Spatially Homogeneous Models Stäckel Spaces of Type (2.1)

Abstract

All classes of spatially homogeneous space-time models are found that allow the integration of the equations of motion of test particles and the eikonal equation by the method of complete separation of variables according to type (2.1). Four classes of model data are obtained. The resulting models can be applied in any modified metric theories of gravity. Two of the above models allow solutions of the Einstein equations with a cosmological constant and radiation. For the models of a spatially homogeneous Universe with a cosmological constant and radiation obtained in Einstein's theory of gravity, the Hamilton-Jacobi equations of motion of the test particles and the eikonal equation for radiation are integrated by the method of separation of variables.

Wave-Like Exact Models with Symmetry of Spatial Homogeneity in the Quadratic Theory of Gravity with a Scalar Field

“…For Shapovalov's wave-like spacetimes to be spatially homogeneous, it is necessary that they admit three-parameter groups of motion with space-like orbits. As we showed earlier [36], for Shapovalov spaces of type II there are two types of spatially homogeneous models-subtypes B1 and B2.…”

Wave-like exact models with symmetry of spatial homogeneity in the quadratic theory of gravity with a scalar field

“…so that they admit three-parameter groups of motion with space-like orbits. As we showed earlier [31], for Shapovalov spaces of type II there are two types of spatially homogeneous models -subtypes B1 and B2.…”

“…The ability to accurately integrate the equations of motion of test particles makes Stäckel spaces an important tool in gravity problems, including modified theories. We also investigated both the general properties of Stäckel spaces with dust matter [25] and radiation [26], and spatially homogeneous models of Stäckel spaces that are of interest for cosmology [27], [28], [29], [30], [31].…”

“…Then from the equations ( 25), ( 27), (31) there follow two solutions depending on the value of the constant σ. For σ = −1/2 -one solution, for σ = −1/2another solution.…”

Shapovalov Wave-Like Spacetimes

“…Stäckel and conformal Stäckel spaces are of great interest for metric theories of gravity, since they allow explicitly, in quadratures, to integrate the equations for the motion of test particles and for radiation propagation and, thereby, determine the form of geodesic lines of spacetime along which test particles move in gravitational field. The application of these mathematical tools is possible in various metric theories of gravity, including modified theories of gravity with various types of matter (see [12][13][14][15][16][17][18][19][20][21][22][23][24][25]).…”