De Donder Form for Second Order Gravity

TL;DR

This paper demonstrates that the De Donder form for second order gravity can be globally defined using local coordinate descriptions, providing a natural differential operator for the theory's invariant Lagrangian.

Contribution

It introduces a globally defined De Donder form for second order gravity based on Ostrogradski's Legendre transformation, enhancing the geometric understanding of the theory.

Findings

De Donder form is globally defined for second order gravity.

The form is constructed via Ostrogradski's Legendre transformation.

It provides a natural differential operator for the invariant Lagrangian.

Abstract

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.

Equations

$${\sigma }_{\mid U}:U\to {\pi }^{-1}(U):x\mapsto (y_{\mu \nu })=g_{\mu \nu }(x^{\lambda }),$$

$${\pi }^1$$

$${\pi }_0^1$$

$$j^1{\sigma }_{\mid U}:U\to ({\pi }^1{)}^{-1}(U)\subseteq J^1(M,N):x\mapsto (x^{\mu },y_{\mu \nu },z_{\mu \nu \lambda })=(x^{\mu },g_{\mu \nu }(x),g_{\mu \nu ,\lambda }(x)),$$

$$g_{\mu \nu ,\lambda }=\frac{\partial }{\partial x^{\lambda }}{g}_{\mu \nu }={\partial }_{\lambda }{g}_{\mu \nu }.$$

$$j^k{\sigma }_{\mid U}$$

$$d{y}_{\mu \nu }-z_{\mu \nu \lambda }d{x}^{\lambda },d{z}_{\mu \nu \lambda }-z_{\mu \nu \lambda \rho }d{x}^{\rho },...,d{z}_{\mu \nu {\lambda }_1...{\lambda }_{k-1}}-z_{\mu \nu {\lambda }_1...{\lambda }_k}d{x}^{{\lambda }_k},$$

$$\begin{array}{ccc}{J}^k(M,N)& \stackrel{e^{Y^{k}t}}{\to }& J^k(M,N)\\ \downarrow {\pi }_l^k& & \downarrow {\pi }_l^k\\ J^l(M,N)& \stackrel{e^{Y^{l}t}}{\to }& J^l(M,N)\end{array}$$

$$\Lambda =L(x^{\mu },y_{\mu \nu },z_{\mu \nu \lambda },z_{\mu \nu \lambda \rho })d{x}^1\wedge ...\wedge d{x}^4,$$

$$L(x^{\mu },y_{\mu \nu },z_{\mu \nu \lambda },z_{\mu \nu \lambda \rho })=L(x,y,z)\text{ and }d{x}^1\wedge ...\wedge d{x}^4=d_{4}x,$$

$$A_U[\sigma ]={\int }_U(j^2\sigma {)}^{\ast }\Lambda .$$

$${\int }_U(j^2\sigma {)}^{\ast }{\pounds }_{Y^2}\Lambda =0,$$

$$\Theta$$

$$(j^3\sigma {)}^{\ast }{p}^{\mu \nu \alpha \beta }$$

$$(j^3\sigma {)}^{\ast }{p}^{\mu \nu \alpha }$$

$$A_U[\sigma ]={\int }_U(j^2\sigma {)}^{\ast }\Lambda ={\int }_U(j^2\sigma {)}^{\ast }L{d}_{4}x={\int }_U(j^3\sigma {)}^{\ast }({\pi }_2^{3\ast }L{d}_{4}x)={\int }_U(j^3\sigma {)}^{\ast }\Theta .$$

$$(j^{3}\sigma)^{\ast}(X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta)=0.$$

$$\displaystyle d\Theta=\pi_{2}^{3\ast}dL\wedge d_{4}x-p^{\mu\nu\alpha\beta}dz_{\mu\nu\alpha\gamma}\wedge dx^{\gamma}\wedge(\partial_{\beta}{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d_{4}x)$$

$$\displaystyle-p^{\mu\nu\alpha}dz_{\mu\nu\beta}\wedge dx^{\beta}\wedge(\partial_{\alpha}{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d_{4}x)+dp^{\mu\nu\alpha\beta}\wedge(dz_{\mu\nu\alpha}-z_{\mu\nu\alpha\gamma}dx^{\gamma})\wedge(\partial_{\beta}{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d_{4}x)$$

$$\displaystyle+dp^{\mu\nu\alpha}\wedge(dy_{\mu\nu}-z_{\mu\nu\beta}dx^{\beta})\wedge(\partial_{\alpha}{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d_{4}x).$$

