Marco Ferraris

Nonlinear gravitational Lagrangians

Abstract“Alternative gravitational theories” based on Lagrangian densities that depend in a nonlinear way on the Ricci tensor of a metric are considered. It is shown that, provided certain weak regularity conditions are met, any such theory is equivalent, from the Hamiltonian point of view, to the standard Einstein theory for a new metric (which, roughly speaking, coincides with the momentum canonically conjugated to the original metric), interacting with external matterfields whose nature depends on the original Lagrangian density.

Variational formulation of general relativity from 1915 to 1925 “Palatini's method” discovered by Einstein in 1925

Among the three basic variational approaches to general relativity, the metric-affine variational principle, according to which the metric and the affine connection are varied independently, is commonly known as the “Palatini method.” In this paper we revisit the history of the “golden age” of general relativity, through a discussion of the papers involving a variational formulation of the field problem. In particular we find that the original Palatini paper of 1919 was rather far from what is usually meant by “Palatinis method,” which was instead formulated, to our knowledge, by Einstein in 1925.

Universality of the Einstein equations for Ricci squared Lagrangians

It has been shown recently that, in the first-order (Palatini) formalism, there is universality of the Einstein equations and the Komar energy-momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets the Einstein equations and Komars expression for the energy-momentum complex. In this paper a similar analysis (also within the framework of the first-order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of the Einstein equations and Komars energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).

Unified geometric theory of electromagnetic and gravitational interactions

A geometric unification of the electromagnetic and gravitational fields is presented. The unified field is described by a linear connection Γ on the space-time. Field equations for the unified field Γ are equivalent to Einstein-Maxwell equations. Field equations for matter interacting with the unified field are the usual ones. The interaction of the unified field with a charged scalar field is studied in detail.

A covariant formalism for Chern?Simons gravity

Chern–Simons type Lagrangians in d = 3 dimensions are analysed from the point of view of their covariance and globality. We use the transgression formula to find out a new fully covariant and global Lagrangian for Chern–Simons gravity: the price for establishing globality is hidden in a bimetric (or biconnection) structure. Such a formulation allows us to calculate from a global and simpler viewpoint the energy–momentum complex and the superpotential both for Yang–Mills and gravitational examples.

Gauge formalism for general relativity and fermionic matter

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures. It does not require any background metric fixing and it applies to quite general manifolds, i.e. it does not need particular symmetries requirement or global frames. A global Lagrangian framework for Dirac spinors is also provided; conserved quantities and superpotentials are given. The interaction between gravity and spinors is described in a minimal coupling fashion with respect to the new variables and the Hilbert stress tensor of spinor fields is computed, providing the gravitational field generated by spinors. Finally differences and analogies between this formalism and gauge theories are discussed.

Almost-complex and almost-product Einstein manifolds from a variational principle

It is shown that the first-order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kahler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterization of anti-Kahler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered.

AUGMENTED VARIATIONAL PRINCIPLES AND RELATIVE CONSERVATION LAWS IN CLASSICAL FIELD THEORY

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities which measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental Physics are considered in detail, starting from classical Mechanics.Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities that measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well-known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms that have recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental physics are considered in detail, starting from classical mechanics.

Generalized symmetries in mechanics and field theories

Generalized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are characterized in a Lagrangian context by means of the transformation rules of the Poincare–Cartan form and the (generalized) Nother theorem is applied to obtain conserved quantities (first integrals in mechanics). In the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of Hamilton–Jacobi equations recovering standard results by means of this new method. Supersymmetries (Wess–Zumino model) are considered as an intriguing example in field theory.

On the equivalence of the relativistic theories of gravitation

Einsteins equations are rewritten in terms of a certain torsionless linear connection Γαβμ which differs, in general, from the Levi-Civita metric connection γαβμ. The torsionless connection Γαβμ appears in a natural way as the canonical momentum of the gravitational field gμν. Einsteins equations have a simple interpretation in terms of the connection Γαβμ. The equivalence of the so-calledpurely metric, purely affine, andmetricaffine theories of gravitation is proved.