## Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance

*This paper dealing with Colombeau extension of the Einstein ﬁeld equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [5]-[30] and [15][31]. The regularizations of singularities present in some Colombeau solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein ﬁld equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates. However the Tolman formula for the total energy E _{T} of a static and asymptotically ﬂat spacetime, gives E_{T} = m, as it should be. The vacuum energy density of free scalar quantum ﬁeld Φ with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the inﬂuence of gravity on quantum ﬁelds should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well behaved spacetime evolutions where the vacuum energy density of free quantum ﬁelds is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity eﬀect ﬁnds its roots in the singular behavior of quantum ﬁelds on curved distributional spacetimes. In particular we obtain that the vacuum ﬂuctuations *

*⟨*

*Φ*

^{2}*⟩*

*has a singular behavior on BHs horizon . r*

_{+}:*⟨*

*Φ*

^{2}(r)*⟩*

*˜ |r − r+|*

^{−2}. A CHALLENGE TO THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained successfully.**For more information contact author**

**Jaykov Foukzon
**Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel.

**Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia.**

**All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia.**

**E-mail: jaykovfoukzon@list.ru**

**View Volume: **http://bp.bookpi.org/index.php/bpi/catalog/book/94

Books Colombeau nonlinear generalized functions curved spacetime Distributional Riemannian Distributional Schwarzschild Geometry Geometry mathematics nonsmooth regularization quantum elds Schwarzschild Horizon Schwarzschild singularity Science smooth regularization vacuum dominance vacuum uctuations