Vector fields in multidimensional cosmology

Abstract

Vector fields in the expanding Universe are considered within the multidimensional theory of General Relativity. Vector fields in general relativity form a three-parametric variety. Our consideration includes the fields with a nonzero covariant divergence. Depending on the relations between the particular parameters and the symmetry of a problem, the vector fields can be longitudinal and/or transverse, ultrarelativistic (i.e. massless) or nonrelativistic (massive), and so on. The longitudinal and transverse vector fields are considered separately in detail in the background of the de Sitter cosmological metric. In most cases the field equations reduce to Bessel equations, and their temporal evolution is analyzed analytically. The energy-momentum tensor of the most simple zero-mass longitudinal vector fields enters the Einstein equations as an additive to the cosmological constant. In this case the de Sitter metric is the exact solution of the Einstein equations. Hence, the most simple zero-mass longitudinal vector field pretends to be an adequate tool for macroscopic description of dark energy as a source of the expansion of the Universe at a constant rate. The zero-mass vector field does not vanish in the process of expansion. On the contrary, massive fields vanish with time. Though their amplitude is falling down, the massive fields make the expansion accelerated. The macroscopic analysis of vector fields in cosmology gives up the hope that the major puzzle -- attraction between individual objects and expansion of the Universe as a whole -- can be solved within the Einstein's theory of general relativity.