Mechanics and kinetics in the Friedmann-Lemaître-Robertson-Walker space-times

Using the standard canonical formalism, the equations of mechanics and kinetics in the Friedmann-Lemaître-Robertson-Walker (FLRW) space-times in Cartesian coordinates have been obtained. The transformation law of the generalized momentum under the shift of the origin of the coordinate system has been found, and the form invariance of the Hamiltonian function relative to the shift transformation has been proved. The derived equations allow one to shift the origin of the coordinate system to the point of location of the observer. The space in the vicinity of this point can be considered as a Euclidean one which makes straightforward the interpretation of calculations. For the distribution function in the phase space, the general solution of the collisionless Boltzmann equation has been found. The results of this work can be used for treatment of evolution of the distribution function of particles arriving from the cosmologically distant objects. We discuss, in particular, two important cases of astrophysical interest: (i) the homogenous distribution particles taking into account energy losses, and (ii) the spherically symmetric case with arbitrary angular distribution. While the first problem is linked to the diffuse distributions of particles produced at cosmological epochs, the second one is relevant to the discrete astrophysical objects.

Figure 1

The dependence of proper distance on time.