µ i1i2···in
M i1i2···inj1j2···jn . (4)
In the following we will often use T and T evaluated for products of tensors. In order
to clearly distinguish between tensors and their contracted counterparts, we will use
∗ indices to indicate the rank of a tensor. For example, R∗∗∗∗ denotes the rank (2, 2)
Riemann tensor, and �∗∗φ denotes the rank (1, 1) second derivative of the dilaton,
while R is the Ricci scalar and �φ the D’Alembertian acting on the dilaton. Thus, for
example,
(R∗∗∗∗)
(�∗∗φ)
= δσ1σ2σ3σ4σ5σ6ρ1ρ2ρ3ρ4ρ5ρ6 R
Rρ3ρ3σ3σ4 �
φ�ρ6σ6φ , (5)
where we used the notation Rρ1ρ2σ1σ2 = R
σ1σ2 and �
σφ = ∇ρ∂σφ to make the formula
more compact. It is easy to see that the sequence of tensors appearing in the product
argument of T is not important. Changing such an order is equivalent to interchang-
ing the appropriate columns of indices in the generalized Kronecker delta. On the
other hand, interchanging two such columns of indices is equivalent to interchanging
the corresponding 2 rows and 2 columns in the determinant in Definition (1). Each
interchange of two columns (or two rows) changes the sign of the determinant, hence
an even number of interchanges leaves the determinant unchanged.
2.2 Conditions
Now we want to construct the n–th order dilaton gravity equations of motion. They
are to be of the form
T (n)µν = 0 ,
W (n) = 0 , (6)
where the tensor T
µν and the scalar W
(n) satisfy the following conditions
(i) They are combinations of terms with exactly 2n derivatives acting on the metric
tensor gµν and on the dilaton field φ. There are no derivatives higher than second
acting on one object;
(ii) Tensor T
µν is symmetric in its indices;
(iii) The covariant derivative of the tensor is proportional to the scalar:
µ = const · (∂µφ)W (n) (the energy–momentum tensor is covariantly con-
served if the dilaton equation of motion is fulfilled).
It is clear that the above conditions are not sufficient to determine something which
could be regarded as an extension of the higher order gravity theory to the dilaton
gravity case. For example, all the above conditions are fulfilled by the Einstein–Gauss–
Bonnet gravity with only the lowest order terms for the dilaton. We are interested in
a theory where the dilaton and the metric are treated in a more symmetric way. It
is not obvious how such a symmetry should be defined, because it ought to relate a
scalar to a second rank tensor. Or, more precisely, it is supposed to relate the first
and second derivatives of the scalar field to the Riemann tensor and its contractions.
A simple observation concerning the gravity part is that it contains even–rank tensors
only. On the other hand, the first derivative of a scalar is a rank–1 tensor. Hence
one can expect that in a gravity–dilaton symmetric theory, the first derivative of the
dilaton appears only as a 0–rank tensor: gµν∂µφ∂νφ. However, the feature mentioned
above is not invariant under change of variables. Thus, we should specify in which
frame it is fulfilled. The theory which relates dilaton to gravity is the string theory so
the string frame seems to be a natural choice. Hence our last condition reads:
(iv) In the string–like frame, in which the pure gravity term is multiplied by exp(−φ),
the first order derivatives of the dilaton appear in the combination (∂µφ)(∂
only.
The relation of this condition to the O(d, d) symmetry present in many string–inspired
theories will be discussed in Section 5.
2.3 Construction
We start our construction with a term in T
µ where all 2n derivatives act on the metric
tensors. The only pure gravity tensor satisfying Conditions (i)–(iii) (with W (n) = 0) is,
up to normalization, equal to the n-th order Lovelock tensor [2]. Because of Condition
(iv), it is most natural to work in the frame in which the gravity term is multiplied by
exp(−φ). Consequently, the tensor T ν(n)µ starts with
T ν (n)µ = −2
−(n+1)e−φδν σ1...σ2nµ ρ1...ρ2nR
· · ·Rρ2n−1ρ2nσ2n−1σ2n + . . . (7)
The reason for such a normalization will be explained in the next section. Calculating
the divergence of (7), we get
∇νT ν (n)µ = 2
−(n+1)e−φ(∂νφ)δ
ν σ1...σ2n
µρ1...ρ2n
Rρ1ρ2σ1σ2 · · ·R
ρ2n−1ρ2n
σ2n−1σ2n
+ . . . (8)
The above term is produced when the derivative acts on e−φ (derivatives of the Riemann
tensor do not contribute due to the Bianchi identity).
The first term in T
µ shown explicitly in (7) can not be the only one. The reason
is that the r.h.s. of (8) is not a product of ∂µφ and a scalar, as Condition (iii) requires.
Using Eq. (2) we can rewrite the r.h.s. of (8) as a combination of (2n + 1) terms.
The one containing the first term from the r.h.s. of (2) is of the desired form but the
remaining 2n terms have different structures of the index contractions. It turns out
that similar terms are also present in the following covariant derivative
e−φδν σ1...σ2n−1µρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2�
ρ2n−1
σ2n−1
= −e−φ(∂νφ)δν σ1...σ2n−1µ ρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2�
ρ2n−1
σ2n−1
+e−φδν σ1...σ2n−1µρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2
∇ν∇ρ2n−1∂σ2n−1φ
. (9)
The second term on the r.h.s. may be rewritten as
e−φδν σ1...σ2n−1µ ρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2
δσ2nρ2nR
ρ2n−1ρ2n
σ2n−1ν
∂σ2nφ
e−φ(∂νφ)δ
δσ1...σ2n−1σ2nρ1...ρ2n−1µ R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2R
ρ2n−1ρ2n
σ2n−1σ2n
, (10)
where in the last step we interchanged the names of the contracted indices ν and
σ2n and rearranged the indices in the generalized Kronecker delta. A term exactly of
this structure must be added to (8) in order to obtain an expression proportional to
∂µφ. From Eqs. (2) and (8) it follows that the coefficient should be equal to (−n2−n)
instead of the (−1/2) present in (10). This fixes the coefficient of the term in T ν(n)µ
which contains (n− 1) Riemann tensors and one second derivative of the dilaton. Now
we know the first two terms of the tensor T
µ . Using the notation introduced in (3)
and (4), they can be written as:
T ν (n)µ = −2
−(n+1)e−φT νµ ((R
)− 2−(n−1)ne−φT νµ
(R∗∗∗∗)
(n−1)
+ . . . (11)
Their covariant derivative reads
∇νT ν (n)µ = 2
−(n+1)e−φ(∂µφ)T ((R∗∗∗∗)
+2−(n−1)ne−φ(∂νφ)T
(R∗∗∗∗)
(n−1)
+ . . . (12)
The first term has the structure required by Condition (iii) and determines the first
term of the scalar equation of motion2 W (n). However, the second term in (12) is not
of the appropriate structure. It means that some additional terms, whose covariant
derivatives are products of (n− 1) Riemann tensors with one second derivative of the
dilaton, are necessary in T
µ . Two such terms are possible:
−φT νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
+ c4e
−φT νµ
(R∗∗∗∗)
(n−1)
(∂φ)2 . (13)
However, it is not enough to have terms with appropriate powers of the Riemann
tensor and the dilaton, because their covariant divergences must contain the correct
combinations of the generalized Kronecker deltas. To check whether this is possible, we
calculate the covariant divergence of (13). When the derivative acts on �∗∗φ in the first
term in (13), it gives an additional Riemann tensor multiplied by ∂φ and a pair of new
indices. Those new indices are contracted with just one ordinary Kronecker delta and
are not under the overall antisymmetrization. Similarly, when the covariant derivative
acts on (∂φ)2 in the second term in (13), it gives the second derivative of the dilaton
multiplied by ∂φ and a pair of new indices. Those two covariant derivatives should
combine with the second term on the r.h.s. of (12) to give an expression proportional
to ∂µφ. This fixes the numerical coefficients c3 and c4. The explicit calculation gives
c3 = −2(2−n)n(n− 1), c4 = 2−nn. Thus, we have found the first four terms of T
T ν (n)µ = −2
−(n+1)e−φ
T νµ ((R
) + 4nT νµ
(R∗∗∗∗)
(n−1)
+8n(n− 1)T νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
−2nT νµ
(R∗∗∗∗)
(n−1)
(∂φ)2
+ . . . (14)
2up to an overall normalization. The choice of the relative normalizations of T
µν and W
(n) shall
become clear when the Lagrangian is introduced in Section 3.
