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      Palatini Variational Principle for an Extended Einstein-Hilbert Action

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          Abstract

          We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation.

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          Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance

          In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group \({A}(4,R)\) and of its subgroup \({GL}(4,R)\) in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the \(\overline{SL}(4,R)\) covering subgroup. --- We set up a (first--order) Lagrangian formalism and build up the corresponding Noether machinery. For an arbitrary gauge Lagrangian, the three gauge field equations come out in a suggestive Yang-Mills like form. The conservation--type differential identities for energy--momentum and hypermomentum and the corresponding complexes and superpotentials are derived. Limiting cases such as the Einstein--Cartan theory are discussed. In particular we show, how the \({A}(4,R)\) may ``break down'' to the Poincar\'e (inhomogeneous Lorentz) group. In this context, we present explicit models for a symmetry breakdown in the cases of the Weyl (or homothetic) group, the \({SL}(4,R)\), or the \({GL}(4,R)\).
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            Mechanism of Generation of Black Hole Entropy in Sakharov's Induced Gravity

            The mechanism of generation of the Bekenstein-Hawking entropy \(S^{BH}\) of a black hole in the Sakharov's induced gravity is proposed. It is suggested that the "physical" degrees of freedom, which explain the entropy \(S^{BH}\), form only a finite subset of the standard Rindler-like modes defined outside the black hole horizon. The entropy \(S_R\) of the Rindler modes, or entanglement entropy, is always ultraviolet divergent, while the entropy of the "physical" modes is finite and it coincides in the induced gravity with \(S^{BH}\). The two entropies \(S^{BH}\) and \(S_R\) differ by a surface integral Q interpreted as a Noether charge of non-minimally coupled scalar constituents of the model. We demonstrate that energy E and Hamiltonian H of the fields localized in a part of space-time, restricted by the Killing horizon \(\Sigma\), differ by the quantity \(T_H Q\), where \(T_H\) is the temperature of a black hole. The first law of the black hole thermodynamics enables one to relate the probability distribution of fluctuations of the black hole mass, caused by the quantum fluctuations of the fields, to the probability distribution of "physical" modes over energy E. The latter turns out to be different from the distribution of the Rindler modes. We show that the probability distribution of the "physical" degrees of freedom has a sharp peak at E=0 with the width proportional to the Planck mass. The logarithm of number of "physical" states at the peak coincides exactly with the black hole entropy \(S^{BH}\). It enables us to argue that the energy distribution of the "physical" modes and distribution of the black hole mass are equivalent in the induced gravity. Finally it is shown that the Noether charge Q is related to the entropy of the low frequency modes propagating in the vicinity of the bifurcation surface \(\Sigma\) of the horizon.
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              Theories of gravitation in two dimensions

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                Author and article information

                Journal
                03 November 1997
                Article
                10.1103/PhysRevD.57.4754
                gr-qc/9711003
                e7273e3c-e5f6-42a0-9c77-eab8e76dc193
                History
                Custom metadata
                WATPHYS TH-97/17
                Phys.Rev. D57 (1998) 4754-4759
                14 pages, LaTeX
                gr-qc

                General relativity & Quantum cosmology
                General relativity & Quantum cosmology

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