On Linear Topological Classification of Spaces of Continuous Functions in the Topology of Pointwise Convergence

Abstract

We study and report on the class of vacuum Maxwell fields in Minkowski space that possess a non-degenerate, diverging, principal null vector field (null eigenvector field of the Maxwell tensor) that is tangent to a shear-free null geodesics congruence. These congruences can be either surface forming (the tangent vectors being proportional to gradients) or not, i.e., the twisting congruences. In the non-twisting case, the associated Maxwell fields are precisely the Lienard-Wiechert fields, i.e., those Maxwell fields arising from an electric monopole moving on an arbitrary worldline. The null geodesic congruence is given by the generators of the light-cones with apex on the worldline. The twisting case is much richer, more interesting and far more complicated.In a twisting subcase, where our main interests lie, the following strange interpretation can be given. If we allow the real Minkowski space to be complexified so that the real Minkowski coordinates x a take complex values, i.e., x a → z a = x a + iy a with complex metric g = η ab dz a dz b , the real vacuum Maxwell equations can be extended into the complex space and rewritten asThis subcase of Maxwell fields can then be extended into the complex space so as to have as source, a complex analytic worldline, i.e., to now become complex Lienard-Wiechart fields. When viewed as real fields on the real Minkowski space (z a = x a ), they possess a real principal null vector that is shear-free but twisting and diverging. The twist is a measure of how far the complex worldline is from the real 'slice'. Most Maxwell fields in this subcase are asymptotically flat with a time-varying set of electric and magnetic moments, all depending on the complex displacements and the complex velocities.

On uniformly continuous maps between function spaces

“…On the other hand, we infer from (3) that |g j (y j ) − g k (y j )| = 2p. The obtained contradiction proves (2).…”

“…[16, page 361], [12]. It was shown by Arhangel'skii [2], Baars and de Groot [4] that for an infinite Polish zero-dimensional space X which is either compact or not σ-compact, 1 A normal space X is strongly countable-dimensional if X can be represented as a countable union of closed finite-dimensional subspaces.…”

Solvable Lie algebras with naturally graded nilradicals and their invariants

“…For non-algebraic Lie algebras, specially for those that are solvable, we find rational or even transcendental invariants. These also find applications in representation theory or in classical integrable Hamiltonian systems [5,15]. In fact, the algorithm usually applied to compute these invariants [16,17], based on a system of linear first order partial differential equations, does not exclude the existence of irrational invariants, nor there is any physical reason for the invariants to be polynomials.…”

“…For high dimensional Lie algebras, it is sometimes convenient to work with the analogue of formula (5) in terms of differential forms. Let L(g) = R {dω i } 1≤i≤dim g be the linear subspace of 2 g * generated by the Maurer-Cartan forms dω i of g. If ω = a i dω i (a i ∈ R) is a generic element of L(g), there always exists an integer j 0 (ω) ∈ N such that…”

“…Although semisimple Lie algebras occupy a central position whitin the Lie algebras appearing in physical models (Lorentz algebra, su(N) and so(p, q) series, symplectic algebras sp(N, R), etc. ), as well as various semidirect products, like the Poincaré or the inhomogeneous Lie algebras, the class of solvable algebras has shown to be of considerable interest, as follows from their applicability to the theory of completely integrable Hamiltonian systems or non-abelian gauge theories [5,6]. While the classification of semisimple Lie algebras constitutes a classical result, solvable Lie algebras over the real field R have been classified only up to dimension six ( [7,8,9] and references therein).…”