$$X=X_{\mu \nu \alpha \beta \gamma }\frac{\partial }{{\partial z}_{\mu \nu \alpha \beta \gamma }}+X_{\mu \nu \alpha \beta }\frac{\partial }{{\partial z}_{\mu \nu \alpha \beta }}+X_{\mu \nu \alpha }\frac{\partial }{{\partial z}_{\mu \nu \alpha }},$$

$$\displaystyle(j^{3}\sigma)^{\ast}(X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta)$$

$$(j^3\sigma {)}^{\ast }\left[{\pi }_2^{3\ast }dL\right]=(j^2\sigma {)}^{\ast }dL.$$

$$\left((j^{3}\sigma)^{\ast}dp^{\mu\nu\alpha\beta}\right)(j^{3}\sigma)^{\ast}X_{\mu\nu\alpha}(\partial_{\beta}{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d_{4}x)=\partial_{\beta}\left((j^{3}\sigma)^{\ast}p^{\mu\nu\alpha\beta}\right)(j^{3}\sigma)^{\ast}X_{\mu\nu\alpha}d_{4}x$$

$$\displaystyle(j^{3}\sigma)^{\ast}(X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta)$$

$${\left(j^3\sigma \right)}^{\ast }\left({\pounds }_{Y^3}\left[\Theta -{\pi }_2^{3\ast }L{d}_{4}x\right]\right)=0,$$

$${\int }_U(j^2\sigma {)}^{\ast }{\pounds }_{Y^2}\Lambda$$

$$\int_{U}(j^{3}\sigma)^{\ast}\left[X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta\right]=0,$$

$$(j^{3}\sigma_{\mid U})^{\ast}\left[X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta\right]=0$$

$$X=X_{\mu \nu \alpha \beta \gamma }\frac{\partial }{\partial z_{\mu \nu \alpha \beta \gamma }}+X_{\mu \nu \alpha \beta }\frac{\partial }{\partial z_{\mu \nu \alpha \beta }}+X_{\mu \nu \alpha }\frac{\partial }{\partial z_{\mu \nu \alpha }}+X_{\mu \nu }\frac{\partial }{\partial y_{\mu \nu }}$$

$$P^{\mu \nu \alpha \beta }=(j^3\sigma {)}^{\ast }{p}^{\mu \nu \alpha \beta }\text{ and }{P}^{\mu \nu \alpha }=(j^3\sigma {)}^{\ast }{p}^{\mu \nu \alpha }.$$

$$\displaystyle(j^{3}\sigma)^{\ast}\left[X{\mbox{$\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt}\,$}}d\Theta\right]$$

Full text

1 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 $\Theta$, which is given by a natural differential operator applied to the invariant Lagrangian $L$. Therefore, we can use $\Theta$ 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.

2 Geometric background

2.1 Jets

Let $M$ be a $4$-dimensional manifold representing the space-time of general relativity, and $N\subset\otimes_{\mathrm{sym}}^{2}T^{\ast}M$ be the bundle of Lorentzian frames on $M$. We denote the canonical projection by $\pi:N\rightarrow M.$ A Lorentzian metric on $M$ is a section $\sigma:M\rightarrow N$ of $\pi:N\rightarrow M$. If $(x^{\mu})=(x^{1},...,x^{4})$ are local coordinates on $M$ with domain $U$, we denote by $(x^{\mu},y_{\mu\nu})$ the induced coordinates on $\pi^{-1}(U)\subset N$. In these coordinates, the section $\sigma$ restricted to $U$ is given by

[TABLE]

where $g_{\mu\nu}(x^{\lambda})=g_{\mu\nu}(x^{1},...,x^{4})$ are smooth functions of the coordinates $(x^{\lambda})$.111Here, we work in the smooth category. In applicatioon to concrete cases, one has to choose a suitable function space.

