Universality of Einstein Equations for the Ricci Squared Lagrangians

Abstract

It has been recently shown that, in the first order (Palatini) formalism,
there is universality of Einstein equations and Komar energy-momentum complex,
in the sense that for a generic nonlinear Lagrangian depending only on the
scalar curvature of a metric and a torsionless connection one always gets
Einstein equations and Komar's expression for the energy-momentum complex. In
this paper a similar analysis (also in the framework of the first order
formalism) is performed for all nonlinear Lagrangians depending on the
(symmetrized) Ricci square invariant. The main result is that the universality
of Einstein equations and Komar energy-momentum complex also extends to this
case (modulo a conformal transformation of the metric).

Full text

Page 1

arXiv:gr-qc/9611067v1 28 Nov 1996
UNIVERSALITY OF EINSTEIN EQUATIONS
FOR THE RICCI SQUARED LAGRANGIANS
Andrzej BOROWIEC∗
Marco FERRARIS
Mauro FRANCAVIGLIA
Igor VOLOVICH†
Istituto di Fisica Matematica “J.–L. Lagrange”
Universit` a di Torino
Via C. Alberto 10, 10123 TORINO (ITALY)
February 7, 2008
Abstract
It has been recently shown that, in the first order (Palatini) for-
malism, there is universality of Einstein equations and Komar energy–
momentum complex, in the sense that for a generic nonlinear La-
grangian depending only on the scalar curvature of a metric and a
torsionless connection one always gets Einstein equations and Komar’s
expression for the energy–momentum complex. In this paper a similar
analysis (also in the framework of the first order formalism) is per-
formed for all nonlinear Lagrangians depending on the (symmetrized)
Ricci square invariant. The main result is that the universality of Ein-
stein equations and Komar energy–momentum complex also extends
to this case (modulo a conformal transformation of the metric).
∗On leave from the Institute of Theoretical Physics, University of Wroc? law, pl. Maksa
Borna 9, 50-204 WROC? LAW (POLAND).
†Permanent address: Steklov Mathematical Institute, Russian Academy of Sciences,
Vavilov St. 42, GSP–1, 117966 MOSCOW (RUSSIA).
1

Page 2

1Introduction
It has been recently shown [1, 2] that Einstein equations are “univer-
sal”, in the sense that for a generic nonlinear Lagrangian f(R) depending
only on the scalar curvature R of a metric and a (torsionless) connection
one always gets Einstein equations if one relies on the first order (“ Pala-
tini ”) formalism. The importance of considering nonlinear gravitational
Lagrangians is particularly stressed by the fact that they provide a simple
but general approach to governing topology [3]; applications to string theory
have also been considered [4]; for a review see [5, 6]. Locally, in fact, one
gets standard Einstein equations from a nonlinear Lagrangian, but there are
global topological effects which are described by nonlinear Lagrangians and
which are absent in the linear case. It is therefore important to investigate
the universality property not only at the level of the equations of motion, but
also for the energy of the gravitational field. This problem was addressed by
us in a previous paper [7], where it was shown that universality holds also for
the energy–momentum complex. We calculated in the first order formalism
the covariant energy–density flow for a nonlinear Lagrangian f(R) and we
showed that, in a generic case, it equals in fact the Komar expression already
known in the purely metric formalism for the linear Hilbert Lagrangian.
In this paper we shall extend our discussion to the case of Lagrangians
with an arbitrary dependence on the square of the symmetric Ricci tensor
constructed out of a metric and a (torsionless) connection. We shall first
suitably extend the universality theorems of [1, 2] concerning field equations,
by showing that also in this case field equations generically reduce to Einstein
equations (modulo a conformal change of metric); also the non–generic cases
will be shortly considered (as in [1, 2]). Subsequently we shall calculate the
energy–momentum complex for this family of Lagrangians and we shall show
that it generically reduces to the standard Komar expression, provided the
metric undergoes a conformal rescaling. In view of further investigations,
we shall finally provide some calculations concerning functions of the square
of the full Ricci tensor.
Our results can be relevant also for quantum gravity. As is well known, in
fact, to remove divergences in quantum gravity one has to add counterterms
to the Lagrangian which depend not only on the scalar curvature but also
on the Ricci and the Riemann tensors invariants. It follows from our results
that ”on-shell”, in the first order formalism, counterterms depending on the
scalar curvature and the Ricci square invariant generically do not change
the semiclassical limit, since we still have the standard Einstein equations.
2

