arXiv:2107.08846v1 [math-ph] 19 Jul 2021 e words Key ocuin n outlook and Conclusions 6 Relativity) (General field Gravitational 5 field Electromagnetic 4 symmetries gauge and laws conservation Symmetries, 3 bundles jet in theories field Lagrangian 2 Introduction 1 Contents S 00codes: 2020 MSC om,smere,guesmere,cnevto as N laws, conservation symmetries, gauge symmetries, forms, isenPltn approach. Einstein-Palatini UTSMLCI IS N SECOND AND FIRST MULTISYMPLECTIC IL THEORIES FIELD * † . h ibr-isenato ...... model) . (metric-affine action . Einstein-Palatini The . action Hilbert-Einstein 5.2 The 5.1 equivale gauge and fields vector . gauge fiel symmetries, Lagrangian . Gauge for quantities . conserved and . 3.2 Symmetries . . 3.1 theories field Lagrangian bund second-order jet and in First fields Multivector bundles. jet 2.2 Higher-order 2.1 e e - - 2 mail mail eateto ahmtc,UiesttPolit Universitat Mathematics, of Department r ple osuyNehradguesmere o h mul t particular, the approaches. in for theory; symmetries gravitational gauge the discu and and tromagnetic also Noether study are to meaning applied geometrical are its and symmetry stated gauge are of theorems formulati Noether-type multisymplectic theories. the field for classical studied are laws) vation 1 [email protected].(RI:0000-0001-8796-3149). (ORCID: [email protected]. : [email protected].(RI:0000-0003-3663-9861). (ORCID: [email protected]. : eateto hsc,UiesttAut Universitat Physics, of Department ymtisad npriua,Cra Nehr symmetrie (Noether) Cartan particular, in and, Symmetries : 1 st S and MERE N AG YMTISIN SYMMETRIES GAUGE AND YMMETRIES Primary 2 dodrLgaga edtere,Hge-re e bundl jet Higher-order theories, field Lagrangian nd-order 34,7S5 83C05; 70S05, 53D42, : 1 J : ORDI LCRMGEI N GRAVITATIONAL AND ELECTROMAGNETIC G ASET uy2,2021 20, July * , FIELDS 2 Abstract N ARCISO nm eBreoa eltra Spain. Bellaterra, Barcelona, de onoma ` Secondary ciad aauy,Breoa Spain. Barcelona, Catalunya, de ecnica ` eEnti–ibr n h Einstein–Palatini the and Einstein–Hilbert he e 3 ...... les nti emti rmwr.Teconcept The framework. geometric this in R hois...... 6 ...... theories d no rtadscn re Lagrangian order second and first of on 15 ...... OM c 9 ...... nce ehrterm ibr-isenaction, Hilbert-Einstein theorem, oether 57,3Q6 30,7H0 83C99. 70H50, 53Z05, 35Q76, 35Q75, : 4 ...... sdi hsfruain h results The formulation. this in ssed iypetcdsrpino h elec- the of description tisymplectic n osre uniis(conser- quantities conserved and s AN ´ - ORDER -R OY 13 ...... † L AGRANGIAN s Multisymplectic es, 18 13 12 6 3 2 J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.2

1 Introduction

It is well known that symmetries have enormous relevance in physical theories and, in general, in the treatment and resolution of differential equations modelling them. This is due to the fact that the presence of symmetries leads to the existence of conservation laws or conserved quantities which, in addition to helping the integration of these equations, highlight fundamental properties of physical systems. In this sense, the work of E. Noether at the beginning of the 20th century provides fundamental results on this topic [27]. From a geometric perspective, symmetries of mechanical systems and classical field theories are usually stated by demanding the invariance of the underlying geometric structures and/or the dynamical elements which characterize these systems (for a review in the case of mechanics see, for instance, [35] and the references cited therein). In the case of the multisymplectic description of field theories, these are the multisymplectic forms which are defined in the jet bundles and the multimomentum bundles where the theory is developed. These forms are constructed from the Lagrangian which describes the system, using the canonical elements which these bundles are endowed with [1, 12, 15, 22, 25, 30, 31, 39]. The study of symmetries and their associated conservation laws in this framework has been carried out in many papers (see, for instance, [11, 14, 18, 19, 38] and the references therein). One of the main characteristics of certain kinds of physical theories is the so-called gauge invariance, which is a consequence of the existence of a particular type of symmetry called gauge symmetry. This is a property which is associated with physical systems described by singular Lagrangians. Gauge symme- tries have their own geometric characterization which is related to the fact that the multisymplectic forms constructed from the (singular) Lagrangians are degenerated (and then they are called premultisymplectic forms). In this review paper, our aim is to present an accurate geometric description of symmetries in classi- cal field theories of first and second-order type. First, we introduce the standard classical symmetries of Noether type and state the Noether theorem which gives the way of obtaining the corresponding associ- ated conservation laws. Second, we discuss in detail the geometrical meaning and the characteristics of gauge symmetries; clarifying several geometric aspects that are not usually analysed in most treaties on this subject. Finally, we apply the results to the case of the two fundamental classical theories: Electro- magnetism (Maxwell theory) and Gravitation (General Relativity); in this last case, considering the two more basic models; i.e., the Einstein–Hilbert and the Einstein–Palatini approaches (the first and third are first-order field theories, but the second is second-order). The organization of the paper is as follows: Section 2 is devoted to summarize the multisymplectic Lagrangian formulation of first and second-order classical field theories. In Section 3 we study symme- tries, conservation laws and gauge symmetries in this multisymplectic setting. The analysis of symme- tries of Electromagnetism is analyzed in Section 4 and, finally, in Section 5 we study Noether and gauge symmetries of both models in General Relativity.

All the manifolds are real, second countable and C∞. The maps and the structures are C∞. Sum over repeated indices is understood. Along this paper, we use the notation of multi-indices: a multi-index I is an element of Zm where every component is positive, the ith position of the multi-index is denoted m I(i), and |I| = I(i) is the length of the multi-index. The equality |I| = k means that the expression i=1 X Zm j is taken for every multi-index of length k. Furthermore, the element 1i ∈ is defined as 1i(j) = δi . Finally, n(ij) is a combinatorial factor with n(ij) = 1 for i = j, and n(ij) = 2 for i 6= j. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.3

2 Lagrangian field theories in jet bundles

2.1 Higher-order jet bundles. Multivector fields in jet bundles

(See [13, 26, 39] for details). π Let E −→ M be a fiber bundle over an orientable m-dimensional manifold M, with dim E = m+n. The kth-order jet bundle of the projection π is the manifold of the k-jets (equivalence classes) of local k k k sections of π, φ ∈ Γ(π), and is denoted J π. Points in J π are denoted by jxφ, with x ∈ M and φ ∈ Γ(π) being a representative of the equivalence class. J kπ is endowed with the following natural projections: if 1 6 r 6 k,

k k r k k k k πr : J π −→ J π π : J π −→ E π¯ : J π −→ M k r k k jxφ 7−→ jxφ jxφ 7−→ φ(x) jxφ 7−→ x

s k k k k k k k where πr ◦ πs = πr , π0 = π , πk = IdJkπ, and π¯ = π ◦ π . We denote ω the volume form in M and all its pull-backs to every J rπ. If (xi,yα), 1 6 i 6 m, 1 6 α 6 n, are local coordinates in E adapted to the bundle structure, such that ω =dx1 ∧ . . . ∧ dxm ≡ dmx; then local coordinates in J kπ are denoted i α (x ,yI ), with 0 6 |I| 6 k. If φ ∈ Γ(π), the kth prolongation of φ to J kπ is denoted jkφ ∈ Γ(¯πk). Then, a section ψ ∈ Γ(¯πk) is holonomic if jk(πk ◦ ψ)= ψ; that is, ψ is the kth prolongation of the section φ = πk ◦ ψ ∈ Γ(π).