The covariant divergence of those terms reads
∇νT ν (n)µ = ∂µφ
2−(n+1)e−φ
T ((R∗∗∗∗)
) + 4nT
(R∗∗∗∗)
(n−1)
+∂νφ 2
(2−n)n(n− 1)e−φT νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
−∂νφ 2−nne−φT
(R∗∗∗∗)
(n−1)
(∂φ)2 + . . . (15)
The terms in the curly bracket above are the first two terms of the scalar W (n) we are
looking for.
Equation (15) shows that the procedure of finding T
µ and W
(n) must be contin-
ued. The last two terms on the r.h.s. of (15) do not have the required form, so more
terms must be added to T
µ . From the steps described so far, it should be clear that
each of such new terms must contain exactly 3 (first or second order) derivatives of the
dilaton. There are two such terms:
−φT νµ
(R∗∗∗∗)
(n−3)
(�∗∗φ)
+ c6e
−φT νµ
(R∗∗∗∗)
(n−2)
(∂φ)2 . (16)
The coefficients c5 and c6 can be fixed in the same way as c3 and c4.
This procedure can be continued step by step for the terms containing higher and
higher powers of the dilaton field with the derivatives acting on it. Eventually, one
obtains the term with the maximal number of dilaton fields, namely c e−φδνµ [(∂φ)
This is the first term in T
µ , the covariant derivative of which need not to be corrected
by contributions from any additional terms. This covariant derivative reads
e−φδνµ
(∂φ)2
= −(∂µφ)e−φ
(∂φ)2
+ 2ne−φ(∇µ∂σφ)(∂σφ)
(∂φ)2
](n−1)
. (17)
The second term on the r.h.s. is used to cancel some unwanted part of
e−φT νµ (�∗∗φ) [(∂φ)2]
(n−1)
, which fixes c to be equal to 1
(−1)(n+1). The first term
on the r.h.s. of (17) has already the required structure of the product of ∂µφ and a
scalar. Thus, the procedure can stop here.
The above iterative procedure gives T
µν and W
(n) satisfying all the four imposed
conditions. The resulting gravitational and dilaton equations of motion can be written
in the following relatively simple form:
T (n)µν = −
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
= 0 , (18)
W (n) = −e−φ
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
= 0 . (19)
The existence of T
µν and W
(n) is a non–trivial result, because in our iterative
procedure there are more conditions than available constants. A priori it could happen
that there were no solutions other than a trivial one with vanishing T
µν and W
However, the solution exists and is unique up to an overall normalization. Hence any
dilaton gravity equations of motion, satisfying Conditions (i)–(iv), which contain at
least one term present in (18) and (19) must also contain all the other terms with
uniquely determined coefficients.
3 Lagrangian
It is interesting to check whether the equations of motion constructed in Section 2
can be obtained from some D–dimensional action. In such case, T
µν and W
(n) would
satisfy
δgµνS
(n) = δgµν
−gL(n) =
−g T (n)µν δg
µν , (20)
(n) = δφ
−gL(n) =
−gW (n)δφ . (21)
It turns out that indeed the equations of motion (18) and (19) can be obtained from
the action with the Lagrangian density given by
L(n) = e−φ
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
. (22)
It is important to underline that for Conditions (i)–(iv) not to be violated, the terms
coming from the n-th Lagrangian can appear only in the space–times with dimension-
ality D ≥ 2n. Moreover, one should be careful when calculating (20) for D = 2n, as
the generalized Kronecker delta (1) can not be employed in (18) for the term of the
highest order in the Riemann tensor. The coefficient of that term should be replaced
δν σ1σ2...σ2nµ ρ1ρ2...ρ2n −→D=2n δ
σ1σ2...σ2n
ρ1ρ2...ρ2n
− δνρ1δ
σ1σ2...σ2n
µ ρ2...ρ2n
− δνρ2δ
σ1σ2...σ2n
ρ1µ ...ρ2n
− . . .− δνρ2nδ
σ1σ2...σ2n
ρ1ρ2...µ
. (23)
Now we can comment on the overall normalization of the tensors T
µν . The reason
for this particular normalization is that the term e−φRn (with R being the Ricci scalar)
appears in the Lagrangian with the coefficient 1. This corresponds to the standard
normalizations of the Hilbert–Einstein and Gauss–Bonnet Lagrangians.
Proving that the equations of motion derived from the Lagrangian (22) really have
the form (6) with T
µν and W
(n) as given in (18) and (19) is a straightforward but
quite tedious calculation. One of the reasons is that apparently several integrations
by parts are required. This can be somewhat simplified if one observes that not all
those integrations by parts have to be performed explicitly. In case of (21), the reason
is as follows. Under the integral (21) there are first (second) derivatives of δφ coming
from the variation of the first (second) derivatives of the dilaton. In general, the terms
containing second derivatives of δφ should be integrated by parts twice. However, one
can notice that the result of a single integration and the terms containing the first
derivatives of δφ cancel each other exactly.
The situation is a little bit more complicated in case of the gravitational equation
of motion. Under the integral (20), there are second derivatives of δgµν coming from
the variation of the Riemann tensor and first derivatives of δgµν coming from the
variation of the second covariant derivative of the dilaton. Similarly as in the case of
the dilaton equation of motion, the terms containing second derivatives of δgµν have to
be integrated by parts only once. And although the cancellation of the resulting terms
is not complete this time, only some residual integration by parts has to be performed
additionally.
Of course, the Lagrangian density (22) is not unique. First, one can rewrite L(n)
changing the variables gµν and φ. Second, one can add to L(n) any total divergence
without changing the resulting equations of motion. However, the form given in Eq.
(22) is especially simple and interesting. It is very similar to the form of T
µν and
W (n). The energy momentum tensor T
µν can be obtained from L(n) by replacing
the generalized trace T with its tensor extension T µν and multiplying the result by
−1/2. In case of the dilaton equation of motion, the analogous relation is even simpler:
W (n) = −L(n).
We were not able to find any other similarly simple form of the Lagrangian by
adding total derivative terms or by changing the variables. For example, we examined
the form of the Lagrangian and of the equations of motion in the Einstein–like frame
in which the common factor e−φ is absorbed by a suitable Weyl transformation. The
results are very complicated and will not be presented here. One of the reasons for
such complications is that the Weyl transformation depends on the dimensionality D
of the space–time. Thus, many different functions of D appear in the Einstein frame,
while there is no explicit dependence on D in our string–like frame.
4 Quasi-linearity
It is easy to show that the equations of motion (18)–(19) are quasi–linear in the second
derivatives of the metric and the dilaton. Let us introduce in the D–dimensional space–
time a (D− 1)–dimensional hypersurface Σ defined by its unit normal vector nµ. The
metric induced at this hypersurface is given by
hµν = gµν −
, (24)
where n2 = nρn
ρ. The components of the D–dimensional Riemann tensor R∗∗∗∗ corre-
sponding to the full metric gµν can be expressed as
Rρσµν = R
µν −n
+ 4n[ρD[µK
+ 4n[µD
4n[µn
Kτν] − 4n[µn
, (25)
where: R is the (D − 1)–dimensional Riemann tensor corresponding to the induced
metric hµν ; K is the extrinsic curvature given by
Kµν =
£nhµν ; (26)
Dµ is the covariant derivative with respect to the induced metric hµν ; £n is the Lie
derivative along the vector field nµ. Similarly we can write the D–dimensional second
covariant (with respect to the metric gµν) derivative of the dilaton
∇µ∇νφ = DµDνφ +n−2
Kµν£nφ+ 2n(µDν)£nφ− 2n(µKτν)Dτφ
+n−4 nµnν
nφ− (n
ρ∇ρnτ )∇τφ
. (27)
We want to check how the second Lie derivatives of the metric hµν (present in
£nKµν) and of the dilaton φ appear in the equations of motion (18) and (19). Such
second derivatives are present in (25) and (27) but in both cases they are multiplied
by coefficients bilinear in the vector n. After substituting the decompositions (25) and
(27) into (18) and (19), one can immediately see that, due to the antisymmetrization
present in T
µν and W
(n), the equations of motion contain terms at most bilinear in n.
Thus, the equations of motion (18) and (19) contain terms at most linear in the second
Lie derivatives of hµν and φ.
We have shown that the equations of motion are quasi–linear in the second Lie
derivatives “perpendicular” to the hypersurface Σ. This quasi–linearity has very im-
portant consequences. For Σ with a time–like n, this allows us to define a standard
Cauchy problem with the initial conditions (values and first Lie derivatives of hµν and
φ) given at Σ. For a space–like n, the quasi–linearity is necessary to have non–singular
brane solutions even in the thin wall limit.
