Distributed Sources and Cosmology

Equation for relativistic dynamics of distributed sources is obtained. It is shown that distributed sources, in certain simple case, produce the effect of cosmological constant in the accelerated expansion of the Universe. Relativistic distributed sources can be used also for local phenomena in cosmology. The singular vortices, sources (sinks) and vortex-sinks are well known models in fluid dynamics (including geophysical fluid dynamics), magnetized plasma, superfluidity and superconductivity (see, for example, Refs. 1-3 and references therein). Dynamics of distributed vortices is a well developed area of research. However, to our knowledge, dynamics of distributed sources (DS) has not been considered until recently [4]. One of the applications of DS, indicated in Ref. 4, is cosmology. Particularly, solution of corresponding equation with constant intensity of DS [4] is similar to homogeneous solution of general relativity with the cosmological constant (CC). However, analysis of DS in Ref. 4 was non-relativistic. In this Note we consider relativistic generalization of DS. We will show that DS, in certain simple case, produce the effect of CC in the accelerated expansion of the Universe. Relativistic DS can be used also for local phenomena in cosmology. Local intensity of nonrelativistic DS is characterized by the divergency of the velocity field ∂v α /∂x α (summation over the repeated Greek indexes is assumed from 1 to 3). Dynamical equation for DS was obtained by considering superposition of localized sources, which move each other with induced velocity field [2-4]. For relativistic DS we need four-dimensional velocity field (see, for example, Refs. 5, 6): u i = dx i ds = γ dx i dτ , γ = (1 − v 2) −1/2 , d ds ≡ u k ∂ ∂x k (1) Here position of fluid element is characterized by the 4-vector x i with components (τ , x α), where τ = ct and c is the velocity of light. Components x α are assumed to be functions of τ and some identification parameters, for example, initial positions x α o. 4-vector u i has components (γ, γv α) with v α normalized by c, γ is the Lorentz factor and summation over repeated Latin indexes is assumed from 1 to 4. Components u i can be considered as functions of x i or as functions of (τ , x α o). For simplicity, we will use the covariant differentiation only at the end of this Note, where it …