In an analogous way, let Φ: E → E be a diffeomorphism and and ΦM : M → M the diffeomor- phism induced on the basis; then the canonical lift of Φ to J kπ is the map jkΦ: J kπ −→ J kπ defined by k k k −1 k k (j Φ)(jx φ) := j (Φ ◦ φ ◦ ΦM )(ΦM (x)) ; for jx φ ∈ J π . Then, if Y ∈ X(E), the canonical lift of Y to J kπ is the vector field j1Y ∈ X(J kπ) whose associated local one-parameter groups of diffeomorphisms are the canonical lifts of the local one-parameter groups ∂ ∂ of diffeomorphisms of Y . In coordinates, if Y = f i + gα ; for instance, for k = 1 we have that ∂xi ∂yα

∂ ∂ ∂gα ∂f j ∂f j ∂gα ∂ j1Y = f i + gα + − yα + yβ + yβ . ∂xi ∂yα ∂xi j ∂xi i ∂yβ i ∂yβ ∂yα     i

Another special kind of vector fields are the coordinate total derivatives [34, 39]:

k ∂ α ∂ Di = i + yI+1i α , ∂x ∂yI |XI|=0 ∞ k and for f ∈ C (J π), we have that Dif = L(Di)f. An m-multivector field in J kπ is a skew-symmetric contravariant tensor field of order m in J kπ. The set of m-multivector fields in J kπ is denoted Xm(J kπ). A multivector field X ∈ Xm(J kπ) is locally k k k k decomposable if, for every jxφ ∈ J π, there is an open neighbourhood U ⊂ J π, with jx φ ∈ U, and X1,...,Xm ∈ X(U) such that X|U = X1 ∧ . . . ∧ Xm. Locally decomposable m-multivector fields are locally associated with m-dimensional distributions D ⊂ TJ kπ, and multivector fields associated with the same distribution make an equivalence class {X} in the set Xm(J kπ). Then, X is integrable if its associated distribution is integrable. In particular, X is holonomic if it is integrable and its integral sections are holonomic sections of π¯k. X Xm k k k k A multivector field ∈ (J π) is π¯ -transverse if, at every point jxφ ∈ J π, we have that k∗ m m k ( (X)(¯π β)) k 6= 0; for every β ∈ Ω (M) such that β k k 6= 0. If X ∈ X (J π) is integrable, i jx φ π¯ (jx φ) J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.4 then it is π¯k-transverse if, and only if, its integral manifolds are local sections of π¯k : J kπ → M. In this k k X case, if ψ : U ⊂ M → J π is a local section with ψ(x) = jx φ and ψ(U) is the integral manifold of k at j φ; then T k (Im ψ)= D k (X) and ψ is an integral section of X. (See [14] for more details). x jx φ jxφ m k For every X ∈ X (J π), there exist X1,...,Xr ∈ X(U) such that

X i1...im |U = f Xi1 ∧ . . . ∧ Xim , i1<...

m k−1 ∂ ∂ ∂ ∂ X = f + yα + yα + F α , (|K| = k) . (1) ∂xi i ∂yα I+1i ∂yα K,i ∂yα  i I K ^=1 |XI|=1   If Ω ∈ Ωp(J kπ) and X ∈ Xm(J kπ), the contraction between X and Ω is the natural contraction between tensor fields; in particular, it gives zero when p

i1...im i1...im i(X)Ω |U := f i(X1 ∧ . . . ∧ Xm)Ω = f i(X1) . . . i(Xm)Ω . i1<...

L(X)Ω := [d, i(X)]Ω = (d i(X) − (−1)m i(X)d)Ω . If X ∈ Xi(M) and Y ∈ Xj(J kπ), the Schouten-Nijenhuis bracket of X, Y is the bilinear assignment X, Y 7→ [X, Y], where [X, Y] is a (i + j − 1)-multivector field obtained as the graded commutator of L(X) and L(Y) (which is an operation of degree i + j − 2),

L([X, Y]) := [L(X), L(Y)] .

It is also called the Lie derivative of Y with respect to X, and is denoted as L(X)Y := [X, Y].

2.2 First and second-order Lagrangian field theories

(See [1, 12, 13, 15, 22, 34, 39] for details).

Let π : E → M be the configuration bundle of a first or second order classical field theory. For first-order field theories, we have a first-order Lagrangian density L ∈ Ωm(J 1π), which is a π1-semibasic m-form and then L = L (π1)∗η, where L ∈ C∞(J 1π) is the Lagrangian function. Using the canonical structures of the bundle J 1π, we can construct the Poincare-Cartan´ m-form associated m 1 with the Lagrangian density L, denoted by ΘL ∈ Ω (J π), whose local expression is ∂L ∂L Θ = dyα ∧ dm−1x − yα − L dmx , L ∂yα i ∂yα i i  i  J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.5

m−1 ∂ m m+1 1 (where d xi ≡ i d x), and the Poincare-Cartan´ (m+1)-form Ω := −dΘ ∈ Ω (J π). ∂xi L L 1   The couple (J π, ΩL) is a first-order Lagrangian system which is said to be regular when ΩL is 1- nondegenerate (that is, a multisymplectic form) and singular elsewhere (then ΩL is premultisymplectic). ∂2L This regularity condition is locally equivalent to det (j1φ) 6= 0, for all j1φ ∈ J 1π. α β x x ∂yi ∂yj ! If the theory is second-order and L ∈ Ωm(J 2π) is a second-order Lagrangian density, then it is a π2-semibasic m-form and L = L (π2)∗η, where L ∈ C∞(J 2π) is the Lagrangian function. As it is well-known, in this case the Lagrangian phase bundle is J 3π and natural coordinates on it adapted i α α α α to the fibration are (x , u , ui , uI , uJ ); 1 ≤ i ≤ m, 1 ≤ α ≤ n, and I, J are multiindices with m 3 |I| = 2, |J| = 3. The Poincare-Cartan´ m-form ΘL ∈ Ω (J π) for these kinds of theories can be unambiguously constructed using again the canonical structures of J kπ [1, 17, 28, 31, 34] and it is locally given by

m ∂L 1 d ∂L α m−1 α m Θ = − (dy ∧ d xi − y d x) L ∂yα n(ij) dxj ∂yα  i i j 1i+1j X=1  1 ∂L α m−1 α m m + α (dyi ∧ d xj − y1i+1j d x)+ Ld x n(ij) ∂y1i+1j i α m−1 ij α m−1 i α ij α m ≡ Lαdy ∧ d xi + Lα dyi ∧ d xj + L − Lαyi − Lα y1i+1j d x ,

i ij ∞ 3   where Lα,Lα ∈ C (J π) are m ∂L 1 ∂L Li = − D Lij ; Lij = . α ∂yα j α α n(ij) ∂yα i j 1i+1j X=1 m+1 3 3 As above, the Poincare-Cartan´ (m + 1)-form is ΩL := −dΘL ∈ Ω (J π), so (J π, ΩL) is a second-order Lagrangian system and it is regular or not depending on the 1-degeneracy of ΩL. In this ∂2L case the regularity condition is locally equivalent to det (j3φ) 6= 0, for every j3φ ∈ J 3π, β α x x ∂yI ∂yJ ! where |I| = |J| = 2. The solutions to the Lagrangian variational problem posed by a first or a second-order Lagrangian L are holonomic sections jkφ: M → J kπ, k = 1, 3, verifying that k ∗ k (j φ) i(X)ΩL = 0 , for every X ∈ X(J π) , (2) or, what is equivalent, they are the integral sections of a class of locally decomposable non-vanishing, m k holonomic multivector fields {XL}⊂ X (J π), such that

i(XL)ΩL = 0 . (3) Holonomic multivector fields are necessarily transverse to the projection π¯k (k = 1, 3) and this condition can be written as i(X)ω 6= 0 . (4) It is usual to fix this condition by taking a representative in the class {X} such that i(X)ω = 1; which implies that f = 1 in (1).