5 Symmetries
The equations of motion presented in Section 2 were obtained assuming some kind of
symmetry between the metric and the dilaton. Now we are in a position to investigate
such a symmetry in more detail.
It is quite amazing that the Lagrangian (22) as well as the equations of motion
(18) and (19) can be expressed as functions of n–th perfect “power” of one simple
n–independent quantity. Namely:
L(n) = −W (n) = e−φT
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
, (28)
T (n)µν = −
e−φT µν
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
. (29)
One can treat these equations as just a new notation allowing us to rewrite the double
sums from (18), (19) and (22) in a compact way3. Yet, on the other hand, it helps
to show that the action and the equations of motion depend on some combinations
of the dilaton derivatives and tensors obtained from the metric only. In each round
parenthesis in Eqs. (28) and (29), there are the rank–4 Riemann tensor, the rank–2
tensor of the second derivatives of the dilaton and the rank–0 tensor built from the
first derivatives of the dilaton:
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
. (30)
All those tensors are under the generalized traces T and T µν . Some of the terms
present in these mappings contain traces of the tensors from (30). There are two
different rank–2 tensors coming from (30). The first is just �∗∗φ. The second is the
Ricci tensor R∗∗ which can be obtained from the Riemann tensor by contraction of its
two indices. There are four different ways to contract one pair of indices in R∗∗∗∗, thus
in the final result the rank–2 tensors appear always in the combination (2R∗∗ + 2�∗∗φ).
There are three different scalars originating from (30): (∂φ)
, �φ and the curvature
scalar R. There are two different constructions giving R, so the final results depend
on a single following scalar combination: R+ 2�φ− (∂φ)2.
The above observation allows us to relate our dilaton gravity equations to the
corresponding equations in the pure Einstein–Lovelock gravity:
L(n) = −W (n) = e−φL(n)E−L
R∗∗∗∗ , (R
∗φ) ,
R+ 2�φ− (∂φ)2
, (31)
T (n)µν = e
gµν ,R∗∗∗∗ , (R
∗φ) ,
R+ 2�φ− (∂φ)2
. (32)
The recipe for the higher order dilaton gravity can be as follows: Start with the higher
order pure gravity Einstein–Lovelock theory. Write the Lagrangian density L(n)E−L and
the equations of motion (T
E−L)µν in terms of the Riemann tensor, Ricci tensor and the
curvature scalar by performing all internal (within a given Riemann tensor) contrac-
tions of indices. Then make the substitutions:
Rσρ →
Rσρ +�
, (33)
R+ 2�φ− (∂φ)2
. (34)
Finally, multiply the result by exp(−φ). The dilaton equation of motion, absent in the
pure gravity case, is simply L(n) = 0.
It occurs that the form of the Lagrangian and the tensor T
µν given in (31) and (32)
is very closely related to the string O(d, d) symmetry4. To show this, we consider the
3One could say that Eqs. (28) and (29) make no sense because they contain a sum of tensors of
different ranks. To make this mathematically sensible, we should consider a simple sum of spaces of
tensors of a given rank. Then the tensors in (28) and (29) should be understood as elements of such a
sum space with all but one components set to zero. Finally, the generalized traces T and T µν should
be further extended in such a way that when acting on an element of this big space they give the
result being the sum of generalized traces of all components.
4For a review on O(d, d) symmetry, see e.g. [29] and the references therein.
D–dimensional block–diagonal metric of the form
gµν =
g̃αβ 0
0 Gmn
, (35)
where α, β = 1, . . . , (D − d); m,n = (D − d + 1), . . . , D. We assume that the metric
components g̃αβ andGmn and the dilaton field φ do not depend on the last d coordinates
xm. In such a case, we obtain the following expressions for the second derivatives of
the dilaton
αφ = �̃
αφ , (36)
G−1∂αG
∂αφ , (37)
�φ = �̃φ+
(∂α ln detG) ∂
αφ , (38)
and for the Ricci tensor and the curvature scalar
Rβα = R̃
α ln detG−
G−1 (∂αG)G
, (39)
Rnm = −
(∂α ln detG)
G−1∂αG
G−1�̃G
G−1 (∂αG)G
−1 (∂αG)
, (40)
R = R̃ −
(∂α ln detG) (∂
α ln detG)− �̃ ln detG
G−1 (∂αG)G
−1 (∂αG)
, (41)
where tilde denotes quantities related to the (D−d)–dimensional metric g̃αβ, G should
be understood as a d×d matrix (and not its determinant) and Tr and det are the trace
and the determinant (acting on d× d matrices).
A necessary condition for the O(d, d) symmetry is that the dilaton field φ appear
only in the O(d, d) invariant combination
Φ = φ−
ln detG . (42)
Hence any derivative of the dilaton φmust be accompanied by an appropriate derivative
of [ln detG]. It is easy to see that there are only three combinations of Eqs. (36)–
(41) and the first derivatives of φ which depend on φ and [ln detG] only through the
combination Φ:
R+ 2�φ− (∂φ)2 = R̃+ 2�̃Φ− ∂αΦ∂αΦ−
G−1 (∂αG)G
−1 (∂αG)
, (43)
Rβα +�
αφ = R̃
α + �̃
G−1 (∂αG)G
, (44)
Rnm +�
(∂αΦ)
G−1∂αG
G−1∂αG
. (45)
These are exactly the combinations which, together with the Riemann tensor with
uncontracted indices, appear in the formulation given in Eqs. (31) and (32). Hence
the higher derivative contributions to the dilaton gravity theory found in the present
paper fulfill the necessary condition for the O(d, d) symmetry formulated before Eq.
(42). This does not mean yet that our theory is a part of some O(d, d) symmetric
theory. One should check whether all terms depending on G other than [ln detG] form
only O(d, d) invariant combinations. Actually, one can calculate that it is really the
case for n = 1 and n = 2. The lowest order theory was analyzed from this point of
view for the first time in [30]. Our second order Lagrangian L(2) differs from the one
found in [31] (for a vanishing tensor field H) by some total derivatives only. Thus, for
n = 1, 2 the equations of motion presented in Section 2 are the same as the dilaton
and gravity part of the equations obtained as appropriate approximations from the
superstring theories. The relation to the O(d, d) symmetry for n > 2 will be discussed
elsewhere [32].
The above discussion shows that Condition (iv) from Section 2.2 can be treated as
a necessary one for the dilaton gravity model to be part of some O(d, d) symmetric
theory. The reason is that there are no O(d, d) invariant expressions containing the
first derivatives of the dilaton other than the combination (∂µφ)(∂
6 Conclusions
We have generalized the Einstein–Lovelock theory by adding interactions with the
dilaton. The corresponding Einstein and dilaton equations of motion can be written
as series in the number of derivatives acting on the fields:
Tµν =
µν = 0 , (46)
(n) = 0 . (47)
The n–th contributions T
µν and W
(n) are sums of terms containing products of the
Riemann tensor and the first and second derivatives of the dilaton field. There are 2n
derivatives in each such term. We have found the most general equations of motion
satisfying Conditions (i)–(iv) given in Section 2.2. The first three conditions are the
standard properties of the dilaton gravity theories. The last one was added in order to
find the theories in which the dilaton and the metric are treated, as much as possible,
on the same footing. Accordingly, we assumed that the rank–1 tensor containing the
first derivatives of the dilaton can appear only in the scalar combination (∂µφ)(∂
µφ), as
there is no way to build an odd–rank tensor from the metric and the Riemann tensor.
It is necessary to specify the frame in which such a condition is to be fulfilled. We have
chosen the string frame where the n–th order term from the Einstein–Lovelock theory
is multiplied by e−φ. The reason is quite simple: symmetries relating the dilaton and
the metric are present in string–motivated theories.
We have shown that at each order T
µν and W
(n) are unique up to a normalization.
General expressions for T
µν and W
(n) for arbitrary n are given in Section 2.3. The
explicite formulae for n ≤ 3 are presented in the Appendix. It occurs that the higher
order dilaton gravity equations of motion have properties similar to those of the pure
Einstein–Lovelock gravity. Namely:
• There is an upper limit on the number of terms in (46)–(47) which can be non–
zero. For a D–dimensional space–time it is given by the inequality 2n ≤ D (the
corresponding limit for pure gravity is 2n < D)
• The equations of motion are quasi–linear in the second derivatives. This allows
us to treat them as a standard Cauchy initial conditions problem. It is crucial
also for the existence of brane–type solutions in the thin wall limit.
There is also another very interesting feature of those equations. The form of the scalar
and Einstein equations is very similar when written with the help of the generalized
Kronecker delta. The tensor T
µν can be obtained from the scalar W
(n) simply by
adding a pair of extra indices µ and ν to each generalized Kronecker delta and dividing
by 2.
