A multisymplectic approach to gravitational theories

The theories of gravity are one of the most important topics in theoretical physics and mathematical physics nowadays. The classical formulation of gravity uses the Hilbert-Einstein Lagrangian, which is a singular second-order Lagrangian; hence it requires a geometric theory for second-order field theories which leads to several difficulties. Another standard formulation is the Einstein-Palatini or Metric-Affine, which uses a singular first order Lagrangian. Much work has been done with the aim of establishing the suitable geometrical structures for describing classical field theories. In particular, the multisymplectic formulation is the most general of all of them and, in recent years, some works have considered a multisymplectic approach to gravity. This formulation allows us to study and better understand several inherent characteristics of the models of gravity. The aim of this thesis is to use the multisymplectic formulation for first and second-order field theories in order to obtain a complete covariant description of the Lagrangian and Hamiltonian formalisms for the Einstein-Hilbert and the Metric-Affine models, and explain their characteristics; in particular: order reduction, constraints, symmetries and gauge freedom. Some properties of multisymplectic field theories have been developed in order to study the models. We have established the constraints generated by the projectability of the Poincaré-Cartan form. These constraints are related to the fact that the higher order velocities are strong gauge vector fields. The concept of gauge freedom for field theories also has been analyzed. We propose to use the term "gauge'' to refer to the non-regularity of the Poincaré-Cartan form. Therefore, the multiple solutions are characterized by two sources: the gauge related one, arising from gauge symmetries and related to the non-regularity; and the non-gauge related one, which arises exclusively from field theories. We studied in detail two models of gravity: the Einstein-Hilbert model and the Metric-Affine (or Einstein-Palatini) model. In both cases, a covariant Hamiltonian multisymplectic formalism has been presented. In every situation, we find the final submanifold where solutions exist, and we explicitly write all semi-holonomic multivector fields solution of the field equations. The natural Lagrangian symmetries are presented aswell. Furthermore, we emphasize different aspects in each model: The Einstein-Hilbert model is a singular second order field theory which, as a consequence of its non-regularity, it is equivalent to a regular first order theory. For this model we have presented the unified Lagrangian-Hamiltonian formalism. We have also considered the presence of energy-matter sources and we show how some relevant geometrical and physical characteristics of the theory depend on the source's type. The Metric-Affine model is a singular first order field theory which has a gauge symmetry. We recover and study this gauge symmetry, showing that there are no more. The constraints of the system are presented and analysed. Using the gauge freedom and the constraints, we establish the geometric relation between the Einstein-Palatini and the Einstein-Hilbert models, including the relation between the holonomic solutions in both formalisms. We also present a Hamiltonian model involving only the connection which is equivalent to the Hamiltonian Metric-Affine formalism.Les teories de la gravetat són un dels temes més importants en física teòrica i física matemàtica avui en dia. La formulació clàssica de la gravetat utilitza el Lagrangià de Hilbert-Einstein, el qual és un Lagrangià singular de segon ordre; per tant requereix una teoria geomètrica per teories de camp de segon ordre, que comporten diverses dificultats. Una altra formulació estàndard és la d'Einstein-Palatini o Mètrica-Afí, la qual utilitza un Lagrangià singular de primer ordre. S'ha treballat molt per establir les estructures geomètriques adients per descriure teories de camps clàssiques. Particularment, la formulació multisimplèctica és la més general de totes i, recentment alguns treballs han considerat la gravetat des de un punt de vista multisimplèctic. Aquesta formulació ens permet estudiar i entendre millor diverses característiques inherents dels models gravitatoris. L'objectiu d'aquesta tesi és utilitzar la formulació multisimplèctica per a teories de camps de primer i segon ordre per obtenir una descripció covariant completa dels formalismes Lagrangià i Hamiltonià per als models d'Einstein-Hilbert i Mètrica-Afí, i explicar les seves característiques. Concretament: reducció de l'ordre, restriccions, simetries i llibertat gauge. Algunes propietats de les teories de camps multisimplèctiques han estat desenvolupades per estudiar els models. S'han establert les restriccions generades per la projectabilitat de la forma de Poincaré-Cartan. Aquestes restriccions tenen relació amb el fet que les velocitats d'ordre superior són camps vectorials gauge forts. El concepte de llibertat gauge per a teories de camps també ha estat analitzat. Es proposa la utilització del terme "gauge" per fer referència a la no regularitat de les formes de Poincaré-Cartan. Per tant, les múltiples solucions es caracteritzen a partir de dues fonts: la relativa al gauge, que està relacionada amb la no regularitat, i altres fonts no relacionades amb el gauge que són exclusives de teories de camps. S'ha estudiat en detall dos models de gravetat: el model d'Einstein-Hilbert i el de Mètrica-Afí (o Einstein-Palatinti). En ambdós casos s'ha presentat una formulació covariant multisimplèctica Hamiltoniana. En tots els casos trobem la subvarietat final on les solucions existeixen, i escrivim explícitament tots els camps multivectorials sem-holònoms solució de les equacions de camp. També presentem les simetries Lagrangianes naturals. A més emfatitzem aspectes diferents en cada model: El model d'Einstein-Hilbert és una teoria de camp singular de segon ordre, la qual, com a conseqüència de la seva no regularitat, és equivalent a una teoria regular de primer ordre. Per aquest model hem presentat el formalisme unificat Lagrangià-Hamiltonià. També hem considerat la presència de fonts d'energia-matèria i es mostra com algunes característiques físiques i geomètriques rellevants de la teoria depenen del tipus de font. El model Mètrica-Afí és una teoria de camps singular de primer ordre que té una simetria gauge. Es recupera i s'estudia aquesta simetria gauge mostrant que és única. Les lligadures del sistema són presentades i analitzades. Utilitzant la llibertat gauge i les lligadures, s'estableix la relació geomètrica entre els models d'Einstein-Palatini i d'Einstein-Hilbert, inclosa la relació entre les solucions holònomes en ambdós formalismes. També es presenta un model Hamiltonià, que conté únicament la connexió, equivalent al formalisme Mètrica-Afí HamiltoniàPostprint (published version

