Conformal Galilean-type algebras, massless particles and gravitation

After defining conformal Galilean-type algebras for arbitrary dynamical exponent z we consider the particular cases of the conformal Galilei algebra (CGA) and the Schrödinger Lie algebra (sch). Galilei massless particles moving with arbitrary, finite velocity are introduced i) in d = 2 as a realization of the centrally extended CGA in 6 dimensional phase space, ii) in arbitrary spatial dimension d as a realization of the unextended sch in 4d dimensional phase space. A particle system, minimally coupled to gravity, shows, besides Galilei symmetry, also invariance with respect to arbitrary time dependent translations and to dilations with z = (d+2)/3. The most important physical property of such a self-gravitating system is the appearance of a dynamically generated gravitational mass density of either sign. Therefore, this property may serve as a model for the dark sector of the universe. The cosmological solutions of the corresponding hydrodynamical equations show a deceleration phase for the early universe and an acceleration phase for the late universe. This paper is based, in large part, on a recent work with W.J. Zakrzewski: Can cosmic acceleration be caused by exotic massless particles? arXiv: 0904.1375 (astro-ph.CO) [1]. ∗Based on a Plenary Lecture delivered at the workshop “Symètries non-relativistes: thèorie mathèmatique et applications physiques”, LMPT Tours, 23–24 Juin 2009