The Hamiltonian of Einstein affine-metric formulation of General Relativity

It is shown that the Hamiltonian of the Einstein affine-metric (first order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as is the case for the second order formulation. In the second order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables [arXiv: 0809.0097]. For the first order formulation, the necessity of such a redefinition "to correspond to diffeomorphism invariance" (reported by Ghalati [arXiv: 0901.3344]) is just an artifact of using the Henneaux-Teitelboim-Zanelli ansatz [Nucl. Phys. B 332 (1990) 169], which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani [Ann. Phys. 143 (1982) 357] is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second and first order formulations of metric GR. The first order formulation of Einstein-Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.Comment: 74 page

The Hamiltonian formulation of General Relativity: myths and reality

A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333] and of ADM [Arnowitt, Deser and Misner, in "Gravitation: An Introduction to Current Research" (1962) 227] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. We show that points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric $g_{\mu\nu}$ to lapse and shift functions and the three-metric $g_{km}$, which is not canonical. This proves that point (iv) is incorrect. Points (i-iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein's theory itself.Comment: References are added and updated, Introduction is extended, Subsection 3.5 is added, 83 pages; corresponds to the published versio

Hasemania piatan, a new characid species (Characiformes: Characidae) from headwaters of Rio de Contas, Bahia, Brazil

Hasemania piatan é descrita para a bacia do alto rio de Contas, Bahia, nordeste do Brasil. Esta pode ser facilmente diferenciada das congêneres pela presença de 18 raios principais na nadadeira caudal. A espécie nova difere ainda das congêneres pela combinação de sete raios ramificados na nadadeira dorsal, seis raios ramificados na nadadeira pélvica, base da nadadeira anal sem escamas, presença de apenas cinco infraorbitais e presença de uma mancha umeral. Pode ainda ser diferenciada por ter 1013 raios ramificados na nadadeira anal, 27-32 escamas na linha longitudinal, 10-12 escamas ao redor do pedúnculo caudal e um a três dentes maxilares.Hasemania piatan is described from the upper rio de Contas drainage, Bahia, northeastern Brazil. It can be easily distinguished from its congeners by having 18 principal caudal-fin rays. The new species differs further from congeners by a combination of seven branched dorsal-fin rays, six branched pelvic-fin rays, anal-fin base not covered by scales, presence of only five infraorbitals, and presence of a humeral blotch. It also can be distinguished by having 10-13 branched anal-fin rays, 27-32 scales on longitudinal series, 10-12 circumpeduncular scales, and one to three maxillary teeth