  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 interesting 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 covariant 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 multimomentum
(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 geometrical 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.