The first derivatives of sections form the first jet bundle $J^{1}(M,N)$ with the source projection $\pi^{1}:J^{1}(M,N)\rightarrow M,$ the target projection $\pi_{0}^{1}:J^{1}(M,N)\rightarrow N$ and the induced coordinates $(x^{\mu},y_{\mu\nu},z_{\mu\nu\lambda})$ such that

[TABLE]

The first jet extension of the section $\sigma_{\mid U}:U\rightarrow\pi^{-1}(U)\subseteq N$ is

[TABLE]

where

[TABLE]

Similarly, for $k=2,3,...$,  we have the $k$-jet bundle $J^{k}(M,N)$ with local coordinates $(x^{\mu},y_{\mu\nu},z_{\mu\nu\lambda_{1}},...,z_{\mu\nu\lambda_{1}...\lambda_{k}})$, source map $\pi^{k}:J^{k}(M,N)\rightarrow M$, target map $\pi_{0}^{k}:J^{k}(M,N)\rightarrow N$ and forgetful maps $\pi_{l}^{k}:J^{k}(M,N)\rightarrow J^{l}(M,N)$ defined for $k>l\geq 0$. The $k$-jet extension of a section $\sigma_{\mid U}:U\rightarrow\pi^{-1}(U)\subseteq N$ is

[TABLE]

Local contact forms are

[TABLE]

where summation over repeated indices is assumed. A section $\rho:M\rightarrow J^{k}(M,N)$ of the source projection $\pi^{k}$ is said to be holonomic if it is the $k$-jet extension of $\sigma=\pi_{0}^{k}\circ\rho:M\rightarrow N$. The importance of local contact forms stems from the fact that a section $\rho:M\rightarrow J^{k}(M,N)$ is holonomic if and only if the pull-back of every local contact form by $\rho$ vanishes.

Let $Y$ be a vector field on $N$. For every $k\geq 0$, the local 1-parameter group $\mathrm{e}^{Yt}$ of local diffeomorphisms of $N$ preserving the projection map $\pi:N\rightarrow M$ gives rise to a local 1-parameter group $\mathrm{e}^{Y^{k}t}$ 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

[TABLE]

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].

2.2 Variational problem

Let $\Lambda$ be the Lagrange form of the second order gravity. This means that $\Lambda$ is a semi-basic $4$-form on $J^{2}(M,N)$. In local coordinates,

[TABLE]

where $L(x^{\mu},y_{\mu\nu},z_{\mu\nu\lambda},z_{\mu\nu\lambda\rho})$ 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

[TABLE]

so that equation (6) reads $\Lambda=L(x,y,z)d_{4}x$.

Let $\sigma:M\rightarrow N$ be a section of $\pi:N\rightarrow M$. If $U\subseteq M$ has compact closure $\overline{U}$, the action $A_{U}$ on $\Lambda$ on $\sigma$ is the integral

[TABLE]

The section $\sigma$ 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 $\pi^{2}:J^{2}(M,N)\rightarrow N$ that vanishes on the boundary of $\pi^{-1}(U)$ up to second order,

[TABLE]

where $Y^{2}$ is the prolongation of $Y$ to $J^{2}(M,N)$ and $\pounds_{Y^{2}}\Lambda$ is the Lie derivative of $\Lambda$ 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.

3 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 $\Lambda=Ld_{4}x$ on $J^{2}(M,N)$ is a form $\Theta$ on $J^{3}(M,N)$ such that, in local coordinates $(x^{\mu})$ on $M$,

[TABLE]

where $p^{\mu\nu\alpha\beta}$ and $p^{\mu\nu\alpha}$ are functions on $J^{3}(M,N)$ such that, for every local section $\sigma:M\rightarrow N:x\mapsto(x,g_{\mu\nu}(x))$ of $\pi,$

[TABLE]

Equations (1) and (11) define a $4$-form $\Theta$ on the domain of coordinates on $J^{3}(M,N)$ defined by local coordinates on $M.$ For a generic Lagrangian, these local forms do not define a global form on $J^{3}(M,N).$The main result of our paper is the following theorem

Theorem 2

For second order gravity with an invariant Lagrangian form $Ld_{4}x$ on $J^{2}(M,N)$, equations (1) and (11) define a global De Donder form $\Theta$ on $J^{3}(M,N)$, given by a natural differential operator applied to $Ld_{4}x$.

Proof of Theorem 2 is given in Section 4.

3.1 Field equations

Since $\Theta$ differs from $\pi_{2}^{3\ast}Ld_{4}x$ by terms proportional to contact forms, for every section $\sigma$ of $\pi:N\rightarrow M$,

[TABLE]

Thus, replacing in the variational principle the Lagrangian form $\Lambda=Ld_{4}x$ by the corresponding De Donder form does not change the value of the action $A_{U}[\sigma]$.

