De Donder Form for Second Order Gravity

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory.


Introduction
In 1929, De Donder formulated an approach to study first order variational problems for several independent variables in terms of a differential form obtained by the Legendre transformation in each independent variable [6] and [7]. The De Donder form is a field theory analogue of the Poincaré-Cartan form, which was introduced for a single independent variable. It is a basis of the multisymplectic formulation of field theory, which is called also a polysymplectic theory or De Donder-Weyl theory. The first application of the De Donder form to general relativity in the Palatini formulation [15] was given in [16]. For further developments see [3], [9], [10], [19] and references quoted there.
In 1936, Lepage [11] constructed a family of forms, each of which can be used in the same way as the De Donder form to reduce the original variational problem to a system of equations in exterior differential forms. In 1977, Aldaya and Azcárraga [1] studied generalizations of the Lepage construction to higher order Lagrangians, for which they used the term Poincaré-Cartan forms. Here, we use the term De Donder form for the Poincaré-Cartan form of Aldaya and Azcárraga which is obtained from the Lagrangian by Ostrogradski's generalization of the Legendre transformation [14] in all independent variables.
The usual expression for a De Donder form is given in terms of coordinates on an appropriate jet bundle induced by a coordinate patch on the space of variables. For a generic Lagrangian, if the number of independent and the number of dependent variables are greater than 1, this expression depends on the choice of coordinates. Therefore, it does not define a global form. This leads to a search for additional geometric structures, which would ensure global existence of such forms, see [4], [8] and references cited there.
The aim of this paper is to show that, for second order general relativity with a diffeomorphism invariant Lagrangian L, the coordinate expression for De Donder form is independent of the choice of coordinates. This implies that the De Donder construction for second order gravity yields a unique form Θ, which is given by a natural differential operator applied to the invariant Lagrangian L. Therefore, we can use Θ to obtain an invariant multisymplectic formulation of second order gravity for any choice of invariant Lagrangian.
The paper is organized as follows. In Section 2, we present a brief review of some fundamentals of jet bundles. We exhibit the results obtained in this work in Section 3, where we also discuss multisymplectic formulation of the second order gravity. Since our proofs are mainly computational and require a lot of attention to details, they are presented in Section 4.
The first jet extension of the section σ |U : U → π −1 (U) ⊆ N is Similarly, for k = 2, 3, ..., we have the k-jet bundle J k (M, N) with local coordinates (x µ , y µν , z µνλ 1 , ..., z µνλ 1 ...λ k ), source map π k : J k (M, N) → M, target map π k 0 : J k (M, N) → N and forgetful maps π k l : : Local contact forms are where summation over repeated indices is assumed. A section ρ : M → J k (M, N) of the source projection π k is said to be holonomic if it is the k-jet extension of σ = π k 0 • ρ : M → N. The importance of local contact forms stems from the fact that a section ρ : M → J k (M, N) is holonomic if and only if the pull-back of every local contact form by ρ vanishes.
Let Y be a vector field on N. For every k ≥ 0, the local 1-parameter group e Y t of local diffeomorphisms of N preserving the projection map π : N → M gives rise to a local 1-parameter group e Y k t of local diffeomorphisms of J k (M, N), which preserve the ideal generated by contact forms, and intertwine forgetful maps. In other words, the following diagram commutes for k > l. The vector field X k on J k (M, N) is called the prolongation of X to J k (M, N). For more details on jet bundles see [13].

Variational problem
Let Λ be the Lagrange form of the second order gravity. This means that Λ is a semi-basic 4-form on J 2 (M, N). In local coordinates, where L(x µ , y µν , z µνλ , z µνλρ ) is a scalar density with respect to the transformations of J 2 (M, N) induced by coordinate transformations in M. For the sake of simplicity, we set L(x µ , y µν , z µνλ , z µνλρ ) = L(x, y, z) and so that equation (6) reads Λ = L(x, y, z)d 4 x.
Let σ : M → N be a section of π : N → M. If U ⊆ M has compact closure U, the action A U on Λ on σ is the integral The section σ is a critical point of the action A U if, for every vector field Y on N tangent to the fibres of the source map π 2 : J 2 (M, N) → N that vanishes on the boundary of π −1 (U) up to second order, where Y 2 is the prolongation of Y to J 2 (M, N) and £ Y 2 Λ is the Lie derivative of Λ with respect to Y 2 . The condition that Y 2 is the prolongation of Y is equivalent to the classical condition that variations and derivatives commute.

De Donder form
Following references [17] and [18], we present here the geometric description of the De Donder construction adapted to the second order gravity.

