Summary. -The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on any spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity. More recently, the work by Ashtekar, Rovelli, Smolin and their collaborators on connection dynamics and loop variables has made it possible to cast the constraint equations of general relativity in polynomial form, and then find a large class of solutions to the quantum version of constraints [7][8][9][10][11][12][13][14]. However, the quantum theory via the Rovelli-Smolin transform still suffers from severe mathematical problems in 3+1 space-time dimensions [15], and there appear to be reasons for studying non-perturbative quantum gravity also from a Lagrangian, rather than Hamiltonian, point of view (see below). The aim of this paper is therefore to provide a multisymplectic, Lagrangian framework for general relativity [16][17][18], to complement the present attempts to quantize general relativity in a non-perturbative way. The motivations of our analysis are as follows.(i) In the case of field theories, there is not a unique prescription for taking duals, on passing to the Hamiltonian formalism. For example, algebraic and topological duals are different. In turn, this may lead to inequivalent quantum theories. SPACE-TIME COVARIANT FORM OF ASHTEKAR'S CONSTRAINTS(ii) The 3+1 split of the Lorentzian space-time geometry, with the corresponding Σ×R topology, appears to violate the manifestly covariant nature of general relativity, as well as rely on a very restrictive assumption on the topology [19].(iii) In the Lagrangian formalism, explicit covariance is instead recovered. The first constraints one actually evaluates correspond to the secondary first-class constraints of the Hamiltonian formalism. At least at a classical level, the Lagrangian theory of constrained systems is by now a rich branch of modern mathematical physics [4,20,21], although the majority of general relativists are more familiar with the Hamiltonian framework.(iv) In the ADM formalism [5,6], the invariance group is not the whole diffeomorphism group, but a subgroup given by the Cartesian product of diffeomorphisms on the real line with the diffeomorphism group on spacelike three-surfaces. By contrast, in the Lagrangian approach, the invariance group of the theory is the full diffeomorphism group of fourdimensional Lorentzian space-time...

Potentials in Backgrounds with Boundary

A discrete theory of gravity, locally invariant under the Poincaré group, is considered as in a companion paper. We define a first order theory, in the sense of Palatini, on the metric-dual Voronoi complex of a simplicial complex. We follow the same spirit of the continuum theory of General Relativity in the Cartan formalism. The field equations are carefully derived taking in account the constraints of the theory. They look very similar to first order Einstein continuum equations in the Cartan formalism. It is shown that in the limit of small deficit angles these equations have Regge Calculus, locally, as the only solution. A quantum measure is easly defined which does not suffer the ambiguities of Regge Calculus, and a coupling with fermionic matter is easily introduced.

We introduce the basic elements of SO(n)-local theory of Regge Calculus. A first order formalism, in the sense of Palatini, is defined on the metric-dual Voronoi complex of a simplicial complex.The Quantum Measure exhibits an expansion, in four dimensions, in characters of irreducible representation of SO(4) which has close resemblance and differences as well with the Spin Foam Formalism. The coupling with fermionic matter is easily introduced which could have consequences for the Spin Foam Formalism and Loop Quantum Gravity.

We deal with the duality symmetry of the Dilaton field in cosmology and specifically with the socalled Gasperini-Veneziano duality transformation. In particular, we determine two conformal equivalent theories to the Dilaton field, and we show that under conformal transformations Gasperini-Veneziano duality symmetry does not survive. Moreover, we show that those theories share a common conservation law, of Noetherian kind, while the symmetry vector which generates the conservation law is an isometry only for the Dilaton field. Finally, we show that the Lagrangian of the Dilaton field is equivalent with that of the two-dimensional "hyperbolic oscillator" in a Lorentzian space whose O(d, d) invariance is transformed to the Gasperini-Veneziano duality invariance in the original coordinates. 95.35.+d, 95.36.+x

We discuss the elongated phase of four-dimensional simplicial quantum gravity by exploiting recent analytical results. In particular using Walkup’s theorem we prove that the dominating configurations in the elongated phase are treelike structures called “stacked spheres.” Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyze the critical behavior of theory in the weak coupling regime.

Summary. -Previous work in the literature has studied the Hamiltonian structure of an R 2 model of gravity with torsion in a closed Friedmann-Robertson-Walker universe.Within the framework of Dirac's theory, torsion is found to lead to a second-class primary constraint linear in the momenta and a second-class secondary constraint quadratic in the momenta. This paper studies in detail the same problem at a Lagrangian level, i.e. working on the tangent bundle rather than on phase space. The corresponding analysis is motivated by a more general program, aiming to obtain a manifestly covariant, multisymplectic framework for the analysis of relativistic theories of gravitation regarded as constrained systems.After an application of the Gotay-Nester Lagrangian analysis, the paper deals with the generalized method, which has the advantage of being applicable to any system of differential equations in implicit form. Multiplication of the second-order Lagrange equations 1

We investigate the motion of uncharged particles scattered by a binary system consisting of extremely charged black holes in equilibrium as described by the Majumdar-Papapetrou solution. We focus on unbound orbits confined to the plane containing both black holes. We consider the two complementary situations of particles approaching the system along a direction parallel to the axis where the black holes are displaced and orthogonal to it. We numerically compute the scattering angle as a function of the particle's conserved energy parameter, which provides a gauge-invariant information of the scattering process. We also study the precession of a test gyroscope along such orbits and evaluate the accumulated precession angle after a full scattering, which is another gauge-invariant quantity.