In the spherically symmetric case the requirements of regularity of density and pressures and finiteness of the ADM mass m, together with the weak energy condition, define the family of asymptotically flat globally regular solutions to the Einstein minimally coupled equations which includes the class of metrics asymptotically de Sitter as r → 0. A source term connects smoothly de Sitter vacuum in the origin with the Minkowski vacuum at infinity and corresponds to anisotropic vacuum defined macroscopically by the algebraic structure of its stress-energy tensor which is invariant under boosts in the radial direction. Dependently on parameters, geometry describes vacuum nonsingular black holes, and self-gravitating particle-like structures whose ADM mass is related to both de Sitter vacuum trapped in the origin and smooth breaking of space-time symmetry. The geometry with the regular de Sitter center has been applied to estimate geometrical limits on sizes of fundamental particles, and to evaluate the gravito-electroweak unification scale from the measured mass-squared differences for neutrino. * Talk
I. THE EINSTEIN COSMOLOGICAL TERMWhich arrow does fly forever? An arrow which hits the goal.Vladimir NabokovIn 1917 Einstein introduced a cosmological term into his equations describing gravity as space-time geometry (G-field) generated by matterto make them consistent with Mach's principle, one of his basic motivations [1], which reads: some matter has the property of inertia only because there exists also some other matter in the Universe [2]. When Einstein found that Minkowski geometry is the regular solution to (1) without source termperfectly describing inertial motion in the absence of a matter, he modified his equations by adding the cosmological term Λg µν in the hope that modified equationswill have reasonable regular solutions only when mattaer is present (if matter is the source of inertia, then in case of its absence there should not be any inertia [3]). The by-product of this hypothesis was the static Einstein cosmology in which the task of Λ was to make a universe The story of abandoning Λ by Einstein is typically told as dominated by successes of FRW cosmology confirmed by Hubble's discovery of the Universe expansion.In reality the first reason was de Sitter solution: soon after introducing Λg µν , in the same year 1917, de Sitter found quite reasonable solution to the equation (3) with T µν = 0,which made evident that a matter is not necessary to produce the property of inertia [4]. In de Sitter geometry [5]Λ must be constant by virtue of the contracted Bianchi identitiesIt plays the role of a universal repulsion whose physical sense remained obscure during several decades when de Sitter geometry has been mainly used as a simple testing ground for developing the quantum field technics in a curved space-time. In 1965 two papers shed some light on the physical nature of the de Sitter geometry. In the first Sakharov suggested that gravitational effects can dominate an equation of state at superhigh densi...