Definition 1 De Donder form corresponding to a Lagrangian
where p µναβ and p µνα are functions on J 3 (M, N) such that, for every local section Equations (10) and (11)  Proof of Theorem 2 is given in Section 4.

Field equations
Since Θ differs from π 3 * 2 Ld 4 x by terms proportional to contact forms, for every section σ of π : N → M, Proof. Equation (10) yields Consider a vector field in form and compute the the following Here, we used the fact that (j 3 σ) * π 3 * 2 dL = (j 2 σ) * dL. Observe that one has by equation (11).
Lemma 4 For each vector field Y on N, which projects to a vector field on M, and every section σ of π : M → N, where Y 3 is the prolongation of Y to J 3 (M, N).
Proof. See Lemma 3 in reference [18]. Taking these results into account and using Stokes' Theorem, we can rewrite equation (9) in the form where ∂U is the boundary of U, and the integral over the boundary vanishes because we assume that Y vanishes on ∂U to second order. Proposition 3 implies that in equation (16) we may replace the prolongation Y 3 of a vector field Y on N tangent to fibres of π : M → N by arbitrary vector field X on J 3 (M, N) tangent to fibres of the source map π 3 : J 3 (M, N) → M and vanishes on (π 3 ) −1 (∂U). Therefore, the variational principle (9) is equivalent to where X is an arbitrary vector field on J 3 (M, N) tangent to fibres of the source map π 3 : J 3 (M, N) → M. The Fundamental Theorem in the Calculus of Variations ensures that the variational principle (17) is equivalent to for every vector field X on J 3 (M, N) tangent to fibres of the source map π 3 : (18) is the De Donder equation for the second order gravity with invariant Lagrangian L.
We can show directly that equation (18) is a system of equations in differential forms equivalent to the Euler-Lagrange equations corresponding to L. Let be a vector field tangent to fibres of the source map, and let σ : (x λ ) → g µν (x λ ) be a section of π : N → M. Introducing the notation P µναβ = (j 3 σ) * p µναβ and P µνα = (j 3 σ) * p µνα .
we can write the left hand side of equation (13) in the from Since components of X are arbitrary, equation (13) reads Equation (20) is the definition of P µναβ , equation (21) is the definition of P µνα , while equation (22) is equivalent to the Euler-Lagrange equations

Example: Hilbert's Lagrangian
Hilbert's Lagrangian of general relativity, expressed in terms of local coordinates, is where R[g] is the scalar curvature of the Lorentzian metric g. Since L Hilbert depends linearly on second derivatives of the metric, the corresponding Euler-Lagrange equations are of second order. The Arnowitt, Deser and Misner Hamiltonian formalism for general relativity, [2], see also [12], is based on the Palatini formalism, [15], in which metric and connection are independent dynamical variables. The De Donder form for the Palatini formulation of general relativity was given in [16].

Proposition 5
The De Donder form for the second order Hilbert Lagrangian L Hilbert , expressed in local coordinates, is Proof of Proposition 5 is given in Section 4.

Example: Matter and Gravitation
In the study of second order gravity, we cannot ignore the interaction of gravity with matter. If L is the Lagrangian for gravity alone and L matter is the Lagrangian for the matter such that the total Lagrangian L total = L + L matter is invariant under the group of diffeomorphisms of M, we conjecture that the statement of Theorem 2 also holds for the De Donder form Θ total associated to the total Lagrangian L total . Here, we illustrate it with the case of second order gravity interacting with a scalar field φ given by the Lagrangian form where V (φ) is a known function. We may consider matter field to be a section φ of the trivial bundle κ : We understand, the Lagrangian L matter , exhibited in (26), as of first order in φ and depends parametrically the Lorentzian metric g. Introducing local coordinates (x µ , t, z µ ) on the first jet bundle J 1 (M, R × M), we write the contact form as dt − z µ dx µ , and g-dependent function L matter as In the coordinate representation, the De Donder form for the present case is defined to be where q µ is a function on J 1 (M, K) such that, for every local section φ of the trivial fibration κ, The total space is the fibre product The total De Donder form Θ total is the pull back of the sum of the De Donder form Θ for the gravity and the De Donder form Θ g matter of the matter to the this Whitney product. In coordinates, the total De Donder form is given by of the fibration π × M κ from the product manifold N × M (R×M) to the base manifold M, the coefficient functions are In equation (31) we did not put any pull-back signs to make the coordinate picture more transparent. Moreover, we replaced the subscript g over L g matter in equation (3.13) by y in order to indicate dependence of L matter on the variable y µν = g µν (x).

