Universality of Einstein Equations for the Ricci Squared Lagrangians

It has been recently shown that, in the first order (Palatini) formalism, there is universality of Einstein equations and 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 Einstein equations and Komar's expression for the energy-momentum complex. In this paper a similar analysis (also in 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 Einstein equations and Komar energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).Comment: 21 pages, Late

The Universality of Einstein Equations

It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.Comment: 15 pages, LaTeX, (Extended Version), TO-JLL-P1/9

Constraining the Physical State by Symmetries

After reviewing the hole argument and its relations with initial value problem and general covariance, we shall discuss how much freedom one has to define the physical state of a system in a generally covariant (or gauge covariant) field theory. We shall show that in gauge covariant theories (and generally covariant theories with a a compact space) one has no freedom and one is forced to declare as physically equivalent two configurations which differ by a gauge transformation (or by a global spacetime diffeomorphism), as it is usually prescribed. On the contrary, when space is not compact, the result proven for the compact case does not hold true and one may have different options to define physically equivalent configurations, still preserving determinism.Comment: 12 page

Entropy of Self-Gravitating Systems from Holst's Lagrangian

We shall prove here that conservation laws from Holst's Lagrangian, often used in LQG, do not agree with the corresponding conservation laws in standard GR. Nevertheless, these differences vanish on-shell, i.e. along solutions, so that they eventually define the same classical conserved quantities. Accordingly, they define in particular the same entropy of solutions, and the standard law S=A/4 is reproduced for systems described by Holst's Lagragian. This provides the classical support to the computation usually done in LQG for the entropy of black holes which is in turn used to fix the Barbero-Immirzi parameter.Comment: 4 pages, no figures; just acknowledgments change

Hamiltonian, Energy and Entropy in General Relativity with Non-Orthogonal Boundaries

A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General Relativity in presence of non-orthogonal boundaries is analysed and the energy is defined as the on-shell value of the Hamiltonian. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is defined by imposing metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with Brown-York original formulation of the first principle of black hole thermodynamics is finally established.Comment: 29 pages with 1 figur

A covariant formalism for Chern-Simons gravity

Chern--Simons type Lagrangians in $d=3$ dimensions are analyzed 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 to calculate from a global and simpler viewpoint the energy-momentum complex and the superpotential both for Yang--Mills and gravitational examples.Comment: 12 pages,LaTeX, to appear in Journal of Physics

Remarks on Conserved Quantities and Entropy of BTZ Black Hole Solutions. Part II: BCEA Theory

The BTZ black hole solution for (2+1)-spacetime is considered as a solution of a triad-affine theory (BCEA) in which topological matter is introduced to replace the cosmological constant in the model. Conserved quantities and entropy are calculated via Noether theorem, reproducing in a geometrical and global framework earlier results found in the literature using local formalisms. Ambiguities in global definitions of conserved quantities are considered in detail. A dual and covariant Legendre transformation is performed to re-formulate BCEA theory as a purely metric (natural) theory (BCG) coupled to topological matter. No ambiguities in the definition of mass and angular momentum arise in BCG theory. Moreover, gravitational and matter contributions to conserved quantities and entropy are isolated. Finally, a comparison of BCEA and BCG theories is carried out by relying on the results obtained in both theories.Comment: PlainTEX, 20 page

Universal field equations for metric-affine theories of gravity

We show that almost all metric--affine theories of gravity yield Einstein equations with a non--null cosmological constant $\Lambda$. Under certain circumstances and for any dimension, it is also possible to incorporate a Weyl vector field $W_\mu$ and therefore the presence of an anisotropy. The viability of these field equations is discussed in view of recent astrophysical observations.Comment: 13 pages. This is a copy of the published paper. We are posting it here because of the increasing interest in f(R) theories of gravit

The dynamical equivalence of modified gravity revisited

We revisit the dynamical equivalence between different representations of vacuum modified gravity models in view of Legendre transformations. The equivalence is discussed for both bulk and boundary space, by including in our analysis the relevant Gibbons-Hawking terms. In the f(R) case, the Legendre transformed action coincides with the usual Einstein frame one. We then re-express the R+f(G) action, where G is the Gauss-Bonnet term, as a second order theory with a new set of field variables, four tensor fields and one scalar and study its dynamics. For completeness, we also calculate the conformal transformation of the full Jordan frame R+f(G) action. All the appropriate Gibbons-Hawking terms are calculated explicitly.Comment: 17 pages; v3: Revised version. New comments added in Sections 3 & 5. New results added in Section 6. Version to appear in Class. Quantum Gravit

Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)

We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles). The contraction of the Euler-Lagrange equations with Lie derivatives of the dynamical fields allows one to derive a variational Lagrangian for any given set of Lagrangian equations. A two steps algorithmical procedure can be thence applied to the variational Lagrangian in order to produce a general expression for the variation of all quantities which are (covariantly) conserved along the given dynamics. As a concrete example we test this new formalism on Einstein's equations: well known and widely accepted formulae for the variation of the Hamiltonian and the variation of Energy for General Relativity are recovered. We also consider the Einstein-Cartan (Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new insight on the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.Comment: Latex file, 31 page