Contravariant Metrics Research Articles

Classical r-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics

A general scheme for the construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to Witten–Dijkgraaf–Verlinde–Verlinde associativity equations is formulated. The advantage is taken from the Rota–Baxter identity and some relation being counterpart of the modified Yang–Baxter identity from the classical r-matrix formalism. The scheme for the construction of Frobenius manifolds is illustrated on the algebras of formal Laurent series and meromorphic functions on Riemann sphere.

Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D

First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics g, and a pair of differential-geometric objects b, . If the determinant of the pencil vanishes for all λ, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.

A simpler method for researching fermions tunneling from black holes

A new simpler mathematic method is proposed to study fermions tunneling from black holes. According to this method, by using semiclassical approximation theory, it simplifies the Dirac equation of curved spacetime and then the relationship of the gamma matrix and the component of contravariant metric is considered in order to transform the set of difficult quantum equations into a simple equation. Finally, the fermion tunneling and Hawking radiation of black holes are obtained. The method is very effective and simple, and we will take the Schwarzschild black hole with global monopole and the higher-dimensional Reissner-Nordstrom de Sitter black hole as two examples to show the fact.

ELLIPTIC CURVES, ALGEBRAIC GEOMETRY APPROACH IN GRAVITY THEORY AND UNIFORMIZATION OF MULTIVARIABLE CUBIC ALGEBRAIC EQUATIONS

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the so called "gravitational theories with covariant and contravariant connection and metrics", it is shown that a wide variety of 3rd, 4th, 5th, 7th, 10th- degree algebraic equations exists in gravity theory. This is important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Since the obtained cubic algebraic equations are multivariable, the standard algebraic geometry approach for parametrization of two-dimensional cubic equations with the elliptic Weierstrass function cannot be applied. Nevertheless, for a previously considered cubic equation for reparametrization invariance of the gravitational Lagrangian and on the base of a newly introduced notion of "embedded sequence of cubic algebraic equations", it is demonstrated that in the multivariable case such a parametrization is also possible, but with complicated irrational and non-elliptic functions. After finding the solutions of a system of first-order nonlinear differential equations, these parametrization functions can be considered also as uniformization ones (depending only on the complex uniformization variable z) for the initial multivariable cubic equation.

Vierbein walls in condensed matter

The effective field, which plays the part of the vierbein in general relativity, can have topologically stable surfaces, vierbein domain walls, at which the effective contravariant metric is degenerate. We consider vierbein walls separating domains with flat spacetime which are not causally connected at the classical level. The possibility of a quantum mechanical connection between the domains is discussed.

Spaces with contravariant and covariant affine connections and metrics

The theory of spaces with contravariant and covariant affine connections, whose components differ not only in sign, and metrics [(L n ,g) spaces] is worked out within the framework of tensoranalysis over differentiable manifolds and in a volume necessary for further consideration of the kinematics of vector fields and the Lagrangiantheory of tensor fields over (L n ,g) spaces. The possibility of introducing affine connections for contravariant and covariant tensor fields, whose components differ not only in sign, over differentiable manifolds with finite dimensions is discussed. The action of the deviation operator, having an important role for deviation equations in gravitational physics, is considered for the case of contravariant and covariant vector fields over differentiable manifolds with different affine connections (called L n spaces). A deviation identity for contravariant vector fields is obtained. The notions of covariant, contravariant, covariant projective, and contravariant projective metric are introduced in (L n ,g) spaces. The action of the covariant and the Lie differential operator on the different types of metric is found. The concepts of symmetric covariant and contravariant (Riemannian) connection are defined and presented by means of the covariant and contravariant metric and the corresponding torsion tensors. The different types of relative tensor fields (tensor densities) as well as the invariant differential operators acting on them are considered. The invariant volume element and its properties under the action of different differential operators are investigated.

Post-Newtonian expansion of an algebraically extended theory of gravity

The post-Newtonian expansion is analyzed for an algebraically extended theory of gravity equivalent to a theory with a real, nonsymmetric metric, previously referred to as Algebraically Extended Hilbert Gravity (AHG). The hermiticity of the algebra-valued covariant and contravariant metrics is found to constrain the form of the expansion of the antisymmetric part of the real metric to one of two possibilities. Both of these display a qualitatively new technical feature: the lowest order equations are nonlinear, depriving the post-Newtonian expansion of its greatest asset. In one case, they lead to a solution with a negative Newtonian mass parameter. In the other case, they lead to a contact-type solution in which some metric components depend directly upon the stress–energy tensor rather than the integral of this distribution. The unphysical nature of these solutions rules out AHG as a physically viable alternative theory of gravitation.

Symmetry and internal time on the superspace of asymptotically flat geometries.

A difficulty with the canonical approach to quantum gravity, leading to attempts at "third quantization," is the absence of symmetry vectors on the superspace of three-metrics: vector fields that generate transformations of superspace leaving the action invariant. We show that on the superspace of asymptotically flat three-metrics, such symmetry vectors exist. They correspond to diffeomorphisms of each three-geometry that behave asymptotically as elements of the symmetry group at spatial infinity. The conserved momentum associated with a symmetry vector has a conjugate variable which can be regarded as an internal time coordinate of an isolated system. In particular, for asymptotic translations, a corresponding internal time is a center-of-mass coordinate. An appendix considers the natural contravariant and covariant metrics on superspace. Because natural contravariant metrics are not everywhere invertible, the associated covariant metrics are not everywhere defined.

Coordination: a vector-matrix description of transformations of overcomplete CNS coordinates and a tensorial solution using the Moore-Penrose generalized inverse

Neuronal organisms express their function, such as a movement, by multicomponental actions. Thus, the problem of how the central nervous system (CNS) coordinates the elements of a single action is fundamental to our understanding of brain function. Coordinated activation of multijointed "limbs" has also become an acute problem in modern multivariable control theory and engineering, such as robotics. Thus, a coherent interdisciplinary approach is expected, one that arrives at concepts and formalisms applicable to this problem both in living and man-made organisms. By treating coordination with coordinates, tensor network theory of the CNS, which explains transformations through the neuronal networks of natural non-orthogonal coordinates that are intrinsic to living organisms, may successfully integrate the diverse approaches to this general problem. A link between tensor network theory of the CNS and multivariable control engineering can be established if the latter is formulated in generalized non-orthogonal coordinates, rather than in conventional Cartesian expressions. In general terms, the problem of coordinating an overcomplete (more than necessary) number of components of an action can be resolved by a three-step tensorial scheme. A key operation is a covariant-to-contravariant transformation executed by the Moore-Penrose generalized inverse when, in an overcomplete manifold, the covariant metric tensor is singular. In the neuronal organization of the CNS, it is assumed that the cerebellum plays this role of acting as a contravariant metric. A quantitative example is also provided, in order to demonstrate the viability of the numerical and network-implementations.

Null hypersurfaces in Lorentzian manifolds: I

This paper is concerned with geometrical properties of null hypersurfaces in Lorentzian manifolds. Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. First, the contravariant metric cannot immediately be defined, so the connection cannot be specified uniquely in the normal way. Secondly, the normal is a null vector lying in the tangent plane, which makes it necessary to look for some other vector to rig the hypersurface, and makes it impossible to normalise the normal in the usual way. These problems are considered in this paper.