Proposition 6
For second order gravity with invariant Lagrangian L on J 2 (M, N), the expressions in equations (31) and (32-34) are independent of the choice of coordinates (x µ ) on M. Hence, they define a global 4-form Θ total on the fibre product Proof of Proposition 6 is given in Section 4.
As in preceeding section, the Euler-Lagrange equations for the total Lagrangian L total are equivalent to the De Donder equations for the total De Donder form Θ total .

Proof of Theorem 2
Recall that De Donder form corresponding to a Lagrangian form Λ = Ld 4 x on for every local section σ of π.
Our aim in this section is to study transformation laws of components of Θ with respect to coordinate transformations in J 3 (M, N) induced by an orientation preserving coordinate transformation on M. It induces a local coordinate transformation on N given by Further, we have the local expressions for transformations for the jet coordinates By assumption the Lagrangian form Λ = Ld 4 x is invariant under the transformations (37) through (40). This implies that under these transformations, L transforms as a scalar density. In other words, Lemma 7 Since the boundary form is defined to be 4-form on J 3 (M, N), under the change of coordinates (37) through (40), the coefficients p µνα and p µναβ transform as follows 2 Proof. Notice that, the boundary term Ξ in (42) is the sum of two four forms, label them as Ξ 1 and Ξ 2 in a respected order. In order to deduce the transformation properties of the coefficients p µνα and p µναβ , we express Ξ 1 and Ξ 2 in primed coordinates using the transformations (37) through (40) under the assumption that Ξ 1 and Ξ 2 are 4-forms on J 3 (M, N) . Consider first the term Ξ 1 = p µναβ (dz µνα − z µναγ dx γ ) ∧ (∂ β d 4 x). Taking exterior differential of the coordinate transformation (39), we get Recalling some basic chain rule operations and using the transformation rule (40) of z µ ′ ν ′ α ′ γ ′ , one computes We subtract the one-form z µ ′ ν ′ α ′ γ ′ dx γ ′ , exhibited in (46), from of the one-from dz µ ′ ν ′ α ′ in (45). While taking the difference, see that the second, the third, and the fourth terms in the first line of (45) cancel with the second, the third, and the fourth terms in the first line of (46), respectively. Notice also that the second and the third terms both in the the second and the third lines of (45) cancel with the second and the third terms of the second and third lines of (45), respectively. Eventually, we arrive at the following expression . On the other hand, it is immediate to see that Hence the first term in the boundary form can be obtained by first taking the exterior product of the one form dz µ ′ ν ′ α ′ − z µ ′ ν ′ α ′ γ ′ dx γ ′ and the three form ∂ β ′ d 4 x ′ then by multiplying the product with p µ ′ ν ′ α ′ β ′ . This shows that Ξ 1 expressed in terms of the primed coordinates is x. So that we have derived the first term in the boundary form (42) in terms of the primed coordinates.
As a second step, we write the one-form dy µν − z µνβ dx β in terms of the primed coordinates. By substituting the transformations of y µ ′ ν ′ in (38) and z µ ′ ν ′ β ′ in (39) into this one-form, we have See that the second and the third terms in the second line cancels with the second and the third terms in the third line, respectively. We take the exterior product of dy µ ′ ν ′ − z µ ′ ν ′ β ′ dx β ′ and ∂ β ′ d 4 x ′ , then multiply this four form by p µ ′ ν ′ β ′ . We obtain the following expression for Ξ 2 in terms of primed coordinates, The sum of the four-forms in (48) and (49) is the boundary form in primed coordinates. Explicitly we have that Since Ξ is a 4-form, its expression Ξ ′ in primed coordinates gives the same form as the expression in the original coordinates. That is Ξ = Ξ ′ , which implies that This completes the proof of Lemma 7. In Lemma 7 we showed that arbitrary smooth functions p µναβ and p µνα on J 3 (M, N) define a 4-form on J 3 (M, N) provided that, under coordinate transformations (37) through (40), they transform according to equations (43) and (44). In the present case, the coefficients p µναβ and p µνα are defined as the Ostrogradski's momenta, which implies equations (36) for every section σ of π : N → M. Note that any function f on J 3 (M, N) is uniquely determined by its pull-backs (j 3 σ) * f for all sections σ of π : N → M. Therefore we may write where D β p µναβ is the total divergence given by In order to simplify computations, we use the notation P µναβ (x γ ) = (j 3 σ) * p µναβ (x γ ) and P µνα (x γ ) = (j 3 σ) * p µνα (x γ ) introduced in equation (19). With this notation, where P µναβ ,β (x λ ) = ∂ ∂x β P µναβ (x λ ). Proof. We start with the second momentum p µναβ and obtain the transformation law (44) as follows,
In order to show that the first momentum p µνα satisfies transformation law (43), start first with the term ∂L/∂z µνα . As in equation (53), In order to compute the divergence term, we work with pull-backs P µνα = (j 3 σ) * p µνα and P µναβ = (j 3 σ) * p µναβ , which allows replacing total derivative by partial derivative, see equation (52). We obtain Note that and Therefore, the second term on the right hand side of equation (55) reads In the light of this, we can rewrite P µναβ ,β in (55) as In order to arrive at the coordinate transformation for Ostrogradski's momentum P µνα , we simply take the difference of (j 2 σ) * (∂L/∂z µνα ) in (54) and P µναβ ,β in (58).
So that, Here, we used notation (19) in the primed coordinates, which yields Equation (59) may be rewritten as Since this equation is valid for every section σ of π : N → M, it follows that where p µνα and p µ ′ ν ′ α ′ β ′ are Ostrogradski's momenta corresponding to the Lagrangian form Ld 4 x, (50). This completes proof of Lemma 8. It follows from Lemma 7 and Lemma 8 that for an invariant Lagrangian form Ld 4 x, the corresponding boundary form has the same expression in the class of coordinate system on M, which differ by orientation preserving transformations. Therefore, the boundary form Ξ is globally defined and is given by a natural differential operator applied to the to the Lagrangian form Ld 4 x. Since Θ = π 3 * 2 Ld 4 x + Ξ, it follows that the De Donder form Θ is globally defined and is also given by a natural differential operator applied to the to the Lagrangian form Ld 4 x. This completes proof of Theorem 2.

