# What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics

## Abstract

**:**

## 1. Introduction

## 2. General Relativity from the Perspective of Sheaf Theory

## 3. Cohomological Conditions for Extending the Smooth Sheaf of Coefficients in General Relativity

- (I)
- For every open set U of X, a set denoted by $\mathbb{F}\left(U\right)$, and
- (II)
- For every inclusion $V\hookrightarrow U$ of open sets of X, a restriction morphism of sets in the opposite direction:$$r\left(U\right|V):\mathbb{F}(U)\to \mathbb{F}(V)$$
- (a)
- $r\left(U\right|U)$ = identity at $\mathbb{F}\left(U\right)$ for all open sets U of X.
- (b)
- $r\left(V\right|W)\circ r(U\left|V\right)=r\left(U\right|W)$ for all open sets $W\hookrightarrow V\hookrightarrow U$. Usually, the following simplifying notation is used: ${r\left(U\right|V)\left(s\right):=s|}_{V}$.

- (1)
- Local identity axiom of sheaf:Given $s,t\in \mathbb{F}\left(V\right)$ with ${s|}_{{V}_{a}}{=t|}_{{V}_{a}}$ for all $a\in I$, then $s=t$.
- (2)
- Gluing axiom of sheaf:Given ${s}_{a}\in \mathbb{F}\left({V}_{a}\right)$, ${s}_{b}\in \mathbb{F}\left({V}_{b}\right)$, $a,b\in I$, such that:$${s}_{a}{{|}_{{V}_{a}\cap {V}_{b}}={s}_{b}|}_{{V}_{a}\cap {V}_{b}},$$

## 4. Coping with Spacetime Singularities: Conceptual and Technical Aspects

## 5. Spacetime Extensions in Terms of Singularity-Free Distributional Algebra Sheaves

## 6. Topological Links in Geometrodynamics

## 7. The Borromean Rings as a Universal Nowhere Dense Singular Link

## 8. Conclusions

“Under the influence of the ideas of Faraday and Maxwell the notion developed that the whole of physical reality could perhaps be represented as a field whose components depend on four space-time parameters. If the laws of this field are in general covariant, that is, are not dependent on a particular choice of coordinate system, then the introduction of an independent (absolute) space is no longer necessary. That which constitutes the spatial character of reality is then simply the four-dimensionality of the field. There is then no “empty” space, that is, there is no space without a field.”[53]

“A field theory is not yet completely determined by the system of field equations. Should one admit the appearance of singularities? … It is my opinion that singularities must be excluded. It does not seem reasonable to me to introduce into a continuum theory points (or lines etc.) for which the field equations do not hold …”[54]

## Conflicts of Interest

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Zafiris, E.
What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics. *Universe* **2016**, *2*, 17.
https://doi.org/10.3390/universe2030017

**AMA Style**

Zafiris E.
What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics. *Universe*. 2016; 2(3):17.
https://doi.org/10.3390/universe2030017

**Chicago/Turabian Style**

Zafiris, Elias.
2016. "What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics" *Universe* 2, no. 3: 17.
https://doi.org/10.3390/universe2030017