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This paper dealing with extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1], [2], [3], [4], [5], [6], [7] and [32]. The regularizations of singularities presented in some 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 field 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 ET of a static and asymptotically flat spacetime, gives ET = m, as it should be. The vacuum energy density of free scalar quantum field Φ with a *Corresponding author: E-mail: jaykovfoukzon@list.ru; Foukzon et al.; BJMCS, 13(6): 1-54, 2016; Article no.BJMCS.19235 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 spacetimes. In particular we obtain that the vacuum fluctuations 〈 Φ 〉 have a singular behavior on BHs horizon r+: 〈 Φ (r) 〉 |r − r+| .

Distributional SAdS BH Spacetime-Induced Vacuum Dominance