Proof of Proposition 5
The outline of the proof is as follows. First, we will write the Hilbert Lagrangian (24) in terms of the metric tensor and its partial derivatives. Such kind of a local presentation of the Hilbert Lagrangian will enable us to prove the Lemma 9 where we shall exhibit the induced Ostrogradski's momenta. Then, we will be ready for the calculation of the De Donder form (25) in an explicit form.
Recall that the Christoffel symbols of the first kind Γ λµν and the Christoffel symbols of the second kind Γ ρ µν are defined and related as where g ρλ is the dual of the metric tensor g ρλ whereas g µλ,ν denotes the partial derivative of g µλ with respect to x ν . It is possible to write the Christoffel symbols in a pure contravariant form Γ λµν = g λα g µβ g νγ Γ αβγ .
For future reference, we define here some symbols by contacting the Christoffel symbols Taking the derivative of the identity g ρλ g λµ = δ ρ µ , one arrives at the relation between g ρλ ,γ and Γ λµν as follows whereas the contraction of this yields Recall also that, the Riemann and the Ricci tensors are respectively. Here, Γ α βδ,γ denotes the partial derivative of Γ α βδ with respect to x γ . In this local representation, the scalar curvature is defined to be Note that the presentation (70) is in terms of the Christoffel symbols of the second kind. It is possible to write R in terms of the Christoffel symbols of the first kind and its partial derivative as well. Simply, by substituting the definition in (60), we compute Notice that, the symbols Γ ρµα contain the first derivative g µν,λ of the metric g µν , so that the partial derivative Γ δβγ,α of the symbols are containing the second partial derivative g µν,λγ of g µν . In accordance with this, we understand the scalar curvature R as the sum of two terms, say R 1 and R 2 by putting all the first order terms that is those involving Γ ρµα into R 1 , and by putting all the second order terms that is those involving Γ δβγ,α into R 2 , that is R = R 1 + R 2 and Therefore, we write the Lagrangian as where R 1 depends linearly on g µν,α linearly while R 2 depends quadratically on g µν,αβ . A simplification is possible for R 1 . See that, R 1 = g βγ −g αµ g δν (Γ µνα + Γ νµα ) Γ δβγ + g αµ g δν (Γ µνγ + Γ νµγ ) Γ δβα +g βγ g αρ g µσ (Γ ρµα Γ σβγ − Γ ρµγ Γ σβα ) = Γ µνλ Γ αβγ −g µλ g να g βγ − g µα g νλ g βγ + g µγ g να g λβ + g µα g νγ g λβ +Γ µνλ Γ αβγ g µλ g να g βγ − g µγ g να g λβ = Γ µνλ Γ αβγ −g µα g νλ g βγ + g µα g νγ g λβ , where we have employed the identities in (66) and (67) in the first line. In the third line, the first term in the parentheses is canceling with the fifth term, and the third term in the parentheses is canceling with the sixth term. We write R 2 in terms of the metric tensor R 2 = g βγ g αδ (Γ δβγ,α − Γ δβα,γ ) = 1 2 g βγ g αδ (g βδ,γα + g γδ,βα − g βγ,δα − g βδ,αγ − g αδ,βγ + g βα,δγ ) = 1 2 g µν,αβ g µα g νβ + g µβ g να − 2g µν g αβ .
In the following Lemma, we are stating the conjugate momenta induced by the Hilbert Lagrangian.
Lemma 9 The Ostrogradski's momenta induced by the Hilbert Lagrangian (71) are where the symbol Γ ν is the one defined in (62).