Page 3

The main result of the present paper is the following:
Theorem Let S ≡ gµνgαβR(µα)(Γ)R(νβ)(Γ) be the (symmetrized) Ricci
square invariant, build out of a metric g and a symmetric connection Γ.
For the action
A(g,Γ) =
?
f(S)√g dx
where f is a generic analytic function of one real variable, the Euler–Lagrange
equations (in the first order formalism) lead always to the Einstein equations
Rµν(h) = γhµν
for a new metric h which is defined as hµν= γ−1R(µν)(Γ) – with a constant
γ ?= 0.
Moreover, a superpotential for the energy–momentum flow along any vec-
torfield ξ is given by the Komar expression
√
h ∇[µξν]
This result will follow from Propositions 2.1, 2.3 and 3.2 below.
It turns out therefore that the action A(g,Γ) leads to two metrics: be-
sides the initial metric g one gets a new Einstein metric h, related to g by
the algebraic equation
(g−1h)2= ± I
This condition provides on spacetime some additional structures, namely:
a Riemannian almost–product structure or an almost–complex structure
with a Norden metric. These aspects will be studied in more detail in our
forthcoming paper [8].
2Field Equations
As is well known gravitational Lagrangians which are nonlinear in the
scalar curvature of a metric give rise to higher derivatives or to the appear-
ance of additional matter fields [9, 10]. This strongly depends on having
taken a metric as basic variable; moreover the equations ensuing from such
Lagrangians show an explicit dependence on the Lagrangian itself. It was
shown in [2] that, in contrast, working in the first order (Palatini) formalism
with independent variations with respect to a metric and a symmetric con-
nection, then, for a large class of Lagrangians of the form f(R)√g, where
3

Page 4

R is the scalar curvature, the equations obtained are almost independent on
the Lagrangian, the only dependence being in fact encoded into constants
(cosmological and Newton’s ones). In this sense the equations obtained are
“universal”. These are in fact Einstein equations in generic cases, while
in degenerate situations and in dimension n = 2 one gets either equations
which express the constancy of the scalar curvature or conformally invariant
equations.
In this paper we shall consider, still in the first order (Palatini) formal-
ism, the family of actions
A(Γ,g) =
?
Mf(S)√g dx(2.1)
where: M is a n–dimensional (connected) manifold endowed with a metric
gµνof an arbitrary signature and a torsionless (i.e., symmetric) connection
Γσ
of one real variable, which we assume to be analytic; the scalar S is the
symmetric part of a Ricci square–invariant, considered as a first order scalar
concomitant of a metric and the (torsionless) connection, i.e.:
µν; the Lagrangian density is Lf= f(S)√g , where f is a given function
S ≡ S(g,Γ) = SµνSµν= gµαgνβSαβSµν
(2.2)
where Sµν≡ Sµν(Γ) = R(µν)(Γ) is the symmetric part of the Ricci tensor.
We adopt the standard notation for the Riemann and Ricci tensors of Γ :
Rλ
µνσ(Γ) ≡ Rλ
Rµσ= Rµσ(Γ) = Rν
µνσ= ∂νΓλ
µσ− ∂σΓλ
µν+ Γλ
ανΓα
µσ− Γλ
ασΓα
µν
µνσ, (α,µ,ν,... = 1,...,n)
and simply write√g instead of |detgµν|1/2. Since the Lagrangian Lfdoes
not depend on derivatives of the metric field, the only momenta are those
conjugated to the connection coefficients:
pµνκ
λ
≡
∂Lf
∂Γλ
µν,κ
= παβ∂Sαβ
∂Γλ
µν,κ
= 2(πµνδκ
λ− πκ(µδν)
λ) (2.3)
where Γλ
µν,κ= ∂κΓλ
µνand
πµν= 2√g f′(S)Sµν
(2.4)
(recall that Sµν≡ gµαgνβSαβbut gµνis the inverse of gµν).
4