Proposition 3

For every vector field $X$ on $J^{3}(M,N)$ tangent to fibres of the target map $\pi_{0}^{3}:J^{3}(M,N)\rightarrow N$ and every section $\sigma$ of $\pi:N\rightarrow M$,

[TABLE]

Proof. Equation (1) yields

[TABLE]

Consider a vector field in form

[TABLE]

and compute the the following

[TABLE]

Here, we used the fact that

[TABLE]

Observe that one has

[TABLE]

so that

[TABLE]

by equation (11).

Lemma 4

For each vector field $Y$ on $N$, which projects to a vector field on $M$, and every section $\sigma$ of $\pi:M\rightarrow N$,

[TABLE]

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

[TABLE]

where $\partial U$ is the boundary of $U$, and the integral over the boundary vanishes because we assume that $Y$ vanishes on $\partial U$ 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 $\pi:M\rightarrow N$ by arbitrary vector field $X$ on $J^{3}(M,N)$ tangent to fibres of the source map $\pi^{3}:J^{3}(M,N)\rightarrow M$ and vanishes on $(\pi^{3})^{-1}(\partial U).$ Therefore, the variational principle (9) is equivalent to

[TABLE]

where $X$ is an arbitrary vector field on $J^{3}(M,N)$ tangent to fibres of the source map $\pi^{3}:J^{3}(M,N)\rightarrow M$. The Fundamental Theorem in the Calculus of Variations ensures that the variational principle (17) is equivalent to

[TABLE]

for every vector field $X$ on $J^{3}(M,N)$ tangent to fibres of the source map $\pi^{3}:J^{3}(M,N)\rightarrow M.$ Equation (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

[TABLE]

be a vector field tangent to fibres of the source map, and let $\sigma:(x^{\lambda})\mapsto g_{\mu\nu}(x^{\lambda})$ be a section of $\pi:N\rightarrow M$. Introducing the notation

[TABLE]

we can write the left hand side of equation (13) in the from

[TABLE]

Since components of $X$ are arbitrary, equation (13) reads

[TABLE]

Equation (20) is the definition of $P^{\mu\nu\alpha\beta}$, equation (21) is the definition of $P^{\mu\nu\alpha}$, while equation (22) is equivalent to the Euler-Lagrange equations

[TABLE]

3.2 Example: Hilbert’s Lagrangian

Hilbert’s Lagrangian of general relativity, expressed in terms of local coordinates, is

[TABLE]

where $R[g]$ is the scalar curvature of the Lorentzian metric $g$. Since $L_{\mathrm{Hilbert}}$ 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 $L_{\mathrm{Hilbert}}$, expressed in local coordinates, is

[TABLE]

where $\Gamma^{\alpha}=g^{\mu\nu}\Gamma_{\mu\nu}^{\alpha}.$

Proof of Proposition 5 is given in Section 4.

3.3 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 $L_{\mathrm{matter}}$ is the Lagrangian for the matter such that the total Lagrangian $L_{\mathrm{total}}=L+L_{\mathrm{matter}}$ is invariant under the group of diffeomorphisms of $M$, we conjecture that the statement of Theorem 2 also holds for the De Donder form $\Theta_{\mathrm{total}}$ associated to the total Lagrangian $L_{\mathrm{total}}$. Here, we illustrate it with the case of second order gravity interacting with a scalar field $\phi$ given by the Lagrangian form

[TABLE]

where $V(\phi)$ is a known function. We may consider matter field to be a section $\phi$ of the trivial bundle

[TABLE]

We understand, the Lagrangian $L_{\mathrm{matter}}$, exhibited in (26), as of first order in $\phi$ and depends parametrically the Lorentzian metric $g$. Introducing local coordinates $(x^{\mu},t,z_{\mu})$ on the first jet bundle $J^{1}(M,\mathbb{R}\times M)$, we write the contact form as $dt-z_{\mu}dx^{\mu}$, and $g$-dependent function $L_{\mathrm{matter}}$ as

[TABLE]

In the coordinate representation, the De Donder form for the present case is defined to be

[TABLE]

where $q^{\mu}$ is a function on $J^{1}(M,K)$ such that, for every local section $\phi$ of the trivial fibration $\kappa$,

[TABLE]

The total space is the fibre product $J^{3}(M,N)\times_{M}J^{1}(M,K)$ of the fibrations $\pi^{3}:J^{3}(M,N)\rightarrow M$ and $\kappa^{1}:J^{1}(M,K)\rightarrow M$. The total De Donder form $\Theta_{\mathrm{total}}$ is the pull back of the sum of the De Donder form $\Theta$ for the gravity and the De Donder form $\Theta_{\mathrm{matter}}^{g}$ of the matter to the this Whitney product. In coordinates, the total De Donder form is given by

