When in general geometric backgrounds the metric is accompanied by torsion,
the metric conformal properties should correspondingly be followed by analogous
torsional conformal properties; however a combined metric torsional conformal
structure has never been found which provides a curvature that is both
containing metric-torsional degree of freedom and conformally invariant: in
this paper we construct such a metric-torsional conformal curvature. We proceed
by building the most general action, then deriving the most general system of
field equations; we check their consistency by showing that both conservation
laws and trace condition are verified. Final considerations and comments are
outlined.Comment: 6 page

Edery

Fabbri

Luca Fabbri

Mannheim

Shapiro

Stelle

Physics Letters B

English

arXiv

AbstractWhen in general geometric backgrounds the metric is accompanied by torsion, the metric conformal properties should correspondingly be followed by analogous torsional conformal properties; however a combined metric torsional conformal structure has never been found which provides a curvature that is both containing metric-torsional degree of freedom and conformally invariant: in this Letter we construct such a metric-torsional conformal curvature. We proceed by building the most general action, then deriving the most general system of field equations; we check their consistency by showing that both conservation laws and trace condition are verified. Final considerations and comments are outlined

Fabbri, Luca

Elsevier - Publisher Connector 

Physics Letters B 707 (2012) 415–417Contents lists available at SciVerse ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Metric-torsional conformal gravity
Luca Fabbri a,b,∗
a DIPTEM Sez. Metodi e Modelli Matematici, Università di Genova, Piazzale Kennedy Pad. D, 16129 Genova, Italy
b INFN & Dipartimento di Fisica, Università di Bologna, Via Irnerio 46, 40126 Bologna, Italy
a r t i c l e i n f o a b s t r a c t
Article history:
Received 17 November 2011
Accepted 4 January 2012
Available online 8 January 2012
Editor: S. Dodelson
When in general geometric backgrounds the metric is accompanied by torsion, the metric conformal
properties should correspondingly be followed by analogous torsional conformal properties; however a
combined metric torsional conformal structure has never been found which provides a curvature that is
both containing metric-torsional degree of freedom and conformally invariant: in this Letter we construct
such a metric-torsional conformal curvature. We proceed by building the most general action, then
deriving the most general system of ﬁeld equations; we check their consistency by showing that both
conservation laws and trace condition are veriﬁed. Final considerations and comments are outlined.
© 2012 Elsevier B.V. Open access under CC BY license.1. Introduction
Of all extended theories of gravitation, one that has a spe-
cial importance is the one displaying conformal invariance: the
theoretical reason for which this extension is elegant is that, of
all extensions that can be constructed by considering Lagrangians
with more curvatures, the special case given by a conformally
symmetric Lagrangian is unique, as discussed by Weyl; the phe-
nomenological argument for which such an extension is important
is that in this generalization the scale symmetry is related to the
property of renormalizability, and thus to the problem of quantiza-
tion, as discussed by Stelle in [1]; the observational fact for which
this extension is interesting is that within this generalization the
projective structure gives rise to the possibility of describing in
terms of background effects the rotation of galaxies, therefore re-
ducing to geometry the problem of dark matter, as discussed by
Mannheim and Kazanas in [2]. There are also motivations against a
theory of gravitation with conformal invariance related to the fact
that the universe appears not to possess such a symmetry, and
so a mechanism of gravitational spontaneous conformal symme-
try breaking must be introduced: if the Ricci scalar has a positive
value in vacuum, at least asymptotically, then such a mechanism
is possible, as it has been discussed in [3]; this mechanism pro-
vides not only conformal but also gauge symmetry breaking, as
it is further discussed in [4]. A basic introduction to gravitational
conformal theories and their developments is for example in [5]
and references therein.
* Correspondence to: INFN & Dipartimento di Fisica, Università di Bologna, Via
Irnerio 46, 40126 Bologna, Italy.
E-mail address: luca.fabbri@bo.infn.it.0370-2693 © 2012 Elsevier B.V.
doi:10.1016/j.physletb.2012.01.008
