The point free Colombeau geometry in singular general relativity

The problem statement. We argue that the canonical interpretation of the Schwarzschild spacetime in contemporary general relativity is wrong and that revision is needed. And we argue that the Schwarzschild solution is impossible to treat classically, since the Levi-Cività connection is not available for the whole Schwarzschild spacetime (Sch,gijSch (t r, , ,θϕ)) ; where Sch=×(({r ≥ 2m} {∪ ≤ ≤0 r 2m})×S2) ; but it can only be treated by using an embedding of the classical Schwarzschild metric tensor gijSch; ,i j =1,2,3,4 into Colombeau algebra δ(4,Σ),Σ= ={r 2m} {∪ =r 0} supergeneralized functions. The classical Schwarzschild spacetime could be extended up to the distributional semi-Riemannian manifold endowed on the tangent bundle with the Colombeau distributional metric tensor. The aim. The development of new physical interpretation for the distributional curvature scalar (Rε( )r )ε and square scalar (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )εis aimed. Results. The Schwarzschild solution using Colombeau distributional geometry without leaving Schwarzschild coordinates (t r, , ,θϕ) is studied. We obtain that the distributional Ricci tensor and the curvature scalar are δ-type, (R rε( ))ε=−m rδ( − 2m) ,>0 . The practical value. As distributional square scalars are essentially nonclassical Colombeau type distributions: (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρσμνε, ( )r )ε∈(3 )\ ′(3 ), this provides a new physical interpretation for the distributional curvature scalar (R rε( ))ε and square scalars (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )ε.

Maximal closed ideals of the Colombeau Algebra of Generalized functions

In this paper we investigate the structure of the set of maximal ideals of $${{\mathcal {G}}}(\Omega )$$ G ( Ω ) . The method of investigation passes through the use of the $$m-$$ m - reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.

Regular generalized solutions to semilinear wave equations

The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

On Products of Distributions in Colombeau Algebra

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.