This is the first of series of papers in which we investigate stability of the spherically symmetric space-time with de Sitter center. Geometry, asymptotically Schwarzschild for large r and asymptotically de Sitter as r → 0, describes a vacuum nonsingular black hole for m ≥ mcr and particle-like self-gravitating structure for m < mcr where a critical value mcr depends on the scale of the symmetry restoration to de Sitter group in the origin. In this paper we address the question of stability of a vacuum non-singular black hole with de Sitter center to external perturbations. We specify first two types of geometries with and without changes of topology. Then we derive the general equations for an arbitrary density profile and show that in the whole range of the mass parameter m objects described by geometries with de Sitter center remain stable under axial perturbations. In the case of the polar perturbations we find criteria of stability and study in detail the case of the density profile ρ(r) = ρ0e −r 3 /r 2 0 rg where ρ0 is the density of de Sitter vacuum at the center, r0 = 3/κρ0 is de Sitter radius and rg is the Schwarzschild radius.

In the spherically symmetric case the dominant energy condition, together with the requirement of regularity at the center, asymptotic flatness and finiteness of the ADM mass, defines the family of asymptotically flat globally regular solutions to the Einstein equations which includes the class of metrics asymptotically de Sitter as r → 0. The source term corresponds to an r−dependent cosmological term Λµν invariant under boosts in the radial direction and evolving from the de Sitter vacuum Λgµν in the origin to the Minkowski vacuum at infinity. The ADM mass is related to cosmological term by m = (2G) −1 ∞ 0 Λ t t r 2 dr, with de Sitter vacuum replacing a central singularity at the scale of symmetry restoration. Space-time symmetry changes smoothly from the de Sitter group near the center to the Lorentz group at infinity through radial boosts in between. In the range of masses m ≥ mcrit, de Sitter-Schwarzschild geometry describes a vacuum nonsingular black hole (ΛBH), and for m < mcrit it describes G-lump -a vacuum selfgravitating particlelike structure without horizons. Quantum energy spectrum of G-lump is shifted down by the binding energy, and zero-point vacuum mode is fixed at the value corresponding (up to the coefficient) to the Hawking temperature from the de Sitter horizon.

We address the question of existence of regular spherically symmetric electrically charged solutions in Nonlinear Electrodynamics coupled to General Relativity. Stress-energy tensor of the electromagnetic field has the algebraic structure T 0 0 = T 1 1 . In this case the Weak Energy Condition leads to the de Sitter asymptotic at approaching a regular center. In de Sitter center of an electrically charged NED structure, electric field, geometry and stress-energy tensor are regular without Maxwell limit which is replaced by de Sitter limit: energy density of a field is maximal and gives an effective cut-off on self-energy density, produced by NED coupled to gravity and related to cosmological constant Λ. Regular electric solutions satisfying WEC, suffer from one cusp in the Lagrangian L(F ), which creates the problem in an effective geometry whose geodesics are world lines of NED photons. We investigate propagation of photons and show that their world lines never terminate which suggests absence of singularities in the effective geometry.To illustrate these results we present the particular example of the new exact analytic spherically symmetric electric solution with the de Sitter center.

We analyze the globally regular solution of the Einstein equations describing a black hole whose singularity is replaced by the de Sitter core. The de Sitter—Schwarzschild black hole (SSBH) has two horizons. Inside of it there exists a particlelike structure hidden under the external horizon. The critical value of mass parameter M cr1 exists corresponding to the degenerate horizon. It represents the lower limit for a black-hole mass. Below M cr1 there is no black hole, and the de Sitter-Schwarzschild solution describes a recovered particlelike structure. We calculate the Hawking temperature of SSBH and show that specific heat is broken and changes its sign at the value of mass M cr 2>M cr 1 which means that a second-order phase transition occurs at that point. We show that the Hawking temperature drops to zero when a mass approaches the lower limit M cr1 .

The requirements are formulated which lead to the existence of the class of globally regular solutions to the minimally coupled GR equations asymptotically de Sitter at the center. 1 The source term for this class, invariant under boosts in the radial direction, is classified as spherically symmetric vacuum with variable density and pressure T vac µν associated with an r−dependent cosmological term Λµν = 8πGT vac µν , whose asymptotic in the origin, dictated by the weak energy condition, is the Einstein cosmological term Λgµν , while asymptotic at infinity is de Sitter vacuum with λ < Λ or Minkowski vacuum. For this class of metrics the mass m defined by the standard ADM formula is related to both de Sitter vacuum trapped in the origin and to breaking of space-time symmetry. In the case of the flat asymptotic, space-time symmetry changes smoothly from the de Sitter group at the center to the Lorentz group at infinity through radial boosts in between. Geometry is asymptotically de Sitter as r → 0 and asymptotically Schwarzschild at large r. In the range of masses m ≥ mcrit, de Sitter-Schwarzschild geometry describes a vacuum nonsingular black hole (ΛBH), and for m < mcrit it describes G-lump -a vacuum selfgravitating particlelike structure without horizons. In the case of de Sitter asymptotic at infinity, geometry is asymptotically de Sitter as r → 0 and asymptotically Schwarzschild-de Sitter at large r. Λµν geometry describes, dependently on parameters m and q = Λ/λ and choice of coordinates, a vacuum nonsingular cosmological black hole, selfgravitating particlelike structure at the de Sitter background λgµν, and regular cosmological models with cosmological constant evolving smoothly from Λ to λ.

