N-dimensional geometries and Einstein equations from systems of PDE's

The aim of the present work is twofold: first, we show how all the $n$-dimensional Riemannian and Lorentzian metrics can be constructed from a certain class of systems of second-order PDE's which are in duality to the Hamilton-Jacobi equation and second we impose the Einstein equations to these PDE's

Quantum Mechanics in Non-Inertial Frames with a Multi-Temporal Quantization Scheme: II) Non-Relativistic Particles

The non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid non-inertial frames, containing the standard rigidly linear accelereted and rotating ones, is given in the framework of parametrized Galilei theories. Then the multi-temporal quantization, in which the gauge variables, describing the non-inertial effects, are not quantized but considered as c-number generalized times, is applied to non relativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of center of mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid non-inertial frames are recovered as special cases

Torsion, Dirac Field, Dark Matter and Dark Radiation

The role of torsion and a scalar field $\phi$ in gravitation, especially, in the presence of a Dirac field in the background of a particular class of the Riemann-Cartan geometry is considered here. Recently, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4,1) Pontryagin density, where the scalar field $\phi$ causes the de Sitter connection to have the proper dimension of a gauge field. In this article the scalar field has been linked to the dimension of the Dirac field. Here we get the field equations for the Dirac field and the scalar field in such a way that both of them appear to be mutually non-interacting. In this scenario the scalar field appears to be a natural candidate for the dark matter and the dark radiation

Complex structures and the Elie Cartan approach to the theory of spinors

Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.Comment: 10

On the Einstein-Weyl and conformal self-duality equations

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar PDEs for a general anti-self-dual conformal structure in neutral signature.This is the accepted manuscript. The final version is available at http://scitation.aip.org/content/aip/journal/jmp/56/8/10.1063/1.4927251

Gravitational Constant and Torsion

Riemann-Cartan space time $U_{4}$ is considered here. It has been shown that when we link topological Nieh-Yan density with the gravitational constant then we get Einstein-Hilbert Lagrangian as a consequence.Comment: 8 page

On N=8 attractors

We derive and solve the black hole attractor conditions of N=8 supergravity by finding the critical points of the corresponding black hole potential. This is achieved by a simple generalization of the symplectic structure of the special geometry to all extended supergravities with $N>2$. There are two solutions for regular black holes, one for 1/8 BPS ones and one for the non-BPS. We discuss the solutions of the moduli at the horizon for BPS attractors using N=2 language. An interpretation of some of these results in N=2 STU black hole context helps to clarify the general features of the black hole attractors.Comment: 15 page

E_7 and the tripartite entanglement of seven qubits

In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group [SL(2,C)]^3 and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain N=2 supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to N=8. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group E_7(C) and that the entanglement measure is given by Cartan's quartic E_7 invariant.Comment: Minor improvements. 15 page late

Weyl's Lagrangian in teleparallel form

The main result of the paper is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that this gives the Weyl Lagrangian up to a nonlinear change of dynamical variable. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J. B. Griffiths and R. A. Newing

Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.Comment: 20 Pages, 3 Figure