Proof. First recall the definition of the Ostrogradski's momenta By substituting the exhibition of the Hilbert Lagrangian given in (71), we can rewrite the momenta as where R 1 and R 2 as the ones in (72) and (73), respectively. It is immediate to observe that the second momenta is Notice from (76) that, in order to determine the first momenta P µνα , we need to take the divergence of the second momenta P µναβ , given in (77), with respect to x β . For this, we start with taking the partial derivative of √ − det g with respect to x β as follows where the symbol ∆ β , in (65), has been substituted in the last line of the calculation.
On the other hand, we take the divergence g µα g νβ + g µβ g να − 2g µν g αβ ,β = g µα ,β g νβ + g µα g νβ ,β + g µβ ,β g να + g µβ g να ,β − 2g µν g αβ ,β − 2g µν ,β g αβ (79) where, the identities (66) and (67) have been used in the second line, and the symbols Γ α and ∆ α , defined in (62) and (63) have been substituted in the third line. In the light of the calculations in (78) and (80), the divergence of the second momenta P µναβ turns out to be where the identity g νβ ∆ β = ∆ ν has been used. Notice that all the terms involving ∆ ν canceling each other in the calculation. Let us now concentrate on the first term ∂R 1 /∂g µν,α √ − det g in the momenta P µνα , applying the chain rule, we have that Notice that, the partial derivative of R 1 with respect to the Christoffel symbol of the first kind Γ λµν is computed to be ∂R 1 ∂Γ λµν = 2Γ αβγ −g λα g µν g βγ + g λα g µγ g νβ , whereas the partial derivative of Γ λµν with respect to g αβ,γ is Here, the factor 1/4 is the manifestation of the symmetry of the metric tensor. We multiply the expressions (82) and (83) and arrange the terms, so that we arrive at Now we are ready to write the first momenta P µνα , for this simply take the difference of (84) and (81), this gives where the first and fourth terms in the second and the third lines are canceling each other, respectively.
We are now ready to prove the Proposition 5. In the present framework, the De Donder form turns out to be where Ξ Hilbert is the boundary form induced by the Hilbert Lagrangian. Explicitly, the boundary form is Ξ Hilbert = (P αβµ dg αβ + P αβµν dg αβ,ν ) ∧ ( ∂ ∂x µ d 4 x) − (P αβµ g αβ,µ + P αβµν g αβ,νµ )d 4 x.
By substituting the conjugate momenta P µνα and P µναβ , respectively given in (74) and (75), one has Substitution of the boundary form Ξ Hilbert and the terms R 1 and R 2 in (72) and (73) leads to the following expression of the De Donder form Θ Hilbert = 1 2 g µν,αβ g µα g νβ + g µβ g να − 2g µν g αβ − det gd 4 x +Γ λµν Γ αβγ −g λα g µν g βγ + g λα g µγ g νβ − det gd 4 x Notice that, the first and the last terms are canceling since they are minus of the each other. So that there remain + 1 2 g αµ g βν + g αν g βµ − 2g αβ g µν − det gdg αβ,ν ∧ ( ∂ ∂x µ d 4 x).

Proof of Proposition 6
Referring to the Proposition 5, to prove the Proposition 6 we need to just focus on the De Donder form (29) for the matter. See that it is composed of a Lagrangian term and the boundary term. We label the boundary term as where the coefficient function q µ reads (37) for a local section. Let us first show that, Ξ 3 is invariant under a coordinate transformation on the base manifold M given in (37). See that For the basis we recall the transformation in (47), and compute Collecting all these, one sees that formulation of the boundary form remains the same under coordinate transformation On the other hand, the Lagrangian term is where we have employed the fact that √ − det gd 4 x is invariant under the coordinate transformation. Notice that, assumption of the invariance of the function V leads to the invariance of L g matter d 4 x.