[TABLE]

where, for every local section

[TABLE]

of the fibration $\pi\times_{M}\kappa$ from the product manifold $N\times_{M}(\mathbb{R}\times M)$ to the base manifold $M$, the coefficient functions are

[TABLE]

In equation (3.3) we did not put any pull-back signs to make the coordinate picture more transparent. Moreover, we replaced the subscript $g$ over $L_{\mathrm{matter}}^{g}$ in equation (3.13) by $y$ in order to indicate dependence of $L_{\mathrm{matter}}$ on the variable $y_{\mu\nu}=g_{\mu\nu}(x)$.

Proposition 6

For second order gravity with invariant Lagrangian $L$ on $J^{2}(M,N)$, the expressions in  equations (3.3) and (32-34) are independent of the choice of coordinates $(x^{\mu})$ on $M$. Hence, they define a global $4$-form $\Theta_{\mathrm{total}}$ on the fibre product $J^{3}(M,N)\times_{M}J^{1}(M,K).$

**Proof **of Proposition 6 is given in Section 4.

As in preceeding section, the Euler-Lagrange equations for the total Lagrangian $L_{\mathrm{total}}$ are equivalent to the De Donder equations for the total De Donder form $\Theta_{\mathrm{total}}$.

4 Proofs

4.1 Proof of Theorem 2

Recall that De Donder form corresponding to a Lagrangian form $\Lambda=Ld_{4}x$ on $J^{2}(M,N)$ is a form $\Theta$ on $J^{3}(M,N),$ defined by

[TABLE]

in the domain of the coordinate chart on $J^{3}(M,N)$ induced by a coordinate chart $(x^{\mu})$ on $M,$ where $\pi_{2}^{3\ast}L\mathrm{d}_{4}x$ is the pull-back of the Lagrangian form by the forgetful map $\pi_{2}^{3}:J^{3}(M,N)\rightarrow J^{2}(M,N)$, while $p^{\mu\nu\alpha\beta}$ and $p^{\mu\nu\alpha}$ are Ostrogradski’s momenta. In other words, $p^{\mu\nu\alpha\beta}$ and $p^{\mu\nu\alpha}$ are functions on $J^{3}(M,N)$ such that

[TABLE]

for every local section $\sigma$ of $\pi$.

Our aim in this section is to study transformation laws of components of $\Theta$ with respect to coordinate transformations in $J^{3}(M,N)$ induced by an orientation preserving coordinate transformation

[TABLE]

on $M$. It induces a local coordinate transformation on $N$ given by

[TABLE]

Further, we have the local expressions for transformations for the jet coordinates

[TABLE]

By assumption the Lagrangian form $\Lambda=Ld_{4}x$ is invariant under the transformations (37) through (40). This implies that under these transformations, $L$ transforms as a scalar density. In other words,

[TABLE]

Lemma 7

Since the boundary form

[TABLE]

is defined to be 4-form on $J^{3}(M,N),$ under the change of coordinates (37) through (40), the coefficients $p^{\mu\nu\alpha}$ and $p^{\mu\nu\alpha\beta}$ transform as follows222Since $\Xi$ depends on the third jet variables only through $p^{\mu\nu\alpha},$ we need not write explicit transformation rules for the third jets.

[TABLE]

Proof. Notice that, the boundary term $\Xi$ in (42) is the sum of two four forms, label them as $\Xi_{1}$ and $\Xi_{2}$ in a respected order. In order to deduce the transformation properties of the coefficients $p^{\mu\nu\alpha}$ and $p^{\mu\nu\alpha\beta}$, we express $\Xi_{1}$ and $\Xi_{2}$ in primed coordinates using the transformations (37) through (40) under the assumption that $\Xi_{1}$ and $\Xi_{2}$ are 4-forms on $J^{3}(M,N)$ .

