Perfect fluids and Ashtekar variables, with applications to Kantowski-Sachs models

Starting from a variational principle for perfect fluids, we develop a Hamiltonian formulation for perfect fluids coupled to gravity expressed in Ashtekar’s spinorial variables. The constraint and evolution equations for the gravitational variables are at most quadratic in these variables, as in the vacuum case and in the coupling of gravity to other matter fields, while some of the matter evolution equations are in general non-polynomial. We specialize the formalism to harotropic fluids and spherically symmetric spacetimes, and, within this class, to Kantowski-Sachs spacetimes. We find explicitly the Kantowski-Sachs solutions corresponding to ’stiff matter’, which we use as examples to look at the behaviour of the Ashtekar variables when the spatial metric becomes degenerate on one hypersurface. We find that in these solutions the coordinate time arising in the present treatment is singularly related to proper time, and the singularities are only reached at infinite values of the former. We obtain some simple necessary conditions that have to be satisfied if one wants to evolve data past singularities of this kind. None of the barotropic-fluid-filled Kantowski-Sachs spacetimes satisfy these conditions.