m m We establish the following notation: let ker ΩL := {X ∈ X (M) | i(X)ΩL = 0}, and let m k kerω ΩL be the set of m-multivector fields satisfying the equation (3) and the π¯ -transversality condition m (4), but being not necessarily locally decomposable. Finally, denote by kerω(I) ΩL the set of integrable m-multivector fields satisfying that they are solutions to (3) and they are integrable (and holonomic). m m m Oviously we have kerω(I) ΩL ⊂ kerω ΩL ⊂ ker ΩL. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.6

k Remark 1. • In general, if (J π, ΩL) (k = 1, 3) is a singular Lagrangian system (i.e., ΩL is a premultisymplectic form), then locally decomposable, non-vanishing, π¯k-transverse multivector fields which are solutions to the equation (3) could not exist and, in the best of cases, they exist k k only in some submanifold S : S ֒→ J π. This submanifold is necessarily π¯ -transverse, as a consequence of condition (4). Furthermore the multivector fields solutions to (3) could not be integrable necessarily (even in the regular case), but maybe in some submanifold. (An algorithmic procedure in order to find these submanifolds has been proposed in [10]).

• Notice that, even in the regular case, the equation (3) does not determine a unique class of mul- tivector fields or distributions but a multiplicity of them, since the solutions depend on arbitrary functions [13, 14]. This means that there is not an unique distribution or a class of multivector fields solution to (3) on J kπ and then, for every point in J kπ, there is a multiplicity of integral submanifolds or integral sections solution to (2) (field states) passing through it. If the Lagrangian system is singular, there is another arbitrariness which comes from the degeneracy of the form ΩL and is related to the existence of gauge symmetries, as we will see in Section 3.2.

• Finally, it is important to point out that integrable and π¯k-transverse multivector fields X solution to the equation(3) may not be necessarily holonomic, even in the regular case for the second-order case [34], although this condition holds in the first-order regular case [13].

3 Symmetries, conservation laws and gauge symmetries

3.1 Symmetries and conserved quantities for Lagrangian field theories

(See [14, 19] for the proofs of all the results in this section. See also [11, 38]).

k Let (J π, ΩL) be a Lagrangian system. Definition 1. A conserved quantity is a form ξ ∈ Ωm−1(J kπ) such that L(X)ξ := (−1)m+1 i(X)dξ = X m 0, for every ∈ kerω ΩL.

The following results characterize conserved quantities:

Theorem 1. 1. A form ξ ∈ Ωm−1(J kπ) is a conserved quantity if, and only if, L(Z)ξ = 0, for every m Z ∈ ker ΩL. m−1 k X m 2. If ξ ∈ Ω (J π) is a conserved quantity and ∈ kerω(I) ΩL, then ξ is closed on the integral X k ∗ .submanifolds of ; that is, if jS : S ֒→ J π is an integral submanifold, then djSξ = 0 Remark 2. Given ξ ∈ Ωm−1(J kπ) and X ∈ Xm(J kπ), for every integral section ψ : M → J kπ of X, ∗ ∗ m−1 there is a unique Xψ∗ξ ∈ X(M) such that i(Xψ∗ξ)η = ψ ξ. This ψ ξ ∈ Ω (M) is the so-called ∗ m−1 form of flux associated with the vector field Xψ∗ξ wich is ψ ξ ∈ Ω (M) and, if divXψ∗ξ denotes ∗ the divergence of Xψ∗ξ, we have that (divXψ∗ξ) η =dψ ξ. Then, as a consequence of Proposition 2, ξ is a conserved quantity if, and only if, divXψ∗ξ = 0, and hence, by Stokes theorem, in every bounded domain U ⊂ M, ∗ ∗ ψ ξ = (divXψ∗ξ) η = dψ ξ = 0 . Z∂U ZU ZU The form ψ∗ξ is called the current associated with the conserved quantity ξ, and this result allows to associate a conservation law in M to every conserved quantity in J kπ.

k k m m Definition 2. 1. A symmetry is a diffeomorphism Φ: J π → J π such that Φ∗(ker ΩL) ⊂ ker ΩL. If Φ= jkϕ for a diffeormorphism ϕ: E → E, the symmetry is called natural. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.7

2. An infinitesimal symmetry is a vector field Y ∈ X(M) whose local flows are local symmetries or, m m what is equivalent, such that [Y, ker ΩL] ⊂ ker ΩL. If Y = jkZ for some Z ∈ X(M), then the infinitesimal symmetry is called natural.

For infinitesimal symmetries we also have the following characterization:

k m m Theorem 2. Y ∈ X(J π) is an infinitesimal symmetry if, and only if, [Y, ker ΩL] ⊂ ker ΩL.

Observe that, if Y1,Y2 ∈ X(M) are infinitesimal symmetries, then so is [Y1,Y2]. Furthermore, if Y ∈ X(M) is an infinitesimal symmetry then, for every Z ∈ ker ΩL, Y + Z is also an infinitesimal symmetry. The Lagrangian field equations are EDP’s and symmetries transform solutions into solutions. In fact:

Theorem 3. Let Φ ∈ Diff(J kπ) be a symmetry. Then:

m 1. For every integrable multivector field X ∈ ker ΩL, the map Φ transforms integral submanifolds of X into integral submanifolds of Φ∗X. 2. In the particular case that Φ ∈ Diff(J kπ) restricts to a diffeormorphism ϕ: M → M (which k k X m means that ϕ ◦ π¯ =π ¯ ◦ Φ); then, for every ∈ kerω(I) ΩL, the map Φ transforms integral X X X m submanifolds of into integral submanifolds of Φ∗ , and hence Φ∗ ∈ kerω(I) ΩL.

As a straightforward consequence of this, we obtain that:

k Theorem 4. Let Y ∈ X(J π) be an infinitesimal symmetry and Ft the local flow of Y . Then:

m 1. For every integrable multivector field X ∈ ker ΩL, the map Ft transforms integral submanifolds of X into integral submanifolds of Ft∗X. 2. In the particular case that Y ∈ X(J kπ) is π¯k-projectable (which means that there exists Z ∈ X k X m (M) such that the local flows of Z and Y are π¯ -related); then, for every ∈ kerω(I) ΩL, Ft transforms integral submanifolds of X into integral submanifolds of Ft∗X, and hence Ft∗X ∈ m kerω(I) ΩL.

If Φ ∈ Diff(J kπ) is a symmetry and ξ ∈ Ωm−1(J kπ) is a conserved quantity, then Φ∗ξ is also a conserved quantity. In the same way, if Y ∈ X(J kπ) is an infinitesimal symmetry and ξ ∈ Ωm−1(J kπ) is a conserved quantity, then L(Y )ξ is also a conserved quantity. k The most relevant kinds of symmetries a Lagrangian system (J π, ΩL) are the following: Definition 3. 1. A Cartan or Noether symmetry is a diffeomorphism Φ: J kπ → J kπ such that, ∗ ∗ Φ ΩL = ΩL. If, in addition, Φ ΘL = ΘL, then Φ is said to be an exact Cartan or Noether symmetry. If Φ= jkϕ for a diffeormorphism ϕ: E → E, the Cartan symmetry is called natural.