Consider first the term \Xi_{1}=p^{\mu\nu\alpha\beta}(\mathrm{d}z_{\mu\nu\alpha}-z_{\mu\nu\alpha\gamma}\mathrm{d}x^{\gamma})\wedge(\partial_{\beta}{\mbox{\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt},}}\mathrm{d}_{4}x). Taking exterior differential of the coordinate transformation (39), we get

[TABLE]

Recalling some basic chain rule operations

[TABLE]

and using the transformation rule (40) of $z_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\gamma^{\prime}}$, one computes

[TABLE]

We subtract the one-form $z_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\gamma^{\prime}}\mathrm{d}x^{\gamma^{\prime}}$, exhibited in (46), from of the one-from $\mathrm{d}z_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}}$ 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

[TABLE]

On the other hand, it is immediate to see that

[TABLE]

Hence the first term in the boundary form can be obtained by first taking the exterior product of the one form $dz_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}}-z_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\gamma^{\prime}}dx^{\gamma^{\prime}}$ and the three form \partial_{\beta^{\prime}}{\mbox{\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt},}}d_{4}x^{\prime} then by multiplying the product with $p^{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\beta^{\prime}}$. This shows that $\Xi_{1}$ expressed in terms of the primed coordinates is

[TABLE]

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 $\mathrm{d}y_{\mu\nu}-z_{\mu\nu\beta}\mathrm{d}x^{\beta}$ in terms of the primed coordinates. By substituting the transformations of $y_{\mu^{\prime}\nu^{\prime}}$ in (38) and $z_{\mu^{\prime}\nu^{\prime}\beta^{\prime}}$ in (39) into this one-form, we have

[TABLE]

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 $\mathrm{d}y_{\mu^{\prime}\nu^{\prime}}-z_{\mu^{\prime}\nu^{\prime}\beta^{\prime}}\mathrm{d}x^{\beta^{\prime}}$ and \partial_{\beta^{\prime}}{\mbox{\rule{5.0pt}{0.5pt}\rule{0.5pt}{6.0pt},}}d_{4}x^{\prime}, then multiply this four form by $p^{\mu^{\prime}\nu^{\prime}\beta^{\prime}}$. We obtain the following expression for $\Xi_{2}$ in terms of primed coordinates,

[TABLE]

The sum of the four-forms in (4.1) and (49) is the boundary form in primed coordinates. Explicitly we have that

[TABLE]

Since $\Xi$ is a 4-form, its expression $\Xi^{\prime}$ in primed coordinates gives the same form as the expression in the original coordinates. That is $\Xi=\Xi^{\prime},$ which implies that

[TABLE]

This completes the proof of Lemma 7.

In Lemma 7 we showed that arbitrary smooth functions $p^{\mu\nu\alpha\beta}$ and $p^{\mu\nu\alpha}$ on $J^{3}(M,N)$ define a 4-form

[TABLE]

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 $p^{\mu\nu\alpha\beta}$ and $p^{\mu\nu\alpha}$ are defined as the Ostrogradski’s momenta, which implies equations (36) for every section $\sigma$ of $\pi:N\rightarrow M$. Note that any function $f$ on $J^{3}(M,N)$ is uniquely determined by its pull-backs $(j^{3}\sigma)^{\ast}f$ for all sections $\sigma$ of $\pi:N\rightarrow M$. Therefore we may write

[TABLE]

where $D_{\beta}p^{\mu\nu\alpha\beta}$ is the total divergence given by

[TABLE]

In order to simplify computations, we use the notation $P^{\mu\nu\alpha\beta}(x^{\gamma})=(j^{3}\sigma)^{\ast}p^{\mu\nu\alpha\beta}(x^{\gamma})$ and $P^{\mu\nu\alpha}(x^{\gamma})=(j^{3}\sigma)^{\ast}p^{\mu\nu\alpha}(x^{\gamma})$ introduced in equation (19). With this notation,

[TABLE]

where $P_{,\beta}^{\mu\nu\alpha\beta}(x^{\lambda})=\frac{\partial}{\partial x^{\beta}}P^{\mu\nu\alpha\beta}(x^{\lambda}).$

Lemma 8

If $Ld_{4}x$ is a second order Lagrangian form on $J^{2}(M,N)$, invariant under the coordinate transformations (37) through (40), Ostrogradski’s momenta given by equation (50) satisfy the transformation rules (43) and (44).

Proof. We start with the second momentum $p^{\mu\nu\alpha\beta}$ and obtain the transformation law (44) as follows,

[TABLE]

where we have used the chain rule, the prolonged coordinate transformation for $z_{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\beta^{\prime}}$ given in (40), and equation (41).

