Routh Reduction of Palatini Gravity in Vacuum
An interpretation of Einstein-Hilbert gravity equations as a Lagrangian reduction of Palatini gravity is made. The main technique involved in this task consists of representing the equations of motion as a set of differential forms on a suitable bundle. In this setting, Einstein-Hilbert gravity can...
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| description | An interpretation of Einstein-Hilbert gravity equations as a Lagrangian reduction of Palatini gravity is made. The main technique involved in this task consists of representing the equations of motion as a set of differential forms on a suitable bundle. In this setting, Einstein-Hilbert gravity can be considered as a kind of Routh reduction of the underlying field theory for Palatini gravity. As a byproduct of this approach, a novel set of conditions for the existence of a vielbein for a given metric is found.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 046, 50 pages
Routh Reduction of Palatini Gravity in Vacuum
Santiago CAPRIOTTI
Departamento de Matemática, Instituto de Matemática de Bah́ıa Blanca (INMABB),
CONICET, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bah́ıa Blanca, Argentina
E-mail: santiago.capriotti@uns.edu.ar
Received September 24, 2019, in final form May 11, 2020; Published online May 30, 2020
https://doi.org/10.3842/SIGMA.2020.046
Abstract. An interpretation of Einstein–Hilbert gravity equations as Lagrangian reduction
of Palatini gravity is made. The main technique involved in this task consists in representing
the equations of motion as a set of differential forms on a suitable bundle. In this setting
Einstein–Hilbert gravity can be considered as a kind of Routh reduction of the underlying
field theory for Palatini gravity. As a byproduct of this approach, a novel set of conditions
for the existence of a vielbein for a given metric is found.
Key words: symmetry reduction; Palatini gravity; frame bundle
2020 Mathematics Subject Classification: 53C80; 53C05; 83C05; 70S05; 70S10
1 Introduction
The relationship between Einstein–Hilbert and Palatini formulation of gravity has been studied
by several authors. It could be established in the Lagrangian formulation by comparing the
action functionals for each case – see for example [37, 40] and references therein. From the
Hamiltonian point of view, the main theoretical tool used in the discussion of the connection
between these formulations of gravity appears to be some flavor of Hamiltonian reduction. For
instance, [15] and [37] use ADM formalism [1] in order to establish the connection; it has also
been explored in [13, 27], where the correspondence is set by using a Hamiltonian structure on
the set of fields at the boundary.
From this viewpoint, it becomes interesting to find a reduction scheme relating the Lagrangian
formulation of Palatini and Einstein–Hilbert gravity directly, without the detour through Hamil-
tonian formalism. So far, there exist two ways to implement reduction at the Lagrangian level,
namely Lagrange–Poincaré reduction [8, 11, 12, 17] and Routh reduction [4, 6, 14, 18, 31, 32].
Moreover, there are physical considerations that can be said in support of this kind of reduction:
They deal with not only the reduction, but also the reconstruction problem, and it is argued
in [33] that reconstruction can be relevant from the physical point of view.
Routh reduction was originally designed as a technique to get rid of cyclic variables in a La-
grangian function by using a fixed value of the corresponding momentum [34]. The mechanical
system thus obtained has a configuration space where the cyclic variables are absent and whose
Lagrangian has a term forcing the momentum to have the prescribed value; the new Lagrangian
function is called Routh function or Routhian. This reduction scheme can be further generalized
to symmetries characterized by a non abelian Lie group – see [32] and references therein. In the
general case, the equations of motion of the system obtained by quotient out the symmetry are
modified by the addition of a new term, called force term, which is related to the curvature of
a principal bundle associated to the symmetry. It should be noted that the Routhian reduces
to the Lagrangian and the force term annihilates when the momentum value is set to zero.
These versions of Routh reduction require some sort of regularity to the Lagrangian in order to
construct the Routhian; this regularity is used to translate the constraints into the momentum
mailto:santiago.capriotti@uns.edu.ar
https://doi.org/10.3842/SIGMA.2020.046
2 S. Capriotti
variables to constraints into the velocities. Therefore, it is possible to avoid this regularity
condition by working in a unified setting, where variables associated to both velocities and
momenta are taken into account; this approach can be used to set a Routh reduction procedure
not suffering this limitation [18]. This viewpoint proves its usefulness in [6], where unified
formalism was employed in the generalization of Routh reduction to the field theory realm. In
fact, in order to improve the understanding of the tools and strategies used in the present work,
it could be helpful to give an account of both the problem addressed in this reference and the
techniques used to solve it:
• Result in [6]: For the moment, we will represent a variational problem with a pair
(π : E → M,L), where π : E → M is a bundle and L : J1π → ∧m(T ∗M) is a Lagrangian
density; in this setting, we can define the action
Γπ 3 s 7→
∫
U
L ◦ j1s
for any compact U ⊂M , and the variational problem becomes
δ
∫
U
L ◦ j1s = 0
for arbitrary variations δs. Let us suppose that there exists a Lie group acting freely
and vertically on E, such that the Lagrangian density is invariant by this action, yielding
to a momentum map J : J1π → ∧m−1
(
T ∗J1π
)
⊗ g∗. In order to construct the reduced
Lagrangian theory, it is necessary to fix a value for the momentum map through the
specification of a g∗-valued (m− 1)-form
µ ∈ Ωm−1(M, g∗),
and also to choose a connection ω on the principal bundle pEG : E → Σ := E/G. Let
us indicate by Gµ ⊂ G the stabilizer of the momentum µ. Then the main result in [6]
established a correspondence between the extremals of the variational problem (π : E →
M,L) and(
π : E/Gµ ×M Lin (π∗TM, g̃)→M,Rµ
)
.
This variational problem is the reduced variational problem for Routh reduction in field
theory. As we warned before, when dealing with variational problems obtained through
Routh reduction, it could be necessary to add a force term, in order to take care of the
non trivial nature of the bundles involved [28]. In short, the local Routhian functions
constructed through Routh original procedure cannot be pasted into a global Routhian
function unless the bundle obtained by the symmetry quotient is trivial. It is thus necessary
to use a connection in order to define this global function; as a by-product, a force term
associated to the curvature of the chosen connection should be taken into account. In
any case, the force term is in this context represented by a (m + 1)-form φµ on J1π
constructed from ω and the momentum µ. Therefore, the underlying variational problem
becomes modified by an external force
δ
∫
U
Rµ ◦ j1s =
∫
U
〈φµ, δs〉.
• Techniques: As was said above, it is useful to add momentum variables in order to avoid
regularity issues when dealing with Routh reduction. It leads us to work with the unified
formulation of field theory [16], where the space of velocities J1π is enlarged with the space
Routh Reduction of Palatini Gravity in Vacuum 3
of multimomentum – see equation (1.5) below for an explanation of the symbols involved
in this formula:
J1π◦ := ∧m2 (E)/ ∧m1 (E)
to form a bundle
Wcl := J1π ×M J1π◦. (1.1)
In this space, a variational problem can be set. A word of caution should be given here:
The variational problem on Wcl is different to those described by a pair (π : E → M,L)
above. Concretely, in this new setting we have a Lagrangian ΘL that is not a density, but
a m-form on Wcl; the associated action becomes
Γ(Wcl →M) 3 σ 7→
∫
U
σ∗ΘL.
A crucial difference with the variational problems of Lagrangian field theory arises from
this action: When performing variations, they should be free because in Wcl there are
no variables that can be identified as velocities. It claims for a definition of variational
problem capable to associate geometrical structures to these differences; this is captured
by the notion of Griffiths variational problem considered in [22, 26] and will be crucial in
our approach to reduction – see Definition 2.1 below.
Beyond better regularity properties, the unified setting has several advantages stemming
from the linear structures that come equipped with it, and the formulation of relevant
data in terms of differential forms, on which pullback maps allow for straightforward
comparisons of equations and other quantities.
Now, we are looking for a Routh reduction scheme similar to the one described above, in
order to relate the variational problems for Palatini and Einstein–Hilbert gravity. This forces
us to discuss how we will represent the flavors of gravity to work with. For Palatini gravity we
will use the formulation given in [3], where it was interpreted as an example of the concept of
Griffiths variational problem. In order to describe this formulation, let us summarize the main
characteristics of a Griffiths variational problem. It consists into three kind of data: A bundle
p : W → M , whose sections will be the fields of the theory, a Lagrangian form λ ∈ Ωm(W )
setting the dynamics, and a set of forms I ⊂ Ω•(W ) (more precisely, an exterior differential
system) describing the set of differential restrictions on the fields. This additional geometric
structure replaces the contact structure of the jet bundle; therefore, the usual constraints
uAµ =
∂uA
∂xµ
are absent. It should be stressed that this change has profound consequences on the equations
governing the extremals of the underlying variational problem: In general, when performing
variations, it results that
δuAµ 6=
(
δuA
)
µ
.
In fact, because the set of sections verifying the differential restrictions represented by I must
be invariant by variations, this relationship should be replaced by conditions describing the
infinitesimal symmetries of this set.
Accordingly, a Griffiths variational problem will be described with the symbol
(p : W →M,λ, I), (1.2)
4 S. Capriotti
where the additional data I has been added. The variational problem underlying such triple
consists in finding the extremals of the action
S[s] :=
∫
M
s∗λ,
where the sections s : M →W of the bundle p must be integral for the set of forms in I, namely,
s∗α = 0
for every α ∈ I.
We can use this approach to represent Palatini gravity as a first order field theory. Recall
that for this type of gravity theory, the degrees of freedom are a vielbein and a connection; it
suggests taking the bundle of frames τ : LM →M (whose sections are the vielbein) as the field
bundle. Moreover, its jet space J1τ can be decomposed as
J1τ = LM ×M C(LM),
where C(LM)→M is the connection bundle, that is, a bundle whose sections are the principal
connections of LM – see Section 3 for details. It means that sections of τ1 : J1τ → M are
composed by a pair vielbein + connection and so, they are suitable for the description of the
fundamental degrees of freedom in Palatini gravity. Now, in Palatini formulation, the vielbein
and the connection are not independent; they are related by two constraints: on the one side,
the connection must be torsionless; on the other, the metric associated to a given vielbein should
be invariant respect to the parallel translation associated to the given connection. So, instead
of working with the total jet space J1τ , we will work in a submanifold T0 ⊂ J1τ , namely, the
submanifold corresponding to the torsion zero constraint ; also, the contact structure is changed
by a set of differential constraints implementing the metricity conditions – see equation (3.4)
for a geometrical definition of these conditions, and Section 3.2 for a discussion of its physical
meaning. It should be stressed that the zero torsion condition in this approach becomes part of
the bundle of fields, rather than appearing as part of the equations of motion, as is usually the
case (see for example [24, 35]).
The underlying variational problem is given by the triple
(τ ′1 : T0 →M,LPG, 〈ωp〉),
where LPG is a Lagrangian m-form on T0 and 〈ωp〉 is an exterior differential system on T0
encoding the metricity conditions.
It should be stressed that, when working with such variational problem, the allowed variations
are not those fullfilling the commutativity property
δuAµ =
(
δuA
)
µ
;
instead, they must be thought as infinitesimal symmetries of the metricity conditions. We will
not explore further this topic here; the interested reader can find a detailed study of these
questions in [7]. On the other hand, the usual variational problem for Einstein–Hilbert gravity
is given by the triple(
(τΣ)2 : J2τΣ →M,LEH, IΣ
con
)
where τΣ : Σ → M is the bundle of metrics with a given signature, LEH is the (second order)
Lagrangian density [19] and IΣ
con is the contact structure on the second order bundle J2τΣ.
Although it would be possible in principle to work with a second order jet, it is easier to deal
Routh Reduction of Palatini Gravity in Vacuum 5
with first order jets; so, we are forced to find a first order variational problem describing Einstein–
Hilbert gravity. In fact, one of the results of the present article is to prove that there exists an
m-form (not a density) L(1)
EH on J1τΣ such that(
(τΣ)1 : J1τΣ →M,L(1)
EH, I
Σ
con
)
also describes Einstein–Hilbert gravity – see Section 8, where a correspondence between the
extremals for these problems will be established.
In summary, a large part of the present article consists in repeating for Palatini gravity
what was done for first order field theory in [6]; more concretely and in line with the previous
discussion, we want to answer an specific problem through a particular set of techniques:
• Problem: To establish correspondences between the extremals of the variational problem(
τ ′1 : T0 →M,LPG, 〈ωp〉
)
,
describing Palatini gravity, with the extremals of the variational problem(
(τΣ)2 : J2τΣ →M,LEH, IΣ
con
)
for Einstein–Hilbert gravity.
• Techniques: The following diagram could help us to outline the strategy involved in
proving the correspondences:
(WPG, λPG, 0) (WEH, λEH, 0)
(
J1τΣ,L(1)
EH, IΣ
con
)
(T0,LPG, 〈ωp〉)
(
J2τΣ,LEH, IΣ
con
)
1
2
3
Reconstruction
Reduction
(1.3)
The main idea is to replace the Griffiths variational problems representing Palatini and
Einstein–Hilbert gravity with other kind of variational problems playing the role, in this
generalized context, of the unified formalism. This operation is indicated schematically
by arrows 1 and 2 . As we described above, formulation of the unified version of field
theory requires to pass to a new phase space where multivelocities and multimomentum
variables are represented. Nevertheless, the product bundle construction (1.1) is not versa-
tile enough to establish the correspondences between the variational problems mentioned
above because it depends on the interpretation of the bundle J1π◦ as a dual of the jet
space J1π. Instead, these correspondences could be studied by using a very general set-
ting described in [20], called Lepage-equivalent variational problem. It consists in lifting
the variational problem
(
π1 : J1π → M,L, Icon
)
to a variational problem on an affine
subbundle of the bundle of m-forms on J1π, and this subbundle becomes isomorphic to
(a quotient of) Wcl. Concretely, let us fix an adapted coordinate chart
(
φ =
(
xµ, uA
)
, U
)
on E; then we have a set of (m− 1)-forms defined as follows
η := dx1 ∧ · · · ∧ dxm, ην :=
∂
∂xν
yη.
6 S. Capriotti
In this setting, the Lepage-equivalent problem of Gotay is given by the data
(π0 : W0 →M,λ0)
where
W0 := L+ Imcon ⊂ ∧m
(
T ∗J1π
)
(1.4)
and, locally, the bundle of forms Imcon is given by
Imcon
∣∣
(xµ,uA,uAν )
:=
{
pµA
(
duA − uAν dxν
)
∧ ηµ : pµA ∈ R
}
⊂ ∧m
(
T ∗(xµ,uA,uAν )
(
J1π
))
.
The Lagrangian form λ0 arises from the fact that W0 is a subbundle of the bundle of
m-forms ∧m
(
T ∗J1π
)
; it has a canonical m-form λ, and λ0 is its pullback to W0. The
underlying action becomes
Γπ0 3 σ 7→
∫
U
σ∗λ0
and the variations must be performed in an entirely free way; that is, in this case, the
variations δuA and δuAµ on J1π are independent. As we said above, the coordinates along
the fibers of the map
q : W0 → J1π
correspond to multimomentum variables, and we can see that
W0 ' J1π ×M J1π◦,
namely, the bundle W0 is isomorphic to the total bundle Wcl in the unified formulation
of first order field theory. Thus, it can be readily seen that Lepage-equivalent variational
problems serve as a generalization of the unified formalism to variational problems of
Griffiths type; the interested reader is referred to [20] for details.
Part of the techniques we will use to establish the relationship (reduction and reconstruc-
tion) between the extremals of these variational problems are analogous to those that
were employed in [6]: To transform the variational problems into its Lepage-equivalent
problems, represented by the two-way arrows 1 and 2 in diagram (1.3), and to use the
operations available there in order to set the correspondences. As we indicated above,
in order to apply this construction to Einstein–Hilbert gravity, it will be necessary to
consider an equivalent first-order variational problem instead; this operation is shown as
arrow 3 in the previous diagram. The rest of the techniques involved in the proof of
reduction/reconstruction are represented by the dashed arrow: The basic idea is to take
advantage of the fact that the variational problems we are trying to connect are formu-
lated on an affine subbundles of a bundle of forms. As we said above, the naturality of
forms respect to the pullback operation is crucial to compare the equations governing the
extremals in every variational problem. Although in our present article this characteristic
will not be used, it is still true that the vector bundle structure enhances this comparison
by allowing the translation along a given form.
These considerations set the purposes of the following article: On the one hand, to carry
out a proof of concept for the generalization of Routh reduction to variational problems more
general than those corresponding to first order field theory, generalizing the techniques employed
in [6]; on the other hand, to apply Routh reduction of field theory in the context of a meaningful
Routh Reduction of Palatini Gravity in Vacuum 7
example, namely, a formulation of gravity with basis. In short, the results achieved in the article
could help to put the relationship between Einstein–Hilbert and Palatini gravity in a precise
geometrical framework, which can in principle be extended to more sophisticated field theoretic
analyses.