2. An infinitesimal Cartan or Noether symmetry is a vector field Y ∈ X(J kπ) satisfying that L(Y )ΩL = 0. If, in addition, L(Y )ΘL = 0, then Y is said to be an infinitesimal exact Car- tan or Noether symmetry. If Y = jkZ for some Z ∈ X(E), then the infinitesimal Cartan symmetry is called natural. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.8

k Obviously, if Y1,Y2 ∈ X(J π) are infinitesimal Cartan or Noether symmetries, then so is [Y1,Y2]. Now, if ψ : M → J kπ (k = 1, 3) is a solution to the equation (2) and Φ ∈ Diff(J kπ) is a Cartan or Noether symmetry, then, for every X ∈ X(J kπ), we have (see Theorem 3)

∗ ∗ ∗ ∗ −1 ∗ ∗ ′ (Φ ◦ ψ) i(X)ΩL = ψ Φ i(X)ΩL = ψ i(Φ∗ X)(Φ ΩL)= ψ i(X )ΩL = 0 , (5) ′ −1 X k ∗ ∗ since X = Φ∗ X ∈ (J π), Φ ΩL = ΩL, and Φ ΩL = 0. Therefore Φ ◦ ψ is also a solution to (2); thus Φ transforms solutions into solutions and then it is a symmetry. Furthermore, if ψ = jkφ: M → J kπ (k = 1, 3) is a holonomic solution to the equation (2) and Φ= jkϕ ∈ Diff(J kπ) is a natural Cartan or Noether symmetry, then (5) reads

k ∗ k ∗ k ∗ (j (ϕ ◦ φ)) i(X)ΩL = (j φ) (j ϕ) i(X)ΩL k ∗ k −1 k ∗ k ∗ ′ = (j φ) i((j ϕ)∗ X)((j ϕ) ΩL) = (j φ) i(X )ΩL = 0 , (6) and therefore jk(ϕ ◦ φ) is also a holonomic solution to (2). Thus Φ = jkϕ transforms holonomic solutions into holonomic solutions.

In addition, if Y ∈ X(M) is an infinitesimal (natural) Cartan or Noether symmetry, by definition, its local flows are local (natural) Cartan or Noether symmetries. In this way we have proved that: Proposition 1. Every Cartan or Noether symmetry is a symmetry and, as a consequence, every infinites- imal Cartan or Noether symmetry is an infinitesimal symmetry. Furthermore, every natural (infinitesimal) Cartan symmetry transforms holonomic solutions to the field equations into holonomic solutions.

k The condition L(Y )ΩL = 0 is equivalent to demanding that i(Y )Ω is a closed m-form in J π. Thus, an infinitesimal Cartan or Noether symmetry is a locally Hamiltonian vector field for the multisymplectic form ΩL, and ξY is the corresponding local Hamiltonian form, (in an open neighbourhood of every point in J kπ). Therefore, Noether’s theorem is stated as follows: k Theorem 5. (Noether): Let Y ∈ X(J π) be an infinitesimal Cartan or Noether symmetry with i(Y )ΩL = k X m X m dξY in an open set U ⊂ J π. Then, for every ∈ kerω ΩL (and hence for every ∈ kerω(I) ΩL), we have L(X)ξY = 0 ; that is, any Hamiltonian (m − 1)-form ξY associated with Y is a conserved quantity. Then, in this ∗ context, for every integral submanifold ψ of X, the form ψ ξY is usually called a Noether current.

Observe that the form L(Y )ΘL is closed since

L(Y )ΘL =d i(Y )ΘL + i(Y )dΘL =d i(Y )ΘL − i(Y )ΩL = d(i(Y )ΘL − ξY ) ≡ dζY (in U) .

In particular, if Y is an exact infinitesimal Cartan or Noether symmetry, we can take ξY = i(Y )ΘL. It is well known that canonical liftings of diffeomorphisms and vector fields preserve the canonical k structures of J π. Nevertheless, the (pre)multisymplectic form ΩL is not canonical, since it depends on the choice of the Lagrangian density L, and then it is not invariant by these canonical liftings. Thus, given a diffeomorphism Φ: J kπ → J kπ or a vector field Y ∈ X(J kπ), a sufficient condition to assure this invariance would be to demand that they leave the canonical structures of the jet bundle J kπ (for instance, Φ and Y being the canonical lifting of a diffeomorphism and a vector field in E), and that the Lagrangian density L be also invariant. In this way, ΩL and hence the Euler-Lagrange equations are invariant by Φ or Y . This leads to define the following kind of symmetries: J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.9

Definition 4. 1. A Lagrangian symmetry of the Lagrangian system is a diffeomorphism Φ: J kπ → J kπ such that:

(a) Φ leaves the canonical geometric structures of J kπ invariant. (b) Φ∗L = L (Φ leaves L invariant).

If Φ = jkϕ, for some diffeomorphism ϕ: E → E, then condition (a) holds and the Lagrangian symmetry is called natural.

2. An infinitesimal Lagrangian symmetry is a vector field Y ∈ X(J kπ) such that:

(a) The canonical geometric structures of J kπ are invariant under the action of Y . (b) L(Y )L = 0 (Y leaves L invariant). If Φ = jkϕ, for some diffeomorphism ϕ: E → E, then condition (a) holds and the infinitesimal Lagrangian symmetry is called natural.

As a direct consequence of these definitions we have:

k ∗ Proposition 2. 1. If Φ ∈ Diff(J π) is a Lagrangian symmetry, then Φ ΘL =ΘL, and hence it is an exact Cartan symmetry.

k 2. If Y ∈ X(J π) is an infinitesimal Lagrangian symmetry, then L(Y )ΘL = 0, and hence it is an infinitesimal exact Cartan symmetry.

To demand the invariance of L is really a strong condition, since there are Lagrangian densities or, what is equivalent, Lagrangian functions that, being different and even of different order, give rise to the same Euler-Lagrange equations. These are the so-called gauge equivalent Lagrangians.

3.2 Gauge symmetries, gauge vector fields and gauge equivalence

The term gauge is used in Physics to refer to different situations in relation to certain kinds of symme- tries which do not change physically the system, and this characteristic is known as gauge freedom. For instance, sometimes it is used to refer to the invariance of a Lagrangian system when is described by different Lagrangian functions which lead to the same Euler-Lagrange equations (really these are the La- grangian symmetries introduced in Definition 4). Nevertheless, the standard use of the concept “gauge” is for describing symmetries related to the non-regularity of the Lagrangian and lead to the existence of states that are physically equivalent. From a geometric point of view, these kinds of symmetries are closely related with the degeneracy of the Poincar´e–Cartan forms associated with the Lagrangians. In this section we introduce and analyze the geometric concept of these gauge symmetries for Lagrangian field theories. This discussion is inspired in the geometric treatment made mainly in [2, 24] about gauge vector fields and gauge equivalent states for non-regular dynamical systems.

k Consider a singular Lagrangian system (J π, ΩL) and assume that equations (3) have solutions on k k k a π¯ -transverse submanifold jS : S ֒→ J π (it could be S = J π). As we have said, the existence of gauge symmetries or gauge freedom which we are interested here is closely related with the fact that the Lagrangian theory is non-regular; that is, the Poincar´e-Cartan form ΩL is 1-degenerated and then it is a premultisymplectic form. As a consequence, besides to the non unicity of solutions which is characteristic of classical field theories, there is another family of additional solutions which is associated with this degeneracy. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.10

The local generators of gauge symmetries are called gauge vector fields. In order to do a more accurate geometric definition, let X(S) ⊂ X(J kπ) be the set of vector fields in J kπ which are tangent to the submanifold S, and let X ∗ ker ΩL := {Z ∈ (S) | jS i(Z)ΩL = 0} ,

S S or, what is equivalent, if Z ∈ X(S) is such that jS∗Z = Z|S , for every Z ∈ X(S); then

∗ S ∗ 0=jS i(Z)ΩL = i(Z )(jS ΩL) ,

S V (¯πk) k k k and hence Z ∈ ker ΩL. Denote by X (J π) the set of π¯ -vertical vector fields in J π and let V (¯πk) V (¯πk) k ker ΩL = ker ΩL ∩ X (J π). Finally consider the set

V (¯πk) G = ker ΩL ∩ X(S)

k (the π¯ -vertical vector fields of ker ΩL which are tangent to S). Therefore, gauge vector fields must have the following properties:

• As physical states are sections of the projection π¯k with image on S ⊆ J kπ, gauge vector fields must be πk-vertical. In this way, we assure that the base manifold M does not contain gauge equivalent points and then all the gauge degrees of freedom are in the fibres of J kπ and, therefore, after removing the gauge redundancy, the base manifold M remains unchanged.

