On the properties of stationary configurations of a rotating self-gravitating ideal fluid with a vortex gravitational field

Background and Objectives: Within the framework of Einstein’s general-relativistic theory of gravity, that is, the general theory of relativity (GR), the properties of stationary distributions of a self-gravitating rotating continuous medium in the form of an ideal liquid with a barotropic equation of state p = wε are considered. Here, w = const, p is the pressure, and ε is the energy density of an ideal liquid. Materials and Methods: A stationary space-time compatible with a self-gravitating rotating continuous medium is described by a stationary cylindrical metric ds2 = A(x)dx2 + B(x)dφ2 + C(x)dz2 + 2E(x)dtdφ – D(x)dt2 , 0 ≤ φ ≤ 2π, where the metric coefficients A, B, C, D, E are functions of the radial coordinate x. This metric corresponds to a rotating space-time in which there is a vortex gravitational field. The latter is determined by means of the angular velocity ω of the field of tetrads ei(a) (xk), which are tangent to the considered Riemannian space. Here, the indices i, k are the world indices corresponding to the coordinates of the Riemannian space (base), and the index (a) is a local Lorentz index. For a vortex gravitational field, in contrast to a total gravitational field, it is possible to determine an energy-momentum tensor Tik(ω) satisfying the local conservation law ∇iTik(ω) = 0 relative to the metric of the corresponding static space in which ω = 0 (in the case under consideration, at a coefficient of E = 0). The tensor Tik(ω) has very exotic properties. For example, a weak energy condition is violated in it, since a p(ω) + ε(ω) < 0. For ordinary matter p + ε > 0. This property Tik(ω) contributes to the formation of wormholes in space-time. To study the properties of the considered configuration of a self-gravitating rotating ideal fluid and a vortex gravitational field, the corresponding Einstein gravitational equations are solved. Results: Solutions of Einstein’s gravitational equations in stationary space-time have been obtained with the metric presented above, that is, with a vortex gravitational field and with wormholes in the presence of a self-gravitating rotating ideal fluid with a limiting equation of state p = ε. At the same time, the obtained solutions describe the geometry of space-time of the so-called traversable wormholes, inside which gravitational forces Fg have a finite magnitude. A solution with a passable wormhole, in which Fg = 0,that is, without gravitational force, has also been obtained. In addition, solutions of Einstein’s vacuum equations Rik = 0 in space-time with the metric presented above have been obtained, that is, in the absence of a rotating continuous medium in the presence of only vortex gravitational field. The resulting solution describes the geometry of the wormhole space-time. Conclusion: Since the above-mentioned solution of gravitational equations with a wormhole is a solution to vacuum equations, that is, for empty space without matter, it is possible to suggest the presence of wormholes in outer space that exist a priori and also exist near the Earth.

vortex gravitational field, ideal fluid, rotation, wormholes