In this vein, it should be mentioned that among the objectives in mind while planning the
article there was one of mathematical nature; in short, to contribute to the improvement of the
language in a geometrical sense around the general question on how Einstein–Hilbert and Palatini
versions of gravity are related. In this regard, it appears that the subject has continuously
been developed since these formulations were found, but the use of a more precise geometrical
language in describing the new achievements in the area has grown at a slower pace – see
[10, 19, 38, 41] for examples dealing with the geometrical perspective in gravity. To some
extent, this imbalance could place some obstacles to the natural development of some aspects
of the subject; in these cases, the geometrical viewpoint might be more useful than a mere
calculation in their understanding. Hence, the present article intends to contribute to a better
comprehension of the correspondence between Einstein–Hilbert and Palatini gravity, through
the formulation of this correspondence in a geometrical framework.
The rest of the paper is organized as follows: In Section 2 fundamentals of Griffiths variational
problems and their relationship with the unified formalism in field theory are given. In Section 3
geometrical tools necessary for the construction of the variational problem for Palatini gravity
we will use in this article are reviewed. In Section 4 the actual construction of this variational
problem, as well as the associated unified problem, is done. The symmetry considerations
necessary to carry out the reduction are discussed in Section 5. Section 6 is rather technical,
containing some calculations used in the reduction and reconstruction theorems. In Section 7
the results achieved in the previous section are employed in the search of identifications between
geometrical structures present in both the reduced and unreduced spaces: A remarkable fact in
this vein is that the metricity constraints correspond with the contact structure of a jet bundle
after projection onto the quotient. Construction of the first order formalism for Einstein–Hilbert
gravity (and its correspondence with the usual second order formalism) is delayed until Section 8;
also, a unified formalism for this variational problem is discussed in this section. The choice of
a connection induces a splitting in the contact structure on the jet space of the frame bundle.
In Section 9 the effects of this splitting in the variational formulation of Palatini gravity are
analyzed. In Section 10 the Routhian is constructed, showing that the Routhian for Palatini
gravity is the (first order) Einstein–Hilbert Lagrangian. Finally, in Section 11 the reduction
theorem and the reconstruction theorem are proved. The main result of this section is the notion
of flat condition for a metric, which is a helpful hypothesis in the proof of the reconstruction
theorem. Also, it is proved there that this condition is equivalent to the parallelizability of the
spacetime manifold M ; therefore, the reconstruction scheme can be carried out only locally.
Notations
We are adopting here the notational conventions from [39] when dealing with bundles and its
associated jet spaces. Also, if Q is a manifold, Λp(Q) = ∧p(T ∗Q) denotes the p-th exterior power
of the cotangent bundle of Q. Moreover, for k ≤ l the set of k-horizontal l-forms on the bundle
π : P → N is
∧lk(P ) :=
{
α ∈ ∧l(P ) : v1y · · · vkyα = 0 for any v1, . . . , vk π-vertical vectors
}
. (1.5)
For the same bundle, the set of vectors tangent to P in the kernel of Tπ will be represented with
the symbol V π ⊂ TP . In this regard, the set of vector fields which are vertical for a bundle
map π : P → N will be indicated by XV π(P ). The space of differential p-forms, sections of
Λp(Q) → Q, will be denoted by Ωp(Q). We also write Λ•(Q) =
⊕dimQ
j=1 Λj(Q). If f : P → Q is
8 S. Capriotti
a smooth map and αx is a p-covector on Q, we will sometimes use the notation αf(x) ◦ Txf to
denote its pullback f∗αx. If P1 → Q and P2 → Q are fiber bundles over the same base Q we
will write P1 ×Q P2 for their fibred product, or simply P1 × P2 if there is no risk of confusion.
Unless explicitly stated, the canonical projections onto its factor will be indicated by
pri : P1 × P2 → Pi, i = 1, 2
or, in general
pri1···ik : P1 × · · · × PL → Pi1 × · · · × Pik
for any collection of different indices i1, . . . , ik ∈ {1, . . . , L}.
Given a manifold N and a Lie group G acting on N , the symbol [n]G for n ∈ N will indicate
the G-orbit in N containing n; the canonical projection onto its quotient will be denoted by
pNG : N → N/G.
Also, if g is the Lie algebra for the group G, the symbol ξN will represent the infinitesimal
generator for the G-action associated to ξ ∈ g. Finally, Einstein summation convention will be
used everywhere.
2 Variational problems and unified formalism
As we mentioned in Section 1, the scheme used for Routh reduction relies on the notion of
unified formulation of a variational problem. So, we will devote the present section to describe
the construction of a unified formalism for a particular family of variational problem, the so
called Griffiths variational problems.
Definition 2.1 (Griffiths variational problem). A Griffiths variational problem is a triple
(p : W →M,λ, I)
where p : W → M is a bundle, λ ∈ Ωm(W ) is a form on W (here m is the dimension of the
base manifold M) and I is an exterior differential system. The underlying variational problem
consists in finding the extremals of the action
Γp 3 σ 7→
∫
U
σ∗λ,
where U ⊂M is any compact submanifold and σ is a section integral for I, namely,
σ∗α = 0
for any α ∈ I.
Remark 2.2. A particular instance of a Griffiths variational problem is provided by the so
called classical variational problem, which is the variational problem underlying the first order
classical field theory [21]: In it, the underlying bundle is the jet space π1 : J1π →M associated
to a bundle (E, π,M), the Lagrangian form is induced on Ωm
(
J1π
)
by the Lagrangian density
L : J1π → ∧m(T ∗M)
and the set of differential restrictions is imposed by the so called contact structure on the jet
bundle. Particularly, it means that the sections to be evaluated in the action should be holo-
nomic.
Routh Reduction of Palatini Gravity in Vacuum 9
In view of the previous discussion, there are two crucial differences between a classical vari-
ational problem and a more general Griffiths variational problem that we would like to point
out:
• First of all, a classical variational problem (of first order) is formulated in a first order jet
bundle, whereas a Griffiths variational problem can use any bundle in principle.
• More important is the fact that the sections are integral for the set of forms I, that in the
general case (as in the present article) could be different of the set of forms belonging to
the contact structure.
A construction used in [20] becomes relevant to this article: Given a Griffiths variational
problem (p : W → M,λ, I), it is possible to build another bundle p′ : W ′ → M and a bundle
map q : W ′ →W covering the identity on M . The new bundle becomes part of a new variational
problem (p′ : W ′ → M,λ′, 0), and the extremals of the new variational problem can be set into
a one-to-one correspondence with the extremals of the original variational problem through the
map q. The variational problem (p′ : W ′ → M,λ′, 0) is called a Lepage-equivalent variational
problem.1 The coordinates along the fibers of the map p′ : W ′ →W can be seen as a generaliza-
tion of momentum variables in the context of these variational problems. An important feature
of a Lepage-equivalent variational problem is that it imposes no differential restrictions on the
sections of p′ : W ′ →M , which could be useful when variations are performed.
A particular instance of this construction called canonical Lepage-equivalent problem will
be relevant for the present article. It requires that the differential constraints encoded by the
exterior differential system I be generated by the sections of a subbundle I ⊂ ∧•(T ∗W ); under
such conditions, we can define
W ′|w :=
{
λ(w) + β : β ∈ I ∩ ∧m(T ∗wW )
}
⊂ ∧m(T ∗W )
for all w ∈ W . The projection p′ : W ′ → W is the restriction of the canonical map τmW : ∧m
(T ∗W ) → W to this subbundle, and the m-form λ′ is the pullback of the canonical m-form
Θ ∈ Ωm(∧m(T ∗W )) to W ′.
In order to describe the relevance of this construction, it will be necessary to refer to the
concept of unified formalism for classical first order field theory, as defined in [2, 16, 36] and
references therein. This formulation becomes useful when dealing with variational problems
whose Lagrangian densities have singular Legendre transformations. The trick is to lift the
variational problem to an space where both velocities and momenta are included and to forget
about the differential restrictions on the fields imposed by the contact structure of the jet bundle.
Therefore, variations can be performed without having to identify the independent degrees of
freedom, and the formulas defining Legendre transform become part of the equations of motion;
on the downside, these equations of motion are in general of the algebraic-differential type,
and thus more difficult to deal with. As we mentioned in the Introduction, this underlying
bundle Wcl for the unified formulation of field theory is given by equation (1.1); the Lagrangian
functional L gives rise to an m-form ΘL on Wcl, and the associated action becomes
Γ(Wcl →M) 3 σ 7→
∫
U
σ∗ΘL.
Finally, let us discuss briefly the relationship between Lepage-equivalent problems and the
unified formalism. It can be seen that, when restricted to the particular case of the classical vari-
ational problem, the canonical Lepage-equivalent variational problem devised by Gotay reduces
1There are some subtleties regarding the use of the word “equivalent” in this context; they will not be discussed
here, so we are referring to the readers interested in these questions to the original article of Gotay.
10 S. Capriotti
to the variational problem associated to the unified formalism. In fact, if the fields are sections
of a bundle π : E →M , the triple describing the associated Griffiths variational problem is(
π1 : J1π →M,L, Icon
)
,
where Icon ⊂ Ω•
(
J1τ
)
is the exterior differential system induced by the contact structure. In
this case, we have the correspondences
W ! J1π, λ! L, I! Icon, W ′!Wcl,
allowing us to think on Lepage-equivalent problems as generalizations of the unified formalism
for field theory.
3 Geometrical tools for Palatini gravity
We choose to focus on variational problems of Griffiths type because there exists a description of
Palatini gravity in terms of this kind of variational problems [5]. The present section is devoted
to give a brief account of the geometrical ingredients involved in this construction.
3.1 Geometry of the jet space for the frame bundle
The basic bundle is the frame bundle τ : LM →M on the spacetime manifold M (dimM = m);
because it is a principal bundle with structure group GL(m), we can lift this action to the jet
bundle J1τ , so that we obtain a commutative diagram
J1τ
LM C(LM)
M
τ10
pJ
1τ
GL(m)
τ1
τ τ
where C(LM) := J1τ/GL(m) is the so called connection bundle of LM , whose sections can
be naturally identified with the principal connections of the bundle τ – for details, see [9] and
references therein. It is interesting to note that there exists an affine isomorphism
F : J1τ → LM ×M C(LM) : j1
xs 7→
(
s(x),
[
j1
xs
]
GL(m)
)
and under this correspondence, the GL(m)-action is isolated to the first factor in the product,
namely
F
(
j1
xs · g
)
=
(
s(x) · g,
[
j1
xs
]
GL(m)
)
. (3.1)
It means that a section of the bundle τ1 is equivalent to a connection on LM plus a moving
frame (X1, . . . , Xm) on M ; although this moving frame has no direct physical interpretation, we
can associate a metric to it, namely, in contravariant terms,
g := ηijXi ⊗Xj
Routh Reduction of Palatini Gravity in Vacuum 11
for some nondegenerate symmetric matrix η – see equation (3.2) below. It is the same to
declare that the metric g is the unique metric on M making the moving frame (X1, . . . , Xm)
(pseudo)orthonormal, with the signature given by η.
The tautological form θ̃ ∈ Ω1(LM,Rm) can be pulled back along τ10 to a 1-form θ := τ∗10θ̃
on J1τ ; moreover, the Cartan form ω̃ ∈ Ω1
(
J1τ, V τ
)
, given by the formula
ω̃|j1xs := Tj1xsτ10 − Txs ◦ Tj1xsτ1,
gives rise to a gl(m)-valued 1-form ω on J1τ , by using the identification
V τ ' LM × gl(m).
By means of the canonical basis {ei} on Rm and
{
Eij
}
on gl(m), where(
Eij
)q
p
:= δqj δ
i
p,
we can define the collection of 1-forms
{
θi, ωij
}
on J1τ such that
θ = θiei, ω = ωjiE
i
j .
We also have the formula
ω̃ = ωji
(
Eij
)
J1τ
,
where AJ1τ ∈ X
V pJ
1τ
GL(m)
(
J1τ
)
is the infinitesimal generator associated to A ∈ gl(m) for the lifted
action. It can be proved that ω is a connection form for a principal connection on the bundle
pJ
1τ
GL(m) : J1τ → C(LM).
Remark 3.1 (coordinates on the jet space J1τ and the connection bundle τ : C(LM) → M).
Let (φ = (xµ), U) be a coordinate chart on M ; for u = (X1, . . . , Xm) ∈ τ−1(U), we define the
maps u 7→ eµi (u) such that
Xi = eµi (u)
∂
∂xµ
for 1 ≤ i ≤ m. Then
u 7→
(
xµ(τ(u)), eµi (u)
)
defines a set of adapted coordinates on τ−1(U); let
(
xµ, eµi , e
ν
kσ
)
be the associated coordinates
on τ−1
1 (U) ⊂ J1τ . We can also define the functions
Γµρσ := −ekρe
µ
kσ,
on τ−1
1 (U), and we can prove that they are GL(m)-invariant; so, they define a set of coordinates
on τ−1(U) ⊂ C(LM).
Let us define
θ0 := θ1 ∧ · · · ∧ θm;
as every u ∈ LM is a collection u = (X1, . . . , Xm) of vectors on τ(u) ∈ M , and θi is a τ1-
horizontal 1-form on J1τ , we can define the set of forms
θi1···ik |j1xs := Xi1y · · ·Xikyθ0|j1xs
for 1 ≤ i1, . . . , ik ≤ m, where j1
xs ∈ J1τ is any element such that u = τ10
(
j1
xs
)
.
12 S. Capriotti
Although the reduction scheme we will develop in this article would work for any signature,
let us fix it using the matrix
η :=
−1 0 · · · 0
0 1 0
...
. . .
...
0 · · · 0 1
∈ GL(m). (3.2)
Accordingly, let ηij be its (i, j)-entry; we will represent with the symbol ηij the (i, j)-entry of
its inverse. With these ingredients we can construct the Palatini Lagrangian
LPG := ηipθik ∧ Ωk
p, (3.3)
where Ω := Ωi
jE
j
i is the curvature of the canonical connection ω. This m-form will determine
the dynamics of the vacuum gravity in this formulation.
Finally, let us describe a decomposition of gl(m) induced by η. In fact, this matrix yields to
a real form u in gl (m,C), given by
u =
{
ξ ∈ gl(m,C) : ξ†η + ηξ = 0
}
and thus we have a Cartan decomposition
gl(m,C) = u⊕ s.
Given the inclusion
gl(m) ⊂ gl(m,C),
we obtain the decomposition
gl(m) = k⊕ p.
The subalgebra k is the Lie algebra of the subgroup K ⊂ GL(m), composed of the linear
transformations keeping invariant the matrix η,
K :=
{
A =
(
Aji
)
: ηijA
i
kA
j
l = ηkl
}
.
The canonical action of GL(m) on LM restricts to an free action ofK on this bundle; accordingly,
we have the K-principal bundle
pLMK : LM → Σ := LM/K.
The bundle τΣ : Σ → M induced by this quotient has an immediate physical meaning: Any
section ζ : M → Σ is a metric on M with η-signature. In fact, let us define the map q : LM →
TM ⊗M TM via
q(X1, . . . , Xm) := ηijXi ⊗Xj ;
indicating by Rk : LM → LM the action of an element k ∈ K on LM , it can be proved that
q ◦Rk = q, so that there exists a map q : LM/K → TM ⊗M TM making the following diagram
commutative
LM Σ
TM ⊗M TM
pLMK
q
q
Routh Reduction of Palatini Gravity in Vacuum 13
When restricted to the subbundle of nondegenerate symmetric 2-tensors (with η-signature)
on M , this map becomes a bundle isomorphism on M , allowing us to interpret Σ as the bundle
of metrics with η-signature on M .
Remark 3.2 (adapted coordinates for the principal bundle pLMK : LM → Σ). It will be useful
to introduce a set of coordinates on LM adapted to the quotent map pLMK . In order to proceed,
we need a theorem on generalized polar decomposition [25]. So, let U ⊂ M be a parallelizable
open set in M , let {Z1, . . . , Zm} ⊂ X(U) be a moving basis on U , and {e1, . . . , en} the canonical
basis on Rm; for every x ∈ U denote V = TxM and W = Rm. The matrix η defines a linear
map
η : W →W ∗;
another operator that will be important for the formulation of the polar decomposition theorem
is
I : V →W : Zµ 7→ δiµei.
Also, set
η̃ := I∗ ◦ η ◦ I.
It is clear that, in this setting, a metric is a map
gx : V → V ∗
for every x ∈ U , and a vielbein becomes a linear map
E : V →W.
Then, the following can be proved.
