QMW-PH-95-37?

gr-qc/9510060

Lovelock Tensor as Generalized Einstein Tensor

M. Farhoudi†

Physics Department, Theoretical GroupQueen Mary & Westfield College, University of London

Mile End Rd., London E1 4NS, UK

September, 1995

Abstract

We show that the splitting feature of the Einstein tensor, as the first term of the Love-

lock tensor, into two parts (the Ricci tensor and the term proportional to the curvature

scalar) with the trace relation between them is a common feature of any other homoge-

neous terms in the Lovelock tensor. Motivated by the principle of general invariance,

we find that this property can be generalized, with the aid of a generalized trace oper-

ator which we define, for any inhomogeneous Euler–Lagrange expression which can be

spanned linearly in terms of homogeneous tensors. As an example, we demonstrate this

analogy for the Lovelock tensor.

s1Introduction 1 Introduction

Guided by the principle of general invariance, candidates for the gravitational

action, and hence the gravitational Lagrangian must be invariant quantities, i.e.scalars. Then, combining this principle with the variational principle, it followsthat, by variation with respect to the metric, the Euler–Lagrange expression for any

scalar Lagrangian will be a symmetric tensor with vanishing covariant derivativeas an inner identity. Hence to construct a gravitational theory, one is led to find

the most suitably scalar Lagrangian, which should be a function of the metric andits derivatives.

Among scalar Lagrangians, field equations based on a Lagrangian quadraticin the curvature tensor have had a long history in the theory of gravitation. The

first idea dates back to the early days of general relativity in work by Weyl[1]

and

? Work supported by a grant from the MCHE/Iran.† Home address: Physics Dept., Shaheed Beheshti University, Tehran, Iran.

1

Eddington[2]

in an attempt to unify gravity with electromagnetism, however this ap-

proach was unfruitful.[3]

Perhaps a legitimate mathematical motivation to examinegravitational theories built on non-linear Lagrangians has been the phenomeno-

logical character of Einstein’s theory which leaves room for such amendments, i.e.the dependence of the Einstein tensor/Lagrangian on the derivatives of the metric,

and the dimension.[4]

Actually, the Einstein Lagrangian is not the most general sec-ond order Lagrangian allowed by the principle of general invariance, and indeed,

through this principle the latter generalization can be performed up to any order,and a general scalar Lagrangian is a higher derivative Lagrangian.

Nowadays, it is also well known that Einstein’s gravity when treated as a fun-

damental quantum gravity leads to a non-renormalizable theory, although, thesedifficulties become manifest only when the curvature of space-time is not negligi-ble. In order to permit renormalization of the divergences, quantum gravity has

indicated that the Einstein–Hilbert action should be enlarged by the inclusion of

higher order curvature terms.[5]

In fact, it has been shown�

that the Lagrangian

L = 1κ2

(R + αR2 + βRµνR

µν), which, by the Gauss–Bonnet theorem, is the most

general quadratic Lagrangian in four dimensions, solves the renormalization prob-

lem and is multiplicatively renormalizable and asymptotically free; however it isnot unitary within usual perturbation theory.

The theory of superstrings, in its low energy limits, also suggests the above

inclusions, and in order to be ghost-free it is shown[7]

that it must be in the form of

dimensionally continued Gauss–Bonnet densities, i.e. the Lovelock Lagrangian,[8]

L =1

κ2

∑0<n<D

2

1

2ncn δ

α1...α2n

β1...β2nRα1α2

β1β2 · · ·Rα2n−1 α2nβ2n−1 β2n ≡

∑0<n<D

2

cn L(n) ,‡

(1.1)where δα1...αp

β1...βpis the generalized Kronecker delta symbol (see, e.g., Ref. [9]) which is

identically zero if p > D, and the maximum value of n is related to the dimensionof space-time, D, by:

nmax. =

{D2 − 1 even D

D−12 odd D .

(1.2)

Indeed, the above ghost-free property, and the fact that the Lovelock La-grangian is the most general second order Lagrangian which, the same as the

� See, for example, Ref. [6] and references therein.‡ Our conventions are a metric of signature (−,+, · · · ,+), Rµναβ = −Γµνα,β + · · ·, andRµν ≡ Rαµαν.

2

Einstein Lagrangian, yields the field equations as second order equations, havestimulated interests in Lovelock gravity and its applications in the literature.

Other properties of the Lovelock Lagrangian are as follows. It reduces to the

Einstein Lagrangian in four dimensions. Its Lagrangian terms, L(n)’s, are the

generators of the characteristic Euler class,◦

∫M

κ2L(n)√−g d2nx =(−1)n22+nπnn!