In order to show that the first momentum $p^{\mu\nu\alpha}$ satisfies transformation law (43), start first with the term $\partial L/\partial z_{\mu\nu\alpha}.$ As in equation (4.1),

[TABLE]

In order to compute the divergence term, we work with pull-backs $P^{\mu\nu\alpha}=(j^{3}\sigma)^{\ast}p^{\mu\nu\alpha}$ and $P^{\mu\nu\alpha\beta}=(j^{3}\sigma)^{\ast}p^{\mu\nu\alpha\beta}$, which allows replacing total derivative by partial derivative, see equation (52). We obtain

[TABLE]

Note that

[TABLE]

and

[TABLE]

which implies

[TABLE]

Therefore, the second term on the right hand side of equation (55) reads

[TABLE]

In the light of this, we can rewrite $P_{,\beta}^{\mu\nu\alpha\beta}$ in (55) as

[TABLE]

In order to arrive at the coordinate transformation for Ostrogradski’s momentum $P^{\mu\nu\alpha},$ we simply take the difference of $(j^{2}\sigma)^{\ast}\left(\partial L/\partial z_{\mu\nu\alpha}\right)$ in (54) and $P_{,\beta}^{\mu\nu\alpha\beta}$ in (58). So that,

[TABLE]

Here, we used notation (19) in the primed coordinates, which yields

[TABLE]

Equation (4.1) may be rewritten as

[TABLE]

Since this equation is valid for every section $\sigma$ of $\pi:N\rightarrow M$, it follows that

[TABLE]

where $p^{\mu\nu\alpha}$ and $p^{\mu^{\prime}\nu^{\prime}\alpha^{\prime}\beta^{\prime}}$ 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 $\Xi$ is globally defined and is given by a natural differential operator applied to the to the Lagrangian form $Ld_{4}x$. Since $\Theta=\pi_{2}^{3\ast}L\mathrm{d}_{4}x+\Xi$, it follows that the De Donder form $\Theta$ 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.

4.2 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 (5) in an explicit form.

Recall that the Christoffel symbols of the first kind $\Gamma_{\lambda\mu\nu}$ and the Christoffel symbols of the second kind $\Gamma_{\mu\nu}^{\rho}$ are defined and related as

[TABLE]

where $g^{\rho\lambda}$ is the dual of the metric tensor $g_{\rho\lambda}$ whereas $g_{\mu\lambda,\nu}$ denotes the partial derivative of $g_{\mu\lambda}$ with respect to $x^{\nu}$. It is possible to write the Christoffel symbols in a pure contravariant form

[TABLE]

For future reference, we define here some symbols by contacting the Christoffel symbols

[TABLE]

Taking the derivative of the identity $g^{\rho\lambda}g_{\lambda\mu}=\delta_{\mu}^{\rho}$, one arrives at the relation between $g_{,\gamma}^{\rho\lambda}$ and $\Gamma_{\lambda\mu\nu}$ as follows

[TABLE]

whereas the contraction of this yields

[TABLE]

Recall also that, the Riemann and the Ricci tensors are

[TABLE]

respectively. Here, $\Gamma_{\beta\delta,\gamma}^{\alpha}$ denotes the partial derivative of $\Gamma_{\beta\delta}^{\alpha}$ with respect to $x^{\gamma}$. In this local representation, the scalar curvature is defined to be

[TABLE]

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

[TABLE]

Notice that, the symbols $\Gamma_{\rho\mu\alpha}$ contain the first derivative $g_{\mu\nu,\lambda}$ of the metric $g_{\mu\nu}$, so that the partial derivative $\Gamma_{\delta\beta\gamma,\alpha}$ of the symbols are containing the second partial derivative $g_{\mu\nu,\lambda\gamma}$ of $g_{\mu\nu}$. 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 $\Gamma_{\rho\mu\alpha}$ into $R_{1}$, and by putting all the second order terms that is those involving $\Gamma_{\delta\beta\gamma,\alpha}$ into $R_{2}$, that is $R=R_{1}+R_{2}$ and

[TABLE]

Therefore, we write the Lagrangian as

[TABLE]

where $R_{1}$ depends linearly on $g_{\mu\nu,\alpha}$ linearly while $R_{2}$ depends quadratically on $g_{\mu\nu,\alpha\beta}$. A simplification is possible for $R_{1}$. See that,

[TABLE]

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

[TABLE]

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

[TABLE]

where the symbol $\Gamma^{\nu}$ is the one defined in (62).

