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 address 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 the action principle of Einstein. We then consider the Newtonian limit of these equations and finally obtain the necessary fluid equations required to describe structure formation. These equations are solved for the density perturbation 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 considered, no structures can grow. We therefore conclude that, within the frame work of Einstein's theory of gravity in (2+1) dimensions, formation of structures cannot take place. Finally, we indicate the different possible ways of getting around this difficulty, so that growing structures can be obtained in two-dimensional cosmological gravitational simulations, and discuss their implications.