Open access under CC BY license.On the other hand however, although theories of gravity are
fundamentally metric nevertheless another essential component
is torsion, so that not only metric but also torsion has to un-
dergo to conformal transformations: one of the possibilities is
therefore to consider that like the metric also torsion has confor-
mal transformations, and this circumstance is called strong con-
formal transformation; yet another possibility is to think that
only the metric is genuinely conformally transforming whereas
torsion is left unchanged, and this situation is called weak con-
formal transformation. A discussion about the relationships be-
tween these instances of conformal transformations and further
special cases is for example in [6] and in some of the references
therein.
In this scenario, if we want to consider a gravitational confor-
mal model in which both metric and torsion conformal transfor-
mations are deﬁned, the central idea is to deﬁne a generalized
metric-torsional curvature that is conformally covariant: however
this simple idea is diﬃcult to be realized because if it is known
that, on the one hand, from the Riemann metric curvature tensor
we can take the irreducible part which turns out to be confor-
mally invariant, it is also true that, on the other hand, from the
Riemann–Cartan metric-torsional curvature tensor we may take
the irreducible part which now is not conformally invariant what-
soever. The idea is then to consider the Riemann–Cartan metric-
torsional curvature tensor modiﬁed in such a way that its irre-
ducible part does turn out to be conformally invariant in (1 + 3)-
dimensional spacetimes.
In the present Letter we shall ﬁnd this metric-torsional con-
formal curvature of the (1 + 3)-dimensional spacetime. Then we
shall study the metric-torsional conformal gravitational dynamics
by obtaining the ﬁeld equations, whose consistency is checked in
terms of the conservation laws and trace conditions.
416 L. Fabbri / Physics Letters B 707 (2012) 415–4172. Metric-torsional conformal curvature tensor
In this Letter, the Riemann–Cartan metric-torsional geometry is
deﬁned in terms of a metric gαβ and a metric-compatible con-
nection Γ μασ where metric and metric-compatible connection are
taken to be independent: metric-compatibility means that by ap-
plying to the metric tensor the covariant derivative associated to
the connection the result vanishes; on the other hand, in deﬁning
the covariant derivatives, the two lower indices of a connection
have two different roles, and thus the connection is not symmet-
ric in these two indices and its antisymmetric part in those two
indices is a tensor that does not vanish, known as Cartan torsion
tensor Qσρα decomposable as
Γ σ ρα = 1
2
gσθ
[
Qραθ + Qαρθ + Q θρα
+ (∂ρ gαθ + ∂α gρθ − ∂θ gρα)
]
(1)
showing that because of Cartan torsion the metric and the metric-
compatible connections are independent indeed. An equivalent
formalism can be introduced, in which we consider the con-
stant Minkowskian metric ηi j and a basis of vierbein eiα such
that we have the relationship epαeiνηpi = gαν together with the
spin-connection ωipα and vierbein and spin-connection are again
taken to be independent: then we have the correspondent metric-
compatibilities for which by applying to the Minkowskian metric
and the vierbein the covariant derivative associated to the spin-
connection the results vanish; respectively we have the antisym-
metry of the spin-connection ωipα = −ωpiα with spin-connection
and metric-compatible connection related by the following formula
ωi pα = eiσ
(
Γ σραe
ρ
p + ∂αeσp
)
(2)
showing that the vierbein and the spin-connection are indepen-
dent. The former formalism indicated with Latin letters and the
latter formalism indicated with Greek letters are respectively de-
noted as spacetime formalism and world formalism, and they are
equivalent; in these equivalent formalisms the independence be-
tween metric and connection is equivalent to the independence
between vierbein and spin-connection. For a more extensive intro-
duction we refer to [5].
Now the conformal transformation is given by requiring that
the line element is stretched by a given function σ and therefore
we have that for the metric it is expressed by
gαβ → σ 2gαβ (3)
while by deﬁning lnσ = φ we have that for the torsion tensor it is
given by the following
Q σ ρα → Q σ ρα + q
(
δσρ ∂αφ − δσα ∂ρφ
)
(4)
in terms of the parameter q as the most general possible; hence-
forth from the relationship (1) it is possible to see what is the
conformal transformation for the connection in its most general
form. Given that there is no conformal transformation for the con-