Proof. First recall the definition of the Ostrogradski’s momenta

[TABLE]

By substituting the exhibition of the Hilbert Lagrangian given in (71), we can rewrite the momenta as

[TABLE]

where $R_{1}$ and $R_{2}$ as the ones in (72) and (73), respectively. It is immediate to observe that the second momenta is

[TABLE]

Notice from (76) that, in order to determine the first momenta $P^{\mu\nu\alpha}$, we need to take the divergence of the second momenta $P^{\mu\nu\alpha\beta}$, given in (77), with respect to $x^{\beta}$. For this, we start with taking the partial derivative of $\sqrt{-\det g}$ with respect to $x^{\beta}$ as follows

[TABLE]

where the symbol $\Delta_{\beta}$, in (65), has been substituted in the last line of the calculation. On the other hand, we take the divergence

[TABLE]

where, the identities (66) and (67) have been used in the second line, and the symbols $\Gamma^{\alpha}$ and $\Delta^{\alpha}$, 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 $P^{\mu\nu\alpha\beta}$ turns out to be

[TABLE]

where the identity $g^{\nu\beta}\Delta_{\beta}=\Delta^{\nu}$ has been used. Notice that all the terms involving $\Delta^{\nu}$ canceling each other in the calculation. Let us now concentrate on the first term $\partial R_{1}/\partial g_{\mu\nu,\alpha}\sqrt{-\det g}$ in the momenta $P^{\mu\nu\alpha}$, applying the chain rule, we have that

[TABLE]

Notice that, the partial derivative of $R_{1}$ with respect to the Christoffel symbol of the first kind $\Gamma_{\lambda\mu\nu}$ is computed to be

[TABLE]

whereas the partial derivative of $\Gamma_{\lambda\mu\nu}$ with respect to $g_{\alpha\beta,\gamma}$ is

[TABLE]

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

[TABLE]

Now we are ready to write the first momenta $P^{\mu\nu\alpha}$, for this simply take the difference of (84) and (81), this gives

[TABLE]

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

[TABLE]

where $\Xi_{\mathrm{Hilbert}}$ is the boundary form induced by the Hilbert Lagrangian. Explicitly, the boundary form is

[TABLE]

By substituting the conjugate momenta $P^{\mu\nu\alpha}$ and $P^{\mu\nu\alpha\beta}$, respectively given in (74) and (75), one has

[TABLE]

Substitution of the boundary form $\Xi_{\mathrm{Hilbert}}$ and the terms $R_{1}$ and $R_{2}$ in (72) and (73) leads to the following expression of the De Donder form

[TABLE]

Notice that, the first and the last terms are canceling since they are minus of the each other. So that there remain

[TABLE]

Let us simplify concentrate on the second line of this expression. A simple calculations give

[TABLE]

where we used the identity (66) in the first line, and the identity (61) in the third line, and in the fourth line we sum up the similar terms. This simplification reads that the coefficient of the basis $\sqrt{-\det g}\mathrm{d}_{4}x$ can be written as

[TABLE]

where we have canceled the first and last terms in the first line, and used the symmetry of the Christoffel symbol in the third line. Eventually, the De Donder form for Hilbert Lagrangian becomes

[TABLE]

A comparison of this form and the one in (5) shows that the proof of the Proposition 5 in Section 3.2 is achieved.

4.3 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

[TABLE]

where the coefficient function $q^{\mu}$ reads (37) for a local section. Let us first show that, $\Xi_{3}$ is invariant under a coordinate transformation on the base manifold $M$ given in (37). See that

[TABLE]

For the basis we recall the transformation in (47), and compute

[TABLE]

Collecting all these, one sees that formulation of the boundary form remains the same under coordinate transformation

[TABLE]

On the other hand, the Lagrangian term is

[TABLE]

where we have employed the fact that $\sqrt{-\det g}\mathrm{d}_{4}x$ is invariant under the coordinate transformation. Notice that, assumption of the invariance of the function $V$ leads to the invariance of $L_{\mathrm{matter}}^{g}\mathrm{d}_{4}x$.

5 Acknowledgement

The authors are grateful to Department of Mathematics and Statistics, University of Victoria, for hosting a Workshop on Geometry and Mechanics, 16 -20 July, 2018, where they began collaboration on this paper.

The visit of (OE) was supported by TÜBİTAK (the Scientific and Technological Research Council of Turkey) under the project title "Matched pairs of Lagrangian and Hamiltonian Systems" with the project number 117F426.