Page 5

The Euler–Lagrange equations for the action (2.1) with respect to inde-
pendent variations of g and Γ can be written in the following form
f′(S)gαβSµαSνβ−1
∇απµν≡ 2∇α(f′(S)√g Sµν) = 0
4f(S)gµν= 0(2.5)
(2.6)
where ∇αis the covariant derivative with respect to Γ and we assume n ≥ 2.
In fact, variation of the action (2.1) with respect to Γ gives the equation:
∇απµν− ∇ρπρ(µδν)
α= 0
which, due to the symmetry of πµν, in any dimension n ≥ 2 reduces to (2.6)
by taking a trace (see Appendix I).
The properties of the system of equations (2.5)-(2.6) are governed by the
following equation for the scalar S:
f′(S)S −n
4f(S) = 0(2.7)
which is obtained by transvecting (2.5) with gµν. Following the lines of the-
orem 1 of [2] we shall then distinguish the following three mutually exclusive
cases: (i) eq. (2.7) has no real solutions; (ii) eq. (2.7) has isolated real so-
lutions; (iii) eq. (2.7) is identically satisfied. Their discussion proceeds as
follows:
(Case 1 – no real solutions) If equation (2.7) has no real solution, then
also the system (2.5)–(2.6) has no real solution.
(Case 2 – isolated real solutions) Let us now suppose that eq. (2.7) is
not identically satisfied and has at least one real solution. In this case, since
analytic functions can have at most a countable set of zeroes on the real
line, eq. (2.7) can have no more than a countable set of solutions S = ci
(i = 1,2,...), where ciare constants. Consider then any solution
S = ci
(2.8)
of eq. (2.7). We have now two possibilities, depending on the value of the
first derivative f′(ci) at the point S = ci.
5

Page 6

(Subcase 2.1 – f′(ci) ?= 0) Let us assume that f′(ci) ?= 0. Then eq. (2.6)
takes the form
∇α(√g Sµν(Γ)) = 0
while equation (2.5) reduces to
(2.9)
gαβSµα(Γ)Sνβ(Γ) = Λ(ci)gµν
(2.10)
where a constant Λ = Λ(ci) arises according to:
Λ = Λ(ci) = f(ci)/4f′(ci) = ci/n(2.11)
We shall discuss separately the subcases n > 2 and n = 2.
(Subcase 2.1.1 – f′(ci) ?= 0 and n > 2) As it is known, for n > 2 and any
metric hµν, the general solution of the equation
√
dethhµν) = 0
∇Γ
α(
(2.12)
is the Levi–Civita connection Γ = ΓLC(h) (here hµνis the inverse of hµν):
Γσ
µν(h) =1
2hσα(∂µhνα+ ∂νhµα− ∂αhµν)
The Ricci tensor Rµν(Γ) is automatically symmetric and in fact identical
to the Ricci tensor Rµν(h) of the metric h itself. Hence the following holds
true for the present subcase:
Proposition 2.1 Assume Λ ?= 0 and let us take γ ?= 0 and n > 2. Then:
• (i) If hµνis a metric satisfying the Einstein equations
Rµν(h) = γhµν
(2.13)
and a metric gµνis an arbitrary solution of the algebraic equation
gαβgνρhµαhνβ=Λ
γ2δρ
µ
(2.14)
where gµνdenotes the inverse of gµν, then the pair (g,ΓLC(h)) satisfies
eq.s (2.9)–(2.10).
6