• As the flux of gauge vector fields connect equivalent physical states, they must be tangent to S; that is, elements of X(S).

• As the existence of gauge symmetries is a consequence of the non-regularity of the Lagrangian L (and conversely); gauge vector fields are necessarily related with the premultisymplectic character of the Poincar´e-Cartan form ΩL. Hence, they should be elements of the set ker ΩL. The flux of these vector fields transforms solutions to the Lagrangian field equations into solutions but, in principle, without preserving the holonomy necessarily.

• It is also usual to demand that physical symmetries are natural. This means that they are canonical liftings to the bundle of phase states of symmetries in the configuration space E; that is, canon- ical lifts to J kπ of vector fields in E. Then, by Proposition 1, this condition assures that gauge symmetries transform holonomic solutions to the field equations into holonomic solutions.

These ideas lead to state the following definitions:

Definition 5. The elements Z ∈ G are called gauge vector fields or (infinitesimal) gauge symmetries k of the singular Lagrangian system (J π, ΩL). If Z is the canonical lift of a vector field in E, then it is called a natural gauge vector field.

As G ⊂ ker ΩL, for every Z ∈ G we have that i(Z)ΩL = 0, then L(Z)ΩL = 0. Therefore every gauge vector field is an infinitesimal Cartan symmetry and then a symmetry (Proposition 1), and hence, if it is also a holonomic vector field, it transforms holonomic solutions to the field equations into holonomic solutions (see (6)). Incidentally, any closed m−1-forms can be thought as an associated local Hamiltonian form to any gauge vector field. Furthermore, gauge vector fields are trivially π¯k-projectable (to the null vector field on M); therefore the item 2 in Theorem 4 holds.

Observe that, for every Z1,Z2 ∈ G we have that [Z1,Z2] ∈ G, and hence G generates an involutive distribution. Then, we construct the quotient set S = S/G. Definition 6. • Points of which are in the same class in are said to be gauge equivalent points. S e S e J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.11

• Two sections ψ1, ψ2 : M → S are gauge equivalent if ψ1(x) is gauge equivalent to ψ2(x) for any x ∈ M.

• If ψ1, ψ2 are gauge equivalent and solutions to the field equations (2), they are also called gauge equivalent field states.

Therefore, the following statement is assumed: Statement: (Gauge principle). Gauge equivalent field states are physically equivalent or, what means the same thing, they represent the same physical state of the field. As a consequence of the gauge principle, we can choose any representative in each gauge equivalence class of sections to represent a physical state. This is known as a gauge fixing and the freedom in choosing the representative is referred as the gauge freedom of the theory. Notice that a holonomic section could be gauge equivalent to a non-holonomic one. It may happen that that there is only one holonomic representative in a class. When, as a consequence of the degeneracy of the Lagrangian, a Lagrangian system has gauge sym- metries, a relevant problem consists in removing the unphysical redundant information introduced by the existence of gauge equivalent states. This can be done by implementing the well-known procedure of re- duction by symmetries which rules as follows: since G generates an involutive distribution, we construct the quotient set S = S/G, which is assumed to be a differentiable manifold which is made of the true physical degrees of freedom. In addition, πS is a fiber bundle over M, and the ‘real’ physical states are the sections of thee projection π˜S : S → M. We have the diagram e S / e S ◆ / J kπ . ◆◆◆ ◆◆ πS τ˜S ◆◆ k ◆◆◆ π¯  ◆◆  π˜S ◆' S / M This is what is known as the gauge reduction procedure for removing the (unphysical) gauge degrees e of freedom of the theory. An alternative way to remove the gauge freedom consists in taking a (local) section of the projection τ˜S : S → S, and this is what is called a gauge fixing. Remark 3. To ensure that the base manifold does not contain gauge equivalent points (that is, that e M all the gauge degrees of freedom are in the fibers of J kπ), we have demanded that gauge vector fields are k only the π¯ -vertical vector fields of ker ΩL, and not all the elements of this set. In this way, after doing this reduction procedure or a gauge fixing in order to remove the gauge redundancy, the base manifold M remains unchanged.

As it has been commented, the existence of gauge symmetries and of gauge freedom is a consequence of the non-regularity of the Lagrangian L (and conversely); and then it is related with the premultisym- plectic character of the form ΩL. Then, it is reasonable to think that the gauge reduction procedure, which removes the (unphysical) gauge degrees of freedom, must remove also the degeneracy of the form. In S ∗ order to analyze this question, we have to consider the form ΩL = SΩL. Then we denote S X ker ΩL := {Z ∈ (S) |∃Z ∈ ker ΩL | S∗Z = Z|S } .

S It is obvious that G1 ⊆ ker ΩL since, for every Z ∈ G1 ⊂ ker ΩL,

∗ S S S i(Z)ΩL =0 =⇒ 0= S i(Z)ΩL = i(ZS )ΩL =⇒ ZS ∈ ker ΩL ⇐⇒ Z ∈ ker ΩL .

S S (Observe that, if ΩL is nondegenerate (multisymplectic), then ker ΩL = {0}, which implies that ker ΩL∩ X S (S) = {0} and then G1 = {0}). As ΩL is a closed form, we have also that L(Z)ΩL = 0, for every J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.12

S m Z ∈ G1; then it is τ˜S-projectable to a form ΩL ∈ Ω (S). Therefore, to say that the gauge reduction procedure consisting in making the quotient of S by G1 removes the degeneracy is equivalent to say S S that ΩL is a multisymplectic form, and this happense if, ande only if, G1 = ker ΩL. In general, this last condition does not hold and, in this case, if we want that the gauge reduction removes the degeneracy, we neede to enlarge the set of admisible gauge vector fields. So, by similarity with the case of presmplectic mechanics [2, 24], we can define: S XV (¯πk) k Definition 7. G = ker ΩL ∩ (J π) is the complete set of gauge vector fields for the singular k Lagrangian system (J π, ΩL). Then, the elements Z ∈ G1 are called primary gauge vector fields and those Z ∈ G − G1 are called secondary gauge vector fields.

k S As above we have forced gauge vector fields to be π¯ -vertical. This means that, unless G = ker ΩL, the gauge reduction procedure do not remove entirely the degeneracy of the form ΩL.

k Remark 4. In the particular situation where (J π, ΩL) is a singular Lagrangian system such that the k V (¯πk) equations (3) have solutions on J π, then the gauge vector fields are all the elements of ker ΩL, V (¯πk) V (¯πk) V (¯πk) since [Z1,Z2] ∈ ker ΩL , for every Z1,Z2 ∈ ker ΩL . Then, when ker ΩL = ker ΩL, the reduction procedure removes both the unphysical degrees of freedom and the degeneracy of the V (¯πk) k premultisymplectic structure. If ker ΩL ⊂ ker ΩL, the vector fields of ker ΩL which are not π¯ - vertical would not be gauge vector fields, but just (infinitesimal) Noether symmetries.

4 Electromagnetic field

Given the G-principle bundle P → M, with M a 4-dimensional manifold and group G = U(1), consider α the connection bundle π : C → M, with local adapted coordinates (x , Aα), where α = 0,..., 3. The 1 α induced coordinates in the associated first-order jet bundle J π are (x , Aα, Aα,β), where Aα denote the components of the potential of the electromagnetic field A. The Maxwell Lagrangian at vacuum is (taking the magnetic constant µ0 = 1) 1 1 1 L = F αβF = (Aβ,α − Aα,β)(A − A )= (ηαν ηβµ − ηαµηβν )A A ; 4 αβ 4 β,α α,β 2 α,β µ,ν ηαβ are the components of the inverse of the Minkowski metric. The associated Poincar´e-Cartan form associated with the Lagrangian density L = L d4x is

αµ βν 4 αν βµ αµ βν 3 ΩL = η η FαβdAµ,ν ∧ d x − (η η − η η )dAµ,ν ∧ dAα ∧ d xβ .