(2n)!χ(M) , (1.3)

where the topological invariants χ(M)’s are the Euler–Poincare characteristics of a

compact oriented Riemannian manifoldM , which equal zero for manifolds with odddimensionality. Hence, the dimensionally extended (or continued) Euler densities

L(n)’s lead to identities

δ∫L(n) d2nx ≡ 0 (1.4)

in even dimensions, among which is the well known Gauss–Bonnet theorem,

δ∫ (

R2 − 4RµνRµν +Rαβµν R

αβµν)√−g d4x ≡ 0 , (1.5)

in four dimensions. Also, by the variational principle and because of the above

identities, one generally gets the following identities for the corresponding Euler–

Lagrange expressions:†

G(n)(2n−dim.)αβ

≡ 0 . (1.6)

Among which is another well known identity for the Weyl tensor in four dimensions

i.e., the Bach–Lanczos identity,[11]

Cαρµν Cβρµν −

1

4gαβCσρµν C

σρµν =1

2G

(2)(4−dim.)αβ

≡ 0 . (1.7)

We have noticed that each term of the Lovelock tensor, i.e. G(n)αβ where the

◦ See, for example, Ref. [10] and references therein.

† The G(n)αβ are given in equation (1.8) below.

3

Lovelock tensor is:[8]

Gαβ = −∑

0<n<D2

1

2n+1cn gαµ δ

µα1...α2n

ββ1...β2nRα1α2

β1β2 · · ·Rα2n−1 α2nβ2n−1 β2n

≡∑

0<n<D2

cnG(n)αβ ,§

(1.8)

has also the following remarkable properties. That is, each term of the G(n)αβ can be

rewritten in a form that analogizes the Einstein tensor with respect to the Ricciand the curvature scalar tensors, namely:

G(n)αβ = R

(n)αβ −

1

2gαβ R

(n) , (1.9)

where R(n)αβ is defined as:

R(n)αβ ≡

n

2nδα1α2...α2n

α β2...β2nRα1α2β

β2 Rα3α4β3β4 · · ·Rα2n−1 α2n

β2n−1 β2n (1.10)

and R(n) is defined as:

R(n) ≡1

2nδα1...α2n

β1...β2nRα1α2

β1β2 · · ·Rα2n−1 α2n

β2n−1 β2n , (1.11)

where also R(1)αβ ≡ Rαβ and R(1) ≡ R.

The proof of these can easily be done using the definition of the generalized

Kronecker delta symbol and the properties of the Riemann–Christoffel tensor. Analternative and more basic approach is to notice that it is what one gets in the

process of varying the action, δI = δ∫L(n)√−g dDx, where its Euler–Lagrange

expression will be:

δL(n)

δgαβ−

1

2gαβ L

(n) ≡1

κ2G

(n)αβ . (1.12)

This has been done by Lovelock,[8]

though he then proceeded from this to derive

equation (1.8).

§ We have neglected the cosmological term, and G(1)αβ = Gαβ i.e., the Einstein tensor.

4

Although the above derivation is straightforward, what is not so obvious at the

first sight is that there also exists a relation between R(n)αβ and R(n) analogous to

that which exists between the Ricci tensor and the curvature scalar, namely:

1

ntraceR

(n)αβ = R(n) , (1.13)

where the trace means the standard contraction of any two indices i.e., for example,trace Aµν ≡ gαβAαβ.

Hence, the splitting feature of the Einstein tensor, as the first term of theLovelock tensor, into two parts with the trace relation between them is a common

feature of any other terms in the Lovelock tensor, in which each term alone is ahomogeneous Lagrangian. Thus, motivated by the principle of general invariance,one also needs to consider what might happen if the Lagrangian under considera-tion, and hence its relevant Euler–Lagrange expression is an inhomogeneous tensor,

as for example, the (whole) Lovelock Lagrangian, L, which is constructed of termswith a mixture of different orders.

In this case, the relevant Euler–Lagrange expression can easily be written byanalogy with the form of equation (1.9), for example,

Gαβ = <αβ −1

2gαβ < , (1.14)

where

<αβ ≡∑

0<n<D2

cnR(n)αβ and < ≡

∑0<n<D

2

cnR(n) . (1.15)

But a similar relation to equation (1.13) does not (apparently) hold between <αβand < due to the factor 1

n .

To overcome this issue, in the next section we will introduce a generalized trace

as an extra mathematical tool for Riemannian manifolds, which will slightly alterthe original form of the trace relation and modify it sufficiently to enable us todeal with the above difficulty. Then, in section 3, we will consider the case of theinhomogeneous Lovelock tensor.

5

s2Generalized Trace 2 Generalized Trace

In this section, we will define a generalized trace, which we will denote byTrace, for tensors whose components are homogeneous functions of the metric and

its derivatives.