Page 7

• (ii) Conversely, if (g,Γ) is a solution of eq.s (2.9)–(2.10) and γ ?= 0,
then
hµν= γ−1Sµν(Γ)
has to satisfy the algebraic relations (2.14). Moreover Γ = ΓLC(h) and
one has Einstein equations (2.13).
Proof. Assume that h and g are two metrics satisfying (2.14). In partic-
ular this implies that both metrics are nondegenerate and that deth2=
(Λ
Moreover one finds
(h−1)µν∼ gµαgνβhαβ
which tells us that rising indices of h by means of g produces a matrix which
is proportional to the inverse of h.
Now assume that h is an Einstein metric: Rµν(h) = γ hµν with γ ?= 0
and take Γ = ΓLC(h). Then
γ2)ndetg2, which ensure proportionality by a constant factor deth ∼ detg.
(A)
Sµν(Γ) ≡ Rµν(ΓLC) ≡ Rµν(h) = γ hµν
Therefore, ∇µ(√hhαβ) = 0 implies (2.9). The first claim is thence proved.
Assume conversely that (2.9) and (2.10) are satisfied. Setting hµν =
(γ−1Sµν) it then follows from (2.10) that h satisfies the algebraic rela-
tion (2.14). Now, using (A), equation (2.9) can be rewritten in the form
∇α(√hhµν) = 0. This in turns implies that Γ = ΓLC(h). Therefore, again
by (A), h is an Einstein metric and our claim is proved.
(B)
Q.E.D.
This extends the results previously found by Higgs [11] for the confor-
mally invariant case f(S) = S in dimension n = 4 (see Proposition 2.3), on
the basis of an earlier paper by Stephenson [12]. The above proposition can
be re–phrased as follows. If we define a new metric hµνby setting:
hµν= γ−1Sµν
with an arbitrary nonvanishing constant γ, then eq.s (2.9) and (2.10) imply
that Γ is the Levi–Civita connection of the new metric h and eq. (2.10)
reduces to the Einstein eq.s for the metric h
Rµν(h) = γ hµν
This, in turn, leads to a constant scalar curvature for the new metric h (in
fact, it is R(h) = nγ) and M will be an Einstein manifold with respect to
7

Page 8

the new metric h.
Remark. Notice that for Λ > 0 eq. (2.14) has an obvious global solution
of the form gµν= e−ωhµνwith ω = 1/2ln(Λ/γ2). The case Λ ≤ 0 is more
complicated and it will be considered in a forthcoming publication [8].
(Subcase 2.1.2 – f′(ci) ?= 0 and n = 2) In the case n = 2, instead,
equation (2.12) allows a further degree of freedom (related with conformal
invariance). It has in fact the following general solution [1, 3, 13]:
Γσ
µν= Wσ
µν(h,B) =1
2hσα(∂µhνα+∂νhµα−∂αhµν)+1
2(δσ
νBµ+δσ
µBν−hµνBσ)
(2.15)
where Bαis an arbitrary vectorfield and we set Bµ= hµαBα. This is due
to the fact that only in dimension n = 2 eq. (2.6) cannot be reduced to
∇αhµν = 0 (since only in two dimensions it does not imply that the Rie-
mannian volume element of h is covariantly constant along Γ). A connection
W(h,B) having the form (2.15) is called a Weyl connection [14]. Using eq.
(2.15), from the definition of Sµν(Γ) one has:
Sµν(W(h,B)) ≡1
2(R(h) − DαBα)hµν
(2.16)
where Dαdenotes the covariant derivative with respect to the metric hµν,
so that eq. (2.13) reduces to the following scalar equation
R(h) − DαBα= 2γ (2.17)
Equation (2.17) is the “universal” equation for 2–dimensional space–times
[3, 13]; it replaces Einstein equations, which are the “universal equations” in
dimension n > 2. Equation (2.17) is in fact the equation of constant scalar
curvature for the metric h and the Weyl connection (2.15), because from
(2.16) one has:
R(h,B) = hµνRµν(W(h,B)) = R(h) − DαBα
(2.18)
We remark that eq. (2.17) has always infinitely many local solutions, but
it might have no global analytic solution (depending on the topology of the
2-dimensional manifold M). In any case, the following holds true for the
subcase 2.1.2:
8