We address the question of existence of regular spherically symmetric electrically charged solutions in nonlinear electrodynamics (NED) coupled to general relativity. The stress–energy tensor of the electromagnetic field has the algebraic structure T00 = T11. In this case, the weak energy condition leads to the de Sitter asymptotic on approaching a regular centre. In the de Sitter centre of an electrically charged NED structure, electric field, geometry and stress–energy tensor are regular without the Maxwell limit which is replaced by the de Sitter limit: energy density of a field is maximal and gives an effective cut-off on self-energy density, produced by NED coupled to gravity and related to the cosmological constant Λ. Regular electric solutions, satisfying WEC, suffer from one cusp in the Lagrangian ℒ(F), which creates the problem in an effective geometry whose geodesics are world lines of NED photons. We investigate propagation of photons and show that their world lines never terminate which suggests absence of singularities in the effective geometry. To illustrate these results we present the particular example of the new exact analytic spherically symmetric electrically charged solution with the de Sitter centre.

We propose to describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor Λµν invariant under boosts in the radial direction. This proposal is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state p = −ρ is satisfied by the radial pressure, p Developments in particles and quantum field theory, as well as the confrontation of models with observations in cosmology [1], compellingly favour treating the cosmological constant Λ as a dynamical quantity.The Einstein equations with a cosmological term readwhere G µν is the Einstein tensor, T µν is the stress-energy tensor of a matter, and Λ is the cosmological constant.In the absence of matter described by T µν , Λ must be constant, since the Bianchi identities guarantee vanishing covariant divergence of the Einstein tensor, G µν ;ν = 0. In quantum field theory, the vacuum stress-energy tensor has the form < T µν >=< ρ vac > g µν which behaves like a cosmological term with Λ = 8πGρ vac .The idea that Λ might be variable has been studied for more than two decades (see [2,3] and references therein). In a recent paper on Λ-variability, Overduin and Cooperstock distinguish three approaches [4]. In the first approach Λg µν is shifted onto the right-hand side of the field equations (1) and treated as part of the matter content. This approach, characterized by Overduin and Cooperstock as being connected to dialectic materialism, goes back to Gliner who interpreted Λg µν as corresponding to vacuum stress-energy tensor with the equation of state p = −ρ [5], to Zel'dovich who connected Λ with the gravitational interaction of virtual particles [6], and to Linde who suggested that Λ can vary [7]. In the ref.[8] a cosmological model was proposed with the equation of state varying from p = −ρ to p = ρ/3. In contrast, idealistic approach prefers to keep Λ on the left-hand side of the Eq.(1) and treat it as a constant of nature. The third approach, allowing Λ to vary while keeping it on the lefthand side as a geometrical entity, was first applied by Dolgov in a model in which a classically unstable scalar field, non-minimally coupled to gravity, develops a negative energy density cancelling the initial positive value of a cosmological constant Λ [9]. Whenever variability of Λ is possible, it requires the presence of some matter source other than T µν = (8πG) −1 Λg µν , since the conservation equation G µν ;ν = 0 implies Λ = const in this case. This requirement makes it impossible to introduce a cosmological term as variable in itself. However, it is possible for a stress-energy tensor other than Λg µν .The aim of this letter is to show what the algebraic structure of a cosmological term can be in the spherically symmetric case, as suggested by the Petrov classification scheme [10] and by the Einstein field equations.In the spherically symmetric static case a line element can be written in the form [11]where dΩ 2 is the line element on ...

We present nonsingular cosmological models with a variable cosmological term described by the second-rank symmetric tensor Λ ν µ evolving from Λδ ν µ to λδ ν µ with λ < Λ. All Λ ν µ dominated cosmologies belong to Lemaître type models for an anisotropic perfect fluid. The expansion starts from a nonsingular nonsimultaneous de Sitter bang, with Λ on the scale responsible for the earliest accelerated expansion, which is followed by an anisotropic Kasner type stage. For a certain class of observers these models can be also identified as Kantowski-Sachs models with regular R regions. For Kantowski-Sachs observers the cosmological evolution starts from horizons with a highly anisotropic "null bang" where the volume of the spatial section vanishes. We study in detail the spherically symmetric case and consider the general features of cosmologies with planar and pseudospherical symmetries. Nonsingular Λ ν µ dominated cosmologies are Bianchi type I in the planar case and hyperbolic analogs of the Kantowski-Sachs models in the pseudospherical case. At late times all models approach a de Sitter asymptotic with small λ.