Our dilaton gravity equations of motion can be obtained from an appropriate La-
grangian. Of course, such a Lagrangian can be determined only up to some total
derivatives. However, we have found that there is one particularly interesting form of
L = −W . (48)
Moreover, this Lagrangian is related in a simple way to the Einstein–Lovelock one
(the same is true also for the gravitational equations of motion). First, one has to
write the Einstein–Lovelock Lagrangian as a function of the Riemann tensor, the Ricci
tensor and the curvature scalar by performing all internal (within the same Riemann
tensor) contractions of indices. Next, one should replace the curvature scalar with the
combination R + 2�φ − (∂φ)2, and the Ricci tensor with Rσρ + �σρ . The result is the
dilaton gravity Lagrangian.
The property that the Lagrangian can be written in terms of only three tensors:
one scalar R+2�φ−(∂φ)2, one rank–2 tensor Rσρ+�σρ and the rank–4 Riemann tensor
is quite important. We have shown that this is a necessary condition for the dilaton
gravity to be a part of any string motivated theory with the O(d, d) symmetry. It turns
out that for n = 1, 2 it is also a sufficient one. The contributions L(1) and L(2) to our
Lagrangian are, up to total derivatives, the same as those found from demanding the
O(d, d) symmetry [30], [31]. It would be interesting to investigate the relation of L(n)
to string theories for n > 2 [32].
Most of the interesting features of the Lagrangian and the equations of motion
are visible in the string frame only. The theory looks more complicated in other
frames. For example, in the most often used Einstein frame there are no simple relations
between tensors built from the metric and from the dilaton derivatives and also many
coefficients become explicitly D–dependent. The advantages of the string frame should
not be surprising. For example, much more explicit solutions in the lowest order
dilaton gravity were found in the string frame [33] than in the Einstein one (discussions
concerning the relation between the string and the Einstein frames are reviewed in [34]).
Our results show that the string frame is the most convenient one to investigate dilaton
gravity also at higher orders.
Acknowledgments
The work of D.K. was partially supported by the EC Project MTKD-CT-2005-029466
“Particle Physics and Cosmology: the Interface” and by the Polish MEiN grant 1
P03B 099 29 for years 2005-2007. M.O. acknowledges partial support from the EU
6th Framework Program MRTN-CT-2004-503369 “Quest for Unification” and from
the Polish MNiSW grant N202 176 31/3844 for years 2006-2008. M.O. thanks for
hospitality experienced at Institute of Theoretical Physics of Heidelberg University
where part of this work has been done.
Appendix
The dilaton gravity Lagrangian and the corresponding equations of motion can be
written as a series in the number of derivatives
[D/2]
κnL(n) , (A.1)
T νµ =
[D/2]
µ = 0 , (A.2)
and the dilaton equation of motion W = −L = 0.
The 0–th order terms correspond to the cosmological constant:
eφL(0) = 1 , (A.3)
eφT ν(0)µ = −
δνµ . (A.4)
The 1–st order contribution is the standard Einstein gravity interacting with the dila-
eφL(1) = R+ 2�φ − (∂φ)2 , (A.5)
eφT ν(1)µ = −
φL(1) +
Rνµ +�
. (A.6)
The next two orders are given by the following expressions:
eφL(2) =
eφL(1)
Rρ2ρ1 +�
Rρ1ρ2 +�
+Rρ2ρ4ρ1ρ3R
, (A.7)
eφT ν(2)µ = −
φL(2) + 2
Rνµ +�
eφL(1) − 4
Rρµ +�
Rνρ +�
−4Rνρ2µρ1
Rρ1ρ2 +�
+ 2Rρ1ρ3µ ρ2R
, (A.8)
eφL(3) = 3
eφL(2)
eφL(1)
eφL(1)
Rρ2ρ1 +�
Rρ3ρ2 +�
Rρ1ρ3 +�
Rρ2ρ1 +�
Rρ4ρ3 +�
Rρ1ρ3ρ2ρ4 − 24
Rρ2ρ1 +�
Rρ1ρ5ρ3ρ4R
−8Rρ2ρ4ρ1ρ3R
Rρ3ρ5ρ4ρ6 + 2R
Rρ1ρ3ρ5ρ6R
, (A.9)
eφT ν(3)µ = −
φL(3) + 3
Rνµ +�
eφL(2) − 12R
Rρµ +�
Rνρ +�
−12RRνρ2µ ρ1
Rρ1ρ2 +�
+ 6RRρ1ρ3µ ρ2R
Rρ1µ +�
Rνρ2 +�
Rρ2ρ1 +�
+24Rνρ2µ ρ1
Rρ3ρ1 +�
Rρ2ρ3 +�
+ 24Rνρ2µ ρ1R
Rρ3ρ4 +�
+24Rρ1ρ3µ ρ2
Rνρ1 +�
Rρ2ρ3 +�
− 12Rρ1ρ3µ ρ2R
Rνρ4 +�
+24Rν ρ3ρ1ρ2
Rρ1µ +�
Rρ2ρ3 +�
− 12Rν ρ3ρ1ρ2R
Rρ4µ +�
−24Rρ1ρ3µ ρ2R
Rρ4ρ3 +�
− 12Rρ1ρ3µ ρ2R
Rρ4ρ3 +�
−12Rνρ2µ ρ1R
Rρ3ρ4ρ2ρ5 + 6R
Rν ρ2ρ4ρ5R
− 24Rρ1ρ3µ ρ2R
Rρ4ρ2ρ1ρ5 . (A.10)
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  A higher order theory of dilaton gravity is constructed as a generalization
of the Einstein-Lovelock theory of pure gravity. Its Lagrangian contains terms
with higher powers of the Riemann tensor and of the first two derivatives of
the dilaton. Nevertheless, the resulting equations of motion are quasi-linear
in the second derivatives of the metric and of the dilaton. This property is
crucial for the existence of brane solutions in the thin wall limit. At each
order in derivatives the contribution to the Lagrangian is unique up to an
overall normalization. Relations between symmetries of this theory and the
O(d,d) symmetry of the string-inspired models are discussed.

Introduction
The equations of motion in the Einstein theory of gravity in 4 space–time dimensions
are the most general divergence–free tensor (rank 2) equations bilinear in the first
derivatives and linear in the second derivatives of the metric. They can be obtained
from the Hilbert–Einstein action which is linear in the Riemann tensor. In more than
4 space–time dimensions, this theory can be generalized to contain higher powers of
the Riemann tensor in the action. The corresponding equations of motion involve
higher powers of the first derivatives of the metric and are quasi–linear in the second
derivatives (all terms are at most linear in the second derivatives, while multiplied by
powers of the first derivatives). It has been shown that the contribution to the action
of a given order in the Riemann tensor is unique up to an overall normalization. The
quadratic contribution is called the Gauss–Bonnet action or the Lanczos action [1]. It
has been generalized to higher orders by Lovelock [2]. The quasi–linearity is a very
important feature of the Einstein–Lovelock equations of motion. It guarantees that
they can be formulated as a Cauchy problem with some constraints on the initial data
[3]. On the other hand, it is crucial for the existence of non–singular domain wall
solutions in the thin wall limit. This problem for arbitrary order in derivatives was
discussed in [4]. Many aspects of the Einstein–Lovelock gravity were discussed in the
literature1.
Higher derivative corrections to the gravity interactions are present in effective
Lagrangians obtained from string theories. The first correction has exactly the form
of the Gauss–Bonnet term [9], [10]. The lowest order dilaton interactions were added
to the Gauss–Bonnet theory in [11]. However, the α′ expansion in string theories
predicts higher derivative corrections not only for the gravitational interactions, such
corrections appear also for the dilaton. The effective action for the dilaton gravity
with terms up to four derivatives was given in [12], [13]. The effective action with six
derivatives was presented in [14], but its gravitational part has a form different from
that of the corresponding Einstein–Lovelock action.
The dilaton gravity at the field theory level has been investigated by many authors.
Some of them included also certain higher order corrections. Yet in most cases such
corrections were considered only for gravitational interactions and not for the dilaton.
Some higher derivative corrections for both the dilaton and the gravitational interac-
tions were considered in [15]–[21] (certain Riemann tensor combinations with dilaton
dependent coefficients were analyzed in [22]–[25]). The terms predicted by superstrings
up to four derivatives have also been considered in [26]–[28].