Page 9

Proposition 2.2 Assume Λ ?= 0, γ ?= 0 and n = 2. Then:
• (i) If hµν is a metric satisfying equation (2.17) where Bαis an arbi-
trary vectorfield and a metric gµνis an arbitrary solution of the alge-
braic eq. (2.14), then the pair (g,W(h,B)) satisfies eq.s (2.9)–(2.10).
• (ii) Conversely, if (g,Γ) is a solution of eq.s (2.9)–(2.10), then
hµν= γ−1Sµν(Γ)
has to satisfy the algebraic relations (2.14). Moreover Γ = W(h,B)
for some vectorfield Bαand one has eq.s (2.17).
Proof . The proof goes along the same lines as in Proposition 2.1, with the
only difference that Einstein eq.s are now replaced by equations (2.17).
Q.E.D.
(Subcase 2.2 – f′(ci) = 0) Suppose now f′(ci) = 0. Then eq. (2.7) implies
that also f(ci) = 0, i.e., ciis a zero of order at least two of f(S). In this
case, eq.s (2.5)–(2.6) are identically satisfied and the only relation between
g and Γ is contained in the following equation
S(g,Γ) = ci
(2.19)
This equation represents a genuine dynamical relation between the metric
and the connection of M, although it is not enough to single out a connection
Γ for any given metric g (as it happened, instead, in subcase (2.1) above,
where Γ turns out to be the Levi–Civita connection of h and h is algebraically
related to g). In fact, defining a tensorfield ∆λ
µνby:
∆λ
µν≡ ∆λ
µν(g,Γ) = Γλ
µν− Γλ
µν(g)(2.20)
equation (2.19) can be turned into a quasi–linear first order PDE for the
unknown ∆, having the term S(g)−cias a source. The space of solutions of
this last equation, as functions of the metric g together with its first deriva-
tives and a number of auxiliary fields, has a complicated structure. In any
case, it is easy to see that this space contains as a subspace the space of all
pairs (g,Γ) satisfying eq. (2.10) for Λ = ci/n. We also remark that for ci= 0
this equation covers the case Λ = 0 which was excluded from Proposition
2.1 (see above).
9

Page 10

(Case 3 – the conformally invariant case) We consider now the case in
which eq. (2.7) is identically satisfied. Under this hypothesis the Lagrangian
is proportional to:
f(S) = Sn/4
(2.21)
We shall then consider for simplicity the case S ≥ 0 (analogous results will
be valid for S ≤ 0 and they may extend across S = 0 at least if n = 4k).
We stress that for n = 4 this is exactly the linear Lagrangian f(S) = S
considered in [11, 12].
In this case equations (2.5) and (2.6) read as follows:
S
n−4
4 (gαβSµαSνβ−4
nSgµν) = 0(2.22)
∇α(S
n−4
4√gSµν) = 0 (2.23)
Notice first of all that, under conformal transformations
˜ gµν= eωgµν,˜Γ = Γ
(2.24)
where ω is an arbitrary function on manifold M, one has
˜S=e−2ωS
˜Sµν
=Sµν
Sn/4√g
4√gSµν
˜Sn/4?˜ g
n−4
4
?˜ g˜Sµν
=
˜S
=S
n−4
Therefore, the action
A(g,Γ) =
?
MSn/4√g dx
as well as the equations (2.22) and (2.23) are invariant under the transfor-
mation (2.24), i.e., A(˜ g,˜Γ) = A(g,Γ).
Also in this case we shall consider separately the two cases n > 2 and
n = 2.
(Subcase 3.1 – conformally invariant for n > 2) If n > 2 there are two
possibilities. If S = 0 everywhere on our manifold then we have only the
equation
S(g,Γ) = 0
10

Page 11

and the discussion proceeds as for eq. (2.19) above. When S ?= 0 there is
instead an additional conformal degree of freedom, as noticed earlier in [11]
and [12] (and later exploited in [15]). In what follows we restrict ourselves to
the case when S is a nowhere vanishing function. Then, in fact, eq.s (2.22)
and (2.23) reduce to the following:
gαβSµαSνβ−4
nSgµν= 0(2.25)
∇α(S−1?
detSαβSµν) = 0(2.26)
In order to get (2.26) we have used (2.25) to calculate first the determinant
of g as a function of det(Sαβ) and next replace it into (2.23). The following
proposition is true:
Proposition 2.3 Assume S ?= 0, γ ?= 0 and n > 2. Then:
• (i) If hµνis a metric satisfying the Einsten equations
Rµν(h) = γhµν
where a metric gµνis an arbitrary solution of the algebraic equation
gαβgνρhµαhνβ=
4s
nγ2δρ
µ
(2.27)
with an arbitrary nowhere vanishing function s, then the pair (g,ΓLC(h))
satisfies eq.s (2.25)–(2.26). Moreover one has S(g,ΓLC(h)) = s.
• (ii) Conversely, if (g,Γ) is a solution of eq.s (2.25)–(2.26), then
hµν= γ−1Sµν(Γ)
has to satisfy the algebraic relations (2.27) with s = S(g,Γ). Moreover
one has Γ = ΓLC(h) and Rµν(h) = γhµν.
Proof. Since S is a nowhere vanishing function on a (connected) manifold,
it is strictly positive or strictly negative. Therefore, it can be written in the
form S = ±e2ω. By the conformal change of metric g ?→ eωg, equation (2.25)
transforms into (2.14) and equation (2.26) transforms into (2.12). (Notice
that after the conformal transformation S transforms to 1). Then the same
reasoning as in Proposition 2.1 proves our claim.Q.E.D.
11