A general locally decomposable multivector field has the local expression

3 ∂ ∂ ∂ X = + f + G , ∂xγ α,γ ∂A αβ,γ ∂A γ α α,β ^=0   and the field equation (3) reads as

αν βµ αµ βν αν βµ αµ βν µβ βµ (η η − η η )(Aα,β − fαβ) = 0 , (η η − η η )Gµν,β = Gµ − Gµ = 0 .

The first group of equations implies fαβ = Aα,β + Tαβ, where Tαβ − Tβα = 0. When we impose holonomy we deduce Tαβ = 0. The second group of equations, for the integral sections of X, leads to Maxwell’s equations ∂2Aβ ∂2Aµ µ − µ = 0 ; ∂x ∂xµ ∂x ∂xβ J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.13 and there are no constraints, so S = J 1π. The gauge vector fields are

∂ G = S | S ∈ C∞(J 1π) , such that S − S = 0 . αβ ∂A αβ αβ βα  α,β 

µ µ ′ ′ Two sections are gauge equivalent if (x , Aα(x), Aα,β (x)) = (x , Aα(x), Aα,β (x)+ sαβ(x)), for some set of functions sαβ(x) which are symmetric by the interchange of α and β. In particular, Aα(x)= ′ Aα(x); therefore, if both sections are holonomic, sαβ(x) = 0. In other words, there is only one holo- nomic section in every gauge equivalent class.

µ This result may confront the well know physical result: for every section Bα(x ) solution to the field ∂f(xµ) equation (2), we can find another solution by the transformation B′ (xµ)= B (xµ)+ , for any α α ∂xα function f. This induces the transformation

Ψ : J 1π → J 1π ∂f ∂2f (xµ, A , A ) 7→ xµ, A + , A + . α α,β α ∂xα α,β ∂xα∂xβ   This transformation is actually a Lagrangian symmetry.

5 Gravitational field (General Relativity)

The multisymplectic approach to the Einstein–Hilbert and the Einstein–Palatini models of General rela- tivity has been done, for instance, in [7, 4, 5, 20, 21, 23, 40] (see also the references therein).

5.1 The Hilbert-Einstein action

Fist we consider the Hilbert Lagrangian for the Einstein equations of gravity without sources (no matter- energy is present).

The configuration bundle for the system is π : E → M, where M is a connected 4-dimensional mani- fold representing space-time and E is the manifold of Lorentzian metrics on M; that is, for every x ∈ M, −1 the fiber π (x) is the set of metrics acting on TxM, with signature (1, 3) (i.e.; (− +++)). Local coor- µ dinates in E are denoted (x , gαβ ), with 0 ≤ α ≤ β ≤ 3. As g is symmetric, gαβ = gβα, actually there are 10 independent variables and, hence, the dimension of the fibers is 10 and dim E = 14. (The fact that g is a Lorentz metric is not explicitly shown and is included requiring that the Lagrangian is invariant 3 µ under Lorentz transformations). The induced coordinates in J π are (x , gαβ, gαβ,µ, gαβ,µν , gαβ,µνρ). Using these coordinates, the local expression of the Hilbert-Einstein Lagrangian is

αβ αβ LEH = |det(g)| R = |det(g)| g Rαβ ≡ ̺g Rαβ = ̺R ,

p αβ p γ γ γ δ where ̺ ≡ |det(gαβ )|, R = g Rαβ is the scalar curvature, Rαβ = DγΓαβ − DαΓγβ +ΓαβΓδγ − 1 ∂g ∂g ∂g 1 Γγ Γδ arep the components of the Ricci tensor, Γρ = gρλ νλ + λµ − µν = gρλ(g + δβ αγ µν 2 ∂xµ ∂xν ∂xλ 2 νλ,µ  αβ  gλµ,ν −gµν,λ) are the Christoffel symbols of the Levi-Civita connection of g, and g denotes the inverse αβ α matrix of g, namely: g gβγ = δγ . It is useful to consider the following decomposition [7, 36]:

αβ,µν LEH = L gαβ,µν + L0 , αX≤β J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.14 where 1 ∂L n(αβ) Lαβ,µν = = ̺(gαµgβν + gαν gβµ − 2gαβgµν ) , n(µν) ∂gαβ,µν 2 αβ γδ µ µ δ γ δ γ L0 = ̺g {g (gδµ,βΓαγ − gδµ,γ Γαβ)+ΓαβΓγδ − ΓαγΓβδ} .

αβ,µν ∞ The key point on this decomposition is that L and L0 project onto functions of C (E) and C∞(J 1π), respectively.

The Poincar´e-Cartan 3-form ΘLEH associated with the Hilbert-Einstein Lagrangian density LEH = 3 ∗ 4 LEH (π ) η = LEH d x is

4 αβ,µ m−1 αβ,µν m−1 ΘLEH = −H d x + L dgαβ ∧ d xµ + L dgαβ,µ ∧ d xν ; αX≤β αX≤β where 3 ∂L 1 ∂L Lαβ,µ = − D ∂g n(µν) ν ∂g αβ,µ ν=0 αβ,µν X   αβ,µ αβ,I H = L gαβ,µ + L gαβ,I − L . αX≤β αX≤β αX≤β

Finally, the corresponding Poincar´e-Cartan 4-form is ΩLEH = −dΘLEH . X As ΩL is a premultisymplectic form, the field equations i( )ΩLEH = 0 have no solution everywhere [in J 3π, but in a final constraint submanifold S ֒→ J 3π which is locally defined by the constraints [20 1 Lαβ := −̺n(αβ)(Rαβ − gαβR) = 0 . 2 1 D Lαβ = D (−̺n(αβ)(Rαβ − gαβR)) = 0 . τ τ 2 In particular,

3 ∂ ∂ ∂ X = + g + g + L ∂xτ αβ,τ ∂g αβ,µτ ∂g τ αβ αβ,µ ^=0 αX≤β µ≤Xν≤λ  ∂ λ σ λ σ ∂ gαβ,µντ + Dτ Dλ(gλσ(ΓναΓµβ +ΓνβΓµα)) ∂gαβ,µν ∂gαβ,µνλ  is a holonomic multivector field solution to the equation in S and tangent to S. Then, their integral µ sections ψ(x) = (x , gαβ(x), gαβ,µ(x), gαβ,µν(x), gαβ,µνλ(x)) are the solutions to the equations ∂g g − αβ = 0 (holonomy conditions) , αβ,µ ∂xµ 1 ∂g ∂g g − αβ,µ + αβ,ν = 0 (holonomy conditions) , αβ,µν n(µν) ∂xν ∂xµ  1  ̺n(αβ)(Rαβ − gαβR) = 0 (Einstein equations) . 2

3 Regarding the gauge vector fields, notice that, as ΩLEH is π1-projectable [7, 29, 30, 36, 37], the 3 3 π1-vertical vector fields in J π are gauge symmetries. It can be show that they are the only ones [20]. In particular, there only exists one holonomic section in each gauge class. We can analyze the Cartan or Noether symmetries for this system. First, we need to state some previous concepts [29, 32, 33]. Remember that π : E → M is a bundle of metrics and hence, if p ≡ (x, gx) ∈ E, then x ∈ M and gx is a Lorentzian metric. Then: J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.15

Definition 8. 1. Let F : M → M be a diffeomorphism. The canonical lift of F to the bundle of metrics E is the diffeomorphism F : E → E defined as follows: for every (x, gx) ∈ E, then −1 ∗ F(x, gx) := (F (x), (F ) (gx)). (Thus π ◦ F = F ◦ π). The canonical lift of F to the jet bundle J kπ is the diffeomorphism jkF : J kπ → J kπ defined as k k k k −1 follows: for every jx φ ∈ J π, then F(jx φ) := j (F ◦ φ ◦ F )(x). 2. Let Z ∈ X(M). The canonical lift of Z to the bundle of metrics E is the vector field Y ∈ X(E) whose associated local one-parameter groups of diffeomorphisms Ft are the canonical lifts to the bundle of metrics E of the local one-parameter groups of diffeomorphisms Ft of Z. The canonical lift of Y ∈ X(E) to the jet bundle J kπ is the vector field Y k ≡ jkY ∈ X(J kπ) 1 whose associated local one-parameter groups of diffeomorphisms are the canonical lifts j Ft of the local one-parameter groups of diffeomorphisms Ft of Y .

