Abstract
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.
Highlights
In recent years, there is an increasing effort in understanding the covariant description of gravitational theories (General Relativity and other derived from it) using different kinds of geometric frameworks such as the multisymplectic or the polysymplectic manifolds
In our geometric formalism, these are really the natural symmetries that we have studied and they are mathematically different from the geometric gauge symmetries that we have analysed in the previous Section
We have presented a multisymplectic covariant description of the Lagrangian and Hamiltonian formalisms of the Einstein-Palatini model of General Relativity
Summary
There is an increasing effort in understanding the covariant description of gravitational theories (General Relativity and other derived from it) using different kinds of geometric frameworks such as the multisymplectic or the polysymplectic manifolds. 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 This unified framework had been previously stated to do a covariant multisymplectic formulation of the Hilbert-Einstein model in General Relativity [25]. Several authors, like [11], pointed out that this property is related to the existence of a particular gauge symmetry Another objective of this work is to make a geometrical analysis of this gauge freedom and to recover the Einstein-Hilbert model for General Relativity by means of a partial gauge fixing.
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