Theorem 3.3 (generalized polar decomposition). Let g : V → V ∗ be an invertible linear map
such that η̃ ◦ g−1 has no eigenvalues on the nonpositive real axis. Then g can be uniquely
descomposed as
g = Q ◦ s,
where Q : W → V ∗ satisfies
Q∗ ◦ η̃ ◦Q = η
and for s : V →W the following property
I∗ ◦ η ◦ s = (I∗ ◦ η ◦ s)∗
holds.
Therefore, the condition on the factor Q tells us that the composite map
I∗ ◦Q : W →W ∗
belongs to K. Thus, let V ⊂ τ−1
Σ (U) be an open set in Σ such that η̃ ◦ g−1 has no eigenvalues
on the nonpositive real axis for every g ∈ V . Using this theorem, we can define the K-bundle
isomorphism
u ∈
(
pLMK
)−1
(V ) 7→
(
pLMK (u), I∗ ◦Q
)
∈ V ×K,
where Q is the corresponding factor in the polar decomposition of g = pLMK (u). Finally, given
a set of coordinates
(
xµ, gµν
)
on τ−1
Σ (U) and
(
kji
)
on K, we can introduce the set of coordinates
u 7→
(
xµ, gµν , kji
)
on τ−1
1 (U) ⊂ LM , where
(
xµ, gµν
)
are the coordinates for pLMK (u) and
(
kji
)
are the coordinates
on K for I∗ ◦Q.
14 S. Capriotti
3.2 Restrictions in Palatini gravity: Zero torsion submanifold
and metricity forms
It is time to discuss the restrictions we must impose on the sections of τ1 in order to have
a characterization of a gravity field in this description. Our aim is to describe a metric and
a connection on the spacetime, and the restrictions to be considered will establish the relationship
between them; this approach has been extensively discussed in the references [3, 5].
Recall that, according to the identification performed by the map F – see equation (3.1)
above – a section of J1τ can be seen as a pair composed by a frame plus a connection. Corre-
spondingly, there are two types of conditions to be imposed to a section of J1τ , each of them
motivated on physical grounds which we will not discuss here:
1) the connection which is a solution for the field equations of Palatini gravity must be
torsionless, and
2) this connection must be metric for the solution metric.
The canonical forms defined in the previous section allow us to set the torsion form
T :=
(
dθj + ωjk ∧ θ
k
)
⊗ ej ∈ Ω2
(
J1τ,Rm
)
.
Now, every connection Γ: M → C(LM) gives rise to a section σΓ : LM → J1τ of the bundle
τ10 : J1τ → LM , as the equivariance of the following diagram shows
J1τ
LM C(LM)
M
τ10
pJ
1τ
GL(m)
τ
σΓ
τ
Γ
The interesting fact is that the pullback form σ∗ΓT coincides with the torsion of the connection Γ.
Additionally, it can be proved that T is a 1-horizontal form on τ1 : J1τ →M , so that there exists
a maximal (respect to the inclusion) submanifold i0 : T0 ↪→ J1τ such that
1) T0 is transversal to the fibers of τ1 : J1τ → M (namely, Tj1xs(T0) ⊕ Vj1xsτ1 = Tj1xs
(
J1τ
)
),
and
2) it annihilates the torsion, i.e.,
i∗0T = 0.
The transformation properties of the form T allow us to conclude that T0 is GL(m)-invariant.
The connections associated to sections of J1τ taking values in T0 are torsionless, so that the
zero torsion restriction can be achieved through the requirement that these sections take values
in this submanifold. Accordingly, we can use the affine isomorphism F : J1τ → LM × C(LM),
to define the bundle of torsionless connections as the bundle τ ′ : C0(LM) → M obtained by
restricting F to T0
C0(LM) := pr2(F (T0)).
Moreover, the following lemma can be proved using standard facts about principal bundles [29].
Routh Reduction of Palatini Gravity in Vacuum 15
Lemma 3.4. The submanifold T0 ⊂ J1τ is a subbundle of the GL(m)-bundle pJ
1τ
GL(m) : J1τ →
C(LM), associated to the isomorphism id : GL(m) → GL(m). Also, it is a GL(m)-principal
bundle with respect to the restriction of the GL(m)-action.
These considerations give rise to the commutative diagram
T0
LM C0(LM)
M
τ ′10
p
T0
GL(m)
=pJ
1τ
GL(m)
∣∣
T0
τ ′
τ ′
Remark 3.5 (local description for T0). Let (xµ, eνk) be a set of adapted coordinates for LM
induced on τ−1(U) by a set of coordinates (xµ) on U ⊂ M ; as was shown in Remark 3.1, it
induces coordinates (xµ, eνk, e
ν
kσ) on τ−1
1 (U). On this open set we have
T = eiσ
(
ekµe
σ
kνdx
µ ∧ dxν
)
⊗ ei,
where (ekµ) is the inverse matrix of (eµk), so that the set T0∩τ−1
1 (U) is described by the constraints
ekµe
σ
kν = ekνe
σ
kµ.
On the other hand, the metricity condition has a differential nature: As we mentioned above,
matrix η determines a factorization of gl(m) in a subalgebra k (the subalgebra of η-Lorentz
transformations) and an invariant subspace p. The explicit formulas for this decomposition are
given by the projectors
Ak :=
1
2
(
A− ηAT η
)
, Ap :=
1
2
(
A+ ηAT η
)
for every A ∈ gl(m). The metricity condition is imposed on a section s : M → J1τ by requiring
that
s∗ωp = 0, (3.4)
where ωp is the p-component of the canonical connection ω respect to this decomposition. Taking
into account the affine isomorphism F : J1τ → LM ×M C(LM), this constraint means that
the parallel transport of the connection pr2 ◦ F ◦ s : M → C(LM) leaves invariant the metric
associated to the vielbein pr1 ◦ F ◦ s : M → LM (see equation (7.4) below).
Remark 3.6 (local expression for the metricity conditions). Let us provide a description of the
metricity conditions (3.4) in terms of the adapted coordinates (xµ, eµk , e
µ
kν) on τ−1
1 (U) ⊂ J1τ . It
results that for a section
s : (xµ) 7→
(
xµ, eµk(x), eµkν(x)
)
,
these conditions can be written as [3]
dgµν +
(
gµσΓνγσ + gνσΓµγσ
)
dxγ = 0, (3.5)
16 S. Capriotti
where gµν(x) := ηkleµk(x)eνl (x) is the metric associated to this section and
Γµρσ(x) := −ekρ(x)eµkσ(x)
are the Christoffel symbols for the associated connection. As it is well-known, these conditions
plus zero torsion imply the metric and the symbols become related by Levi-Civita formula
Γσµν =
1
2
gσρ
(
∂gρµ
∂xν
+
∂gρν
∂xµ
− ∂gµν
∂xρ
)
.
Remark 3.7 (on the nature of the metricity conditions). In some approaches to Palatini gravity,
it is customary to work with a SO(3, 1) connection, without a clear reference to the bundle
to which this connection should be associated. It gives rise to an ambiguity that could be
problematic in some cases. Namely, when in local terms it is referred to a so(3, 1)-valued 1-form
α : TM → so(3, 1),
it is not clear whether it is a local version of a connection on a principal bundle with structure
group SO(3, 1) or instead, a local version of a connection on a principal bundle with a larger
structure group G ⊃ SO(3, 1) taking values in the smaller Lie algebra so(3, 1) ⊂ g. It is not
uncommon to find articles in the literature where the information provided to the reader to
decide in which case one is working is not enough, because in general, the underlying principal
bundle is not associated to quantities of physical nature. It should remain clear that in general,
these connections are not equivalent, unless a reduction of the G-bundle to a subbundle with
structure group SO(3, 1) is admitted. When G = GL(4), it is equivalent to have a metric with
signature 3 + 1; in this case, the SO(3, 1)-bundle is the bundle of pseudo orthogonal frames. It
could be argued that to have a metric at our disposal is a feature of working with metric-affine
theories of gravitation; but then it should be pointed out in this case that variations of the
metric have the undesired side effect of changing the SO(3, 1)-bundle. In order to avoid this
behavior, we have adopted metricity conditions (3.4): A connection verifying these conditions
will be a connection on the frame bundle, whose structure group is GL(m), and will have a local
version taking values in the subalgebra k ⊂ gl(m) associated to the matrix η.
4 Griffiths variational problem for Palatini gravity
The variational problem we will consider here for the Palatini gravity is not a classical one; it
will differ from a variational problem of this kind in both of the aspects mentioned in Section 2:
• The relevant bundle is not the first order jet J1τ ; instead, it is the subset T0 consisting in
the jets associated to torsionless connections. Due to this fact, we will consider the pullback
of the canonical forms and the restriction of maps from J1τ to T0; unless explicitly stated,
the new forms and maps will be indicated with the same symbols. An exception to this
rule will be the restriction of the bundle maps τ10 and τ1, which will be indicated as
τ ′10 : T0 → LM and τ ′1 : T0 →M.
• The forms we will use for the restriction of the sections of τ ′1 : T0 → M are not the whole
set of contact forms
{
ωij
}
, but a geometrically relevant subset, namely, the components of
the metricity forms ωp.
Using these considerations, we will introduce the following definition.
Routh Reduction of Palatini Gravity in Vacuum 17
Definition 4.1 (Griffiths variational problem for Palatini gravity). The variational problem for
Palatini gravity is given by the action
S[σ] :=
∫
U
σ∗LPG,
where σ : U ⊂ M → T0 is any section of τ ′1 such that σ∗ωp = 0. According to equation (1.2), it
is described by the triple (τ ′1 : T0 → M,LPG, 〈ωp〉), where 〈·〉 indicates the exterior differential
system generated by the set of forms enclosed by the brackets.
Remark 4.2. The considerations made in Section 3 allow us to compare the variational problem
given by Definition 4.1 with the classical variational problem, as in [8]. In fact, Proposition 4
in this reference tells us that when holonomic sections are factorized using the isomorphism F
(in [8] it is called Ψ), the connection
pr2 ◦ F ◦ s : M → C(LM)
is flat, a restriction far more stringent than those imposed by the metricity conditions, that only
require the associated connection form to have values in k. For a description of these admissible
variations in the case of the metricity constraints, see [7, Section 4.1].
It is interesting to point out the effect that the replacement of the contact structure with
the differential system 〈ωp〉 has on the way that Euler–Lagrange equations are calculated. The
key change has to do with the characterization of the admissible infinitesimal variations: Given
that the metricity constraints are a part of the constraints imposed by the contact structure,
it results that the set of admissible infinitesimal variations considered in [8] contain them as
a subset.
In order to establish the unified version of the equations of motion for Palatini gravity, it will
be necessary to represent the metricity constraints as a subbundle of the set of m-forms on T0.
To this end, let us define the metricity subbundle ImPG on T0,
ImPG :=
{
ηikβkp ∧ ωpk : βij ∈ ∧m−1
1 (T ∗T0), βij − βji = 0
}
⊂ ∧m2 (T ∗T0),
where ∧m−1
1 (T ∗T0) indicates the set of τ ′1-horizontal (m− 1)-covectors on T0.
Remark 4.3 (local description for the metricity subbundle). Given a nondegenerate symmetric
matrix (gµν), there exists a matrix
(
eµk
)
such that
gµν = ηkleµke
ν
l ;
also, any two matrices (eµk), (fµk ) giving rise to the same symmetric matrix (gµν) differ in an
element (kij) ∈ GL(m) such that
kijk
l
mη
jm = ηil.
Using the factorization
T0 = LM ×M C0(LM),
we can set the coordinates
(
xµ, gµν , kij ,Γ
µ
ρσ
)
on T0; the functions Γµρσ are not independent; rather,
they are constrained by the relations
Γµρσ = Γµσρ.
18 S. Capriotti
On the other hand, metricity constraints are given by equation (3.5)); therefore
ImPG
∣∣
(xµ,gµν ,kij ,Γ
µ
ρσ)
=
{
pρµν
[
dgµν +
(
gµσΓνγσ + gνσΓµγσ
)
dxγ
]
∧ ηρ : pρµν = pρνµ
}
,
where, as before
ηρ :=
∂
∂xρ
yη, η = dx1 ∧ · · · ∧ dxm.
It is a description of the fibers of ImPG in local terms; the independence of these forms respect
to the variables
(
kji
)
is due to the fact that the group K describes a fundamental symmetry of
this description of gravity.
Remark 4.4 (metricity conditions and the annihilation of the metricity tensor). It could be
useful to give an explanation of the relevance of this subbundle. To this end, recall the notion
of metricity tensor of a connection respect to a metric [23]; in short, given a manifold with
a metric and a connection in it, this tensor is essentially the covariant derivative of the metric.
When working in the jet space of the frame bundle, where a section induced both a metric and
a connection, this tensor can be encoded in terms of the canonical connection. Concretely, let
us suppose that s : M → T0 is a section with the following property
s∗
(
ηikβip ∧ ωpk
)
= 0
for any collection of l-forms βip such that
βip − βpi = 0.
Then the connection on LM induced by s has zero metricity tensor; therefore, the metricity
subbundle allows us to introduce the set of restrictions imposed by the annihilation of the
metricity tensor into the Lagrangian. From this viewpoint, forms βij in the definition of the
metricity subbundle play the role of Lagrange multipliers.
With the metricity subbundle in mind, and following the prescriptions made in [20] for
the construction of the canonical Lepage-equivalent variational problem, we define the affine
subbundle
WPG := LPG + ImPG ⊂ ∧m2 (T ∗T0), (4.1)
which comes with the projection
τPG : WPG → T0 : α ∈ ∧m2
(
T ∗j1xsT0
)
7→ j1
xs.
We are referring to the interested reader to the article of Gotay for details; for the purposes
of the present work, it should be enough to mention that the bundle defined by equation (4.1)
will become the underlying bundle for the Lepage-equivalent problem associated to the Griffiths
variational problem describing Palatini gravity (see Definition 4.1 above).
Because this is a subbundle in the set of m-forms on T0, it has a canonical m-form λPG on it
given by
λPG|α(w1, . . . , wm) := α(TατPG(w1), . . . , TατPG(wm)), w1, . . . , wm ∈ Tα(WPG).
This m-form will be the Lagrangian form for the Lepage-equivalent problem associated to the
Griffiths variational problem for Palatini gravity, namely
(τ ′1 ◦ τPG : WPG →M,λPG, 0). (4.2)
Routh Reduction of Palatini Gravity in Vacuum 19
Because no differential restrictions must be considered when performing variations of the varia-
tional problem
Γ
(
τ ′1 ◦ τPG
)
3 Γ 7→
∫
U
Γ∗λPG,
and assuming that the variations annihilate at the boundary, we obtain that the Euler-Lagrange
equations for the Lepage-equivalent variational problem (4.2) are
Γ∗(XydλPG) = 0 for all X ∈ XV (τ ′1◦τPG)(WPG).
The relevance of the unified formalism in dealing with the variational problems posed by Defi-
nition 4.1 is guaranteed by the following result [5].
Proposition 4.5. A section s : U ⊂ M → T0 is critical for the variational problem established
in Definition 4.1 if and only if there exists a section Γ: U ⊂M →WPG such that
1) Γ covers s, i.e., τPG ◦ Γ = s, and
2) Γ∗(XydλPG) = 0, for all X ∈ XV (τ ′1◦τPG)(WPG).
Γ is called a solution of the Palatini gravity equations of motion.
Remark 4.6. Although the proof in [5] refers to sections of τ1 : J1τ →M , it can be also readily
adapted to cover this case; in this regard, see Appendix B.
The situation described by Proposition 4.5 is summarized in the following diagram:
WPG T0
M
τ ′1◦τPG
τPG
τ ′1
Γ s
Accordingly, any section s that is extremal for the Griffiths variational problem
(τ ′1 : T0 →M,LPG, 〈ωp〉)
can be lifted to a section Γ that is extremal for the Lepage-equivalent variational problem
(τ ′1 ◦ τPG : WPG →M,λPG, 0)
and viceversa.
Remark 4.7 (local version of Proposition 4.5). According to Remark 4.3, section Γ will be
obtained from section s by providing the set of functions
(pσµν) : U → R
m2(m+1)
2 .
These momentum variables pσµν are determined by the equations of motion in Proposition 4.5,
concretely, through the contractions
Γ∗
(
∂
∂Γσµν
ydλPG
)
= 0.
We will see below (Section 8) that an unified formalism for (first order) Einstein–Hilbert
variational problem can also be given; the reduction and reconstruction theorems (see Section 11)
will be proved using these lifted systems.
20 S. Capriotti
5 Symmetry and reduction
As we described in the introduction, a crucial ingredient in Routh reduction is the restriction
of the dynamics to a level set of the momentum map. It forces us to discuss the presence of
natural symmetries in our formulation of gravity, and to construct their momentum maps. Also,
this procedure requires the choice of a connection on a bundle obtained by quotient out the
symmetries of the variational problem. This section is devoted to these tasks.