Observe that the canonical lifts Y ∈ X(E) of vector fields Z ∈ X(M) to the bundle of metrics E are π-projectable vector fields and that Y k ∈ X(J kπ) are πk and π¯k projectable vector fields. ∂ In natural coordinates, if Z = uµ(x) ∈ X(M), then the canonical lift of Z to the bundle of ∂xµ metrics, Y ∈ X(E), is given by

µ µ µ ∂ ∂u ∂u ∂ Y = u µ − α gµβ + β gµα , ∂x ∂x ∂x ∂gαβ α β X≤   and then we can lift this vector field Y to the higher-order jet bundles J kπ; for instance,

1 1 µ ∂ ∂ ∂ Y = j Y = u µ + Yαβ + Yαβµ ∂x ∂gαβ ∂gαβ,µ αX≤β αX≤β ∂ ∂uµ ∂uµ ∂ = uµ − g + g ∂xµ ∂xα µβ ∂xβ µα ∂g α β αβ X≤   ∂2uν ∂2uν ∂uν ∂uν ∂uν ∂ − α µ gνβ + β µ gαν + α gνβ,µ + β gαν,µ + µ gαβ,ν . ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂gαβ,µ α β X≤   Every Z ∈ X(M) is an infinitessimal generator of diffeomorphisms in M. Then, if Y 3 = j3Y , we 3 3 have that L(Y )LEH = 0, because LEH is invariant under diffeomorphisms. Furthermore, as Y is a canonical lift, it is an infinitesimal Lagrangian symmetry and thus, by Proposition 2, Y 3 it is an exact 3 3 infinitesimal Cartan symmetry. The conserved quantity associated to Y is ξY = i(Y )ΘLEH and, as 3 ΘLEH is a π1-basic form, we have that

3 1 µ αβ,µ αβ,νµ 3 ξY = i(Y )ΘLEH = i(Y )ΘLEH = u H + YαβL + Yαβν L d xµ

ν αβ,µ µ αβ,ν  2 ν αβ,λµ µ αβ,λν 2 + u L − u L dgαβ ∧ d xµν + u L − u L dgαβ,λ ∧ d xµν ,     2 ∂ ∂ 4 where d xµν = i i d x. The vector fields of the form Y are the only natural infinitesi- ∂xν ∂xµ mal Lagrangian symmetries   [29, 37].

5.2 The Einstein-Palatini action (metric-affine model)

Now we consider the Einstein-Palatini (or metric-affine) action for the Einstein equations without sources. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.16

The configuration bundle for this system is π : E → M, where M is a connected 4-dimensional manifold representing space-time and E is Σ×M C(LM), where Σ is the manifold of Lorentzian metrics on M and C(LM) is the bundle of connections on M; that is, linear connections in TM. We use the µ α local coordinates in E (x , gαβ, Γβγ), with 0 ≤ α ≤ β ≤ 3. We do not assume torsionless connections; α α 1 µ α α thus, in general, Γβγ 6=Γγβ. The induced coordinates in J π are (x , gαβ, gαβ,µ, Γβγ , Γβγ,µ). Using this coordinates, the local expression of the Palatini-Einstein Lagrangian is

αβ αβ LPE = |det(g)| g Rαβ ≡ ̺g Rαβ p γ γ γ σ γ σ where, as above, ̺ ≡ |det(gαβ )|, Rαβ =Γβα,γ − Γγα,β +ΓβαΓσγ − ΓβσΓγα are the components of the Ricci tensor, which now depend only on the connection. We consider the auxiliary functions p βγ,µ ∂LPE 1 µ βγ β µγ Lα := α = ̺ (δαg − δαg ) , ∂Γβγ,µ 2 βγ,µ α αβ γ σ γ σ H := Lα Γβγ,µ − LPE = ̺g ΓβσΓγα − ΓβαΓσγ .   Then, the Poincar´e-Cartan 4-form ΩLEP associated with the Einstein-Palatini Lagrangian density LEP = 2 ∗ 4 LEP (π ) ω = LEp d x is

4 βγ,µ α m−1 ΩLEP =dH ∧ d x − dLα ∧ dΓβγ ∧ d xµ ;

X 3 As above, the field equations i( )ΩLEP = 0 have no solution everywhere in J π. Now, the premul- tisymplectic constraint algorithm leads to the constraints (see [21]):

βγ,σ ∂H ∂Lα α µν 0 = − Γβγ,σ ≡ c , ∂gµν ∂gµν 2 0 = g − g Γλ − g Γλ − g T λ ≡ m , ρσ,µ σλ µρ ρλ µσ 3 ρσ λµ ρσ,µ 1 1 0 = T α − δαT µ + δαT µ ≡ tα , βγ 3 β µγ 3 γ µβ βγ 1 1 0 = T α − δαT µ + δαT µ ≡ rα , βγ,ν 3 β µγ,ν 3 γ µβ,ν βγ,ν 2 0 = g Γγ Γλ + g Γγ Γλ + g Γλ + g Γλ + g T λ ≡ i . ργ [νλ µ]σ σγ [νλ µ]ρ ρλ [µσ,ν] σλ [µρ,ν] 3 ρσ λ[µ,ν] ρσ,µν α α α 1 where Tβγ ≡ Γβγ − Γγβ. They define the submanifold S : S ֒→ J π, where there are holonomic multivector fields solution to the field equations in S and tangent to S. The holonomic multivector fields which are solutions to the field equations on S are

3 ∂ ∂ ∂ α ∂ α ∂ X = + gρσ,ν + fρσµ,ν +Γ + f . ∂xν ∂g ∂g βγ,ν ∂Γα βγµ,ν ∂Γα  ν ρ σ ρσ ρσ,µ βγ βγ,µ ^=0 X≤     As the Einstein-Palatini action only has solutions on S, to get the gauge fields of the theory is more laborious. After some computation it can be shown that

V (¯π1) α ∂ α ∂ ∂ ∂ ker ΩL = δγ α , Kβγ α , , α , * ∂Γβγ ∂Γβγ ∂gαβ,µ ∂Γβγ,µ +

ν ν ν α ∂ where Kνγ = 0 and Kβγ + Kγβ = 0. The vector fields Kβγ α are not tangent to S, thus they are not ∂Γβγ α ∂ gauge vector fields. The vectors field of the form δγ α are tangent to S and therefore they are gauge ∂Γβγ J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.17

∂ ∂ vector fields. The other vector fields, , α , arise from the projectability of the theory. Only ∂gαβ,µ ∂Γβγ,µ α ∂ one representative is holonomic and consistent with a gauge fixing for δγ α . ∂Γβγ Finally, it is relevant to point out that a gauge fixing in the Einstein-Palatini model leads to recover the Einstein-Hilbert model (see [9, 21]).