Page 12

Remark. Notice that for s > 0 eq. (2.27) has an obvious global solution of
the form gµν= e−ωhµνwith a conformal factor 2ω = ln(4s/nγ2).
We can also restate the result as follows: if (g,Γ) is a solution of eq.s
(2.25) and (2.26), then there exists a scalar field ψ such that the conformally
related metric ψgµνsatisfies Einstein equations and the connection Γ is the
Levi–Civita connection of ψgµν; moreover, the scalar curvature of the origi-
nal metric g and the connection Γ equals ψ. The origin of this extra scalar
field ψ = e−ωwill be discussed elsewhere [16] in the framework of Legendre
transformation for metric–affine theories.
(Subcase 3.2 – conformally invariant and n = 2) If n = 2 equations
(2.18) and (2.19) simplify to
gαβSµαSνβ(Γ) −1
2S(g,Γ)gµν= 0(2.28)
∇α(S−1
2√gSµν) = 0(2.29)
Then the following holds true:
Proposition 2.4 Assume S ?= 0, γ ?= 0 and n = 2. Then:
• (i) If hµν is a metric satisfying equation (2.17) where Bαis an arbi-
trary vectorfield and a metric gµν is an arbitrary solution of the al-
gebraic eq.(2.27), with an arbitrary nowhere vanishing function s ∈
F(M), then the pair (g,W(h,B)) satisfies eq.s (2.28)–(2.29). More-
over S(g,W(h,B)) = s.
• (ii) Conversely, if (g,Γ) is a solution of eq.s (2.28)–(2.29), then
γ hµν= Sµν(Γ)
has to satisfy the algebraic relations (2.27) with s = S(g,Γ). Moreover
Γ = W(h,B) for some vectorfield Bαand one has (2.29).
Proof. The proof is an obvious combination of those already given for Propo-
sitions 2.2 and 2.3 . Q.E.D.
12

Page 13

3Energy–Momentum Complex
We are now ready to apply general formulae for the energy–momentum
complex to the Lagrangian density
L = L(gµν,Γλ
µν,Γλ
µν,α) = f(S)√g(3.1)
These general formulae are considered in Appendix II and follow closely the
earlier works of ours [17, 18, 19]; see also [20, 21]. Our results will extend
those of our previous paper [7].
Our field variables are now g and Γ, while the only derivatives entering
the Lagrangian are the first order derivatives of Γ. Therefore, from (II-1)
one has the following definition of the energy–density flow
Eα= pµνα
λ
LξΓλ
µν− L ξα
(3.2)
where pµνα
λ
are given by (2.3)–(2.4). By using the decomposition (see II–6):
LξΓκ
µν= −ξαRκ
(µν)α+ ∇(µ∇ν)ξκ
and taking the symmetry of pµνλ
κ
into account, one has
pµνλ
κ
LξΓκ
µν= −pµνλ
κ
Rκ
µναξα+ pµνλ
α
∇µ∇νξα
where ∇µis the covariant derivative with respect to the connection Γ. From
this (see (II–8)) one can easily recognize that:
Tλσ
α
= 0,Tλσρ
α
= pσρλ
α
Substituting these into (II–13) and making use of (2.3)–(2.4) one gets finally
Uµν(ξ) = 2πσ[µ∇σξν]+ (∇σπσ[µ)ξν]
(3.3)
The second term of (3.3) vanishes on shell by virtue of (2.6). Therefore we
have proved the following:
Proposition 3.1 The energy–density flow (3.2) for the Lagrangian (3.1)
can be represented on shell under the form Eµ= dνUµν, where the superpo-
tential Uµνis given by:
Uµν= 2 πσ[µ∇σξν]
(3.4)
Q.E.D.
13