5.1 Momentum map and connection
As we said above (see Lemma 3.4), there exists a GL(m)-action on T0; nevertheless, the La-
grangian LPG is preserved by the action of the subgroup K ⊂ GL(m) composed of the linear
transformations keeping invariant the matrix η,
K :=
{
A =
(
Aji
)
: ηijA
i
kA
j
l = ηkl
}
.
We can lift the GL(m)-action to ∧m(T0); it results that the subbundle ImPG is also preserved by
the action of K, and so
K ·WPG ⊂WPG.
Our aim is to find a momentum map for this action, in the sense of the following definition.
Definition 5.1. A momentum map for the action of K on WPG is a map
J : WPG → Λm−1(T ∗WPG)⊗ k∗
over the identity in WPG such that
ξWPG
ydλPG = −dJξ,
where Jξ is the (m− 1)-form on WPG whose value at α ∈WPG is Jξ(α) = 〈J(α), ξ〉.
A momentum map is Ad∗-equivariant if it satisfies
〈J(gα),Adg−1ξ〉 = g〈J(α), ξ〉.
Also, it is said that J is conserved along a section Γ: M →WPG if and only if Γ∗(dJ) = 0.
Remark 5.2. A clarification about the nomenclature seems necessary here: Suppose M = R×N
for some (m− 1)-dimensional manifold N , we can take U ⊂ N and Γ: [T1, T2]×U →WPG such
that J is conserved along Γ and moreover
J(Γ(t, q)) = 0 for all T1 ≤ t ≤ T2 and q ∈ ∂U ;
then, denoting Γt : U →WPG : r 7→ Γ(t, r) for any t ∈ [T1, T2], Stokes’ theorem will tell us that∫
U
Γ∗T1
J =
∫
U
Γ∗T2
J.
It is in this sense that the momentum map is “conserved”, namely, the integral of the (m− 1)-
form Γ∗tJ is independent of t.
Thus, we obtain Noether’s theorem in this setting:
Proposition 5.3. The momentum map J is conserved along solutions of the Palatini gravity
equations of motion.
Routh Reduction of Palatini Gravity in Vacuum 21
Proof. Recall that Γ: U ⊂M →WPG is a solution for the Palatini gravity equations of motion
if and only if
Γ∗(ZydλPG) = 0
for any τ ′1 ◦ τPG-vertical vector field Z. Then for each ξ ∈ k we have
d(Γ∗Jξ) = Γ∗(dJξ) = Γ∗(−ξWPG
ydλPG) = 0,
and therefore the momentum is conserved along solutions. �
Accordingly, we think of a “momentum” µ̂ as an element µ̂ ∈ Ωm−1(WPG, gl(m)∗), i.e., as
a gl(m)∗-valued (m − 1)-form on WPG; a conserved value µ̂ of the momentum map is a closed
one, i.e., dµ̂ = 0.
The construction of a momentum map for the action on WPG is standard [21]:
Lemma 5.4. The map J : WPG → Λm−1(T ∗WPG)⊗ k∗ defined by
〈J(α), ξ〉 = ξWPG
(α)yλPG|α,
for each ξ ∈ k, is an Ad∗-equivariant momentum map for the GL(m)-action on WPG.
Now, because
TτPG ◦ ξWPG
= ξT0 ◦ τPG,
then for every α ∈WPG it results that
〈J(α), ξ〉 = ξWPG
(α)yλPG|α = ξT0yα
= i∗0
[
ξJ1τy
(
ηipθpk ∧ Ωk
i + ηipβpq ∧ ωqi
)]
= 0
for all ξ ∈ k. It means that the unique allowed momentum level set for this symmetry is J = 0;
accordingly, the isotropy group of this level set is K, and
J−1(0) = WPG.
The other ingredient needed in Routh reduction is the factorization of the metricity bun-
dle ImPG induced by a connection ωK on the underlying bundle pLMK : LM → Σ, where
τΣ : Σ := LM/K →M
is the bundle of metrics of signature η. We will carry out this task in Section 9; here we will
construct this connection. To this end, consider the decomposition associated to the matrix η (see
Section 3). The connection ωK on the bundle pLMK : LM → Σ is induced by this decomposition,
namely
ωK := πk ◦ ω0,
where πk : gl(m) → k is the canonical projector onto the k-factor in the Cartan decomposition
and ω0 is a connection form on the principal bundle τ : LM → M . The K-invariance of the
factor p,
AdAp ⊂ p ∀A ∈ K
ensures us that it has the expected properties of a connection.
22 S. Capriotti
5.2 Reduced bundle for Palatini gravity
We have singled out the symmetries of our formulation of Palatini gravity; they are described by
the Lie group K. On the other hand, Palatini gravity can be formulated through the variational
problem(
τ ′1 ◦ τPG : WPG →M,λPG, 0
)
.
The relevant bundles in this triple fit in the following diagram
WPG T0
M
τ ′1◦τPG
τPG
τ ′1
It should be stressed that the link between this diagram and the original variational problem
depends on the facts that T0 ⊂ J1τ and WPG ⊂ ∧m
(
T ∗
(
J1τ
))
: The first inclusion allows us to
interpret some fields as derivatives, while the second one tells us that other degrees of freedom
behave like (multi)momenta. These correspondences could be lost when symmetries are singled
out; by performing the quotients, the corresponding diagram becomes
WPG/K T0/K
M
The immediate problem is to find a variational problem associated to this diagram. As we said
above, this is difficult to achieve, because neither T0/K is a subset of a jet bundle nor WPG/K
is a subbundle of a space of forms. One of the objectives of Routh reduction is to identify in
these quotient spaces the degrees of freedom that can be described in this way, and to deal with
those that cannot be fitted in this classification. In order to proceed with this identification,
the connection ωK defined in Section 5.1 is used. Here we will carry out this operation on the
quotient space T0/K, leaving the discussion of the splitting of WPG/K for later (see Section 9).
Now, using the adjoint bundle τk : k̃→ Σ, the following result holds.
Proposition 5.5. The map
Υω : J1τ −→
(
pLMK
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
))
,
j1
xs 7−→
(
s(x), j1
x[s]K , [s(x), ωK ◦ Txs]K
)
.
is a bundle isomorphism.
The inverse of Υω is given by
Υ−1
ω
(
e, j1
xs,
[
e, ξ̂
]
K
)
=
[
vx ∈ TxM 7−−−→ (Txs(vx))He +
(
ξ̂(vx)
)
LM
(e)
]
,
where (·)He , e ∈ LM, is the horizontal lift associated to ωK .
The map Υω enjoys a useful property: under this identification, the action of K on J1τ is
simply
g ·
(
e, j1
xs, [e, ξ̂]K
)
=
(
g · e, j1
xs, [e, ξ̂]K
)
.
This is a direct consequence of the equivariance of the principal connection ωK . As a result,
we get the following corollary, which is well-known in the literature on Lagrangian reduction
[11, 12, 17].
Routh Reduction of Palatini Gravity in Vacuum 23
Corollary 5.6. There is an identification
J1τ/K ' J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
.
Remark 5.7. The choice of a connection on the bundle pLMK allows us to establish a relationship
between the quotient space J1τ/K and the jet bundle of the metric bundle J1τΣ, the latter being
the relevant bundle in the Einstein–Hilbert approach to relativity, which will be studied in detail
in Section 7.
Motivated by these considerations, we are in position to split the quotient bundle J1τ/K
into a jet bundle part and a set of complementary degrees of freedom. It suggests the following
definition.
Definition 5.8 (quotient bundle for Palatini gravity). The bundle J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
is
the quotient bundle for Palatini gravity.
In the next sections, we will explore a further simplification for this bundle as well as a
reduction for the Lagrangian responsible of the dynamics on these bundles.
5.3 Routh reduction scheme for Palatini gravity
Our aim is to interpret usual Einstein–Hilbert variational problem as a Routh reduction of the
Griffiths variational problem for Palatini gravity. As far as I know, there is no formulation of
this type of reduction that could be used in dealing with a Griffiths variational problem, so it is
necessary to generalize the techniques employed in [6] for Routh reduction in field theory to cover
this case. The variational problems to be related by this procedure are the Griffiths variational
problem for Palatini gravity, described in Definition 4.1, and a variational problem which has not
been determined yet, but whose underlying bundle would be related to the quotient bundle given
by Definition 5.8. Now, in Routh reduction the equivalence between the extremals is restricted
to those having a particular value of the momentum map;2 hence, it has some advantages to
work in the unified formalism, where momentum level sets have a straightforward meaning (this
fact was first recognized in [18]). Moreover, as the setting of the unified formalism is an affine
subbundle of a bundle of forms (see equation (1.4)), the proof of the equivalence between the
equations of motion is less involved. A partially filled diagram could clarify these considerations:
(WPG, λPG, 0) (W ′EH, λ
′
EH, 0) (WEH, λEH, 0)
(T0,LPG, 〈ωp〉)
(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
, F , F
) (
J1τΣ, F , F
)
A
1
1 + 2
B
2
C
3
Stars (F) refer to geometrical structures (Lagrangians forms and differential constraints) not
identified yet. Arrows A , B and C correspond to the equivalence given by the Lepage-
equivalent construction detailed at the end of Section 2; Theorems 11.1 and 11.12 below proved
the equivalence indicated by the composition of the horizontal arrows 1 + 2 . In order to carry
out this operation, we will see in the next sections that the reduced variational problem on
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
can be further simplified to a variational problem on J1τΣ; this fact is
depicted in the diagram by arrow 3 , and will be carried out in Section 7.
2In our case this consideration is superfluous, as the momentum map assumes just one value, but it is the way
the method proceeds in the general case.
24 S. Capriotti
6 Local coordinates expressions
Here we will obtain some identities allowing us to write down the isomorphism Υ−1
ω in local
terms. In order to proceed, we fix a coordinate chart on M , inducing coordinates (xµ, eµk)
on LM . As usual, we will indicate with (xµ, eµk , e
µ
kσ) the coordinates induced on J1τ . As shown
above, there exists a set of coordinates (xµ, gµν ,Γσµν) on
J1τ/K = Σ× C(LM)
and adapted to this decomposition, namely
pLMK
(
xµ, eµk
)
=
(
xµ, ηkleµke
ν
l
)
.
In terms of these coordinates, we have
pJ
1τ
K
(
xµ, eµk , e
µ
kσ
)
=
(
xi, ηijeµi e
ν
j ,−ekµeσkν
)
.
It means in particular that
TpLMK
(
∂
∂xµ
)
=
∂
∂xµ
and
TpLMK
(
∂
∂eµk
)
=
(
ηkqeρqδ
σ
µ + ηkpeσpδ
ρ
µ
) ∂
∂gσρ
. (6.1)
On the other hand, a principal connection on LM can be written as
ω0 = −elµ
(
deµk − f
µ
kσdxσ
)
Ekl ,
where (fµkσ) is a collection of local functions on M ; its Christoffel symbols will be
Γ
σ
ρµ = −ekρfσkµ.
Given our definition of the connection ωK on the K-bundle pLMK : LM → Σ, its components
become
[(ω0)k]
l
k = −ηkp
(
ηpqelµ − ηlqepµ
)(
deµq − fµqσdxσ
)
.
Now we will find the horizontal lift defined by ωK for vector fields on Σ:
Proposition 6.1. The horizontal lift of vector fields on Σ associated to the connection ωK is
locally given by(
∂
∂xµ
)H
=
∂
∂xµ
+
1
2
gβρe
ρ
k
(
gασΓ
β
αµ − gαβΓ
σ
αµ
) ∂
∂eσk
,(
∂
∂gµν
)H
=
1
4
gρβe
β
k
(
δαµδ
ρ
ν + δαν δ
ρ
µ
) ∂
∂eαk
.
Proof. See Appendix C. �
This proposition has the following consequence, that will be important to work with the
reduction of the Palatini variational problem.
Routh Reduction of Palatini Gravity in Vacuum 25
Corollary 6.2. Let (xµ, gµν , gµνσ ) be the induced coordinates on J1τΣ. Then(
∂
∂xσ
+ gµνσ
∂
∂gµν
)H
=
∂
∂xσ
+
1
2
gβρe
ρ
k
[
gκβσ +
(
gακΓ
β
ασ − gαβΓ
κ
ασ
)] ∂
∂eκk
, (6.2)
where (·)H is the horizontal lift in the K-principal bundle pLMK : LM → Σ for the connection ωK .
Proof. According to Proposition 6.1, we have that(
∂
∂xσ
+ gµνσ
∂
∂gµν
)H
=
∂
∂xσ
+
1
2
gβρe
ρ
k
(
gακΓ
β
ασ − gαβΓ
κ
ασ
) ∂
∂eκk
+
1
4
gµνσ gρβe
ρ
k
(
δαµδ
β
ν + δαν δ
β
µ
) ∂
∂eαk
=
∂
∂xσ
+
1
2
gβρe
ρ
k
[
gκβσ +
(
gακΓ
β
ασ − gαβΓ
κ
ασ
)] ∂
∂eκk
,
as required. �
Let us now introduce coordinates on the vector bundle k̃. In order to do this, let us suppose
that (φ = (xµ) , U) is a coordinate chart on M ; then it is also a trivializing domain for the
principal bundle LM , where
tU : τ−1(U)→ U ×GL(m) : u = (X1, . . . , Xm) 7→
(
xµ(τ(u)), eµk(u)
)
if and only if
Xk = eµk(u)
∂
∂xµ
.
Hence, we can define the coordinate chart
(
φk, τ
−1
k (U)
)
[9]. In order to proceed, we use the
correspondence between the space of sections of the adjoint bundle Γτk and the set of pLMK -
vertical K-invariant vector fields on LM .
Therefore, taking the base
{
Eρσ
}
on gl(m) such that(
Eρσ
)β
α
= δβσδ
ρ
α,
we can define the set of GL(m)-invariant τ -vertical vector fields Ẽρσ whose flow ΦẼρσ
t : τ−1(U)→
τ−1(U) is given by
ΦẼρσ
t (u) := t−1
U
(
τ(u),
[
exp
(
tEρσ
)]α
β
eβi (u)
)
;
it means that, locally, these vector fields are such that
TutU
(
Ẽρσ(u)
)
= eρi
∂
∂eσi
. (6.3)
In the following, we will adopt the usual convention according to which the map TtU is not
explicitly written, namely, where
∂
∂eσi
and Tt−1
U
(
∂
∂eσi
)
are identified. We can write down any pLMK -vertical K-invariant vector field Z on LM as
Z = AρσẼ
σ
ρ ;
then, using equation (6.1), we obtain the following result.
26 S. Capriotti
Lemma 6.3. The vector field on τ−1(U) given by
Z = AρσẼ
σ
ρ
is pLMK -vertical if and only if
gσαAρα + gραAσα = 0.
Proof. In fact, we have that
0 = Tup
LM
K
(
AρσẼ
σ
ρ (u)
)
= AρσTup
LM
K
(
Ẽσρ (u)
)
= Aρσe
σ
i (u)Tup
LM
K
(
∂
∂eρi
)
= Aρσe
σ
i (u)ηiq
[
eαq (u)δβρ + eβq (u)δαρ
] ∂
∂gαβ
,
and the identity follows. �
Therefore, we will have that
φk([u,B]K) :=
(
xµ(u), ηkleµk(u)eνl (u), Aρσ([u,B]K)
)
if and only if gσαAρα + gραAσα = 0 and
[u,B]K = AρσẼ
σ
ρ (u).
In order to relate the coordinates Aρσ with the element [u,B]K , we need to look closely to the
identification between Γk̃ and the set of pLMK -vertical K-invariant vector fields on LM . It uses
the correspondence
V τ ' LM × gl(m)
given by
(u,B) 7→
~d
dt
∣∣∣∣
t=0
(u · exp (−tB)).
In coordinates it reads(
u = (X1, . . . , Xm), B =
(
Bj
i
))
7→ −Bi
je
ρ
i
∂
∂eρj
,
and using equation (6.3) it becomes(
u = (X1, . . . , Xm), B =
(
Bj
i
))
7→ −Bi
je
ρ
i e
j
σẼ
σ
ρ .
Therefore, it results that
Âσρ (u,B) = −eiρB
j
i e
σ
j
is a GL(m)-invariant function on LM × gl(m) when GL(m) acts on gl(m) by the adjoint action;
then, it gives us the set of functions Aσρ on τ−1
k (U) ⊂ k̃ that completes the coordinates φk.
Lemma 6.4. The map φk : τ−1
k (U)→ U × Rm2+
m(m+1)
2 given by
φk([u,B]K) =
(
xµ(u), ηkleµk(u)eνl (u),−eiρ(u)Bj
i e
σ
j (u)
)
defines a set of coordinates on τ−1
k (U).
Routh Reduction of Palatini Gravity in Vacuum 27
Proof. According to the previous discussion, it is only necessary to prove that for any B ∈ k,
i.e., such that
ηikBj
k + ηjkBi
k = 0,
the corresponding element on TuLM ,
Z = −eiρ(u)Bj
i e
σ
j (u)Ẽρσ
verifies the constraint
Tup
LM
K (Z) = 0.
