3,780 research outputs found
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
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
Noether 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 introduced to deal with the (vacuum)
gravitational field on a reference background. The two Lagrangians are compared
and we show that the first one can be obtained from the latter under suitable
hypotheses. The induced variational principles are also compared and discussed.
A conditioned correspondence among Noether conserved quantities, quasilocal
energy and the standard Hamiltonian obtained by 3+1 decomposition is also
established. As a result, it turns out that the covariant first order
Lagrangian is better suited whenever a reference background field has to be
taken into account, as it is commonly accepted when dealing with conserved
quantities in non-asymptotically flat spacetimes. As a further advantage of the
use of a covariant first order Lagrangian, we show that all the quantities
computed are manifestly covariant, as it is appropriate in General Relativity.Comment: 43 pages, 3 figures, PlainTeX fil
On-shell symmetries
We define on-shell symmetries and characterize them for Lagrangian systems.
The terms appearing in the variation of the Poincare'-Cartan form, which vanish
because of field equations, are found to be strongly constrained if the space
of solutions has to be preserved. The behaviour with respect to solution
dragging is also investigated in order to discuss relations with the theory of
internal symmetries of a PDE.Comment: 13 page
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
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
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 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
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.
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