M. Raiteri

Remarks on conserved quantities and entropy of BTZ black hole solutions. 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.

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 equation 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-step algorithmical procedure can thence be 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 Einsteins 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 into the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.

Charges and energy in Chern–Simons theories and Lovelock gravity

Starting from the SO(2, 2n) Chern–Simons form in (2n + 1) dimensions, we calculate the variation of conserved quantities in Lovelock gravity and Lovelock–Maxwell gravity through the covariant formalism developed in [23]. Despite the technical complexity of the Lovelock Lagrangian, we obtain a remarkably simple expression for the variation of the charges ensuing from the diffeomorphism covariance of the theory. The viability of the result is tested in specific applications and the formal expression for the entropy of Lovelock black holes is recovered.

Nöther charges, Brown–York quasilocal energy, and related topics

The Lagrangian proposed by York et al. and the covariant first-order Lagrangian for general relativity are reviewed. They both deal with the (vacuum) gravitational field on a reference background and were conjectured to be equivalent. The two corresponding actions are compared and we show that the first one can in fact be obtained from the latter under suitable hypotheses. A conditioned correspondence among Nother conserved quantities of covariant first-order Lagrangian, Brown–York quasilocal energy and the standard ADM Hamiltonian is also established.

Remarks on Nöther Charges and Black Holes Entropy

Abstract We criticize and generalize some properties of Nother charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge–natural field theory. As an application charged Kerr–Newman solutions are considered. As a further example we consider the (1+2) BTZ black hole solution.

Hamiltonian, energy and entropy in general relativity with non-orthogonal boundaries

A general recipe to define, via the 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 the Noether-conserved quantities. The Hamiltonian for general relativity in the 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 the metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with the Brown–York original formulation of the first principle of black hole thermodynamics is finally established.

The Entropy of the Taub–Bolt Solution

Abstract A geometrical framework for the definition of entropy in general relativity via the Nother theorem is briefly recalled, and the entropy of Taub–Bolt Euclidean solutions of Einstein equations is then obtained as an application. The computed entropy agrees with previously known results, obtained by statistical methods. It was generally believed that the entropy of a Taub–Bolt solution could not be computed via the Nother theorem, due to the particular structure of the singularities of this solution. We show here that this is not true. The Misner string singularity is, in fact, considered, and its contribution to the entropy is analyzed. As a result, in our framework entropy does not obey the “one-quarter area law” and it is not directly related to horizons, as is sometimes erroneously suggested in the current literature on the subject.

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 consid- ered 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 Nother 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.

Covariant charges in Chern–Simons AdS3 gravity

We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern–Simons theory, with particular reference to Chern–Simons AdS3 gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern–Simons Lagrangian and using a recipe developed by Allemandi et al (2002 Class. Quantum Grav. 19 2633–55, 237–58) to calculate the variation of conserved quantities. The problem of giving a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black-hole mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern–Simons AdS3 gravity theory and the variation of conserved quantities in general relativity is analysed in detail.

Dual Lagrangian field theories

We investigate how, under suitable regularity conditions, first-order Lagrangian field theories can be recasted in terms of a second-order Lagrangian, called the dual Lagrangian of the theory, depending on canonical conjugate momenta together with their derivatives. The necessary and sufficient conditions which allow such a (local) reformulation, obtained through a suitable generalization of the Legendre transformation, are analyzed. The global geometric framework is also investigated in detail. As an example, we apply the dual Lagrangian formulation to the Hilbert Lagrangian and to Euclidean self-dual gravity.