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

Remarks on Conserved Quantities and Entropy of BTZ Black Hole Solutions. Part I: the General Setting

The BTZ stationary black hole solution is considered and its mass and angular momentum are calculated by means of Noether theorem. In particular, relative conserved quantities with respect to a suitably fixed background are discussed. Entropy is then computed in a geometric and macroscopic framework, so that it satisfies the first principle of thermodynamics. In order to compare this more general framework to the prescription by Wald et al. we construct the maximal extension of the BTZ horizon by means of Kruskal-like coordinates. A discussion about the different features of the two methods for computing entropy is finally developed.Comment: PlainTEX, 16 pages. Revised version 1.

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

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

Gauge Formalism for General Relativity and Fermionic Matter

A new formalism for spinors on curved spaces is developed in the framework of variational calculus on fibre bundles. The theory has the same structure of a gauge theory and describes the interaction between the gravitational field and spinors. An appropriate gauge structure is also given to General Relativity, replacing the metric field with spin frames. Finally, conserved quantities and superpotentials are calculated under a general covariant form.Comment: 18 pages, Plain TEX, revision, explicit expression for superpotential has been adde

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

Newtonian limit of the singular f(R) gravity in the Palatini formalism

Recently D. Vollick [Phys. Rev. D68, 063510 (2003)] has shown that the inclusion of the 1/R curvature terms in the gravitational action and the use of the Palatini formalism offer an alternative explanation for cosmological acceleration. In this work we show not only that this model of Vollick does not have a good Newtonian limit, but also that any f(R) theory with a pole of order n in R=0 and its second derivative respect to R evaluated at Ro is not zero, where Ro is the scalar curvature of background, does not have a good Newtonian limit.Comment: 9 page

Boundary Conditions, Energies and Gravitational Heat in General Relativity (a Classical Analysis)

The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric framework, a formal analogy between gravitation and the laws governing the evolution of a thermodynamic system. The analogy applies to stationary spacetimes admitting multiple causal horizons as well as to AdS Taub-bolt solutions.Comment: Latex file, 31 pages; one reference and two comments added, misprints correcte

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