Superpotentials from variational derivatives rather than Lagrangians in relativistic theories of gravity

The prescription of Silva to derive superpotential equations from variational derivatives rather than from Lagrangian densities is applied to theories of gravity derived from Lovelock Lagrangians in the Palatini representation. Spacetimes are without torsion and isolated sources of gravity are minimally coupled. On a closed boundary of spacetime, the metric is given and the connection coefficients are those of Christoffel. We derive equations for the superpotentials in these conditions. The equations are easily integrated and we give the general expression for all superpotentials associated with Lovelock Lagrangians. We find, in particular, that in Einstein's theory, in any number of dimensions, the superpotential, valid at spatial and at null infinity, is that of Katz, Bicak and Lynden-Bell, the KBL superpotential. We also give explicitly the superpotential for Gauss-Bonnet theories of gravity. Finally, we find a simple expression for the superpotential of Einstein-Gauss-Bonnet theories with an anti-de Sitter background: it is minus the KBL superpotential, confirming, as it should, the calculation of the total mass-energy of spacetime at spatial infinity by Deser and Tekin.Comment: Scheduled to appear in Class. Quantum Grav. August 200

Universal features of dimensional reduction schemes from general covariance breaking

Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors, that successfully generalize to one side Kaluza–Klein gauge fields and to the other side extrinsic curvature and torsion of embedded spaces, thus fully characterizing the geometry of dimensional reduction. We obtain general formulas for the reduction of the main tensors and operators of Riemannian geometry. In particular, we provide what is probably the maximal possible generalization of Gauss, Codazzi and Ricci equations and various other standard formulas in Kaluza–Klein and embedded spacetimes theories. After general covariance breaking, part of the residual covariance is perceived by effective lower dimensional observers as an infinite dimensional gauge group. This reduces to finite dimensions in Kaluza–Klein and other few remarkable backgrounds, all characterized by the vanishing of appropriate lower dimensional tensors

Geometry of Brane-Worlds

The most general geometrical scenario in which the brane-world program can be implemented is investigated. The basic requirement is that it should be consistent with the confinement of gauge interaction, the existence of quantum states and the embedding in a bulk with arbitrary dimensions, signature and topology. It is found that the embedding equations are compatible with a wide class of Lagrangians, starting with a modified Einstein-Hilbert Lagrangian as the simplest one, provided minimal boundaries are added to the bulk. A non-trivial canonical structure is derived, suggesting a canonical quantization of the brane-world geometry relative to the extra dimensions, where the quantum states are set in correspondence with high frequency gravitational waves. It is shown that in the cases of at least six dimensions, there exists a confined gauge field included in the embedding structure. The size of extra dimensions compatible with the embedding is calculated and found to be different from the one derived with product topology.Comment: Minor changes and a correction to equation (22). 9 pages twocolumn Revte

On Global Conservation Laws at Null Infinity

The ``standard'' expressions for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are also obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity to use or not to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to N\oe ther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, the conserved quantities are in principle explicitly related to the sources of gravity through Einstein's equations, while standard definitions are not. The relations depend, however, on a rule for mapping spacetimes on backgrounds

Affine Gravity, Palatini Formalism and Charges

Affine gravity and the Palatini formalism contribute both to produce a simple and unique formula for calculating charges at spatial and null infinity for Lovelock type Lagrangians whose variational derivatives do not depend on second-order derivatives of the field components. The method is based on the covariant generalization due to Julia and Silva of the Regge-Teitelboim procedure that was used to define properly the mass in the classical formulation of Einstein's theory of gravity. Numerous applications reproduce standard results obtained by other secure but mostly specialized methods. As a novel application we calculate the Bondi energy loss in five dimensional gravity, based on the asymptotic solution given by Tanabe, Tanahashi and Shiromizu, and obtain, as expected, the same result. We also give the superpotential for Einstein-Gauss-Bonnet gravity and find the superpotential for Lovelock theories of gravity when the number of dimensions tends to infinity with maximally symmetrical boundaries. The paper is written in standard component formalism.Comment: The work is dedicated to Joshua Goldberg from whom I learned and got interested in conservation laws in General Relativity (J.K

Perturbations of brane worlds

We consider cosmological models where the universe, governed by Einstein's equations, is a piece of a five dimensional double-sided anti-de Sitter spacetime (that is, a "$Z_2$-symmetric bulk") with matter confined to its four dimensional Robertson-Walker boundary or "brane". We study the perturbations of such models. We use conformally minkowskian coordinates to disentangle the contributions of the bulk gravitons and of the motion of the brane. We find the restrictions put on the bulk gravitons when matter on the brane is taken to be a scalar field and we solve in that case the brane perturbation equations.Comment: 19 pages, no figures, RevTex, version to appear in Phys.Rev.D; minor changes in chap.V, polarisation tensor at page 13 correcte

Opportunities and challenges of China’s inquiry-based education reform in middle and high schools: Perspectives of science teachers and teacher educators

Consistent with international trends, an emergent interest in inquiry-based science teaching and learning in K-12 schools is also occurring in China. This study investigates the possibilities for and the barriers to enactment of inquiry-based science education in Chinese schools. Altogether 220 Chinese science teachers, science teacher educators and researchers (primarily from the field of chemistry education) participated in this study in August 2001. Participants represented 13 cities and provinces in China. We administered two questionnaires, one preceding and one following a 3-hour presentation by a US science educator and researcher about inquiry-based teaching and learning theories and practices. In each of three sites in which the study was conducted (Shanghai, Guangzhou and Beijing), questionnaires were administered, and four representative participants were interviewed. Our coding and analysis of quantifiable questionnaire responses (using a Likert scale), of open-ended responses, and of interview transcripts revealed enthusiastic interest in incorporating inquiry-based teaching and learning approaches in Chinese schools. However, Chinese educators face several challenges: (a) the national college entrance exam needs to align with the goals of inquiry-based teaching; (b) systemic reform needs to happen in order for inquiry-based science to be beneficial to students, including a change in the curriculum, curriculum materials, relevant resources, and teacher professional development; (c) class size needs to be reduced; and (d) an equitable distribution of resources in urban and rural schools needs to occur.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42933/1/10763_2005_Article_1517.pd