Concerning to Noether symmetries, the Einstein-Palatini Lagrangian LEP is invariant under diffeo- morphisms in M (as it can be checked using the constraints cµν ). Then we have to consider the canonical ∂ lift of vector fields Z = f µ(x) ∈ X(M) to the bundle E → M, which is now ∂xµ λ λ µ ∂ ∂f ∂f ∂ YZ = f µ − α gλβ + β gλα ∂x ∂x ∂x ∂gαβ αX≤β   ∂f α ∂f λ ∂f λ ∂2f α ∂ + Γλ − Γα − Γα − ∈ X(E) . ∂xλ βγ ∂xβ λγ ∂xγ βλ ∂xβ∂xγ ∂Γα   βγ and then λ λ 1 µ ∂ ∂f ∂f ∂ j YZ = f µ − α gλβ + β gλα − ∂x ∂x ∂x ∂gαβ α β X≤   ∂2f ν ∂2f ν ∂f ν ∂f ν ∂f ν ∂ α µ gνβ + β µ gαν + α gνβ,µ + β gαν,µ + µ gαβ,ν + ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂gαβ,µ αX≤β   ∂f α ∂f λ ∂f λ ∂2f α ∂ Γλ − Γα − Γα − + ∂xλ βγ ∂xβ λγ ∂xγ βλ ∂xβ∂xγ ∂Γα   βγ ∂f α ∂f λ ∂f λ ∂f λ Γλ − Γα − Γα − Γα + ∂xλ βγ,µ ∂xβ λγ,µ ∂xγ βλ,µ ∂xµ βγ,λ  ∂2f α ∂2f λ ∂2f λ ∂3f α ∂ Γλ − Γα − Γα − ∂xλ∂xµ βγ ∂xβ∂xµ λγ ∂xγ∂xµ βλ ∂xβ∂xγ∂xµ ∂Γα  βγ,µ µ ∂ ∂ ∂ α ∂ α ∂ X 1 ≡ f µ + Yαβ + Yαβµ + Yβγ α + Yβγµ α ∈ (J π) . ∂x ∂gαβ ∂gαβ,µ ∂Γ ∂Γ α β α β βγ βγ,µ X≤ X≤ 1 As a long calculation shows, j YZ are tangent to S since

µ ν βγ,λ 1 µν ∂f ν ∂f µ ∂H ∂Lα α L(j YZ )c = − δ − δ − Γ = 0; (on S) , ∂xρ σ ∂xσ ρ ∂g ∂g βγ,λ   ρσ ρσ ! α β ν 1 ∂f β ν ∂f α ν ∂f α β L(j YZ )mρσ,µ = − δ δ − δ δ − δ δ mαβ,ν = 0; (on S) , ∂xρ σ µ ∂xσ ρ µ ∂xµ ρ σ  α ρ σ  1 α ∂f ρ σ ∂f α σ ∂f α ρ λ L(j YZ)t = δ δ − δ δ − δ δ t = 0; (on S) , βγ ∂xλ β γ ∂xβ λ γ ∂xγ λ β ρσ α ρ  σ τ  1 α ∂f ρ σ τ ∂f α σ τ ∂f α ρ τ ∂f α ρ σ λ L(j YZ)r = δ δ δ − δ δ δ − δ δ δ − δ δ δ r = 0; (on S) , βγ,ν ∂xλ β γ ν ∂xβ λ γ ν ∂xγ λ β ν ∂xν λ β γ ρσ,τ   α β λ γ 1 ∂f β λ γ ∂f α λ γ ∂f α β γ ∂f α β λ L(j YZ )iρσ,µν = − δ δ δ − δ δ δ − δ δ δ − δ δ δ iαβ,λγ = 0; (on S) . ∂xρ σ µ ν ∂xσ ρ µ ν ∂xµ ρ σ ν ∂xν ρ σ µ   1 Furthermore, L(j YZ)LEP|S = 0, for every Z ∈ X(M), and as these are natural vector fields, the Euler- Lagrange equations are also invariant. Thus, they are natural infinitesimal Lagrangian symmetries and, 1 hence, natural infinitesimal Noether symmetries. The associated conserved quantity to each j YZ is 1 βγ,µ α µ 3 µ βγ,ν α 2 ξYZ = i(j Y )ΘLEP = (Lα Yβγ − Hf )d xµ + f Lα dΓβγ ∧ d xµν ; J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.18

1 and, given a section ψ solution to the field equations, the current associated with j YZ is

∗ ∗ βγ,µ α α λ µ 3 ψ ξYZ = ψ (Lα (Yβγ − Γβγ,λf ) − f LEP)d xµ .

6 Conclusions and outlook

In this work a careful review has been made on the geometric meaning of symmetries for Lagrangian field theories. The discussion has been done for first and second-order Lagrangians because our aim was to apply the concepts and results to different cases of classical field theories; namely, the Maxwell theory of Electromagnetism (first-order) and the Einstein–Hilbert (second-order) and the Einstein–Palatini (first- order) models in General Relativity. First, we have stated the geometrical meaning of symmetry in the ambient of the multisymplectic framework of field theories. Then, we have introduced the so-called Cartan or Noether symmetries, which are associated with conservation laws by means of Noether’s theorem. These kinds of symmetries are geometrically characterized by the fact that they let the multisymplectic Poincar´e–Cartan (n+1)-form invariant, and this property allows us to obtain conserved quantities as (n − 1)-forms satisfying certain properties which have been carefully analyzed and that lead to state the corresponding conservation laws. (Most of these are well-known results, previously studied in [11, 14, 19, 38], as well as in other works cited therein). We have also studied gauge symmetries from a pure geometric perspective. These symmetries gen- erate transformations that have no physical relevance and are associated to singular Lagrangians and, hence, to premultisymplectic Poincar´e–Cartan forms. The definition and characteristic properties of vec- tor fields generating gauge transformations have been discussed and justified, and this leads to take the vertical vector fields in the kernel of the Poincar´e–Cartan form as infinitesimal generators of these sym- metries. The concept of gauge equivalent solutions to the field equations and of gauge equivalent states has been also discussed. Finally, we have explained the guidelines of gauge reduction which allows us to eliminate the non-physical degrees of freedom of the theory associated with gauge symmetries. All these ideas are the generalization to premultisymplectic field theories of the analysis made on this topic for presymplectic mechanical systems, mainly in [2, 24]. As an application of the above ideas, we have analyzed the symmetries, conservation laws and gauge content of the aforementioned cases: Electromagnetism, and the Einstein–Hilbert and the Einstein– Palatini models of Gravitation; recovering the already well-known results of these theories (see, for instance, [20, 21, 23, 29, 37]). Following this guidelines, all this methods could be applied to investigate Noether symmetries and conservation laws, as well as gauge symmetries for other theories of gravity (for instance, Chern-Simons gravity) and other extended models of General Relativity [3, 6, 8, 16]. The geometric interpretation of gauge symmetries in the multisymplectic context is clearly incom- plete. Looking at the case of Electromagnetism, one could argue that the gauge freedom in the physical sense is better understood as a Lagrangian symmetry than as a gauge symmetry. This points out that it would be relevant to study the interplay between this two kind of symmetries, and the role of the holon- omy condition. Moreover, the classical works [2, 24] for mechanical systems show a more complex structure of gauge vector fields and a more consistent reduction. It would be interesting to generalize these results to the multisymplectic case, if possible. Finally, there is the important problem on how to use conserved currents to integrate the field equa- tions. We expect that the complete characterization of all the types of symmetries of multisymplectic systems could be relevant for this problem. J. Gaset, N. Rom´an-Roy: Symmetries in Lagrangian field theories: Electromagnetism and Gravitation.19

Acknowledgments

We acknowledge the financial support of the Ministerio de Ciencia, Innovaci´on y Universidades (Spain), project PGC2018-098265-B-C33, and of Generalitat de Catalunya, project 2017–SGR–932.

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