But it follows that
gραAσα + gσαAρα = −gραeiαB
j
i e
σ
j − gσαeiαB
j
i e
ρ
j = −ηikBj
i
(
eσke
ρ
j + eρke
σ
j
)
= −
(
ηikBj
k + ηjkBi
k
)
eρje
σ
k = 0,
as required. �
Then, let us point out that Lemma 6.4 allows us to set coordinates on the bundle
p : Lin
(
τ∗ΣTM, k̃
)
→ Σ.
In fact, any element (gx, α) ∈ Lin
(
τ∗ΣTM, k̃
)
admits coordinates
(
xµ, gµν , Aµσρ
)
if and only if(
xµ, gµν
)
are the corresponding coordinates for gx ∈ Σ and
α
(
∂
∂xρ
)
= AµσρẼ
σ
µ(ex),
where ex ∈ LM is any element in
(
pLMK
)−1
(gx).
7 Torsion, metricity and contact structures
on the quotient space
There are two tasks to carry out in order to understand the Routh reduction of Palatini gravity:
We need to characterize the effects produced by the fact that we are working on the subbundle
T0 ⊂ J1τ instead of the full jet space J1τ ; additionally, we want to find the differential constraints
for the reduced system. Accordingly, in this section we will prove that
• When restricted to T0, the quotient map
J1τ → J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
will reduce to
T0 → J1τΣ,
that is, we can forget about the “vertical” degrees of freedom related to the factor
Lin
(
τ∗ΣTM, k̃
)
. This is achieved in Propositions 7.1 and 7.2, and in Corollary 7.3.
• The metricity conditions are the pullback along the quotient map of the contact forms
on J1τΣ. This result is very interesting because it tells us that the reduction scheme
implemented relates a Griffiths variational problem (Palatini gravity, see Definition 4.1)
with a classical variational problem (Einstein–Hilbert gravity as described in Section 8
below). This is accomplished in Proposition 7.4.
28 S. Capriotti
So, let us use the following diagram
J1τ
(
pLMK
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
))
J1τ/K J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
Σ× C(LM)
Υω
pJ
1τ
K
pr23
Υω
∼
Gω
in order to define the diffeomorphism Gω; here Υω is the map induced by Υω. Therefore, let us
construct the pullback bundle pr1 :
(
pLMK
)∗(
J1τΣ
)
→ LM by means of the commutative diagram
(
pLMK
)∗(
J1τΣ
)
J1τΣ
LM Σ
pr2
pr1 (τΣ)10
pLMK
In this setting, we can prove that the zero torsion submanifold T0 has some nice properties
regarding the decomposition induced by the connection ωK .
Proposition 7.1. The canonical projection
prΣ :
(
pLMK
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
))
−→
(
pLMK
)∗(
J1τΣ
)(
e, j1
xs,
[
e, ξ̂
]
K
)
7−−−−−−−−−−→
(
e, j1
xs
)
restricted to the submanifold
T ′0 := Υω(T0) ⊂
(
pLMK
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
))
is a diffeomorphism between T ′0 and
(
pLMK
)∗(
J1τΣ
)
.
Proof. The proof of this proposition will be local. Using equation (6.2) and the coordinates
introduced above, we have that
∂
∂xσ
+ eµkσ
∂
∂eµk
=
(
∂
∂xσ
+ gµνσ
∂
∂gµν
)H
+AµρσẼ
ρ
µ(ex)
=
∂
∂xσ
+
1
2
gβρe
ρ
k
[
gµβσ +
(
gαµΓ
β
ασ − gαβΓ
µ
ασ
)] ∂
∂eµk
−Aµρσe
ρ
k
∂
∂eµk
,
namely
eµkσ =
1
2
gβρe
ρ
k
[
gµβσ +
(
gαµΓ
β
ασ − gαβΓ
µ
ασ
)]
−Aµρσe
ρ
k.
Then it follows that, for the K-invariant functions Γµνσ,
Γµρσ = −ekρe
µ
kσ = −1
2
gβρ
[
gµβσ +
(
gαµΓ
β
ασ − gαβΓ
µ
ασ
)]
+Aµρσ. (7.1)
Routh Reduction of Palatini Gravity in Vacuum 29
It means that the set T ′0 is locally given by the equation
1
2
gβσ
[
gµβρ +
(
gαµΓ
β
αρ − gαβΓ
µ
αρ
)]
− 1
2
gβρ
[
gµβσ +
(
gαµΓ
β
ασ − gαβΓ
µ
ασ
)]
+Aµρσ −Aµσρ = 0.
Let us define the set of quantities
Aµνσ := gµρA
ρ
νσ;
then using this equation and the fact that
Aµνσ +Aνµσ = gµρA
ρ
νσ + gνρA
ρ
µσ = 0,
we can conclude, from Proposition A.1, that the elements Aµνσ are uniquely determined by the
fact that they belong to T ′0 . In other words, the set
(prΣ)−1
(
e, j1
xs
)
∩ T ′0
consists in a single element. �
Proposition 7.1 can be geometrically interpreted: Recall that, through isomorphism
Gω : J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
→ Σ× C(LM),
any section of the reduced bundle J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
can be seen as a pair “metric” plus
“connection”. With this interpretation in mind, the previous proposition tells us that, when
projected to the quotient, the “connection part” of the section corresponds to Levi-Civita con-
nection, and so, it is uniquely determined by its “metric part”. The following result summarizes
it.
Proposition 7.2. Let
σ : M → J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
be a section of the composite map
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
−→Σ
τΣ−→M
such that pr1 ◦ σ : M → J1τΣ is a holonomic section and
Imσ ⊂ pr23(T ′0 ).
Then
Γσ := pr2 ◦Gω ◦ σ : M → C0(LM)
is the Levi-Civita connection associated to the metric gσ := pr1 ◦Gω ◦ σ.
Proof. Locally, the map Gω is given by equation (7.1). Therefore, from the proof of the previous
Proposition and using Proposition A.1, we will have that the elements
Γµνσ := gµρΓ
ρ
νσ
are uniquely determined by the set of equations
Γµρσ − Γρµσ = 0, Γµρσ + Γρµσ = −gµαgρβgαβσ . (7.2)
30 S. Capriotti
It means that
Γµρσ = −1
2
(
gµαgρβg
αβ
σ + gραgσβg
αβ
µ − gσαgµβgαβµ
)
.
Now, using the definition
gµν,σ := −gµαgνβgαβσ
we obtain
Γµρσ =
1
2
gµα
(
gαν,σ + gρσ,α − gσα,ν
)
. (7.3)
Because pr1 ◦ σ is holonomic, we have that
gµνσ =
∂gµν
∂xσ
,
as required. �
Let us define
T ′′0 := G−1
ω (Σ× C0(LM)) = Υω
(
pJ
1τ
K (T0)
)
;
then, we need to draw our attention to the diagram in Fig. 1.
T0 T ′0
T0/K J1τ
(
pLMk
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
))
Σ× C0(LM)
(
pLMk
)∗(
J1τΣ
)
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
T ′′0 J1τΣ
Υω |T0
pJ
1τ
K
∣∣
T0
prΣ|T ′0
∼
Υω |T0/K
Υω
Υω◦pJ
1τ
K pr23
prΣ
G−1
ω
∣∣
Σ×C0(LM)
pr2
pr1
pr1|T ′′0
Figure 1. Maps involved in the Routh reduction of Palatini gravity.
As a consequence of formula (7.3), we obtain the following corollary; in short, it says that
in the reduced bundle J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
, the degrees of freedom associated to the factor
Lin
(
τ∗ΣTM, k̃
)
are superfluous.
Corollary 7.3. The map
pr1|T ′′0 : T ′′0 → J1τΣ
is a bundle isomorphism over the identity on Σ.
Proof. Locally, composite map Υω ◦ pJ
1τ
K is given by
Υω ◦ pJ
1τ
K
([
xµ, eνk, e
σ
kρ
]
K
)
=
(
xµ, ηijeµi e
ν
j , g
µν
σ
)
,
where coordinates gµνσ are calculated using equation (7.2). �
Routh Reduction of Palatini Gravity in Vacuum 31
For the last result of the section, we need any of the composite maps
J1τ
(
pLMK
)∗(
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)) (
pLMK
)∗(
J1τΣ
)
J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
J1τΣ
Υω
Υω◦pJ
1τ
K
prΣ
pr2
pr1
So far, we have obtained a result allowing us to reduce the bundle J1τΣ ×Σ Lin
(
τ∗ΣTM, k̃
)
further down to J1τΣ; thus, we are halfway to connect the reduced variational problem defined
on this bundle with the Einstein–Hilbert variational problem. Now, as we mentioned above,
the splitting induced by the connection form ωK allows us to relate the metricity forms with
a contact structure on the quotient bundle; it will make possible to complete this connection
by showing that the contact structure on J1τΣ is a sort of reduction structure for metricity
conditions.
Proposition 7.4. The metricity forms are (pr2 ◦ prΣ ◦Υω)-horizontal (also
(
pr1 ◦Υω ◦ pJ
1τ
K
)
-
horizontal). In fact,
TpLMK ◦ ωp = (pr2 ◦ prΣ ◦Υω)∗ω =
(
pr1 ◦Υω ◦ pJ
1τ
K
)∗
ω
where ω is the contact form on J1τΣ.
Proof. In local coordinates, we have that
(pr2 ◦ prΣ ◦Υω)
(
xµ, eνk, e
ν
kσ
)
=
(
xµ, gµν , gµνσ
)
,
where gµνσ is calculated using equation (7.1). On the other hand, the metricity forms have the
following local expression [3]
ηikωjk + ηjkωik = eiµe
j
ν
[
dgµν +
(
gµσΓνσρ + gνσΓµσρ
)
dxρ
]
. (7.4)
Using equation (7.1), it follows that
ηikωjk + ηjkωik = eiµe
j
ν
(
dgµν − gµνσ dxσ
)
,
namely, the metricity condition is horizontal with respect to the projection
pr2 ◦ prΣ ◦Υω : J1τ −→ J1τΣ,
and the form in the base manifold is nothing, but the generator of the contact structure. �
8 First order variational problem for Einstein–Hilbert gravity
Einstein–Hilbert variational problem is a classical second order variational problem on the bundle
of metrics τΣ : Σ→M of signature η on M – see [10, 19] and references therein. It means that
its dynamics is dictated by a Lagrangian density
LEH : J2τΣ → ∧m(T ∗M),
given essentially by the scalar curvature of the Levi-Civita connection associated to a metric. In
terms of our nomenclature regarding variational problems, this variational problem is prescribed
by the triple(
(τΣ)2 : J2τΣ →M,LEH, IΣ
con
)
,
with IΣ
con referring to the exterior differential system on J2τΣ associated to its contact structure.
32 S. Capriotti
The main objective of this section is to provide a first order variational problem for Einstein–
Hilbert gravity. Namely, we have proved in Section 7 that the quotient bundle given by Defini-
tion 5.8 can be further reduced to J1τΣ; also, it was shown (see Proposition 7.4) that contact
structure on this jet space is related to metricity conditions via reduction map. Therefore, we
will construct a variational problem on this bundle J1τΣ and we will prove that this variational
problem describes Einstein–Hilbert gravity. Later on, we will prove that this variational problem
can be interpreted as Routh reduction of the variational problem for Palatini gravity as defined
in Section 4.1.
Definition 8.1 (Einstein–Hilbert Lagrangian form). The Einstein–Hilbert Lagrangian form is
the unique 2-horizontal m-form L(1)
EH on J1τΣ such that(
pr1 ◦Υω ◦ pJ
1τ
K
)∗L(1)
EH = i∗0LPG.
Recall also that in local terms, Palatini Lagrangian (3.3) can be written as
LPG = εµ1···µn−2γκ
√
| det g|gκφdxµ1 ∧· · ·∧ dxµn−2∧
(
dΓγρφ ∧ dxρ + ΓσδφΓγβσdxβ ∧ dxδ
)
. (8.1)
Using equation (7.3) we see that L(1)
EH has the same form than LPG, but replacing Γσµν by
their expressions in terms of the jet variables gµνσ . Nevertheless, it is not yet Einstein–Hilbert
Lagrangian density because is it neither a density nor a functional on J2τΣ.
We are pursuing here to establish the equivalence between the classical variational problem
associated to the Lagrangian density LEH : J2τΣ → ∧m (T ∗M) and the variational problem(
J1τΣ,L(1)
EH, IΣ
con
)
. As we have said above, the main difference between these variational problems
is related to the nature of the Lagrangian form. In the latter, this form is not a horizontal
form on J1τΣ, whereas in the former case, the Lagrangian form on J2τΣ is specified through
a Lagrangian density, giving rise to a horizontal form on this jet bundle. The following lemma
tells us how these Lagrangians are related. In order to formulate this result, the definition of
the horizontalization operator h : Ωm
(
J1τΣ
)
→ Ωm
(
J2τΣ
)
should be kept in mind [30]; in fact,
defining the map
hj2xs := Txj
1s ◦ Tj2xs(τΣ)2 : Tj2xs
(
J2τΣ
)
→ Tj1xs
(
J1τΣ
)
,
where (τΣ)2 : J2τΣ → M is the canonical projection of the 2-jet bundle of the metric bundle
onto M , we have
h(α)
∣∣
j2xs
:= α
∣∣
j1xs
◦ hj2xs
for every α ∈ Ωm
(
J1τΣ
)
and j2
xs ∈ J2τΣ.
Lemma 8.2. It results that
LEH = h
(
L(1)
EH
)
for h : Ωm
(
J1τΣ
)
→ Ωm
(
J2τΣ
)
the horizontalization operator.
Proof. In terms of the coordinates
(
xµ, gµν , gµνα , gµναβ
)
on J2τΣ, we have that
h
(
dgµν
)
= gµνα dxα, h
(
dgµνα
)
= gµναβdxβ.
The result follows from a (rather lenghty) calculation, using expression (8.1) and the formula
for the Christoffel symbols (7.3). �
Routh Reduction of Palatini Gravity in Vacuum 33
The occurrence of the horizontalization operator in this lemma is crucial for our purposes, as
the following proposition shows.
Theorem 8.3. Let π : E → M be a bundle on a (compact) manifold M of dimension m. For
any α ∈ Ωm
(
Jkπ
)
and any section s ∈ Γπ, we have that∫
M
(
jks
)∗
α =
∫
M
(
jk+1s
)∗
h(α).
Proof. It follows from the formula
Txj
ks = Txj
ks ◦ Tjk+1
x sπk+1 ◦ Txjk+1s,
that holds for every x ∈Mand s ∈ Γπ. �
It is immediate to prove the desired equivalence.
Corollary 8.4. The classical variational problem specified by the Lagrangian density LEH
on J2τΣ and the variational problem
(
J1τΣ,L(1)
EH, IΣ
con
)
have the same set of extremals.
Proof. From Lemma 8.2 and using Theorem 8.3, we see that g : M → Σ is an extremal for the
action integral
g 7→
∫
M
(
j2g
)∗LEH
if and only if it is an extremal for the action integral
g 7→
∫
M
(
j1g
)∗L(1)
EH,
as required. �
As usual [20], the equations of motion of this variational problem can be lifted to a space of
forms on J1τΣ. Let us define the affine subbundle
WEH := L(1)
EH + IΣ
con,2 ⊂ ∧m
(
J1τΣ
)
.
Here, for every j1
xs ∈ J1τΣ,
IΣ
con,2
∣∣
j1xs
= L
{
αs(x) ◦
(
Tj1xs(τΣ)10 − Txs ◦ Tj1xs(τΣ)1
)
∧ β :
αs(x) ∈ T ∗s(x)Σ, β ∈
(
Λm−1
1
(
J1τΣ
))
j1xs
}
is the corresponding fiber for the contact subbundle on J1τΣ. The canonical map will be denoted
by
τEH : WEH → J1τΣ.
We will indicate with λEH the pullback of the canonical m-form on ∧m
(
J1τΣ
)
to WEH. Then
we have a result analogous to Proposition 4.5 in the context of (first order) Einstein–Hilbert
formulation.
Proposition 8.5. A section s : U ⊂M → J1τΣ is a critical holonomic section for the variational
problem
(
J1τΣ,L(1)
EH, IΣ
con
)
if and only if there exists a section Γ: U ⊂M →WEH such that
1) Γ covers s, i.e., τEH ◦ Γ = s, and
2) Γ∗(XydλEH) = 0, for all X ∈ XV ((τΣ )1◦τEH)(WEH).
Remark 8.6. This proposition provides us with a unified formalism for Einstein–Hilbert gravity,
based on the first order formulation. For the corresponding formalism associated to the second
order formulation, see [19].
