Griffiths variational multisymplectic formulation for Lovelock gravity
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Griffiths variational multisymplectic formulation for Lovelock gravity S. Capriotti1
· J. Gaset2 · N. Román-Roy3 · L. Salomone4
Received: 12 May 2020 / Accepted: 1 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This work is mainly devoted to constructing a multisymplectic description of Lovelock’s gravity, which is an extension of General Relativity. We establish the Griffiths variational problem for the Lovelock Lagrangian, obtaining the geometric form of the corresponding field equations. We give the unified Lagrangian–Hamiltonian formulation of this model and we study the correspondence between the unified formulations for the Einstein–Hilbert and the Einstein–Palatini models of gravity. Keywords Field theory · Lagrangian and Hamiltonian formalisms · Jet bundles · Multisymplectic manifolds · Griffiths variational problem · Lovelock gravity · Hilbert–Einstein and Einstein–Palatini actions · Einstein equations Mathematics Subject Classification Primary 49S05 · 70S05 · 83D05; Secondary 35Q75 · 35Q76 · 53D42 · 55R10
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S. Capriotti [email protected] J. Gaset [email protected] N. Román-Roy [email protected] L. Salomone [email protected]
1
Departamento de Matemática and CONICET, Universidad Nacional del Sur, 8000 Bahía Blanca, Argentina
2
Department of Physics, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
3
Department of Mathematics, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
4
Departamento de Matemática-CMaLP, Facultad de Ciencias Exactas, CONICET, UNLP, 1900 La Plata, Argentina 0123456789().: V,-vol
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The frame bundle and its canonical forms . . . . . . . . . . . . . . . . . . . . 2.1 Basic definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The universal principal connection . . . . . . . . . . . . . . . . . . . . . 2.3 The canonical form θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Sparling forms θi 1 ...i p . . . . . . . . . . . . . . . . . . . . . . 3 Variational problem for Lovelock gravity . . . . . . . . . . . . . . . . . . . . 3.1 The Lovelock Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Expressions in terms of the canonical basis of Rm . . . . . . . . . . 3.1.2 Relation with the metric-affine Lagrangian . . . . . . . . . . . . . 4 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Infinitesimal symmetries of IL . . . . . . . . . . . . . . . . . . . . . . . 4.2 Field equations for Lovelock gravity from its Griffiths variational problem 5 Unified formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Tautological form on a bundle of forms . . . . . . . . . . . . . . . . . . . 5.2 The multimomentum bundle . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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