Differential geometry, Palatini gravity and reduction

Capriotti, Santiago

doi:10.1063/1.4862855

Cited by 15 publications

(37 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) To set a Griffiths variational problem for this theory, and (2) to construct the classical Lepage-equivalent variational problem for this variational problem, proving also that it is a contravariant Lepage-equivalent in the sense of Definition 9. As we know, a Griffiths variational problem describing vacuum Palatini gravity exists [Cap14]. In short, it is the variational problem specified by the triple…”

Section: The Unified Formalism For Palatini Gravitymentioning

confidence: 99%

“…Our starting point will be the Griffiths variational problem considered in [Cap14] for Palatini gravity; this variational problem can be considered as an alternative Lagrangian version for the formulation of vacuum GR equations in terms of exterior differential systems, as it is given for example in [Est05]. The method chosen for the construction of the unified formalism comes from the work of Gotay [Got91a,Got91b], and it consists into the employment of a classical Lepageequivalent variational problem (in the sense defined by Gotay in these works) as a replacement for the Griffiths variational problem at hands.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unified formalism for Palatini gravity

Capriotti

2018

Int. J. Geom. Methods Mod. Phys.

Self Cite

View full text Add to dashboard Cite

The present article is devoted to the construction of a unified formalism for Palatini and unimodular gravity. The basic idea is to employ a relationship between unified formalism for a Griffiths variational problem and its classical Lepage-equivalent variational problem. As a way to understand from an intuitive viewpoint the Griffiths variational problem approach considered here, we may say the variations of the Palatini Lagrangian are performed in such a way that the so called metricity condition, i.e. (part of) the condition ensuring that the connection is the Levi-Civita connection for the metric specified by the vielbein, is preserved. From the same perspective, the classical Lepage-equivalent problem is a geometrical implementation of the Lagrange multipliers trick, so that the metricity condition is incorporated directly into the Palatini Lagrangian. The main geometrical tools involved in these constructions are canonical forms living on the first jet of the frame bundle for the spacetime manifold Å . These forms play an essential rôle in providing a global version of the Palatini Lagrangian and expressing the metricity condition in an invariant form; with their help, it was possible to formulate an unified formalism for Palatini gravity in a geometrical fashion. Moreover, we were also able to find the associated equations of motion in invariant terms and, by using previous results from the literature, to prove their involutivity. As a bonus, we showed how this construction can be used to provide a unified formalism for the so called unimodular gravity by employing a reduction of the structure group of the principal bundle ÄÅ to the special linear group ËÄ´Ñµ Ñ dim Å .

show abstract

Section: The Unified Formalism For Palatini Gravitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unified formalism for Palatini gravity

Capriotti

2018

Int. J. Geom. Methods Mod. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We denote by ker 4 π 1 Ω L EP the set of locally decomposable and π 1 -transverse multivector fields satisfying equations (4) but not being (semi)holonomic necessarily. Then, ker 4 SH Ω L EP and ker 4 H Ω L EP denote the sets of semi-holonomic and the holonomic multivector fields which are solutions to the equations (4), respectively. Obviously we have…”

Section: Compatibility and Consistency Constraintsmentioning

confidence: 99%

“…The multisymplectic and polysymplectic techniques have been also applied to treat different aspects of one of the most classical approaches in General Relativity: the Einstein-Palatini or Metric-Affine model [4,5,31,37,38]. In particular, in [5] an exhaustive study of the multisymplectic description of the model has been done, using a unified formalism which joins both the Lagrangian and Hamiltonian formalisms into a single one.…”

Section: Introductionmentioning

confidence: 99%

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

Gaset

Román‐Roy

2019

Journal of Geometric Mechanics

View full text Add to dashboard Cite

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

show abstract

“…As a result we present here a successful approach to the discretization of a variational theory that involves a field theory with several dependent and independent variables on arbitrary manifolds, in a situation that includes non-trivial bundles with non-commutative groups of symmetries, where a reduction procedure allows to describe critical fields through a system of partial differential equations formulated for a reduced field. More precisely, we shall deal with a general problem of Euler-Poincaré reduction for a field theory in a principal bundle [13,14,15], a common situation in several scientific branches, like motion and control of mechanical systems [33,32], elasticity [18,19], liquid crystal theory [27], Palatini gravity [7], and other gauge field theories. We show how to formulate the discrete counterparts of the smooth elements present in the theory and in its reduction, and present the needed objects that allow to relate smooth and discrete fields.…”

Section: Introductionmentioning

confidence: 99%

Discrete formulation for the dynamics of rods deforming in space

Casimiro

Rodrigo

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

The movement of rods in an euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth lagrangian density is invariant by some group of transformations, there exist corresponding conserved Noether currents. Generally numerical schemes dealing with PDEs fail to reflect these conservation properties.We describe the main ingredients needed to create, from the smooth lagrangian density, a variational principle for discrete motions of a discrete rod, with corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle, using a reduced forward difference operator. We show how this introduces a discrete lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.

show abstract

Differential geometry, Palatini gravity and reduction

Cited by 15 publications

References 36 publications

Unified formalism for Palatini gravity

Unified formalism for Palatini gravity

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

Discrete formulation for the dynamics of rods deforming in space

Contact Info

Product

Resources

About