Nonlinear Gravitational Lagrangians revisited

The Legendre transformation method, applied in 1987 to deal with purely metric gravitational Lagrangians with nonlinear dependence on the Ricci tensor, is extended to metric-affine models and is shown to provide a concise and insightful comparison of the dynamical content of the two variational frameworks.Comment: Variational principles and conservation laws in General Relativity - Workshop in memory of Prof. Mauro Francaviglia - Torino, June 25th-26th, 201

Are there metric theories of gravity other than General Relativity?

Current generalizations of the classical Einstein-Hilbert Lagrangian formulation of General Relativity are reviewed. Some alternative variational principles are known to reproduce Einstein's gravitational equations, and should therefore be regarded as equivalent descriptions of the same physical model, while other variational principles ("Scalar-tensor theories" and "Higher-derivative theories") are commonly presented as truly alternative physical theories. Such theories, however, are also known to admit a reformulation which is formally identical to General Relativity (with auxiliary fields). The physical significance of this change of variables has been questioned by several authors in recent years. Here, we investigate to which extent purely affine, metric-affine, scalar-tensor and purely metric theories can be regarded as physically equivalent to GR; we show that in general this depends on which metric tensorfield is assumed to represent the true physical space-time geometry. For purely metric theories where the Lagrangian is a nonlinear function f(R) of the curvature scalar, we present an argument based on the definition of the physical energy, which leads one to regard the rescaled metric (Einstein frame) as the true physical one. As a direct consequence, the physical content of such "alternative" models is reset to coincide with General Relativity, and the "Nonlinear Gravity Theories" become nothing but exotic reformulations of General Relativity in terms of unphysical variables.Comment: 19 pages, plain TeX, no figures. Talk given at the XI Italian Conference on General Relativity and Gravitation (Trieste, Sept 26-30, 1994). improper line break (causing TeX error on line 340) fixe

New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains

We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix. We show that for the case of such twist matrices one can interchange the role of classical separating functions $A(u)$ and $B(u)$ and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin. We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates

On the Generalised Equipartition Law

We observe that the so-called Generalised Equipartition Law for hamiltonian systems is actually valid only under specific hypotheses -- unfortunately omitted in some textbooks -- which limit its applicability when dealing with nonlinear systems. We introduce a new coordinate-independent generalisation which overcomes this problem, and moreover can be applied to a larger set of functions. A simple example of application is discussed

A priori reliability of tests with cut score

The theoretical probability of misclassification in a mastery test is exactly computed using the raw score probability distribution (in the Rasch model) as a function of the examinee???s latent ability. The resulting misclassification probability curve, together with the latent ability distribution in the group of examinees, completely determines the expected rate of classification errors. It is shown that several distinct ability thresholds, playing different roles in connection to classification reliability, can be associated to a test with a single cut score. In particular, it is possible to define (and compute) two relevant ability intervals, which encapsulate the functioning of a mastery test (about and far from the cut score, respectively); the dependence of these intervals on the item difficulty spectrum is investigated. Extension to the 2PL model is also discussed, with emphasis on the effects of weighted scoring

Test-field limit of metric nonlinear gravity theories

In the framework of alternative metric gravity theories, it has been shown by several authors that a generic Lagrangian depending on the Riemann tensor describes a theory with 8 degrees of freedom (which reduce to 3 for f(R) Lagrangians depending only on the curvature scalar). This result is often related to a reformulation of the fourth-order equations for the metric into a set of second-order equations for a multiplet of fields, including a massive scalar field and a massive spin-2 field. In this article we investigate an issue which does not seem to have been addressed so far: in ordinary general-relativistic field theories, all fundamental fields (i.e. fields with definite spin and mass) reduce to test fields in some appropriate limit of the model, where they cease to act as sources for the metric curvature. In this limit, each of the fundamental fields can be excited from its ground state independently from the others. The question is: does higher-derivative gravity admit a test-field limit for its fundamental fields? It is easy to show that for a f(R) theory the test-field limit does exist; then, we consider the case of Lagrangians quadratically depending on the full Ricci tensor. We show that the constraint binding together the scalar field and the massive spin-2 field does not disappear in the limit where they should be expected to act as test fields, except for a particular choice of the Lagrangian, which cause the scalar field to disappear (reducing to 7 DOF). We finally consider the addition of an arbitrary function of the quadratic invariant of the Weyl tensor and show that the resulting model still lacks a proper test-field limit. We argue that the lack of a test-field limit for the fundamental fields may constitute a serious drawback of the full 8 DOF higher-order gravity models, which is not encountered in the restricted 7 DOF or 3 DOF cases.Comment: Title and abstract modified to make the content of the paper more clear and readabl

Chi è Johann Sebastian Mastropiero?

Who is Johann Sebastian Mastropiero? The Hispanic public knows it well: he is the fetish character of Les Luthiers. But who are these Luthiers? Have we ever seen them on television? No, they have rarely been on television, but after a fifty-year history of theatrical success in Latin America and Spain, the shows of the Argentinean comic-musical group Les Luthiers are now accessible to a worldwide audience thanks to YouTube. In this essay we aim to show how the high musical level of their performances, the extraordinary inventive vein and the refined use of irony give an educational value to the surreal parodies of musical genres (from ancient music to opera and Latin American folklore) that the Luthiers have proposed during their artistic career. Chi è Johann Sebastian Mastropiero? Il pubblico ispanico lo sa bene: è il personaggio feticcio dei Les Luthiers. Ma chi sono questi Luthiers? Si sono mai visti in televisione? No, in televisione ci sono stati di rado, ma dopo una storia cinquantennale di successi teatrali in America Latina e in Spagna, gli spettacoli del gruppo comico-musicale argentino Les Luthiers sono ora accessibili a una platea mondiale grazie a YouTube. In questo saggio ci proponiamo dimostrare come l’alto livello musicale delle loro performances, la straordinaria vena inventiva e l’uso raffinato dell’ironia conferiscano una valenza anche educativa alle parodie surreali dei generi musicali (dalla musica antica fino all’opera lirica e al folklore latinoamericano) che i Luthiers hanno proposto nel corso della loro parabola artistica.Chi è Johann Sebastian Mastropiero? Il pubblico ispanico lo sa bene: è il personaggio feticcio dei Les Luthiers. Ma chi sono questi Luthiers? Si sono mai visti in televisione? No, in televisione ci sono stati di rado, ma dopo una storia cinquantennale di successi teatrali in America Latina e in Spagna, gli spettacoli del gruppo comico-musicale argentino Les Luthiers sono ora accessibili a una platea mondiale grazie a YouTube. In questo saggio ci proponiamo dimostrare come l’alto livello musicale delle loro performances, la straordinaria vena inventiva e l’uso raffinato dell’ironia conferiscano una valenza anche educativa alle parodie surreali dei generi musicali (dalla musica antica fino all’opera lirica e al folklore latinoamericano) che i Luthiers hanno proposto nel corso della loro parabola artistica. Who is Johann Sebastian Mastropiero? The Hispanic public knows it well: he is the fetish character of Les Luthiers. But who are these Luthiers? Have we ever seen them on television? No, they have rarely been on television, but after a fifty-year history of theatrical success in Latin America and Spain, the shows of the Argentinean comic-musical group Les Luthiers are now accessible to a worldwide audience thanks to YouTube. In this essay we aim to show how the high musical level of their performances, the extraordinary inventive vein and the refined use of irony give an educational value to the surreal parodies of musical genres (from ancient music to opera and Latin American folklore) that the Luthiers have proposed during their artistic career