stant Minkowskian matrix then the conformal transformation for
the vierbein is simply
ekα → σ ekα (5)
as it is clear; therefore by employing the relationship (2) it is pos-
sible to get the conformal transformation for the spin-connection.
For a general discussion about the most general conformal trans-
formation for the metric-torsional or equivalently the vierbein-
torsional system we refer the reader to reference [6].In this framework, the Riemann–Cartan metric-torsional curva-
ture tensor is
Gikμν = Gρξμνeiρeξk
= (∂μΓ ρξν − ∂νΓ ρξμ + Γ ρσμΓ σξν − Γ ρσνΓ σξμ)eiρeξk
≡ ∂μωikν − ∂νωikμ + ωiaμωakν − ωiaνωakμ (6)
within which there is the implicit presence of the Cartan torsion
tensor; the Riemann–Cartan curvature tensor is antisymmetric in
both the ﬁrst and second couple of indices while Cartan torsion is
antisymmetric in the second couple of indices, and accordingly the
Riemann–Cartan curvature has one independent contraction that
is chosen to be given in the form Gρμρν = Giμρνeρi = Gμν whose
contraction is Gην gην = G while Cartan torsion has one indepen-
dent contraction chosen to be given by Q ρρν = Q ν setting our
convention. The commutator of covariant derivatives can be ex-
pressed in terms of both curvature and torsion and thus the cyclic
permutations of commutators of commutators of covariant deriva-
tives gives an inner relationship between these two tensors as
(
DμG
ν
ισρ − Gν ιβμQ βσρ
)+ (Dσ Gν ιρμ − Gν ιβσ Q βρμ)
+ (DρGν ιμσ − Gν ιβρ Q βμσ )≡ 0 (7)(
Dσ Q
ρ
μν + Q ρσπ Q πμν + Gρσμν
)
+ (Dν Q ρσμ + Q ρνπ Q πσμ + Gρνσμ)
+ (DμQ ρνσ + Q ρμπ Q π νσ + Gρμνσ )≡ 0 (8)
known as Jacobi–Bianchi identities, which will be used in the fol-
lowing.
Now if we were to start from the purely metric curvature
Rρημν and employ its contraction to construct a metric curvature
W ρημν with the same symmetries but irreducible then we would
get a tensor with the property of being conformally covariant, but
as torsion is included within the connection the most straightfor-
ward generalization of the metric-torsional irreducible curvature
would not be conformal invariant any longer; to solve this prob-
lem and get a metric-torsional conformal curvature, the simplest
idea would be to ﬁnd a way in which torsion should be included
through the connection implicitly as for the metric-torsional curva-
ture Gρημν and added explicitly to get a metric-torsional modiﬁed
curvature Mρημν with the same symmetries but such that its ir-
reducible part T ρημν would be conformally covariant: this can
actually be done by starting from the metric-torsional curvature
Gρημν plus torsion deﬁning the metric-torsional modiﬁed curva-
ture tensor as given by the following form
Mαβμν = Gαβμν +
(
1− q
3q
)
(Q β Qαμν − QαQ βμν) (9)
antisymmetric in both the ﬁrst and second couple of indices,
whose contractions are given by Mρμρν = Mμν with contraction
Mην gην = M and from which we construct the metric-torsional ir-
reducible curvature tensor as given by
Tαβμν = Mαβμν − 1
2
(Mα[μgν]β − Mβ[μgν]α)
+ 1
12
M(gα[μgν]β − gβ[μgν]α) (10)
antisymmetric in both the ﬁrst and second couple of indices and
irreducible, and also conformally covariant as desired. In this way
we have constructed the metric-torsional conformal curvature in
(1+ 3)-dimensional spacetimes.
Notice that there are two special cases of the parameter q that
have to be considered: the ﬁrst is given by the fact that for q = 0
L. Fabbri / Physics Letters B 707 (2012) 415–417 417it is not possible to deﬁne a metric-torsional modiﬁed curvature
and therefore the metric-torsional conformal curvature, at least in
this most straightforward manner, telling that torsion cannot have
the simplest weak conformal transformations, but instead torsion
must have the most general strong conformal transformation, if
we want to have any chance to construct, at least in the sim-
plest case, a metric-torsional conformal curvature; the second is
given by q = 1 where the metric-torsional curvature does not need
to be modiﬁed in order for its irreducible part to be conformally
invariant, telling that the torsional strong conformal transforma-
tion is actually a conformal transformation, because the metric and
torsional conformal transformations are able to compensate one
another. A ﬁnal comment is that in the limit for vanishing tor-
sion the present metric-torsional conformal curvature reduces to
the purely metric conformal curvature as expressed by Weyl ten-
sor.
3. Metric-torsional conformal gravity; energy-spin conformal
dynamical systems
Up to now we have been able to obtain the quantity that de-
scribes the metric-torsional conformal gravitational background,
and in the following we shall study the case in which this back-
