Involutive Spacetime Distributions and p-Brane Dynamics

Abstract

We propose a precise definition of multidimensional fluids generated by self-gravitating extended objects such as strings and membranes: a p-dimensional perfect fluid is a smooth involutive p-dimensional distribution on a spacetime, each integral manifold of which is a timelike, connected, immersed submanifold of dimension, p -- representing the history of a (p-1)-dimensional extended object. This geometric formulation of perfect fluids of higher dimensions naturally leads to the associated stress-energy tensor. Furthermore, the laws of temporal evolution and symmetries of such systems are derived, in general, from the Einstein field equations and the integrability conditions. We also present a matter model based on a 2-dimensional involutive distribution, and it is shown that the stress-energy tensor for self-gravitating strings gives rise to a non-trivial spherically symmetric spacetime with a naked singularity.