We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincaré group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity c) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants R and c. The limit transitions lead to the ordinary (R → ∞) or alternative (c → ∞) but nevertheless relativistic kinematics.

We study the properties of fundamental physical constants using the threefold classification of dimensional constants proposed by J.-M. Lévy-Leblond: constants of objects (masses, etc.), constants of phenomena (coupling constants), and “universal constants” (such as c and ℏ). We show that all of the known “natural” systems of units contain at least one non-universal constant. We discuss the possible consequences of such non-universality, e.g., the dependence of some of these systems on the number of spatial dimensions. In the search for a “fully universal” system of units, we propose a set of constants that consists of c, ℏ, and a length parameter and discuss its origins and the connection to the possible kinematic groups discovered by Lévy-Leblond and Bacry. Finally, we give some comments about the interpretation of these constants.

In this paper we consider the conservation laws for classical particles in AdS 4 . At first we parameterize a geodesic line and construct conserved quantities with analog of five dimensional Minkowski space-time M (2,3) .Consequently we change AdS 4 space to AdS-Beltrami space-time and write conserved quantities in the Beltrami coordinates. Finally we take a limit for small velocityẋ ≪ c and we get the conserved quantities in Lorentz-Fock space-time. And finally we give out the energy of the nonrelativistic noncosmological particle under the cosmological limit.

Here we construct new solutions from the Einstein Field equations -some analog of the Schwarzschild metric in anti-de Sitter-Beltrami spacetime in the c → ∞ limit (R-spacetime). In this case we derive an adiabatic invariant for finite movement of the massive point particle and separate variables Hamilton-Jacobi equation. Quasi orbital motion is analyzed and its radius time dependence is obtained.

We study some particular modifications of gravity in search for a natural way to unify the gravitational and electromagnetic interaction. The certain components of connection in the appearing variants of the theory can be identified with electromagnetic potential. The methods of adding matter in the form of scalar and spinor fields are studied. In particular, the expansion of the local symmetry group up to GL(2, C) is explored, in which equations of Einstein, Maxwell and Dirac are reproduced for the theory with Weyl spinor.⋆