Shapovalov wave-like spacetimes

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton-Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric of space depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton-Jacobi equation of motion of test particles.Comment: 17 page

Deviation of geodesics and particle trajectories in a gravitational wave of the Bianchi type VI universe

For the Bianchi type VI universe, exact solutions of the equation of geodesic deviation in a strong primordial gravitational wave in a privileged coordinate system are obtained. The solutions refer to Shapovalov's gravitational-wave models of spacetime and allow the existence of complete integrals of the Hamilton-Jacobi equation for test particles. For all the solutions obtained, the analytical form of the tidal acceleration vector in a strong primordial gravitational wave is obtained. An explicit form of the coordinate transformation, an explicit form of the metric of the primordial gravitational wave of the Bianchi type VI universe, and the form of the tidal acceleration vector in the laboratory synchronous coordinate system are obtained. The synchronous coordinate system is associated with a freely falling observer and allows the observer to separate time and spatial coordinates, as well as to synchronize time at different points in space. The presented mathematical approach can be applied both in the general theory of relativity and in modified theories of gravity.Comment: 26 page

An Exact Model of a Gravitational Wave in the Bianchi III Universe Based on Shapovalov II Wave Spacetime and the Quadratic Theory of Gravity

Exact models of primordial gravitational waves in the Bianchi type-III universe were constructed on the basis of the quadratic theory of gravity with a scalar field and pure radiation in Shapovalov wave spacetimes of type II (subtype 2). Exact solutions of the field equations and scalar equation were obtained. The characteristics of pure radiation were determined. An explicit form of the scalar field functions included in the Lagrangian of the considered quadratic theory of gravity was found. The trajectories of the propagation of light rays in the considered gravitational wave models were obtained

Gravitational wave of the Bianchi VII universe: particle trajectories, geodesic deviation and tidal accelerations

For the gravitational wave model based on the type III Shapovalov wave space-time, test particle trajectories and the exact solution of geodesic deviation equations for the Bianchi type VII universe are obtained. Based on the found 4-vector of deviation, tidal accelerations in a gravitational wave are calculated. For the obtained solution in a privileged coordinate system, an explicit form of transformations into a synchronous reference system is found, which allows time synchronization at any points of space-time with separation of time and spatial coordinates. The synchronous reference system used is associated with a freely falling observer on the base geodesic. In a synchronous coordinate system, an explicit form of the gravitational wave metric, a 4-vector of geodesic deviation, and a 4-vector of tidal accelerations in a gravitational wave are obtained. The exact solution describes a variant of the primordial gravitational wave. The results of the work can be used to study the plasma radiation generated by tidal accelerations of a gravitational wave

Geodesic deviation and tidal acceleration in the gravitational wave of the Bianchi type IV universe

For models of gravitational waves in the Bianchi type IV universe, an exact solution of geodesic deviation equations is obtained. The spacetime models under consideration refer to the Shapovalov wave spaces of type III and type N according to Petrov's classification. The form of test particle trajectories, the geodesic deviation vector, and tidal acceleration in a gravitational wave are found. The results are presented both in the privileged coordinate system and in the laboratory synchronous coordinate system, where the freely falling observer is at rest. The resulting exact models describe primordial gravitational waves. The presented approach can be applied both to Einstein's classical theory of gravity and to modified theories of gravity