The purpose of the present work is to find a generalization of the lowest order
dilaton gravity theory to an arbitrary order in derivatives. We start with the Einstein–
Lovelock higher order gravity and couple it to the dilaton. There are many ways to do
this but we are only interested in the theories where dilaton and gravity interactions
1Quasi–linearity of the Einstein–Gauss–Bonnet theory was reviewed in [5]. A discussion of general
quasi–linear differential equations can be found in [6]. For a review on brane–world gravity see eg. [7].
For a discussion of the Lovelock gravity in the context of the equivalence of the Palatini and metric
formulations see eg. [8].
are as similar to each other as possible. Equations of motion in such a theory are
presented in Section 2. We begin with formulating the conditions which should be
fulfilled by such equations. Most of them are simple generalizations of the conditions
fulfilled by the Einstein–Lovelock equations of motion. One condition is added in order
to eliminate at least some of the possible theories in which the dilaton interactions are
not related to the gravitational ones. The equations of motion satisfying all those
conditions are constructed in Subsection 2.3. It turns out that at each order those
equations are unique up to a numerical normalization. Moreover, they can be obtained
by the standard Euler–Lagrange procedure from the Lagrangian presented in Section 3.
Section 4 contains the proof that our equations of motion are quasi–linear in the second
derivatives of both the metric and the dilaton. The relation between the gravity and
the dilaton interactions is discussed in Section 5. We point out that the Lagrangian of
our higher order dilaton gravity can be obtained in a simple way from the pure gravity
Einstein–Lovelock Lagrangian. We also discuss the relation of the resulting theory to
the O(d, d) symmetric theories. We conclude in Section 6. The Appendix contains the
explicit formulae for the Lagrangian and the equations of motion up to terms of the
sixth order in derivatives.
2 Equations of motion
2.1 Notation
Let us start with introducing certain generalizations of the Kronecker delta and the
trace operator which will be used later to make the formulae more compact. The
generalized Kronecker delta is defined by
j1j2···jn
i1i2···in
= det
· · · δj1in
· · · · · ·
· · · · · ·
· · · δjnin
, (1)
and should be only employed when the spacetime dimensionality D is sufficient: D ≥ n.
Using this definition it is easy to prove some relations among Kronecker deltas of
different order. For example:
ν j1j2...jn
µ i1i2...in
= δνµδ
j1j2...jn
i1i2...in
− δνi1δ
j1j2...jn
µ i2...in
− δνi2δ
j1j2...jn
i1µ ...in
− . . .− δνinδ
j1j2...jn
i1i2...µ
. (2)
The generalized Kronecker delta can be used to define the following trace–like linear
mapping from tenors of rank (n, n) into numbers
T (M) = δj1j2···jni1i2···in M
i1i2···in
j1j2···jn
, (3)
which reduces to the ordinary trace for n = 1. We will also employ an extension of
this operation which maps tensors of rank (n, n) into tensors of rank (1, 1):
T νµ (M) = δ
ν j1j2···jn
µ i1i2···in
M i1i2···inj1j2···jn . (4)
In the following we will often use T and T evaluated for products of tensors. In order
to clearly distinguish between tensors and their contracted counterparts, we will use
∗ indices to indicate the rank of a tensor. For example, R∗∗∗∗ denotes the rank (2, 2)
Riemann tensor, and �∗∗φ denotes the rank (1, 1) second derivative of the dilaton,
while R is the Ricci scalar and �φ the D’Alembertian acting on the dilaton. Thus, for
example,
(R∗∗∗∗)
(�∗∗φ)
= δσ1σ2σ3σ4σ5σ6ρ1ρ2ρ3ρ4ρ5ρ6 R
Rρ3ρ3σ3σ4 �
φ�ρ6σ6φ , (5)
where we used the notation Rρ1ρ2σ1σ2 = R
σ1σ2 and �
σφ = ∇ρ∂σφ to make the formula
more compact. It is easy to see that the sequence of tensors appearing in the product
argument of T is not important. Changing such an order is equivalent to interchang-
ing the appropriate columns of indices in the generalized Kronecker delta. On the
other hand, interchanging two such columns of indices is equivalent to interchanging
the corresponding 2 rows and 2 columns in the determinant in Definition (1). Each
interchange of two columns (or two rows) changes the sign of the determinant, hence
an even number of interchanges leaves the determinant unchanged.
2.2 Conditions
Now we want to construct the n–th order dilaton gravity equations of motion. They
are to be of the form
T (n)µν = 0 ,
W (n) = 0 , (6)
where the tensor T
µν and the scalar W
(n) satisfy the following conditions
(i) They are combinations of terms with exactly 2n derivatives acting on the metric
tensor gµν and on the dilaton field φ. There are no derivatives higher than second
acting on one object;
(ii) Tensor T
µν is symmetric in its indices;
(iii) The covariant derivative of the tensor is proportional to the scalar:
µ = const · (∂µφ)W (n) (the energy–momentum tensor is covariantly con-
served if the dilaton equation of motion is fulfilled).
It is clear that the above conditions are not sufficient to determine something which
could be regarded as an extension of the higher order gravity theory to the dilaton
gravity case. For example, all the above conditions are fulfilled by the Einstein–Gauss–
Bonnet gravity with only the lowest order terms for the dilaton. We are interested in
a theory where the dilaton and the metric are treated in a more symmetric way. It
is not obvious how such a symmetry should be defined, because it ought to relate a
scalar to a second rank tensor. Or, more precisely, it is supposed to relate the first
and second derivatives of the scalar field to the Riemann tensor and its contractions.
A simple observation concerning the gravity part is that it contains even–rank tensors
only. On the other hand, the first derivative of a scalar is a rank–1 tensor. Hence
one can expect that in a gravity–dilaton symmetric theory, the first derivative of the
dilaton appears only as a 0–rank tensor: gµν∂µφ∂νφ. However, the feature mentioned
above is not invariant under change of variables. Thus, we should specify in which
frame it is fulfilled. The theory which relates dilaton to gravity is the string theory so
the string frame seems to be a natural choice. Hence our last condition reads:
(iv) In the string–like frame, in which the pure gravity term is multiplied by exp(−φ),
the first order derivatives of the dilaton appear in the combination (∂µφ)(∂
only.
The relation of this condition to the O(d, d) symmetry present in many string–inspired
theories will be discussed in Section 5.
2.3 Construction
We start our construction with a term in T
µ where all 2n derivatives act on the metric
tensors. The only pure gravity tensor satisfying Conditions (i)–(iii) (with W (n) = 0) is,
up to normalization, equal to the n-th order Lovelock tensor [2]. Because of Condition
(iv), it is most natural to work in the frame in which the gravity term is multiplied by
exp(−φ). Consequently, the tensor T ν(n)µ starts with
T ν (n)µ = −2
−(n+1)e−φδν σ1...σ2nµ ρ1...ρ2nR
· · ·Rρ2n−1ρ2nσ2n−1σ2n + . . . (7)
The reason for such a normalization will be explained in the next section. Calculating
the divergence of (7), we get
∇νT ν (n)µ = 2
−(n+1)e−φ(∂νφ)δ
ν σ1...σ2n
µρ1...ρ2n
Rρ1ρ2σ1σ2 · · ·R
ρ2n−1ρ2n
σ2n−1σ2n
+ . . . (8)
The above term is produced when the derivative acts on e−φ (derivatives of the Riemann
tensor do not contribute due to the Bianchi identity).
The first term in T
µ shown explicitly in (7) can not be the only one. The reason
is that the r.h.s. of (8) is not a product of ∂µφ and a scalar, as Condition (iii) requires.
Using Eq. (2) we can rewrite the r.h.s. of (8) as a combination of (2n + 1) terms.
