Finite N effects on the collapse of fuzzy spheres

Finite N effects on the time evolution of fuzzy 2-spheres moving in flat spacetime are studied using the non-Abelian DBI action for N D 0-branes. Constancy of the speed of light leads to a definition of the physical radius in terms of symmetrised traces of large powers of Lie algebra generators. These traces, which determine the dynamics at finite N , have a surprisingly simple form. The energy function is given by a quotient of a free multi-particle system, where the dynamics of the individual particles are related by a simple scaling of space and time. We show that exotic bounces of the kind seen in the 1 /N expansion do not exist at finite N . The dependence of the time of collapse on N is not monotonic. The time-dependent brane acts as a source for gravity which, in a region of parameter space, violates the dominant energy condition. We find regimes, involving both slowly collapsing and rapidly collapsing branes, where higher derivative corrections to the DBI action can be neglected. We propose some generalised symmetrised trace formulae for higher dimensional fuzzy spheres and observe an application to D -brane charge calculations. for of this effective mass squared indicates that the acts as a gravitational which the dominant energy condition. We extended some of our discussion to the case of higher even fuzzy spheres with SO (2 k + 1) symmetry. The results for symmetrised traces that we obtain can be used in a proposed calculation of charges in the D 1 ⊥ D (2 k + 1) system. They also provide further illustrations of how the correct definition of physical radius using symmetrised traces of large powers of Lie algebra generators gives consistency with a constant speed of light. A more complete discussion of the finite N effects for the higher fuzzy spheres could start from these results.