Theoretical and Mathematical Physics

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

V. G. Krechet^a, V. B. Oshurko^ab, A. E. Kisser<sup>a

a</sup> Moscow State University of Technology "STANKIN"
^b Prokhorov General Physics Institute of the Russian Academy of Sciences

Abstract: 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 $p = w\epsilon$ are considered. Here, $w = \mathrm{const}$, $p$ is the pressure, and $\epsilon$ 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 $ds^2 = A(x)dx^2 + B(x)d\phi^2 + C(x)dz^2 + 2E(x)dtd\phi- D(x)dt^2$, $0 \leq \phi \leq 2\pi$, 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 $\omega$ of the field of tetrads $e^i_{(a)}$ ($x^k$), 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 $T^i_k(\omega)$ satisfying the local conservation law $\triangledown_iT^i_k(\omega) = 0$ relative to the metric of the corresponding static space in which $\omega = 0$ (in the case under consideration, at a coefficient of $E = 0$). The tensor $T^i_k(\omega)$ has very exotic properties. For example, a weak energy condition is violated in it, since a $p(\omega) + \epsilon(\omega) < 0$. For ordinary matter $p + \epsilon > 0$. This property $T^i_k(\omega)$ 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 $p = \epsilon$. At the same time, the obtained solutions describe the geometry of space-time of the so-called traversable wormholes, inside which gravitational forces $\vec{F}_g$ have a finite magnitude. A solution with a passable wormhole, in which $\vec{F}_g = 0$, that is, without gravitational force, has also been obtained. In addition, solutions of Einstein's vacuum equations $R_{ik} = 0$ 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.

Keywords: vortex gravitational field, ideal fluid, rotation, wormholes.

UDC: 530.12:531.51

Received: 27.06.2025
Revised: 28.11.2025
Accepted: 10.09.2025

DOI: 10.18500/1817-3020-2025-25-4-396-407