The one containing the first term from the r.h.s. of (2) is of the desired form but the
remaining 2n terms have different structures of the index contractions. It turns out
that similar terms are also present in the following covariant derivative
e−φδν σ1...σ2n−1µρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2�
ρ2n−1
σ2n−1
= −e−φ(∂νφ)δν σ1...σ2n−1µ ρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2�
ρ2n−1
σ2n−1
+e−φδν σ1...σ2n−1µρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2
∇ν∇ρ2n−1∂σ2n−1φ
. (9)
The second term on the r.h.s. may be rewritten as
e−φδν σ1...σ2n−1µ ρ1...ρ2n−1R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2
δσ2nρ2nR
ρ2n−1ρ2n
σ2n−1ν
∂σ2nφ
e−φ(∂νφ)δ
δσ1...σ2n−1σ2nρ1...ρ2n−1µ R
· · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2R
ρ2n−1ρ2n
σ2n−1σ2n
, (10)
where in the last step we interchanged the names of the contracted indices ν and
σ2n and rearranged the indices in the generalized Kronecker delta. A term exactly of
this structure must be added to (8) in order to obtain an expression proportional to
∂µφ. From Eqs. (2) and (8) it follows that the coefficient should be equal to (−n2−n)
instead of the (−1/2) present in (10). This fixes the coefficient of the term in T ν(n)µ
which contains (n− 1) Riemann tensors and one second derivative of the dilaton. Now
we know the first two terms of the tensor T
µ . Using the notation introduced in (3)
and (4), they can be written as:
T ν (n)µ = −2
−(n+1)e−φT νµ ((R
)− 2−(n−1)ne−φT νµ
(R∗∗∗∗)
(n−1)
+ . . . (11)
Their covariant derivative reads
∇νT ν (n)µ = 2
−(n+1)e−φ(∂µφ)T ((R∗∗∗∗)
+2−(n−1)ne−φ(∂νφ)T
(R∗∗∗∗)
(n−1)
+ . . . (12)
The first term has the structure required by Condition (iii) and determines the first
term of the scalar equation of motion2 W (n). However, the second term in (12) is not
of the appropriate structure. It means that some additional terms, whose covariant
derivatives are products of (n− 1) Riemann tensors with one second derivative of the
dilaton, are necessary in T
µ . Two such terms are possible:
−φT νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
+ c4e
−φT νµ
(R∗∗∗∗)
(n−1)
(∂φ)2 . (13)
However, it is not enough to have terms with appropriate powers of the Riemann
tensor and the dilaton, because their covariant divergences must contain the correct
combinations of the generalized Kronecker deltas. To check whether this is possible, we
calculate the covariant divergence of (13). When the derivative acts on �∗∗φ in the first
term in (13), it gives an additional Riemann tensor multiplied by ∂φ and a pair of new
indices. Those new indices are contracted with just one ordinary Kronecker delta and
are not under the overall antisymmetrization. Similarly, when the covariant derivative
acts on (∂φ)2 in the second term in (13), it gives the second derivative of the dilaton
multiplied by ∂φ and a pair of new indices. Those two covariant derivatives should
combine with the second term on the r.h.s. of (12) to give an expression proportional
to ∂µφ. This fixes the numerical coefficients c3 and c4. The explicit calculation gives
c3 = −2(2−n)n(n− 1), c4 = 2−nn. Thus, we have found the first four terms of T
T ν (n)µ = −2
−(n+1)e−φ
T νµ ((R
) + 4nT νµ
(R∗∗∗∗)
(n−1)
+8n(n− 1)T νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
−2nT νµ
(R∗∗∗∗)
(n−1)
(∂φ)2
+ . . . (14)
2up to an overall normalization. The choice of the relative normalizations of T
µν and W
(n) shall
become clear when the Lagrangian is introduced in Section 3.
The covariant divergence of those terms reads
∇νT ν (n)µ = ∂µφ
2−(n+1)e−φ
T ((R∗∗∗∗)
) + 4nT
(R∗∗∗∗)
(n−1)
+∂νφ 2
(2−n)n(n− 1)e−φT νµ
(R∗∗∗∗)
(n−2)
(�∗∗φ)
−∂νφ 2−nne−φT
(R∗∗∗∗)
(n−1)
(∂φ)2 + . . . (15)
The terms in the curly bracket above are the first two terms of the scalar W (n) we are
looking for.
Equation (15) shows that the procedure of finding T
µ and W
(n) must be contin-
ued. The last two terms on the r.h.s. of (15) do not have the required form, so more
terms must be added to T
µ . From the steps described so far, it should be clear that
each of such new terms must contain exactly 3 (first or second order) derivatives of the
dilaton. There are two such terms:
−φT νµ
(R∗∗∗∗)
(n−3)
(�∗∗φ)
+ c6e
−φT νµ
(R∗∗∗∗)
(n−2)
(∂φ)2 . (16)
The coefficients c5 and c6 can be fixed in the same way as c3 and c4.
This procedure can be continued step by step for the terms containing higher and
higher powers of the dilaton field with the derivatives acting on it. Eventually, one
obtains the term with the maximal number of dilaton fields, namely c e−φδνµ [(∂φ)
This is the first term in T
µ , the covariant derivative of which need not to be corrected
by contributions from any additional terms. This covariant derivative reads
e−φδνµ
(∂φ)2
= −(∂µφ)e−φ
(∂φ)2
+ 2ne−φ(∇µ∂σφ)(∂σφ)
(∂φ)2
](n−1)
. (17)
The second term on the r.h.s. is used to cancel some unwanted part of
e−φT νµ (�∗∗φ) [(∂φ)2]
(n−1)
, which fixes c to be equal to 1
(−1)(n+1). The first term
on the r.h.s. of (17) has already the required structure of the product of ∂µφ and a
scalar. Thus, the procedure can stop here.
The above iterative procedure gives T
µν and W
(n) satisfying all the four imposed
conditions. The resulting gravitational and dilaton equations of motion can be written
in the following relatively simple form:
T (n)µν = −
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
= 0 , (18)
W (n) = −e−φ
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
= 0 . (19)
The existence of T
µν and W
(n) is a non–trivial result, because in our iterative
procedure there are more conditions than available constants. A priori it could happen
that there were no solutions other than a trivial one with vanishing T
µν and W
However, the solution exists and is unique up to an overall normalization. Hence any
dilaton gravity equations of motion, satisfying Conditions (i)–(iv), which contain at
least one term present in (18) and (19) must also contain all the other terms with
uniquely determined coefficients.
3 Lagrangian
It is interesting to check whether the equations of motion constructed in Section 2
can be obtained from some D–dimensional action. In such case, T
µν and W
(n) would
satisfy
δgµνS
(n) = δgµν
−gL(n) =
−g T (n)µν δg
µν , (20)
(n) = δφ
−gL(n) =
−gW (n)δφ . (21)
It turns out that indeed the equations of motion (18) and (19) can be obtained from
the action with the Lagrangian density given by
L(n) = e−φ
2b−an!
a!b!(n− a− b)!
(R∗∗∗∗)
(�∗∗φ)
−(∂φ)2
)n−a−b
. (22)
It is important to underline that for Conditions (i)–(iv) not to be violated, the terms
coming from the n-th Lagrangian can appear only in the space–times with dimension-
ality D ≥ 2n. Moreover, one should be careful when calculating (20) for D = 2n, as
the generalized Kronecker delta (1) can not be employed in (18) for the term of the
highest order in the Riemann tensor. The coefficient of that term should be replaced
δν σ1σ2...σ2nµ ρ1ρ2...ρ2n −→D=2n δ
σ1σ2...σ2n
ρ1ρ2...ρ2n
− δνρ1δ
σ1σ2...σ2n
µ ρ2...ρ2n
− δνρ2δ
σ1σ2...σ2n
ρ1µ ...ρ2n
− . . .− δνρ2nδ
σ1σ2...σ2n
ρ1ρ2...µ
. (23)
Now we can comment on the overall normalization of the tensors T
µν . The reason
for this particular normalization is that the term e−φRn (with R being the Ricci scalar)
appears in the Lagrangian with the coefficient 1. This corresponds to the standard
normalizations of the Hilbert–Einstein and Gauss–Bonnet Lagrangians.
Proving that the equations of motion derived from the Lagrangian (22) really have
the form (6) with T
µν and W
(n) as given in (18) and (19) is a straightforward but
quite tedious calculation. One of the reasons is that apparently several integrations
by parts are required. This can be somewhat simplified if one observes that not all
those integrations by parts have to be performed explicitly. In case of (21), the reason
is as follows. Under the integral (21) there are first (second) derivatives of δφ coming
from the variation of the first (second) derivatives of the dilaton. In general, the terms
containing second derivatives of δφ should be integrated by parts twice. However, one
can notice that the result of a single integration and the terms containing the first
derivatives of δφ cancel each other exactly.
The situation is a little bit more complicated in case of the gravitational equation
of motion. Under the integral (20), there are second derivatives of δgµν coming from
the variation of the Riemann tensor and first derivatives of δgµν coming from the
variation of the second covariant derivative of the dilaton. Similarly as in the case of
the dilaton equation of motion, the terms containing second derivatives of δgµν have to
be integrated by parts only once. And although the cancellation of the resulting terms
is not complete this time, only some residual integration by parts has to be performed
additionally.
Of course, the Lagrangian density (22) is not unique. First, one can rewrite L(n)
changing the variables gµν and φ. Second, one can add to L(n) any total divergence
without changing the resulting equations of motion. However, the form given in Eq.
