It is shown that general relativity coupled to nonlinear electrodynamics (NED) with the Lagrangian L(F ) , F = Fµν F µν having a correct weak field limit, leads to nontrivial spherically symmetric solutions with a globally regular metric if and only if the electric charge is zero and L(F ) tends to a finite limit as F → ∞ . Properties and examples of such solutions, which include magnetic black holes and solitonlike objects (monopoles), are discussed. Magnetic solutions are compared with their electric counterparts. A duality between solutions of different theories specified in two alternative formulations of NED (called the F P duality) is used as a tool for this comparison.

For self-gravitating, static, spherically symmetric, minimally coupled scalar fields with arbitrary potentials and negative kinetic energy (favored by the cosmological observations), we give a classification of possible regular solutions to the field equations with flat, de Sitter and AdS asymptotic behavior. Among the 16 presented classes of regular rsolutions are traversable wormholes, Kantowski-Sachs (KS) cosmologies beginning and ending with de Sitter stages, and asymptotically flat black holes (BHs). The Penrose diagram of a regular BH is Schwarzschild-like, but the singularity at r = 0 is replaced by a de Sitter infinity, which gives a hypothetic BH explorer a chance to survive. Such solutions also lead to the idea that our Universe could be created from a phantomdominated collapse in another universe, with KS expansion and isotropization after crossing the horizon. Explicit examples of regular solutions are built and discussed. Possible generalizations include k -essence type scalar fields (with a potential) and scalar-tensor theories of gravity. Observations provide more and more evidence that the modern accelerated expansion of our Universe is governed by a peculiar kind of matter, called dark energy (DE), characterized by negative values of the pressure to density ratio w . Moreover, by current estimates, even w < −1 seems rather likely [1,2,3,4,5,6], though many of such estimates are model-dependent. Thus, assuming a perfect-fluid DE with w = const implies, using various observational data (CMB, type Ia supernovae, large-scale structure), −1.39 < w < −0.79 at 2σ level [4]. Considerations of a variable DE equation of state [1, 2] also allow highly negative values of w . A model-independent study [5] of data sets containing 172 SNIa showed a preferable range −1.2 < w < −1 for the recent epoch. Similar figures follow from an analysis of the Chandra telescope observations of hot gas in 26 X-ray luminous dynamically relaxed galaxy clusters [6]: w = −1.20 +0.24 −0.28 . Moreover, a highly negative w makes negligible the undesirable DE contribution to the total energy density in the period of structure formation. Thus, even if the cosmological constant, giving precisely w = −1 , is still admitted by observations as possible DE, there is a need for a more general framework allowing w < −1 .The perfect-fluid description of DE is plagued with instability at small scales due to imaginary velocity of sound; more consistent descriptions providing w < −1 use self-interacting scalar fields with negative kinetic energy (phantom scalars) or tachyonic fields [7,8,9] (see also references therein). To avoid the obvious quantum instability, a phantom scalar may perhaps be regarded as an effective field description following from an underlying theory with positive energies [10,11]. Curiously, in a classical setting, a massless phantom field even shows a more stable behavior than its usual couterpart [12,13].A fundamental origin of phantom fields is under discussion, but they naturally appear in some models of string theory ...

The condition R = 0, where R is the four-dimensional scalar curvature, is used for obtaining a large class (with an arbitrary function of r) of static, spherically symmetric Lorentzian wormhole solutions. The wormholes are globally regular and traversable, can have throats of arbitrary size and can be both symmetric and asymmetric. These solutions may be treated as possible wormhole solutions in a brane world since they satisfy the vacuum Einstein equations on the brane where effective stress-energy is induced by interaction with the bulk gravitational field. Some particular examples are discussed.

We use the general solution to the trace of the 4-dimensional Einstein equations for static, spherically symmetric configurations as a basis for finding a general class of black hole (BH) metrics, containing one arbitrary function gtt = A(r) which vanishes at some r = r h > 0, the horizon radius. Under certain reasonable restrictions, BH metrics are found with or without matter and, depending on the boundary conditions, can be asymptotically flat or have any other prescribed asymptotic. It is shown that our procedure generically leads to families of globally regular BHs with a Kerr-like global structure as well as symmetric wormholes. Horizons in space-times with zero scalar curvature are shown to be either simple or double. The same is generically true for horizons inside a matter distribution, but in special cases there can be horizons of any order. A few simple examples are discussed. A natural application of the above results is the brane world concept, in which the trace of the 4D gravity equations is the only unambiguous equation for the 4D metric, and its solutions can be continued into the 5D bulk according to the embedding theorems.Maeda and Sasaki [11] from 5-dimensional gravity with the aid of the Gauss and Codazzi equations [28]:

We study the structure and stability of the recently discussed spherically symmetric Brans-Dicke black-hole type solutions with an infinite horizon area and zero Hawking temperature, existing for negative values of the coupling constant ω . These solutions split into two classes: B1, whose horizon is reached by an infalling particle in a finite proper time, and B2, for which this proper time is infinite. Class B1 metrics are shown to be extendable beyond the horizon only for discrete values of mass and scalar charge, depending on two integers m and n. In the case of even m − n the space-time is globally regular; for odd m the metric changes its signature at the horizon. All spherically symmetric solutions of the Brans-Dicke theory with ω < −3/2 are shown to be linearly stable against spherically symmetric perturbations. This result extends to the generic case of the Bergmann-Wagoner class of scalar-tensor theories of gravity with the coupling function ω(φ) < −3/2 .

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We study the stability of static, spherically symmetric solutions to the Einstein equations with a scalar field as the source. We describe a general methodology of studying small radial perturbations of scalar-vacuum configurations with arbitrary potentials V (φ), and in particular space-times with throats (including wormholes), which are possible if the scalar is phantom. At such a throat, the effective potential for perturbations V eff has a positive pole (a potential wall) that prevents a complete perturbation analysis. We show that, generically, (i) V eff has precisely the form required for regularization by the known S-deformation method, and (ii) a solution with the regularized potential leads to regular scalar field and metric perturbations of the initial configuration. The wellknown conformal mappings make these results also applicable to scalar-tensor and f (R) theories of gravity. As a particular example, we prove the instability of all static solutions with both normal and phantom scalars and V (φ) ≡ 0 under spherical perturbations. We thus confirm the previous results on the unstable nature of anti-Fisher wormholes and Fisher's singular solution and prove the instability of other branches of these solutions including the anti-Fisher "cold black holes".

We give a comparative description of different types of regular static, spherically symmetric black holes (BHs) and discuss in more detail their particular type, which we suggest to call black universes. The latter have a Schwarzschild-like causal structure, but inside the horizon there is an expanding Kantowski-Sachs universe and a de Sitter infinity instead of a singularity. Thus a hypothetic BH explorer gets a chance to survive. Solutions of this kind are naturally obtained if one considers static, spherically symmetric distributions of various (but not all) kinds of phantom matter whose existence is favoured by cosmological observations. It also looks possible that our Universe has originated from phantom-dominated collapse in another universe and underwent isotropization after crossing the horizon. An explicit example of a black-universe solution with positive Schwarzschild mass is discussed.