Ja. Foukzon¹, A.A. Potapov², E.R. Men'kova³

1 Center for Mathematical Sciences, Israel Institute of Technology (Haifa, Israel)

2 V.A. Kotel’nikov Institute of Radioengineering and Electronics of RAS (Moscow, Russia);

3 All-Russian Research Institute for Optical and Physical Measurements (Moscow, Russian)

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.

Foukzon Ja., Potapov A.A., Men'kova E.R. The point free Colombeau geometry in singular general relativity. Nonlinear World. 2021. V. 19. № 1. 2021. P. 58−72. DOI: https://doi.org/10.18127/j20700970-202101-06

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