ground contains energy-spin matter with conformal dynamics
which for the moment will be taken in its most general form.
Since (10) is a conformal tensor then we shall employ it to
work out all possible invariants: because it is irreducible then we
will have to take the product of two of them contracting indices
of one another; and because of its antisymmetries in both the
ﬁrst and second couple of indices then we will have that the in-
dependent invariants are T αβμν Tαβμν , T αβμν Tμναβ , T αβμν Tαμβν
so that it will be in terms of the parameters A, B , C that the
most general invariant is given by AT αβμν Tαβμν +BT αβμν Tμναβ +
CT αβμν Tαμβν as a straightforward analysis may show. Therefore it
is useful to deﬁne the parametric quantity
Pαβμν = ATαβμν + BTμναβ
+ C
4
(Tαμβν − Tβμαν + Tβναμ − Tανβμ) (11)
in terms of the parameters A, B , C , antisymmetric in both the ﬁrst
and second couple of indices, irreducible and conformally covari-
ant: in terms of this parametric conformal tensor Pαβμν the most
general invariant we wrote above reduces to the form given by
T αβμν Pαβμν and so the most general action is
S =
∫ [
kT αβμν Pαβμν + Lmatter
]√|g|dV (12)
with constant k and complemented by the material Lagrangian,
and where it is over the volume of the spacetime that the integral
is taken. By varying this action with respect to metric and connec-
tion or equivalently vierbein and spin-connection one obtains the
ﬁeld equations
2k
[
P θσραTθσρ
μ − 1
4
gαμP θσρβ Tθσρβ + PμσαρMσρ
+
(
1− q
3q
)(
Dν
(
2Pμραν Qρ − gμα Pνθρσ Q θρσ
+ gμν Pαθρσ Q θρσ
)
+ Q ν
(
2Pμραν Qρ − gμα Pνθρσ Q θρσ − Pμνρσ Q αρσ
))]
= 1
2
T αμ (13)4k
[
Dρ P
αβμρ + Qρ Pαβμρ − 1
2
Q μρθ P
αβρθ
−
(
1− q
3q
)(
Qρ P
ρ[αβ]μ − 1
2
Qσρθ g
μ[α Pβ]σρθ
)]
= Sμαβ (14)
in terms of the parameter q and the constant k and where Tμν
and Sρμν are the energy and spin densities of the matter confor-
mal ﬁeld; as Einstein equations are generalized by Sciama–Kibble
equations, similarly Weyl equations are generalized by this system
of equations, with the difference that Einstein equations tell how
energy is the source of curvature while Sciama–Kibble equations
tell how spin is the source of torsion whereas Weyl equations tell
how energy is the source of both curvature and torsion while the
new system of equations tell how spin is the source of both curva-
ture and torsion. As far as we are concerned, this seems a genuine
property of matter ﬁelds in conformal dynamical systems.
Finally by taking into account the Jacobi–Bianchi identities
in their fully contracted form we have that the ﬁeld equations
(13)–(14) are converted into conservation laws that are given in
the following form
DμT
μρ + QμTμρ − Tμσ Q σμρ + Sβμσ Gσμβρ = 0 (15)
Dρ S
ρμν + Qρ Sρμν + 1
2
T [μν] = 0 (16)
with trace condition as another conservation law
(1− q)(DμSννμ + QμSννμ)+ 1
2
Tμ
μ = 0 (17)
and the whole set of conservation laws have to be satisﬁed once
the matter conformal ﬁeld equations are eventually given: it is im-
portant to notice that the general conservation laws (15)–(16) are
now accompanied by an additional conservation law for the trace
(17) because general coordinate transformations are now accom-
panied by general conformal transformations, and the Lagrangian
formalism tells that for any symmetry a conservation law follows;
it is also important to remark that as the energy is not symmet-
ric because its antisymmetric part is related to the spin tensor
through (16) analogously the energy is not traceless because its
trace is related to the spin trace vector through (17), and because
the constrain constituted by the trace condition (17) is a conser-
vation law then this constraint is dynamically implemented within
the model. In following papers we shall consider speciﬁc matter
ﬁelds to better discuss these issues.
4. Conclusion
In this Letter we have been able to construct the metric-
torsional conformal curvature of the (1 + 3)-dimensional space-
time; we have studied the corresponding metric-torsional confor-
mal gravity by obtaining the ﬁeld equations in presence of energy-
spin matter with conformal dynamics, then we have performed a
consistency check in terms of both conservation laws and trace
condition and a discussion about these conservation laws has been
sketched.
References
[1] K.S. Stelle, Phys. Rev. D 16 (1977) 953.
[2] P.D. Mannheim, D. Kazanas, Astrophys. J. 342 (1989) 635.
[3] A. Edery, L. Fabbri, M.B. Paranjape, Class. Quant. Grav. 23 (2006) 6409.
[4] A. Edery, L. Fabbri, M.B. Paranjape, Can. J. Phys. 87 (2009) 251.
[5] L. Fabbri, arXiv:0806.2610 [hep-th].
[6] I.L. Shapiro, Phys. Rept. 357 (2002) 113.


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