But first, as either of gµν or gµν can be chosen as a base for counting the

homogeneity degree numbers, we will choose, without loss of generality and as a

convention, the homogeneity degree number (HDN) of gµν as [+1]; hence, the

HDN of gµν will be [−1] since gµν gµα = δνα. Similarly, we will choose the HDN of

gµν , α as [+1]. Therefore, from gαβ, ρ = −gαµ gβν gµν, ρ, the HDN of gµν, α will be

[−1]. To specify the HDN’s of higher derivatives of the metric, one may consider

∂α as if with the HDN of zero, and hence for ∂α(= gαβ ∂β

)as if with [+1]. For

convenience, the HDN’s, h, of some homogeneous functions of the metric and its

derivatives are given in Table 1.1, and whenever necessary, we will show the HDN

of a function in brackets attached to the upper left hand side, e.g. [+1]gµν and[−1]gµν .

Table 1.1: The HDN’s of useful homogeneous functions.

Function The h number

gµν (our convention) +1

gµν −1

gµν ,α (our convention) +1

gµν,α +2

Operator ∂α as if 0

Operator ∂α as if +1

gµν,α −1

gµν ,αβ... +1

gµν, αβ... −1

g ≡ det(gµν) −D

Γαµν 0

Rαβµν 0

Rαβµν & Cαβµν −1

Rµν 0

Rµν +2

R +1

L(n) & R(n) +n

R(n)µν & G

(n)µν n − 1

Now, for a general(NM

)tensor, e.g., Aα1...αN

β1...βM , that is a homogeneous

6

function of degree h with respect to the metric and its derivatives, we define:?

Trace [h]Aα1...αNβ1...βM :=

1

h−N2 +M2

trace [h]Aα1...αNβ1...βM when h−N

2 +M2 6=0

trace [h]Aα1...αNβ1...βM when h−N

2 +M2 =0 .

(2.1)

Contravariant and covariant components of a tensor obviously have different

HDN’s, however, by the above definition, the equality of their Traces is still re-

tained. For example,

Trace [h]Aµν = TraceAµν ≡ A , (2.2)

regardless of what the HDN, just as for the trace operator i.e., trace Aµν =

trace Aµν ≡ Aαα. Note that using our definition it follows that:{A = 1

h+1 Aαα for h 6= −1

A = Aαα for h = −1 ,(2.3)

where the factor of 1h+1 has entered because we have taken [h]Aµν, and therefore

used the fact that the HDN’s of both A and Aαα are [h+ 1].

In general, the generalized trace has, by its definition, all of the propertiesof the usual trace, especially its invariance under a similarity transformation (for

similar tensors) if the transformation does not change the HDN of the tensor, and

its basis independence for linear operators on a finite dimensional Hilbert space.

However, as we will show in the following, it cannot act as a linear operator†

when

the coefficients of linearity themselves are any scalar homogeneous functions of

degree h′ 6= 0.

By the definition of the Trace, for the case when h′ 6= 0, we have, for example,

Trace(

[h′]C [h]Aµν

)=

1

h′ + h+ 1trace

([h′]C [h]Aµν

)for h′ + h 6= −1

=[h′]C

h′ + h+ 1trace [h]Aµν ,

? These definitions match our HDN conventions of [+1]gµν and [+1]gµν ,α to comply with ourneeds.

† That is, for example, trace(c1Aµν + c2Bµν

)= c1 traceAµν + c2 traceBµν .

7

and using the definition once again, we get:

Trace(

[h′]C [h]Aµν

)=

h+1

h′+h+1[h′]C Trace [h]Aµν for h 6= −1

1h′

[h′]C Trace [h]Aµν for h = −1 .(2.4)

Alternatively, we obtain:

Trace(

[h′]C [h]Aµν

)= [h′]C trace [h]Aµν for h′ + h = −1

=(h + 1

)[h′]C Trace [h]Aµν for h 6= −1 .

(2.5)

Obviously, these extra factors can be made equal to one, only when h = −1 andh′ = +1, or when h = 0 and h′ = −1, as in second equation of (2.4) or inequation (2.5), respectively.

Therefore, due to these extra factors the Trace is not a linear operator asmentioned above.

It is necessary to emphasize that for our immediate purposes (which led usinitially to define a generalized trace), the Trace indeed has the distributive law ofthe usual trace in the cases when there are either no coefficients of linearity, or whencoefficients are included with their associated tensors, and/or when coefficients aremeant to be scalars with h′ = 0.

To justify the way that we have defined a generalized trace, other than that itsatisfies our need for dealing with inhomogeneous Lagrangians, one can show thatthis definition also has a link with Euler’s theorem for homogeneous functions.