(22) is especially simple and interesting. It is very similar to the form of T
µν and
W (n). The energy momentum tensor T
µν can be obtained from L(n) by replacing
the generalized trace T with its tensor extension T µν and multiplying the result by
−1/2. In case of the dilaton equation of motion, the analogous relation is even simpler:
W (n) = −L(n).
We were not able to find any other similarly simple form of the Lagrangian by
adding total derivative terms or by changing the variables. For example, we examined
the form of the Lagrangian and of the equations of motion in the Einstein–like frame
in which the common factor e−φ is absorbed by a suitable Weyl transformation. The
results are very complicated and will not be presented here. One of the reasons for
such complications is that the Weyl transformation depends on the dimensionality D
of the space–time. Thus, many different functions of D appear in the Einstein frame,
while there is no explicit dependence on D in our string–like frame.
4 Quasi-linearity
It is easy to show that the equations of motion (18)–(19) are quasi–linear in the second
derivatives of the metric and the dilaton. Let us introduce in the D–dimensional space–
time a (D− 1)–dimensional hypersurface Σ defined by its unit normal vector nµ. The
metric induced at this hypersurface is given by
hµν = gµν −
, (24)
where n2 = nρn
ρ. The components of the D–dimensional Riemann tensor R∗∗∗∗ corre-
sponding to the full metric gµν can be expressed as
Rρσµν = R
µν −n
+ 4n[ρD[µK
+ 4n[µD
4n[µn
Kτν] − 4n[µn
, (25)
where: R is the (D − 1)–dimensional Riemann tensor corresponding to the induced
metric hµν ; K is the extrinsic curvature given by
Kµν =
£nhµν ; (26)
Dµ is the covariant derivative with respect to the induced metric hµν ; £n is the Lie
derivative along the vector field nµ. Similarly we can write the D–dimensional second
covariant (with respect to the metric gµν) derivative of the dilaton
∇µ∇νφ = DµDνφ +n−2
Kµν£nφ+ 2n(µDν)£nφ− 2n(µKτν)Dτφ
+n−4 nµnν
nφ− (n
ρ∇ρnτ )∇τφ
. (27)
We want to check how the second Lie derivatives of the metric hµν (present in
£nKµν) and of the dilaton φ appear in the equations of motion (18) and (19). Such
second derivatives are present in (25) and (27) but in both cases they are multiplied
by coefficients bilinear in the vector n. After substituting the decompositions (25) and
(27) into (18) and (19), one can immediately see that, due to the antisymmetrization
present in T
µν and W
(n), the equations of motion contain terms at most bilinear in n.
Thus, the equations of motion (18) and (19) contain terms at most linear in the second
Lie derivatives of hµν and φ.
We have shown that the equations of motion are quasi–linear in the second Lie
derivatives “perpendicular” to the hypersurface Σ. This quasi–linearity has very im-
portant consequences. For Σ with a time–like n, this allows us to define a standard
Cauchy problem with the initial conditions (values and first Lie derivatives of hµν and
φ) given at Σ. For a space–like n, the quasi–linearity is necessary to have non–singular
brane solutions even in the thin wall limit.
5 Symmetries
The equations of motion presented in Section 2 were obtained assuming some kind of
symmetry between the metric and the dilaton. Now we are in a position to investigate
such a symmetry in more detail.
It is quite amazing that the Lagrangian (22) as well as the equations of motion
(18) and (19) can be expressed as functions of n–th perfect “power” of one simple
n–independent quantity. Namely:
L(n) = −W (n) = e−φT
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
, (28)
T (n)µν = −
e−φT µν
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
. (29)
One can treat these equations as just a new notation allowing us to rewrite the double
sums from (18), (19) and (22) in a compact way3. Yet, on the other hand, it helps
to show that the action and the equations of motion depend on some combinations
of the dilaton derivatives and tensors obtained from the metric only. In each round
parenthesis in Eqs. (28) and (29), there are the rank–4 Riemann tensor, the rank–2
tensor of the second derivatives of the dilaton and the rank–0 tensor built from the
first derivatives of the dilaton:
R∗∗∗∗ ⊕ 2�
∗φ ⊕ (−1) (∂φ)
. (30)
All those tensors are under the generalized traces T and T µν . Some of the terms
present in these mappings contain traces of the tensors from (30). There are two
different rank–2 tensors coming from (30). The first is just �∗∗φ. The second is the
Ricci tensor R∗∗ which can be obtained from the Riemann tensor by contraction of its
two indices. There are four different ways to contract one pair of indices in R∗∗∗∗, thus
in the final result the rank–2 tensors appear always in the combination (2R∗∗ + 2�∗∗φ).
There are three different scalars originating from (30): (∂φ)
, �φ and the curvature
scalar R. There are two different constructions giving R, so the final results depend
on a single following scalar combination: R+ 2�φ− (∂φ)2.
The above observation allows us to relate our dilaton gravity equations to the
corresponding equations in the pure Einstein–Lovelock gravity:
L(n) = −W (n) = e−φL(n)E−L
R∗∗∗∗ , (R
∗φ) ,
R+ 2�φ− (∂φ)2
, (31)
T (n)µν = e
gµν ,R∗∗∗∗ , (R
∗φ) ,
R+ 2�φ− (∂φ)2
. (32)
The recipe for the higher order dilaton gravity can be as follows: Start with the higher
order pure gravity Einstein–Lovelock theory. Write the Lagrangian density L(n)E−L and
the equations of motion (T
E−L)µν in terms of the Riemann tensor, Ricci tensor and the
curvature scalar by performing all internal (within a given Riemann tensor) contrac-
tions of indices. Then make the substitutions:
Rσρ →
Rσρ +�
, (33)
R+ 2�φ− (∂φ)2
. (34)
Finally, multiply the result by exp(−φ). The dilaton equation of motion, absent in the
pure gravity case, is simply L(n) = 0.
It occurs that the form of the Lagrangian and the tensor T
µν given in (31) and (32)
is very closely related to the string O(d, d) symmetry4. To show this, we consider the
3One could say that Eqs. (28) and (29) make no sense because they contain a sum of tensors of
different ranks. To make this mathematically sensible, we should consider a simple sum of spaces of
tensors of a given rank. Then the tensors in (28) and (29) should be understood as elements of such a
sum space with all but one components set to zero. Finally, the generalized traces T and T µν should
be further extended in such a way that when acting on an element of this big space they give the
result being the sum of generalized traces of all components.
4For a review on O(d, d) symmetry, see e.g. [29] and the references therein.
D–dimensional block–diagonal metric of the form
gµν =
g̃αβ 0
0 Gmn
, (35)
where α, β = 1, . . . , (D − d); m,n = (D − d + 1), . . . , D. We assume that the metric
components g̃αβ andGmn and the dilaton field φ do not depend on the last d coordinates
xm. In such a case, we obtain the following expressions for the second derivatives of
the dilaton
αφ = �̃
αφ , (36)
G−1∂αG
∂αφ , (37)
�φ = �̃φ+
(∂α ln detG) ∂
αφ , (38)
and for the Ricci tensor and the curvature scalar
Rβα = R̃
α ln detG−
G−1 (∂αG)G
, (39)
Rnm = −
(∂α ln detG)
G−1∂αG
G−1�̃G
G−1 (∂αG)G
−1 (∂αG)
, (40)
R = R̃ −
(∂α ln detG) (∂
α ln detG)− �̃ ln detG
G−1 (∂αG)G
−1 (∂αG)
, (41)
where tilde denotes quantities related to the (D−d)–dimensional metric g̃αβ, G should
be understood as a d×d matrix (and not its determinant) and Tr and det are the trace
and the determinant (acting on d× d matrices).
A necessary condition for the O(d, d) symmetry is that the dilaton field φ appear
only in the O(d, d) invariant combination
Φ = φ−
ln detG . (42)
Hence any derivative of the dilaton φmust be accompanied by an appropriate derivative
of [ln detG]. It is easy to see that there are only three combinations of Eqs. (36)–
(41) and the first derivatives of φ which depend on φ and [ln detG] only through the
combination Φ:
R+ 2�φ− (∂φ)2 = R̃+ 2�̃Φ− ∂αΦ∂αΦ−
G−1 (∂αG)G
−1 (∂αG)
, (43)
Rβα +�
αφ = R̃
α + �̃
G−1 (∂αG)G
, (44)
Rnm +�
(∂αΦ)
G−1∂αG
G−1∂αG
. (45)
These are exactly the combinations which, together with the Riemann tensor with
uncontracted indices, appear in the formulation given in Eqs. (31) and (32). Hence
the higher derivative contributions to the dilaton gravity theory found in the present
paper fulfill the necessary condition for the O(d, d) symmetry formulated before Eq.
