A formal analysis of (2+1)-dimensional gravity

Several investigations in the study of cosmological structure formation use numerical simulations in both two and three dimensions. In this paper we a ddress the subtle question of ambiguities in the nature of two-dimensional gravity in an expanding background. We take a detailed and formal approach by deriving the equations describing gravity in (D + 1) dimensions using th e action principle of Einstein. We then consider the Newtonian limit of the se equations and finally obtain the necessary fluid equations required to d escribe structure formation. These equations are solved for the density per turbation in both the linearized form and in the spherical top-hat model of nonlinear growth. We find that, when the special case of D = 2 is consider ed, no structures can grow. We therefore conclude that, within the frame wo rk of Einstein's theory of gravity in (2 + 1) dimensions, formation of stru ctures cannot take place. Finally, we indicate the different possible ways of getting around this difficulty, so that growing structures can be obtain ed in two-dimensional cosmological gravitational simulations, and discuss t heir implications.