Suppose A(gµν) is a homogeneous scalar function of degree [h], i.e. A(λgµν) =λhA(gµν). Euler’s theorem states that:

gµν∂ A

∂gµν= hA . (2.6)

As a rough and ready argument, define ∂ A∂gµν ≡ Aµν, where Aµν is of degree h− 1,

then gµν Aµν denotes its usual trace. Also, define A ≡ Trace Aµν. Then, from

equation (2.6), one can derive trace [h−1]Aµν = hTrace Aµν, or:

Trace [h−1]Aµν =1

htraceAµν when h 6= 0 , (2.7)

which is the same as our definition (2.1). When h = 0 (which means that Adoes not depend on the metric and its derivatives), Euler’s theorem is trivial, i.e.∂ A∂gµν = 0. Therefore, the best and consistent choice is to make no distinction

between Trace and the trace for [0]A.

8

On the other hand, using Table 1.1, it is straightforward to relate the orders

n in any Lagrangian, as in L(n), that represents its HDN.‡

So, one may refer toLagrangians with their HDN’s rather than their orders. For example, in order to

amend the Lagrangian of sixth order gravity,[12]

Berkin et al[13]

discussed that the

Lagrangian term of R R is a third order Lagrangian based on the dimensionalityscale (i.e., two derivatives are dimensionally equivalent to one Riemann–Christoffeltensor or any one of its contractions). However, it can be better justified on accountof the above regard, since it has the HDN [+3].

Using the above aspect, the special case of h = 0, as in equation (2.7), cor-

responds to c0 L(0) ≡ 1

κ2λ, a constant, which produces the cosmological term,

−12λ

[−1]gµν (or 12λ

[+1]gµν ), in the field equations. Hence, the exception value

of the HDN in our definition of generalized trace, equation (2.1), maybe relatedto the cosmological term difficulty. Nevertheless, with our choice of definition forthe generalized trace, we have Trace [+1]gµν = trace gµν = D and Trace [−1]gµν =trace gµν = D.

Finally, as an example, if one applies the definition of generalized trace on

equation (1.10) a relation similar to equation (1.13) will be obtained, but in evenmore analogous form for each order, namely:

TraceR(1)αβ = R(1) = κ2 L(1)

TraceR(2)αβ = R(2) = κ2 L(2)

TraceR(3)αβ = R(3) = κ2 L(3)

...

TraceR(n)αβ = R(n) = κ2 L(n) ,

(2.8)

where, by equation (2.3), we generally have:

R(n) =1

nR(n) ρ

ρ , (2.9)

thus, for example, R(1) = R(1) ρρ ≡ R, as expected.

‡ This choice is as to be consistent with our HDN conventions of [+1]gµν and [+1]gµν ,α, andwith our definition of Trace.

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s3Inhomogeneous Lovelock Tensor 3 Inhomogeneous Lovelock Ten-sor

In the case of inhomogeneous Lovelock Lagrangian, we saw that the Lovelocktensor can be written as in equation (1.14). Furthermore, by substituting for R(n)

from equation (1.11) and using equation (1.1), we have that:

L =1

κ2< , (3.1)

and also by substituting for it from equation (2.8), then using the distributivity ofthe Trace, we get:

< = Trace<αβ , (3.2)

which is exactly in the same form as the equations of (2.8).

Hence, in higher dimensional space-times, the Lovelock tensor, which reducesto the Einstein tensor in four dimensions, analogizes the mathematical form of theEinstein tensor. The field equations in this case are as:

Gαβ =1

2κ2 Tαβ . (3.3)

We therefore classify the Lovelock tensor, as a generalized Einstein tensor, and callL, <αβ and < the generalized Einstein’s gravitational Lagrangian, generalized Riccitensor and generalized curvature scalar, respectively.

Therefore, we showed that the analogy of the Einstein tensor can be generalizedto any inhomogeneous Euler–Lagrange expression if it can be spanned linearly interms of homogeneous tensors.

However, note that wherever the term gαβ R is involved in an equation of

Einstein’s gravity, its analogous counterpart, gαβ <, may not lead to the correctcorresponding equation in generalized Einstein’s gravity. For example, a tracelessRicci tensor is usually defined as:

Qαβ ≡ Rαβ −1

DgαβR . (3.4)

Whereas, the corresponding generalized traceless Ricci tensor can only be definedas:

Qαβ ≡ <αβ −1

Dgαβ

∑0<n<D

2

n cnR(n) , (3.5)

and obviously, it is not the covariant form of the former equation. However, ourmain concern is the analogy in the fundamental equations of the theory.

10

Acknowledgment: The author is grateful to John M. Charap for advice, andthe Physics Department of Queen Mary & Westfield College where the work hasbeen carried out.

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