Phantom solutions with scalar fields in GR with different potentials

Abstract

Domain, wormhole, spherically symmetric and cylindrically symmetric solutions in general relativity created with two phantom scalar fields with potentials of the 4th, 6th and 8th orders are investigated. It has been shown that the solutions with finite energy exist for some specific values of the parameters m₁, m₂ only. Corresponding field equations are solved numerically as nonlinear eigenvalue problem where the parameters m₁, m₂ are eigenvalues and scalar fields are eigenfunctions. The phantom solutions depending on values of scalar fields at the center of phantom domain, wormhole, spherically symmetrical and cylindrically symmetric solutions are obtained. The dependence of parameters m₁, m₂ on initial values χ₀ is investigated. The solutions depending on the value of scalar field at the centers of the domain wall, throat, boson star, and string for different potentials are obtained. The dependence of m₁, m₂ parameters on the initial values χ₀ for different potentials is presented. It is shown that for a phantom cosmic string with parameters χ₀ = 0.7, ϕ₀ =1, λ₁ = 0.15, λ₂ = 1.1 there is no solution with a fourth-order potential term. For a domain wall created by usual scalar field (ε = +1) and having potential term of the 8th order with the parameters χ₀ = 0.7 for ϕ₀ = 1, λ₁ = 0.15, λ₂ = 1.1 solutions do not exist also. This allows us to conclude that the existence of extended solutions essentially depends on the form of the potential term of scalar fields. For each pair of eigenvalues m1, m2, the energy density of the domain wall, wormhole, boson star and cosmic strings T⁰ ₀ is calculated and the dependence of this density on the corresponding coordinate is constructed from the obtained data.