Page 14

Let us now discuss the expression (3.4) we found for the superpotential.
It resembles very much Komar superpotential, which is known to be a good
superpotential in the purely metric formalism [22] (Komar superpotential
is in fact equal also to the Schr¨ odinger–M¨ oller–Mizkievich superpotential;
for a discussion see, e.g., [23, 24]). Our superpotential differs in fact from
Komar’s one by a factor. If the dimension n of space–time is larger than 2
and we are in a generic position (see Section 2), i.e. if equation (2.4) has a
solution S = c such that c ?= 0 and f′(c) ?= 0, then we know that Γ = ΓLC(h)
for some h given by eq. (2.14). Accordingly, from (2.4) and (2.14) one has
πµν= 2γ (γ2
Λ)
n
4√h hµν
(3.5)
and (3.4) reduces to:
Uµν= 4γ (γ2
Λ)
n
4√
h ∇[µξν]
(3.6)
which is proportional to the Komar expression for h. This corresponds to
the subcase 2.1.
At bifurcation points S = c, i.e., in the subcase (2.2), one has instead
f′(c) = 0. Therefore in this case one has πµν= 0 and the energy–density
flow as given by (3.4) is identically vanishing.
For the conformally invariant Lagrangian L = Sn/4√g (which in dimen-
sion n = 4 is just the quadratic Lagrangian S√g) one has finally:
2πµν= nS(g,Γ)
n−4
4
√g Sµν
and one can use a conformal transformation h = eωg (for S ?= 0) to reduce
the system to Einstein equations (this is in fact the subcase 3.1). Otherwise,
one can decompose the connection Γ as in (2.20):
Γρ
µν(h) = Γρ
µν(g) + ∆ρ
µν
where the tensor ∆ ≡ ∆(ω) is symmetric in its lower indices. The superpo-
tential takes then the form
Uµν= 2 πσ[µ˜∇σξν]+ 2 πσ[µ∆ν]
σαξα
(3.7)
where˜∇ is the covariant derivative with respect to the metric g. As we
mentioned above, the tensor ∆ can be eliminated by a suitable conformal
transformation, so that the relevant part of the superpotential turns out to
be again proportional, modulo a conformal rescaling of the metric, to the
Komar superpotential of h. Therefore one gets
14

Page 15

Proposition 3.2 In the cases described by Propositions 2.1 and 2.3 above
the superpotential turns out to be proportional to the Komar superpotential
of h, i.e.
Uµν∼
√h∇[µξν]
Q.E.D.
The two–dimensional case is particularly important (see [3, 4, 13]). In
this case Γ is a Weyl connection with (see (2.7)):
∆ρ
µν= δρ
(µBν)− 1/2hµνBρ
and the superpotential reduces to:
Uµν∼
√
h(D[µξν]− B[µξν]) (3.8)
This covers subcases 2.1.2 (see Proposition 2.2) and 3.2 (see Proposition
2.4).
A discussion of the energy–momentum complex for non–linear Lagrangians
in the first order formalism, based on the use of a background connection as
in [25], will be discussed elsewhere [26].
4 Conclusions
In this paper it was shown that the universality of Einstein equations
and Komar energy-momentum complex, already found in previous investiga-
tions about the first order formalism, for nonlinear Lagrangians depending
on the scalar curvature, extends also to Lagrangians depending on the Ricci
square invariant. We have preferred to limit ourselves here to discuss in
detail various important particular cases and emphasize the general picture.
It seems to us that the universality of Einstein equations and Komar energy-
momentum complex for nonlinear Lagrangians depending on the scalar cur-
vature or the Ricci square invariant is not only interesting and important as
a mathematical results but also in view of furthers applications to concrete
problems in gravity theories (including cosmological models), both at clas-
sical and quantum level. Moreover our results find interesting applications
to signature change in spacetime. We will discuss these matters, as well
as the algebraic relations between the two metrics g and h in forthcoming
papers. It will be in particular shown in [8] that we obtain, in fact, new in-
teresting geometrical structures on manifold, so called almost-product and
almost-complex Einsteinian structures.
15