This article gives a comprehensive review on thick brane solutions and related topics. Such models have attracted much attention from many aspects since the birth of the brane world scenario. In many works, it has been usually assumed that a brane is an infinitely thin object; however in more general situations, one can no longer assume this. It is also widely considered that more fundamental theories such as string theory would have a minimal length scale. Many multidimensional field theories coupled to gravitation have exact solutions of gravitating topological defects, which can represent our brane world. The inclusion of brane thickness can realize a variety of possible brane world models. Given our understanding, the known solutions can be classified into topologically non-trivial solutions and trivial ones. The former class contains solutions of a single scalar (domain walls), multi-scalar, gauge-Higgs (vortices), Weyl gravity and so on. As an example of the latter class, we consider solutions of two interacting scalar fields. Approaches to obtain cosmological equations in the thick brane world are reviewed. Solutions with spatially extended branes (S-branes) and those with an extra time-like direction are also discussed.

In this paper spherically symmetric solutions to 5D Kaluza-Klein theory, with "electric" and/or "magnetic" fields are investigated. It is shown that the global structure of the spacetime depends on the relation between the "electrical" and "magnetic" Kaluza-Klein fields. For small "magnetic" field we find a wormhole-like solution. As the strength of the "magnetic" field is increased relative to the strength of the "electrical" field, the wormholelike solution evolves into a finite or infinite flux tube depending on the strengths of the two fields. For the large "electric" field case we conjecture that this solution can be considered as the mouth of a wormhole, with the G 55 , G 5t and G 5ϕ components of the metric acting as the source of the exotic matter necessary for the formation of the wormhole's mouth. For the large "magnetic" field case a 5D flux tube forms, which is similar to the flux tube between two monopoles in Type-II superconductors, or the hypothesized color field flux tube between two quarks in the QCD vacuum.

We consider a gravitating spherically symmetric configuration consisting of a scalar field nonminimally coupled to ordinary matter in the form of a perfect fluid. For this system we find static, regular, asymptotically flat solutions for both relativistic and nonrelativistic cases. It is shown that the presence of the nonminimal interaction leads to substantial changes both in the radial matter distribution of the star and in the star's total mass. A simple stability test indicates that, for the choice of parameters used in the paper, the solutions are unstable.

It is shown that if a metric in quantum gravity can be decomposed as a sum of classical and quantum parts, then Einstein quantum gravity looks approximately like modified gravity with a nonminimal interaction between gravity and matter.

A thick brane in six dimensions is constructed using two scalar fields. The field equations for 6D gravity plus the scalar fields are solved numerically. This thick brane solution shares some features with previously studied analytic solutions, but has the advantage that the energy-momentum tensor which forms the thick brane comes from the scalar fields rather than being put in by hand. Additionally the scalar fields which form the brane also provide a universal, nongravitational trapping mechanism for test fields of various spins.

We consider rotating Lorentzian wormholes with a phantom field in five dimensions. These wormhole solutions possess equal angular momenta and thus represent cohomogeneity-1 configurations. For a given size of the throat, the angular momenta are bounded by the value of the corresponding extremal Myers-Perry black hole, which represents the limiting configuration. With increasing angular momenta the throat becomes increasingly deformed. At the same time, the violation of the null energy condition decreases to zero, as the limiting configuration is approached. Symmetric wormhole solutions satisfy a Smarr-like relation, which is analogous to the Smarr relation of extremal black holes. A stability analysis shows that the unstable mode of the static wormhole solutions vanishes when the angular momentum exceeds some critical value.

We consider configurations consisting of a gravitating nonlinear spinor field ψ, with a nonlinearity of the type λ ψ ψ 2 , minimally coupled to Maxwell and Proca fields through the coupling constants QM [U(1) electric charge] and QP , respectively. In order to ensure spherical symmetry of the configurations, we use two spin-1/2 fields having opposite spins. By means of numerical computations, we find families of equilibrium configurations with a positive Arnowitt-Deser-Misner (ADM) mass described by regular zero-node asymptotically flat solutions for static Maxwell and Proca fields and for stationary spinor fields. For the case of the Maxwell field, it is shown that, with increasing charge QM , the masses of the objects increase and diverge as the charge tends to a critical value. For negative values of the coupling constant λ, we demonstrate that, by choosing physically reasonable values of this constant, it is possible to obtain configurations with masses comparable to the Chandrasekhar mass and with effective radii of the order of kilometers. It enables us to speak of an astrophysical interpretation of such systems, regarding them as charged Dirac stars. In turn, for the system with the Proca field, it is shown that the mass of the configurations also grows with increasing both |λ| and the coupling constant QP . Although in this case the numerical calculations do not allow us to make a definite conclusion about the possibility of obtaining masses comparable to the Chandrasekhar mass for physically reasonable values of λ, one may expect that such masses can be obtained for certain values of free parameters of the system under consideration.

We consider a configuration consisting of a wormhole filled by a perfect fluid. Such a model can be applied to describe stars as well as neutron stars with a nontrivial topology. The presence of a tunnel allows for motion of the fluid, including oscillations near the core of the system. Choosing the polytropic equation of state for the perfect fluid, we obtain static regular solutions. Based on these solutions, we consider small radial oscillations of the configuration and show that the solutions are stable with respect to linear perturbations in the external region.