Order reduction, projectability and constrainsts of second-order field theories and higuer-order mechanics

The consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation.Peer ReviewedPostprint (author's final draft

Multisymplectic unified formalism for Einstein-Hilbert gravity

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interest- ing when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the co- variant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomen- tum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geo- metrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.Peer ReviewedPostprint (author's final draft

Time-dependent contact mechanics

The version of record is available online at: http://dx.doi.org/10.1007/s00605-022-01767-1Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations.We acknowledge fruitful discussions and comments from our colleague Narciso Román-Roy. MdL acknowledges the financial support of the Ministerio de Ciencia e Innovación (Spain), under grants PID2019-106715GB-C2, “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and EIN2020-112107. JG, XG, MCML and XR acknowledge the financial support of the Ministerio de Ciencia, Innovación y Universidades (Spain), project PGC2018-098265-B-C33.Peer ReviewedPostprint (author's final draft

Multisymplectic formalism for cubic horndeski theories

We present the covariant multisymplectic formalism for the so-called cubic Horndeski theories and discuss the geometrical and physical interpretation of the constraints that arise in the uni¿ed Lagrangian-Hamiltonian approach. We analyse in more detail the covariant Hamiltonian formalism of these theories and we show that there are particular conditions that must be satis¿ed for the Poincar´e-Cartan form of the Lagrangian to project onto J1p. From this result, we study when a formulation using only multimomenta is possible. We further discuss the implications of the general case, in which the projection onto J1p conditions are not met.The authors acknowledge financial support from the Ministerio de Ciencia, Innovación y Universidades (Spain), projects PGC2018-098265-B-C33 and D2021-125515NB-21Peer ReviewedPostprint (author's final draft

Order reduction, projectability and constrainsts of second-order field theories and higuer-order mechanics

The consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation.Peer Reviewe

Multisymplectic unified formalism for Einstein-Hilbert gravity

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interest- ing when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the co- variant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomen- tum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geo- metrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.Peer Reviewe

Variational principles and symmetries on fibered multisymplectic manifolds

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.Peer Reviewe

New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.Peer ReviewedPostprint (author's final draft

A K-contact Lagrangian formulation for nonconservative field theories

Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of k-contact structure, and obtained a modified version of the De Donder–Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a k-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of k-velocities times a k-dimensional Euclidean space with coordinates sa, which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a k-contact Hamiltonian system. We study several types of symmetries for k-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.Financial support from the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932.Peer ReviewedPostprint (author's final draft