34 S. Capriotti
9 Contact bundle decomposition for Palatini gravity
We return here to the discussion initiated in Section 5.2, regarding the splitting of the quo-
tient WPG/K; it will be shown that the connection ωK is useful also for the identification of
elements in WPG/K that can be seen as elements of an space of forms. Intuitively, it means
that the associated degrees of freedom can be interpreted as multimomenta.
In order to perform this identification, we will recall some general facts regarding the decom-
position induced for the connection ωK [6] on the bundle of forms WPG defined in equation (4.1).
The contact structure on J1τ gives rise to the contact subbundle on T0 given by
Imcon,2
∣∣
j1xs
= L
{
αs(x) ◦ (Tj1xsτ
′
10 − Txs ◦ Tj1xsτ
′
1) ∧ β :
αs(x) ∈ T ∗s(x)(LM), β ∈
(
Λm−1
1 (T0)
)
j1xs
}
, (9.1)
where L indicates linear closure. There is an splitting of Imcon,2 induced by the choice of a con-
nection on the principal bundle pLMK : LM → Σ. Its construction proceeds as follows. We denote
by ωK ∈ Ω1(LM, k) the chosen connection and consider the following splitting of the cotangent
bundle:
T ∗(LM) =
(
pLMK
)∗
(T ∗Σ)⊕ (LM × k∗).
The identification is obtained as follows:(
pLMK
)∗
(T ∗Σ)⊕ (LM × k∗)→ T ∗(LM),
(e, α̂[u], σ) 7→ αu = α̂[u] ◦ TupLMK + 〈σ, ωK(·)〉.
Accordingly, we have an splitting of contact bundle (9.1)
Imcon,2 = Ĩmcon,2 ⊕ I
m
k∗,2,
with
Ĩmcon,2
∣∣
j1xs
= L
{
α̂[s(x)] ◦ Ts(x)p
LM
K ◦
(
Tj1xsτ
′
10 − Txs ◦ Tj1xsτ
′
1
)
∧ β :
α̂[s(x)] ∈ T ∗[s(x)]Σ, β ∈
(
Λm−1
1 T0
)
j1xs
}
,
Imk∗,2
∣∣
j1xs
=
{〈
σ ∧, ωK ◦
(
Tj1xsτ
′
10 − Txs ◦ Tj1xsτ
′
1
)〉
: σ ∈
(
Λm−1
1 T0 ⊗ k∗
)
j1xs
}
.
The symbol 〈· ∧, ·〉 denotes the natural contraction, defined as follows: For elements of the form
α1 ⊗ ν, α2 ⊗ η with ν, η ∈ k and α1, α2 forms, we have 〈α1 ⊗ ν ∧, α2 ⊗ η〉 = 〈ν, η〉α1 ∧ α2. For
a general element in the linear closure, the definition extends linearly.
We can split our metricity subbundle ImPG using the inclusion
ImPG ⊂ Imcon,2,
namely
ImPG =
(
ImPG ∩ Ĩmcon,2
)
⊕
(
ImPG ∩ Imk∗,2
)
.
But we have the following fact.
Lemma 9.1. For every j1
xs ∈ T0
ImPG ⊂ Ĩmcon,2.
Routh Reduction of Palatini Gravity in Vacuum 35
Proof. Let us work in the coordinates considered above; therefore, we have equation (6.1) for
the projector TpLMK and also
Tj1xsτ
′
10 − Txs ◦ Tj1xsτ
′
1 =
∂
∂eµk
⊗
(
deµk − e
µ
kαdxα
)
.
Then
Ts(x)p
LM
K ◦
(
Tj1xsτ
′
10 − Txs ◦ Tj1xsτ
′
1
)
= Ts(x)p
LM
K
(
∂
∂eµk
)
⊗
(
deµk − e
µ
kαdxα
)
=
∂
∂gρσ
⊗
[
ηkq
(
eρqde
σ
k + eσq deρk
)
− ηkq
(
eρqe
σ
kα + eσq e
ρ
kα
)
dxα
]
=
∂
∂gρσ
⊗
[
dgρσ −
(
gρβekβe
σ
kα + gσβekβe
ρ
kα
)
dxα
]
=
∂
∂gρσ
⊗
[
dgρσ +
(
gρβΓσβα + gσβΓρβα
)
dxα
]
that will define to a set of generators of the bundle ImPG (see equation (7.4)). �
This result is compatible with the fact that the whole subbundle WPG is in the zero level set
for the momentum map. We will return to that below.
10 First order Einstein–Hilbert Lagrangian as Routhian
As we mentioned in the introductory sections, a crucial role in Routh reduction is played by the
Routhian, which replaces the Lagrangian in determining the dynamics of the mechanical system.
This replacement is unavoidable, because the dynamics for the Routh reduced problem should
happen on a level set for the momentum map of the theory
J : TQ→ g∗,
and the constraints imposed by this requirement must be included in the Lagrangian.
To fix ideas, let us briefly describe how this construction proceeds in the case of a classical
variational problem(
pr1 : R× TQ→ R, Ldt, 〈dq − q̇dt〉
)
for a mechanical system with configuration space Q and a symmetry described by a Lie group G
acting freely on Q. The idea is to incorporate the constraints imposed by the momentum map
into the Lagrangian through a family of Lagrange multipliers; this prescription tells us that the
Routhian becomes
Rµ := L− 〈µ, ωQ〉
where µ ∈ g∗ is the level chosen for the momentum map and ωQ is a connection form for the
principal bundle pQG : Q→ Q/G. A similar construction can be done when working for classical
variational problems describing a first order field theory [6].
Now, let us try to particularize these considerations for the case of Palatini gravity. The
Routhian form is expected to coincide with the Lagrangian LPG because J ≡ 0. Nevertheless,
as we have stressed above, this result would be valid if we were working within the range of [6]; on
the contrary, the variational problem describing Palatini gravity is not reached by these results,
and so this should be properly verified in this particular case.
36 S. Capriotti
First, we write p : Lin
(
τ∗ΣTM, k̃
)
→ Σ for the obvious projection. In principle, the bundle
Lin
(
τ∗ΣTM, k̃
)
would be the field bundle for the reduced system; nevertheless, we will show in
Lemma 10.2 that the Routhian, namely, the Lagrangian form for this reduced system, will be
horizontal for the projection onto the jet space of the base bundle Σ.
In particular, one can consider the map:
q : J1
(
τΣ ◦ p
)
−→ J1τΣ × Lin
(
τ∗ΣTM, k̃
)
,
j1
xσ 7−→
(
j1
x
(
p ◦ σ
)
, σ(x)
)
projecting onto the quotient bundle for Palatini gravity. So, we can formulate the reduced
system as a first order field theory by taking the bundle τΣ ◦ p : Lin
(
τ∗ΣTM, k̃
)
→M as the basic
field bundle. Nevertheless, there are some identifications that will permit us to simplify this
basic bundle further.
In order to proceed, let use the connection ωK to define the maps fitting in the following
diagram:
T0 Lin
(
τ∗ΣTM, k̃
)
J1
(
τΣ ◦ p
)
T0/K J1τΣ × Lin
(
τ∗ΣTM, k̃
)pJ
1τ
K
∣∣
T0
fω
(
τΣ◦p
)
10
q
gω :=Υω |T0/K
The definitions are as follows:
fω : T0 −→ Lin
(
τ∗ΣTM, k̃
)
,
j1
xs 7−→ [s(x), ωK ◦ Txs]K ,
gω : T0/K −→ J1τΣ × Lin
(
τ∗ΣTM, k̃
)
,[
j1
xs
]
K
7−→
(
j1
x
(
pLMK ◦ s
)
, [s(x), ωK ◦ Txs]K
)
.
The map gω is the identification from Corollary 5.6. Since the Lagrangian form LPG is basic
for the projection pKT0 : T0 → T0/K, it defines a reduced form on T0/K which can be seen as
a form on J1τΣ × Lin
(
τ∗ΣTM, k̃
)
. We will denote it by LPG:(
gω ◦ pJ
1τ
K
)∗LPG = LPG, LPG ∈ Ωm
2
(
J1τΣ × Lin
(
τ∗ΣTM, k̃
))
.
Definition 10.1. The m-form LPG ∈ Ωm
(
J1τΣ × Lin
(
τ∗ΣTM, k̃
))
is the Routhian for the vari-
ational problem
(
T0,LPG, ImPG
)
.
Then, we are ready to prove a characteristic property for the Routhian associated to the
reduction of Palatini gravity.
Lemma 10.2. The Routhian LPG is pr1-horizontal, where
pr1 : J1τΣ × Lin
(
τ∗ΣTM, k̃
)
−→ J1τΣ
is the projection onto the first factor of the fibred product.
Proof. It follows from equations (7.1) and (7.3) that
pr∗1L
(1)
EH = LPG, (10.1)
as required. �
Routh Reduction of Palatini Gravity in Vacuum 37
In short, Routhian LPG does not depend on the fiber coordinates Aσµρ of the bundle
p : Lin
(
τ∗ΣTM, k̃
)
→ Σ;
it is just the pullback along pr1 of the first order Lagrangian for Einstein–Hilbert gravity.
In the usual Routh reduction, the reduced Routhian is a m-form on J1
(
τΣ ◦ p
)
; in this
case, Lemma 10.2 allows us to consider the form L(1)
EH on J1τΣ as the Routhian. Therefore, we
can forget about the degrees of freedom associated to the factor Lin
(
τ∗ΣTM, k̃
)
in the quotient
bundle, and take as the quotient bundle for Palatini gravity the jet bundle J1τΣ; this is the way
in which we will proceed from this point.
11 Einstein–Hilbert gravity as Routh reduction
of Palatini gravity
We will devote the present section to establish the two main results of the article, namely, Theo-
rem 11.4 regarding reduction of Palatini gravity and Theorem 11.7 dealing with reconstruction of
metrics verifying Einstein equations of gravity. The strategy, as we mention in the introduction,
is to compare the equations of motion (lifted to the corresponding spaces of forms WEH and WPG)
in a bundle containing every relevant degree of freedom; this role is played below by a pullback
bundle of the bundle WEH along a suitable map. So, let us define
Fω := pr1 ◦ gω ◦ pJ
1τ
K : T0 −→ J1τΣ,
namely
Fω
(
j1
xs
)
= j1
x
(
pLMK ◦ s
)
for every j1
xs ∈ T0. Then we have the diagram
WPG ∧m2 (T ∗T0) F ∗ω (WEH) WEH
T0 J1τΣ
πPG
τmT0 prω1
F̃ω prω2
πEH
Fω
(11.1)
where
F̃ω : F ∗ω(WEH) −→ ∧m2 (T ∗T0) :
(
j1
xs, ρ
)
7→ ρ ◦ Tj1xsFω
and
prω1 : F ∗ω(WEH) −→ T0, prω2 : F ∗ω(WEH) −→WEH
are the canonical projections of the pullback bundle.
Remark 11.1. Let us give a local version of the map Fω; recall that locally T0 is described by
the set of equations
ekσe
ν
kρ = ekρe
ν
kσ,
where
(
xµ, eσk , e
ρ
kµ
)
is a set of adapted coordinates induced by a coordinate chart (φ = (xµ), U)
on M . In these coordinates, the canonical quotient map pLMK : LM → Σ is given by
pLMK
(
xµ, eσk
)
=
(
xµ, ηkleρke
σ
l
)
;
accordingly, map Fω will become
Fω
(
xµ, eσk , e
ρ
kµ
)
=
(
xµ, ηkleσke
ρ
l , η
kl
(
eσkµe
ρ
l + eσke
ρ
lµ
))
.
38 S. Capriotti
Lemma 11.2. The bundle map F̃ω is an affine bundle isomorphism on T0 between WPG and
F ∗ω(WEH).
Proof. It is consequence of equation (10.1) and Proposition 7.4. �
We will use diagram (11.1) as a means to compare the equations of motion of Palatini gravity
and Einstein–Hilbert gravity; the idea is to use Propositions 4.5 and 8.5 in order to represent
these equations in terms of the spaces of forms WPG and WEH respectively, and to pull them
back to the common space F ∗ω(WEH). Crucial to this strategy is the following result.
Proposition 11.3. The following relation holds
F̃ω
∗
λPG =
(
prω2
)∗
λEH.
Proof. Let
(
j1
xs, ρ
)
∈ F ∗ω(WEH) be an arbitrary element in this pullback bundle; then using
diagram (11.1) we will have that
λPG
∣∣
ρ◦T
j1xs
Fω
◦ T(j1xs,ρ)F̃ω =
(
ρ ◦ Tj1xsFω
)
◦ Tρ◦T
j1xs
Fωτ
m
T0 ◦ T(j1xs,ρ)F̃ω
=
(
ρ ◦ Tj1xsFω
)
◦ T(j1xs,ρ)prω1 = ρ ◦ TρπEH ◦ T(j1xs,ρ)prω2
= λEH|ρ ◦ T(j1xs,ρ)prω2 ,
using the fact that πEH : WEH → J1τΣ is the restriction of the canonical projection
τmJ1τΣ
: ∧m2
(
T ∗J1τΣ
)
−→ J1τΣ
to WEH. This identity proves the proposition. �
11.1 Routh reduction of Palatini gravity
We are now ready to prove the first result on Routh reduction of Palatini gravity; essentially, we
will prove that any section obeying the equations of motion for Palatini gravity projects along
the map Fω to a holonomic section obeying Einstein–Hilbert equations of motion.
Theorem 11.4. Let Ẑ : U ⊂ M → T0 be a section that obeys the Palatini gravity equations of
motion. Then the section
Fω ◦ Ẑ : U → J1τΣ
is holonomic and obeys the Einstein–Hilbert gravity equations of motion.
Proof. The idea of the proof is encoded in the following diagram
WPG F ∗ω(WEH) WEH
T0 J1τΣ
M
πPG
F̃ω
prω1
prω2
πEH
Fω
τ1
(τΣ)1
Ẑ
Γ
Γ′
Γ̃
(11.2)
Routh Reduction of Palatini Gravity in Vacuum 39
Using Proposition 4.5, we construct Γ: U → WPG out of Ẑ; the Palatini gravity equations of
motion will become
Γ∗(ZydλPG) = 0
for any Z ∈ XV (τ1◦πPG)(WPG). Using Lemma 11.2 we can define
Γ′ :=
(
F̃ω
)−1 ◦ Γ: U −→ F ∗ω(WEH);
then the Palatini equations of motion translate into
(Γ′)∗
(
Z ′ydF̃ω
∗
λPG
)
= 0
for any Z ′ ∈ XV (τ1◦prω1 )(F ∗(WEH)). Then using Proposition 11.3 and the fact that prω2 : F ∗ω(WEH)
→WEH is a submersion, we can conclude that the section
Γ̃ := prω2 ◦ Γ′ : U →WEH
obeys the equations of motion
Γ̃∗
(
Z̃ydλEH
)
= 0,
where Z̃ ∈ XV ((τΣ)1◦πEH)(WEH) is an arbitrary vertical vector field on WEH. Also, using dia-
gram (11.2), we have that
πEH ◦ Γ̃ = πEH ◦ prω2 ◦ Γ′ = Fω ◦ prω1 ◦ Γ′ = Fω ◦ Ẑ.
The theorem then follows from Proposition 8.5. �
Remark 11.5 (reduction theorem in local coordinates). Let us look at the reduction theorem
in local terms. First, we have that the equations of motion on J1τ can be written as
∂Γβµν
∂xβ
−
∂Γβµβ
∂xν
+ ΓβσβΓσµν − ΓβσνΓσµβ = 0,
Γσµν = −ekµeσkν ,
ekµe
σ
kν − ekνekµ = 0. (11.3)
On the other hand, from Remark 11.1 we have that
Fω
(
xµ, eσk , e
ρ
kµ
)
=
(
xµ, ηkleσke
ρ
l , η
kl
(
eσkµe
ρ
l + eσke
ρ
lµ
))
.
Thus, Theorem 11.4 tells us that any local section
Ẑ
(
xµ
)
=
(
xµ, eµk(x), eµkν(x)
)
obeying equations (11.3) will gives rise to a solution of Einstein equations of motion when
composed with Fω.
40 S. Capriotti
11.2 . . . and reconstruction
We will give now a (somewhat partial) converse to Theorem 11.4. That is, given a section
ζ : U ⊂ M → Σ such that j1ζ : U → J1τΣ is extremal for the Einstein–Hilbert variational
problem, find a section
Ẑ : U → T0
such that Fω ◦ Ẑ = j1ζ and Ẑ is an extremal for the Palatini variational problem. From Fig. 1
it is clear that we need to lift the section j1ζ through the quotient map pJ
1τ
K
∣∣
T0 : T0 → T0/K,
which has the structure of a principal bundle on T0/K. It is clear that any principal bundle
can be trivialized by a convenient restriction of the base space. As discussed in [6], it is not
the way in which this kind of reconstruction problems are solved. Rather, the problem of lifting
sections along the projection in a principal bundle is reduced to the problem of deciding if certain
connection is flat; moreover, it is expected that this connection is related to the connection used
to define the Routhian. We will present in this section a theory of reconstruction along these
lines. With this goal in mind, we will recall here some of the details developed in [6]; for proofs
we refer to the original article. We begin with a pair of diagrams (11.4):
P P/G
M
pPG
π π
ζ
s
ζ∗P P
M P/G
pr2
pr1 pPG
ζ
(11.4)
Then we have the following result.