(42). This does not mean yet that our theory is a part of some O(d, d) symmetric
theory. One should check whether all terms depending on G other than [ln detG] form
only O(d, d) invariant combinations. Actually, one can calculate that it is really the
case for n = 1 and n = 2. The lowest order theory was analyzed from this point of
view for the first time in [30]. Our second order Lagrangian L(2) differs from the one
found in [31] (for a vanishing tensor field H) by some total derivatives only. Thus, for
n = 1, 2 the equations of motion presented in Section 2 are the same as the dilaton
and gravity part of the equations obtained as appropriate approximations from the
superstring theories. The relation to the O(d, d) symmetry for n > 2 will be discussed
elsewhere [32].
The above discussion shows that Condition (iv) from Section 2.2 can be treated as
a necessary one for the dilaton gravity model to be part of some O(d, d) symmetric
theory. The reason is that there are no O(d, d) invariant expressions containing the
first derivatives of the dilaton other than the combination (∂µφ)(∂
6 Conclusions
We have generalized the Einstein–Lovelock theory by adding interactions with the
dilaton. The corresponding Einstein and dilaton equations of motion can be written
as series in the number of derivatives acting on the fields:
Tµν =
µν = 0 , (46)
(n) = 0 . (47)
The n–th contributions T
µν and W
(n) are sums of terms containing products of the
Riemann tensor and the first and second derivatives of the dilaton field. There are 2n
derivatives in each such term. We have found the most general equations of motion
satisfying Conditions (i)–(iv) given in Section 2.2. The first three conditions are the
standard properties of the dilaton gravity theories. The last one was added in order to
find the theories in which the dilaton and the metric are treated, as much as possible,
on the same footing. Accordingly, we assumed that the rank–1 tensor containing the
first derivatives of the dilaton can appear only in the scalar combination (∂µφ)(∂
µφ), as
there is no way to build an odd–rank tensor from the metric and the Riemann tensor.
It is necessary to specify the frame in which such a condition is to be fulfilled. We have
chosen the string frame where the n–th order term from the Einstein–Lovelock theory
is multiplied by e−φ. The reason is quite simple: symmetries relating the dilaton and
the metric are present in string–motivated theories.
We have shown that at each order T
µν and W
(n) are unique up to a normalization.
General expressions for T
µν and W
(n) for arbitrary n are given in Section 2.3. The
explicite formulae for n ≤ 3 are presented in the Appendix. It occurs that the higher
order dilaton gravity equations of motion have properties similar to those of the pure
Einstein–Lovelock gravity. Namely:
• There is an upper limit on the number of terms in (46)–(47) which can be non–
zero. For a D–dimensional space–time it is given by the inequality 2n ≤ D (the
corresponding limit for pure gravity is 2n < D)
• The equations of motion are quasi–linear in the second derivatives. This allows
us to treat them as a standard Cauchy initial conditions problem. It is crucial
also for the existence of brane–type solutions in the thin wall limit.
There is also another very interesting feature of those equations. The form of the scalar
and Einstein equations is very similar when written with the help of the generalized
Kronecker delta. The tensor T
µν can be obtained from the scalar W
(n) simply by
adding a pair of extra indices µ and ν to each generalized Kronecker delta and dividing
by 2.
Our dilaton gravity equations of motion can be obtained from an appropriate La-
grangian. Of course, such a Lagrangian can be determined only up to some total
derivatives. However, we have found that there is one particularly interesting form of
L = −W . (48)
Moreover, this Lagrangian is related in a simple way to the Einstein–Lovelock one
(the same is true also for the gravitational equations of motion). First, one has to
write the Einstein–Lovelock Lagrangian as a function of the Riemann tensor, the Ricci
tensor and the curvature scalar by performing all internal (within the same Riemann
tensor) contractions of indices. Next, one should replace the curvature scalar with the
combination R + 2�φ − (∂φ)2, and the Ricci tensor with Rσρ + �σρ . The result is the
dilaton gravity Lagrangian.
The property that the Lagrangian can be written in terms of only three tensors:
one scalar R+2�φ−(∂φ)2, one rank–2 tensor Rσρ+�σρ and the rank–4 Riemann tensor
is quite important. We have shown that this is a necessary condition for the dilaton
gravity to be a part of any string motivated theory with the O(d, d) symmetry. It turns
out that for n = 1, 2 it is also a sufficient one. The contributions L(1) and L(2) to our
Lagrangian are, up to total derivatives, the same as those found from demanding the
O(d, d) symmetry [30], [31]. It would be interesting to investigate the relation of L(n)
to string theories for n > 2 [32].
Most of the interesting features of the Lagrangian and the equations of motion
are visible in the string frame only. The theory looks more complicated in other
frames. For example, in the most often used Einstein frame there are no simple relations
between tensors built from the metric and from the dilaton derivatives and also many
coefficients become explicitly D–dependent. The advantages of the string frame should
not be surprising. For example, much more explicit solutions in the lowest order
dilaton gravity were found in the string frame [33] than in the Einstein one (discussions
concerning the relation between the string and the Einstein frames are reviewed in [34]).
Our results show that the string frame is the most convenient one to investigate dilaton
gravity also at higher orders.
Acknowledgments
The work of D.K. was partially supported by the EC Project MTKD-CT-2005-029466
“Particle Physics and Cosmology: the Interface” and by the Polish MEiN grant 1
P03B 099 29 for years 2005-2007. M.O. acknowledges partial support from the EU
6th Framework Program MRTN-CT-2004-503369 “Quest for Unification” and from
the Polish MNiSW grant N202 176 31/3844 for years 2006-2008. M.O. thanks for
hospitality experienced at Institute of Theoretical Physics of Heidelberg University
where part of this work has been done.
Appendix
The dilaton gravity Lagrangian and the corresponding equations of motion can be
written as a series in the number of derivatives
[D/2]
κnL(n) , (A.1)
T νµ =
[D/2]
µ = 0 , (A.2)
and the dilaton equation of motion W = −L = 0.
The 0–th order terms correspond to the cosmological constant:
eφL(0) = 1 , (A.3)
eφT ν(0)µ = −
δνµ . (A.4)
The 1–st order contribution is the standard Einstein gravity interacting with the dila-
eφL(1) = R+ 2�φ − (∂φ)2 , (A.5)
eφT ν(1)µ = −
φL(1) +
Rνµ +�
. (A.6)
The next two orders are given by the following expressions:
eφL(2) =
eφL(1)
Rρ2ρ1 +�
Rρ1ρ2 +�
+Rρ2ρ4ρ1ρ3R
, (A.7)
eφT ν(2)µ = −
φL(2) + 2
Rνµ +�
eφL(1) − 4
Rρµ +�
Rνρ +�
−4Rνρ2µρ1
Rρ1ρ2 +�
+ 2Rρ1ρ3µ ρ2R
, (A.8)
eφL(3) = 3
eφL(2)
eφL(1)
eφL(1)
Rρ2ρ1 +�
Rρ3ρ2 +�
Rρ1ρ3 +�
Rρ2ρ1 +�
Rρ4ρ3 +�
Rρ1ρ3ρ2ρ4 − 24
Rρ2ρ1 +�
Rρ1ρ5ρ3ρ4R
−8Rρ2ρ4ρ1ρ3R
Rρ3ρ5ρ4ρ6 + 2R
Rρ1ρ3ρ5ρ6R
, (A.9)
eφT ν(3)µ = −
φL(3) + 3
Rνµ +�
eφL(2) − 12R
Rρµ +�
Rνρ +�
−12RRνρ2µ ρ1
Rρ1ρ2 +�
+ 6RRρ1ρ3µ ρ2R
Rρ1µ +�
Rνρ2 +�
Rρ2ρ1 +�
+24Rνρ2µ ρ1
Rρ3ρ1 +�
Rρ2ρ3 +�
+ 24Rνρ2µ ρ1R
Rρ3ρ4 +�
+24Rρ1ρ3µ ρ2
Rνρ1 +�
Rρ2ρ3 +�
− 12Rρ1ρ3µ ρ2R
Rνρ4 +�
+24Rν ρ3ρ1ρ2
Rρ1µ +�
Rρ2ρ3 +�
− 12Rν ρ3ρ1ρ2R
Rρ4µ +�
−24Rρ1ρ3µ ρ2R
Rρ4ρ3 +�
− 12Rρ1ρ3µ ρ2R
Rρ4ρ3 +�
−12Rνρ2µ ρ1R
Rρ3ρ4ρ2ρ5 + 6R
Rν ρ2ρ4ρ5R
− 24Rρ1ρ3µ ρ2R
Rρ4ρ2ρ1ρ5 . (A.10)
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