Lemma 11.6. There exists a section s : M → P covering the section ζ : M → P/G if and only
if ζ∗P is a trivial bundle.
Using that ζ∗P is a principal bundle, being trivial can be characterized in terms of a flat
connection [29]:
Theorem 11.7. Let π : P → M be a G-principal bundle with M simply connected. Then P is
trivial if and only if there exists a flat connection on P .
If M is not simply connected, then one can ask for a flat connection with trivial holonomy and
obtain a similar result. For the sake of simplicity, we will assume that M is simply connected to
apply Theorem 11.7 when needed. For later use, we also observe that the section constructed
in the proof of Theorem 11.7 has horizontal image with respect to the given connection.
We wish now to apply the previous discussion to the case of the bundle pT0K := pJ
1τ
K
∣∣
T0 : T0 →
T0/K. We have the situation depicted in diagram (11.5) (left): Z : M → T0/K is a given section
and ζ : M → Σ is the induced section. The basic question we want to address is whether there
exists a section Ẑ : M → T0 such that pT0K ◦ Ẑ = Z:
T0 T0/K
LM Σ
M
p
T0
K
τ ′10 τ ′10
pLMK
τ ′ τΣ
ζ
Ẑ Z
ζ∗(LM) LM
M Σ
pr2
pr1 pLMK
ζ
(11.5)
Routh Reduction of Palatini Gravity in Vacuum 41
Now, using the fact that
T0/K ' Σ× C0(LM), T0 ' LM × C0(LM)
we have that Z = (ζ,Γ) is composed by the metric plus the Levi-Civita connection Γ; therefore,
we will have that
Ẑ =
(
ζ̂,Γ
)
where ζ̂ : M → LM is some lift of the section ζ : M → Σ. Then, we can then construct the
pullback bundle ζ∗(LM) (diagram (11.5), right) and particularize Lemma 11.6 to conclude the
following:
Lemma 11.8. Assume that M is simply connected. If ζ∗(LM) admits a flat connection then
there exists a section Ẑ : M → T0 such that(
pLMK ◦ τ ′10
)
◦ Ẑ = ζ
and Ẑ∗ωp = 0. Conversely, every such section gives rise to a flat connection on ζ∗(TM).
Proof. Because ζ∗(LM) is a K-principal bundle, Theorem 11.7 and Lemma 11.6 allow us to
find a section ζ̂ : M → LM iff there exists a flat connection on it. Thus if ωζ is flat, we can
construct a lift ζ̂ : M → LM for ζ and so
Ẑ =
(
ζ̂,Γ
)
is the desired lift to T0, where Γ: M → C(LM) is the Levi-Civita connection for ζ.
Conversely, let us suppose that we have a lift
ζ̂ := τ ′10 ◦ Ẑ : M → LM
for the metric ζ : M → Σ. Recall that, for every (x, u) ∈ ζ∗(LM),
T(x,u)ζ
∗(LM) =
{
(vx, Vu) : Txζ(vx) = Tup
LM
K (Vu)
}
⊂ TxM × Tu(LM).
Then we construct the following K-invariant distribution H on ζ∗(LM): If k ∈ K fullfils the
condition u = ζ̂(x) · k, then
H(x,u) :=
{(
vx, Tζ̂(x)
Rk
(
Txζ̂(vx)
))
: vx ∈ TxM
}
.
It can be shown that it defines a flat connection on ζ∗(LM). �
So, in order to find a lift for the section ζ, it is sufficient to construct a flat connection on
the K-principal bundle ζ∗(LM).
To this end, we will define
ωζ := πk ◦ (pr2)∗ω0 ∈ Ω1(ζ∗(LM), k)
where ω0 ∈ Ω1(LM, gl(m)) is a principal connection on LM and πk : gl(m)→ k is the canonical
projection onto k. Lemma 11.8 allows us to establish the following definition, inspired in the
analogous concept from regular Routh reduction.
Definition 11.9 (flat condition for Palatini gravity). We will say that a metric ζ : M → Σ
satisfies the flat condition regarding the principal connection ω0 ∈ Ω1(LM, gl(m)) if and only if
the associated connection ωζ is flat.
42 S. Capriotti
Remark 11.10 (flat condition and parallelizability). This condition yields to a relationship
between the metric ζ : M → Σ and the principal connection ω0; the physical relevance of this
relationship remains unclear for the author. In order to get this condition mathematically, let
us discuss the meaning of the bundle ζ∗(LM). By definition, (x, u = (X1, . . . , Xm)) ∈M ×LM
belongs to ζ∗(LM) if and only Xi ∈ TxM for all i = 1, . . . ,m and also
ζ(x)] = ηijXi ⊗Xj . (11.6)
Here g] indicates the contravariant 2-tensor associated to the metric g; an equivalent way to
express this is given by the formula
ζ(x) ◦ (Tuτ ⊗ Tuτ) = ηijθ
i
∣∣
u
⊗ θj
∣∣
u
.
It means in particular that we can identify
ι : ζ∗(LM)
∼−→ Oζ(M),
where Oζ(M) ⊂ LM is the orthogonal subbundle associated to the metric ζ. This identification
goes as follows: Given (x, u) ∈ ζ∗(LM), equation (11.6) tells us that u ∈ Oζ(M); conversely,
given u ∈ Oζ(M), the pair (τ(u), u) belongs to ζ∗(LM). Therefore, we have the commutative
diagram
ζ∗(LM) LM
Oζ(M)
ι
pr2
inc
Thus, under the previous identification, the gl(m)-valued 1-form pr∗2ω0 is the pullback of the
form ω0 to the subbundle Oζ(M); because the decomposition
gl(m) = k⊕ p
is K-invariant, the k-valued 1-form πk ◦ (pr2)∗ω0 is a connection form on Oζ(M). Thus, flatness
of this connection is equivalent to Oζ(M) being trivial. Now, it means that M is parallelizable;
therefore, we have proved that the flat condition for Palatini gravity is equivalent to paralleliz-
ability of the spacetime manifold M . In particular, the reconstruction scheme can be carried
out only locally.
Also, it is necessary to establish the following result regarding the map Fω.
Lemma 11.11. The following diagram commutes
T0 J1τΣ
T0/K Σ
p
T0
K
Fω
(τΣ)10
τ ′10
(11.7)
Proof. In fact, for j1
xs ∈ T0 we have
(τΣ)10
(
Fω
(
j1
xs
))
= (τΣ)10
(
j1
x
(
pLMK ◦ s
))
= [s(x)]K
and also
τ ′10
(
pT0K
(
j1
xs
))
= τ ′10
([
j1
xs
]
K
)
= [s(x)]K ,
and the lemma follows. �
Routh Reduction of Palatini Gravity in Vacuum 43
With this in mind, we are ready to formulate the reconstruction side of this version of Routh
reduction for Palatini gravity.
Theorem 11.12 (reconstruction in Palatini gravity). Let ζ : M → Σ be a metric satisfying the
flat condition and the Einstein–Hilbert equations of motion. Then there exists a section
Ẑ : M → T0
that is extremal of the Griffiths variational problem for Palatini gravity.
Proof. The holonomic lift
j1ζ : M → J1τΣ
is extremal for the variational problem
(
J1τΣ,L(1)
EH, cI
Σ
con
)
; then, by Proposition 8.5, there exists
a section
Γ̃ : M →WEH
such that τEH ◦ Γ̃ = j1ζ and
Γ̃∗(XydλEH) = 0 (11.8)
for all Z ∈ XV ((τΣ)1◦τEH)(WEH).
On the other hand, by Lemma 11.8 we have a lift
Ẑ : M → T0
such that
τ ′10 ◦ p
T0
K ◦ Ẑ = ζ;
by diagram (11.7) we have that
ζ = τ ′10 ◦ p
T0
K ◦ Ẑ = (τΣ)10 ◦ Fω ◦ Ẑ. (11.9)
We will define the map
Γ′ :=
(
Ẑ, Γ̃
)
: M → T0 ×WEH
and show that it is a section of prω1 : F ∗ω(WEH)→ T0; namely, we have to show that
Fω ◦ Ẑ = πEH ◦ Γ̃.
It is important to this end to note that the conclusion of Proposition 7.4 can be translated to
this context into
TpLMK ◦ ωp = F ∗ωω;
moreover, by Lemma 11.8 we know that Ẑ∗ωp = 0, so(
Fω ◦ Ẑ
)∗
ω = Ẑ∗
(
F ∗ωω
)
= TpLMK ◦
(
Ẑ∗ωp
)
= 0.
Then the section
Fω ◦ Ẑ : M → J1τΣ
44 S. Capriotti
is holonomic; finally, from equation (11.9) we must conclude that
Fω ◦ Ẑ = j1ζ.
But j1ζ = πEH ◦ Γ̃ by construction of the section Γ̃; then
Fω ◦ Ẑ = πEH ◦ Γ̃
and Γ′ is a section of F ∗ω (WEH), as required.
Now define the section
Γ := F̃ω
(
Ẑ, Γ̃
)
: M →WPG;
Therefore, for any Z ∈ XV (τ1◦πPG)(WPG) that is
(
prω2 ◦ F−1
ω
)
-projectable, we have that
Γ∗(ZydλPG) = (Γ′)∗
((
TF−1
ω ◦ Z
)
ydF ∗ωλPG
)
= (Γ′)∗
((
TF−1
ω ◦ Z
)
yd
(
prω2
)∗
λEH
)
=
(
prω2 ◦ Γ′
)∗((
Tprω2 ◦ TF−1
ω ◦ Z
)
ydλEH
)
= Γ̃∗
((
Tprω2 ◦ TF−1
ω ◦ Z
)
ydλEH
)
= 0
because Γ̃ obeys equation (11.8). �
12 Conclusions and outlook
We adapt the Routh reduction scheme developed in [6] to the case of affine gravity with vielbeins.
It suggests that this formalism could be fit to deal with Griffiths variational problems more
general than the classical, at least with cases when the differential restrictions are a subset of
those imposed by the contact structure. Extensions of this scheme to gravity interacting with
matter fields will be studied elsewhere.
A An important algebraic result
First, we want to state the following algebraic proposition.
Proposition A.1. Let {cijk} be a set of real numbers such that{
cijk ∓ cjik = bijk,
cijk ± cikj = aijk
for some given set of real numbers {aijk} and {bijk} such that bijk∓ bjik = 0 and aijk±aikj = 0.
Then,
cijk =
1
2
(aijk + ajki − akij + bijk + bkij − bjki)
is the unique solution for this linear system.
Proof. From first equation we see that
±cjik = cijk − bijk.
The trick now is to form the following combination
aijk + ajki − akij = cijk ± cikj + cjki ± cjik − (ckij ± ckji) = 2cijk − bijk − bkij + bjki,
where in the permutation of indices was used the remaining condition. �
Routh Reduction of Palatini Gravity in Vacuum 45
B Proof of Proposition 4.5
In order to do this proof, it will be necessary to bring some facts from [5]. First, we have the
bundle isomorphism on T0
WPG ' E2
where p′2 : E2 → T0 is the vector bundle
E2 := ∧m−1
1 (T0)⊗ S∗(m),
with S∗(m) :=
(
Rm
)∗ � (Rm)∗ the set of symmetric forms on Rm, and
∧m−1
1 (T0) :=
{
γ ∈ ∧m−1(T0) : γ is horizontal respect to the projection τ ′1 : T0 →M
}
.
The bundle E2 is a bundle of forms with values in a vector space; therefore, it has a canonical
(m− 1)-form
Θ := Θije
i � ej .
Using the structure equations for the canonical connection on J1τ (pulled back to T0), we have
that the differential of the Lagrangian form λPG is given by
dλPG|ρ =
[
2ηkp(ωp)
i
k ∧ θil − (ωp)
s
s ∧ ηkpθkl + ηipΘil|β
]
∧ Ωl
p
+ ηik
[
dΘij |β + ηrqηliΘrj |β ∧ (ωk)
l
q −Θip|β ∧ (ωk)
p
j
]
∧ (ωp)
j
k.
The equations of motion
Γ∗(XydλPG) = 0, X ∈ XV (τ ′1◦τPG)(WPG) (B.1)
are obtained by choosing a convenient set of vertical vector fields; because of the identification
given above, it is sufficient to give a set of vertical vector fields on T0 and on E2. It results that
a global basis of vertical vector fields on T0 is
B′ :=
{
AJ1τ ,Mrs
(
θr,
(
Esj
)
LM
)V
: A ∈ gl(m),Mpq −Mqp = 0
}
;
in fact, the equation defining T0
ekρe
µ
kσ = ekσe
µ
kρ
is invariant by the GL(m)-action, and also(
θr,
(
Esj
)V
LM
)
·
(
ekρe
µ
kσ − e
k
σe
µ
kρ
)
= eµj
(
esσe
r
ρ − erσesρ
)
.
Given that E2 is a vector bundle on T0, any section β : T0 → E2 gives rise to a vertical vector
field; the equations of motion associated to these kind of vector fields are the metricity conditions
ωp = 0.
Therefore, fixing an Ehresmann connection on the bundle p′2 : E2 → T0, we can produce the set
of vertical vector fields on E2
(AT0)H , M := Mrs
(
θr,
(
Esj
)
LM
)H
;
46 S. Capriotti
the equations of motion associated to M are
ηksΓ∗
(
MrsΘkt ∧ θr
)
= 0.
The unique solution of these equations is Θkt = 0. In fact, by writing
Γ∗Θkt = ηlpNktpθl
and taking into account the symmetry properties of Mpq, we have that the set of quantities Npqr
must satisfy
Npqr −Nqpr = 0, Npqr +Nprq = 0;
by Proposition A.1, it results that Npqr = 0, as desired. The rest of the equations of motion
can be calculated is the same fashion that in the J1τ case; therefore, the equations (B.1) are
equivalent to the equations for the extremals of the Palatini variational problem.
C Proof of Proposition 6.1
First, let us write down(
∂
∂xµ
)H
= Mν
µ
∂
∂xν
+Nν
µk
∂
∂eνk
.
Then from
TpLMK
((
∂
∂xµ
)H)
=
∂
∂xµ
(C.1)
it results
Mν
µ = δνµ;
the condition
ωK
((
∂
∂xµ
)H)
= 0
implies
Nν
µk
(
ηpkelν − ηlkepν
)
+ ηlqepσf
σ
qµ − ηpqelσfσqµ = 0.
In order to understand this equation for the unknowns Nν
µk, let us change of variables through
the formula
Nν
µk = gσρe
ρ
kN
νσ
µ ;
in terms of these new variables, and the Christoffel symbols of the connection ω0
Γ
σ
αµ = −ekαfσkµ,
the previous equations can be expressed as
0 = gαρe
ρ
kN
να
µ
(
ηpkelν − ηlkepν
)
− ηlqepσeαq Γ
σ
αµ + ηpqelσe
α
q Γ
σ
αµ
= Nνα
µ
(
gαρe
ρ
kη
pkelν − gαρe
ρ
kη
lkepν
)
+
(
ηpqeαq e
l
σ − ηlqeαq epσ
)
Γ
σ
αµ
Routh Reduction of Palatini Gravity in Vacuum 47
= Nνα
µ
(
gαρg
ρβepβe
l
ν − gαρgρβelβepν
)
+
(
epβe
l
σ − elβepσ
)
gαβΓ
σ
αµ
= Nσβ
µ
(
epβe
l
σ − elβepσ
)
+
(
epβe
l
σ − elβepσ
)
gαβΓ
σ
αµ
=
(
epβe
l
σ − elβepσ
)(
Nσβ
µ + gαβΓ
σ
αµ
)
.
The operator in the left is essentially an antisymmetrizator because of the formula
eµpe
ν
l
(
epβe
l
σ − elβepσ
)
= δµβδ
ν
σ − δνβδµσ ;
therefore
Nσβ
µ + gαβΓ
σ
αµ = Sσβµ , (C.2)
where
Sσβµ − Sβσµ = 0.
Finally, from the condition (C.1) we obtain
Nσ
µk
(
ηkqeρqδ
α
σ + ηkqeαq δ
ρ
σ
)
= 0
or, in terms of the variables Nνσ
µ
0 = gνβe
β
kN
σν
µ
(
ηkqeρqδ
α
σ + ηkqeαq δ
ρ
σ
)
= Nσν
µ
(
gνβe
β
kη
kqeρqδ
α
σ + gνβe
β
kη
kqeαq δ
ρ
σ
)
= Nσν
µ
(
gνβg
βρδασ + gνβg
βαδρσ
)
= Nσν
µ
(
δρνδ
α
σ + δαν δ
ρ
σ
)
.
From equation (C.2) it results that
gαβΓ
σ
αµ + gασΓ
β
αµ − 2Sσβµ = 0,
or equivalently
Nσβ
µ =
1
2
(
gασΓ
β
αµ − gαβΓ
σ
αµ
)
.
Therefore, we have(
∂
∂xµ
)H
=
∂
∂xµ
+
1
2
gβρe
ρ
k
(
gασΓ
β
αµ − gαβΓ
σ
αµ
) ∂
∂eσk
.
Additionally, we need to construct the horizontal lifts(
∂
∂gµν
)H
= P σµν
∂
∂xσ
+Qσµνk
∂
∂eσk
with P σµν − P σνµ = 0, Qσµνk −Qσνµk = 0. The equation
TpLMK
((
∂
∂gµν
)H)
=
∂
∂gµν
and the identity (6.1) imply
P σµν
∂
∂xσ
+Qσµνk
(
ηkqeρqδ
α
σ + ηkqeαq δ
ρ
σ
) ∂
∂gαρ
=
∂
∂gµν
,
48 S. Capriotti
namely
P σµν = 0
and (given the symmetry properties of gµν)
1
2
(
δαµδ
ρ
ν + δαν δ
ρ
µ
)
= Qσµνk
(
ηkqeρqδ
α
σ + ηkqeαq δ
ρ
σ
)
= ηkqeρqQ
α
µνk + ηkqeαqQ
ρ
µνk. (C.3)
The horizontality condition
ωK
((
∂
∂gµν
)H)
= 0
will be equivalent to(
ηpkelσ − ηlkepσ
)
Qσµνk = 0. (C.4)
These conditions can be understood by introducing the variables
Qαρµν := ηkqeρqQ
α
µνk;
then, equation (C.3) becomes
Qαρµν +Qραµν =
1
2
(
δαµδ
ρ
ν + δαν δ
ρ
µ
)
and equation (C.4) is equivalent to
Qαρµν −Qραµν = 0.
Therefore
Qαρµν +Qραµν =
1
2
(
Qαρµν +Qραµν
)
+
1
2
(
Qαρµν −Qραµν
)
=
1
4
(
δαµδ
ρ
ν + δαν δ
ρ
µ
)
and so(
∂
∂gµν
)H
= Qαµνk
∂
∂eαk
=
1
4
ηkle
l
ρ
(
δαµδ
ρ
ν + δαν δ
ρ
µ
) ∂
∂eαk
=
1
4
gρβe
β
k
(
δαµδ
ρ
ν + δαν δ
ρ
µ
) ∂
∂eαk
.
Acknowledgements
The author thanks the CONICET and UNS for financial support, and Eduardo Garćıa-Toraño
for valuable discussion regarding aspects of Routh reduction contained in this article, as well
as for pointing me out reference [28]. Also, the author would like to warmly thank the referees
for the care they put in reviewing this work. The article has been greatly improved by their
suggestions.
References
[1] Arnowitt R., Deser S., Misner C.W., The dynamics of general relativity, Gen. Relativity Gravitation 40
(2004), 1997–2027, arXiv:gr-qc/0405109.
[2] Barbero-Liñán M., Echeverŕıa-Enŕıquez A., Mart́ın de Diego D., Muñoz Lecanda M.C., Román-Roy N.,
Skinner–Rusk unified formalism for optimal control systems and applications, J. Phys. A: Math. Gen. 40
(2007), 12071–12093, arXiv:0705.2178.
https://doi.org/10.1007/s10714-008-0661-1
https://arxiv.org/abs/gr-qc/0405109
https://doi.org/10.1088/1751-8113/40/40/005
https://arxiv.org/abs/0705.2178
Routh Reduction of Palatini Gravity in Vacuum 49
[3] Capriotti S., Differential geometry, Palatini gravity and reduction, J. Math. Phys. 55 (2014), 012902,
29 pages, arXiv:1209.3596.
[4] Capriotti S., Routh reduction and Cartan mechanics, J. Geom. Phys. 114 (2017), 23–64, arXiv:1606.02630.
[5] Capriotti S., Unified formalism for Palatini gravity, Int. J. Geom. Methods Mod. Phys. 15 (2018), 1850044,
33 pages, arXiv:1707.06057.
[6] Capriotti S., Garćıa-Toraño Andrés E., Routh reduction for first-order Lagrangian field theories, Lett. Math.
Phys. 109 (2019), 1343–1376, arXiv:1909.10088.
[7] Capriotti S., Gaset J., Román-Roy N., Salomone L., Griffiths variational multisymplectic formulation for
Lovelock gravity, arXiv:1911.07278.
[8] Castrillón López M., Garćıa P.L., Rodrigo C., Euler–Poincaré reduction in principal bundles by a subgroup
of the structure group, J. Geom. Phys. 74 (2013), 352–369.
[9] Castrillón López M., Muñoz Masqué J., The geometry of the bundle of connections, Math. Z. 236 (2001),
797–811.
[10] Castrillón López M., Muñoz Masqué J., Rosado Maŕıa E., First-order equivalent to Einstein–Hilbert La-
grangian, J. Math. Phys. 55 (2014), 082501, 9 pages, arXiv:1306.1123.
[11] Castrillón López M., Ratiu T.S., Reduction in principal bundles: covariant Lagrange–Poincaré equations,
Comm. Math. Phys. 236 (2003), 223–250.
[12] Castrillón López M., Ratiu T.S., Shkoller S., Reduction in principal fiber bundles: covariant Euler–Poincaré
equations, Proc. Amer. Math. Soc. 128 (2000), 2155–2164, arXiv:math.DG/9908102.
[13] Cattaneo A.S., Schiavina M., The reduced phase space of Palatini–Cartan–Holst theory, Ann. Henri
Poincaré 20 (2019), 445–480, arXiv:1707.05351.
[14] Crampin M., Mestdag T., Routh’s procedure for non-abelian symmetry groups, J. Math. Phys. 49 (2008),
032901, 28 pages, arXiv:0802.0528.
[15] Dadhich N., Pons J.M., On the equivalence of the Einstein–Hilbert and the Einstein–Palatini formula-
tions of general relativity for an arbitrary connection, Gen. Relativity Gravitation 44 (2012), 2337–2352,
arXiv:1010.0869.
[16] Echeverŕıa-Enŕıquez A., López C., Maŕın-Solano J., Muñoz Lecanda M.C., Román-Roy N., Lagrangian–
Hamiltonian unified formalism for field theory, J. Math. Phys. 45 (2004), 360–380, arXiv:math-ph/0212002.
[17] Ellis D.C.P., Gay-Balmaz F., Holm D.D., Ratiu T.S., Lagrange–Poincaré field equations, J. Geom. Phys.
61 (2011), 2120–2146, arXiv:0910.0874.
[18] Garćıa-Toraño Andrés E., Mestdag T., Yoshimura H., Implicit Lagrange–Routh equations and Dirac reduc-
tion, J. Geom. Phys. 104 (2016), 291–304, arXiv:1509.01946.
[19] Gaset J., Román-Roy N., Multisymplectic unified formalism for Einstein–Hilbert gravity, J. Math. Phys. 59
(2018), 032502, 39 pages, arXiv:1705.00569.
[20] Gotay M.J., An exterior differential systems approach to the Cartan form, in Symplectic Geometry and
Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., Vol. 99, Birkhäuser Boston, Boston, MA,
1991, 160–188.
[21] Gotay M.J., Isenberg J., Marsden J.E., Montgomery R., Momentum Maps And Classical Relativistic Fields.
Part I: Covariant field theory, arXiv:physics/9801019.
[22] Griffiths P.A., Exterior differential systems and the calculus of variations, Progress in Mathematics, Vol. 25,
Birkhäuser, Boston, Mass., 1983.
[23] Hehl F.W., McCrea J.D., Mielke E.W., Ne’eman Y., Metric-affine gauge theory of gravity: field equa-
tions, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rep. 258 (1995), 1–171,
arXiv:gr-qc/9402012.
[24] Hehl F.W., von der Heyde P., Kerlick G.D., Nester J.M., General relativity with spin and torsion: founda-
tions and prospect, Rev. Modern Phys. 48 (1976), 393–416.
[25] Higham N.J., J-orthogonal matrices: properties and generation, SIAM Rev. 45 (2003), 504–519.
[26] Hsu L., Calculus of variations via the Griffiths formalism, J. Differential Geom. 36 (1992), 551–589.
[27] Ibort A., Spivak A., On a covariant Hamiltonian description of Palatini’s gravity on manifolds with boundary,
arXiv:1605.03492.
[28] Kharlamov M.P., Characteristic class of a bundle and the existence of a global Routh function, Funct. Anal.
Appl. 11 (1977), 80–81.
https://doi.org/10.1063/1.4862855
https://arxiv.org/abs/1209.3596
https://doi.org/10.1016/j.geomphys.2016.11.015
https://arxiv.org/abs/1606.02630
https://doi.org/10.1142/S0219887818500445
https://arxiv.org/abs/1707.06057
https://doi.org/10.1007/s11005-018-1140-6
https://doi.org/10.1007/s11005-018-1140-6
https://arxiv.org/abs/1909.10088
https://arxiv.org/abs/1911.07278
https://doi.org/10.1016/j.geomphys.2013.08.008
https://doi.org/10.1007/PL00004852
https://doi.org/10.1063/1.4890555
https://arxiv.org/abs/1306.1123
https://doi.org/10.1007/s00220-003-0797-5
https://doi.org/10.1090/S0002-9939-99-05304-6
https://arxiv.org/abs/math.DG/9908102
https://doi.org/10.1007/s00023-018-0733-z
https://doi.org/10.1007/s00023-018-0733-z
https://arxiv.org/abs/1707.05351
https://doi.org/10.1063/1.2885077
https://arxiv.org/abs/0802.0528
https://doi.org/10.1007/s10714-012-1393-9
https://arxiv.org/abs/1010.0869
https://doi.org/10.1063/1.1628384
https://arxiv.org/abs/math-ph/0212002
https://doi.org/10.1016/j.geomphys.2011.06.007
https://arxiv.org/abs/0910.0874
https://doi.org/10.1016/j.geomphys.2016.02.010
https://arxiv.org/abs/1509.01946
https://doi.org/10.1063/1.4998526
https://arxiv.org/abs/1705.00569
https://arxiv.org/abs/physics/9801019
https://doi.org/10.1007/978-1-4615-8166-6
https://doi.org/10.1016/0370-1573(94)00111-F
https://arxiv.org/abs/gr-qc/9402012
https://doi.org/10.1103/RevModPhys.48.393
https://doi.org/10.1137/S0036144502414930
https://doi.org/10.4310/jdg/1214453181
https://arxiv.org/abs/1605.03492
https://doi.org/10.1007/BF01135548
https://doi.org/10.1007/BF01135548
50 S. Capriotti
[29] Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. I, Interscience Publishers, New York –
London, 1963.
[30] Krupka D., Introduction to global variational geometry, Atlantis Studies in Variational Geometry, Vol. 1,
Atlantis Press, Paris, 2015.
[31] Langerock B., López M.C., Routh reduction for singular Lagrangians, Int. J. Geom. Methods Mod. Phys. 7
(2010), 1451–1489, arXiv:1007.0325.
[32] Marsden J.E., Ratiu T.S., Scheurle J., Reduction theory and the Lagrange–Routh equations, J. Math. Phys.
41 (2000), 3379–3429.
[33] Nawarajan D., Visser M., Global properties of physically interesting Lorentzian spacetimes, Internat. J.
Modern Phys. D 25 (2016), 1650106, 15 pages, arXiv:1601.03355.
[34] Pars L.A., A treatise on analytical dynamics, John Wiley & Sons, Inc., New York, 1965.
[35] Peldán P., Actions for gravity, with generalizations: a review, Classical Quantum Gravity 11 (1994), 1087–
1132, arXiv:gr-qc/9305011.
[36] Prieto-Mart́ınez P.D., Román-Roy N., A new multisymplectic unified formalism for second order classical
field theories, J. Geom. Mech. 7 (2015), 203–253, arXiv:1402.4087.
[37] Romano J.D., Geometrodynamics vs. connection dynamics, Gen. Relativity Gravitation 25 (1993), 759–854,
arXiv:gr-qc/9303032.
[38] Sardanashvily G., Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1869–1895,
arXiv:1110.1176.
[39] Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142,
Cambridge University Press, Cambridge, 1989.
[40] Tsamparlis M., On the Palatini method of variation, J. Math. Phys. 19 (1978), 555–557.
[41] Vey D., Multisymplectic formulation of vielbein gravity: I. De Donder–Weyl formulation, Hamiltonian
(n− 1)-forms, Classical Quantum Gravity 32 (2015), 095005, 50 pages, arXiv:1404.3546.
https://doi.org/10.2991/978-94-6239-073-7
https://doi.org/10.1142/S0219887810004907
https://arxiv.org/abs/1007.0325
https://doi.org/10.1063/1.533317
https://doi.org/10.1142/S0218271816501066
https://doi.org/10.1142/S0218271816501066
https://arxiv.org/abs/1601.03355
https://doi.org/10.1088/0264-9381/11/5/003
https://arxiv.org/abs/gr-qc/9305011
https://doi.org/10.3934/jgm.2015.7.203
https://arxiv.org/abs/1402.4087
https://doi.org/10.1007/BF00758384
https://arxiv.org/abs/gr-qc/9303032
https://doi.org/10.1142/S0219887811005993
https://arxiv.org/abs/1110.1176
https://doi.org/10.1017/CBO9780511526411
https://doi.org/10.1063/1.523699
https://doi.org/10.1088/0264-9381/32/9/095005
https://arxiv.org/abs/1404.3546
1 Introduction
2 Variational problems and unified formalism
3 Geometrical tools for Palatini gravity
3.1 Geometry of the jet space for the frame bundle
3.2 Restrictions in Palatini gravity: Zero torsion submanifold and metricity forms
4 Griffiths variational problem for Palatini gravity
5 Symmetry and reduction
5.1 Momentum map and connection
5.2 Reduced bundle for Palatini gravity
5.3 Routh reduction scheme for Palatini gravity
6 Local coordinates expressions
7 Torsion, metricity and contact structures on the quotient space
8 First order variational problem for Einstein–Hilbert gravity
9 Contact bundle decomposition for Palatini gravity
10 First order Einstein–Hilbert Lagrangian as Routhian
11 Einstein–Hilbert gravity as Routh reduction of Palatini gravity
11.1 Routh reduction of Palatini gravity
11.2 … and reconstruction
12 Conclusions and outlook
A An important algebraic result
B Proof of Proposition 4.5
C Proof of Proposition 6.1
References
|
| id | nasplib_isofts_kiev_ua-123456789-210704 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:32Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Capriotti, Santiago 2025-12-15T15:25:03Z 2020 Routh Reduction of Palatini Gravity in Vacuum. Santiago Capriotti. SIGMA 16 (2020), 046, 50 pages 1815-0659 2020 Mathematics Subject Classification: 53C80; 53C05; 83C05; 70S05; 70S10 arXiv:1909.10088 https://nasplib.isofts.kiev.ua/handle/123456789/210704 https://doi.org/10.3842/SIGMA.2020.046 An interpretation of Einstein-Hilbert gravity equations as a Lagrangian reduction of Palatini gravity is made. The main technique involved in this task consists of representing the equations of motion as a set of differential forms on a suitable bundle. In this setting, Einstein-Hilbert gravity can be considered as a kind of Routh reduction of the underlying field theory for Palatini gravity. As a byproduct of this approach, a novel set of conditions for the existence of a vielbein for a given metric is found. The author thanks the CONICET and UNS for financial support, and Eduardo Garca-Torano for valuable discussion regarding aspects of Routh reduction contained in this article, as well as for pointing me out to reference [28]. Also, the author would like to warmly thank the referees for the care they put into reviewing this work. The article has been greatly improved by their suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Routh Reduction of Palatini Gravity in Vacuum Article published earlier |
| spellingShingle | Routh Reduction of Palatini Gravity in Vacuum Capriotti, Santiago |
| title | Routh Reduction of Palatini Gravity in Vacuum |
| title_full | Routh Reduction of Palatini Gravity in Vacuum |
| title_fullStr | Routh Reduction of Palatini Gravity in Vacuum |
| title_full_unstemmed | Routh Reduction of Palatini Gravity in Vacuum |
| title_short | Routh Reduction of Palatini Gravity in Vacuum |
| title_sort | routh reduction of palatini gravity in vacuum |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210704 |
| work_keys_str_mv | AT capriottisantiago routhreductionofpalatinigravityinvacuum |