Hamiltonian approach to relativistic star models

Abstract

An ADM-like Hamiltonian approach is proposed for static spherically symmetric relativistic star configurations. For a given equation of state the entire information about the model can be encoded in a certain two-dimensional minisuperspace geometry. We derive exact solutions which arise from symmetries corresponding to linear and quadratic geodesic invariants in minisuperspace by exploiting the relation to minisuperspace Killing tensors. A classification of exact solutions having the full number of integration constants is given according to their minisuperspace symmetry properties. In particular it is shown that Schwarzschild's exterior solution and Buchdahl's n=1 polytrope solution correspond to minisuperspaces with a Killing vector symmetry, while Schwarzschild's interior solution, Whittaker's solution and Buchdahl's n=5 polytrope solution correspond to minisuperspaces with a second rank Killing tensor. New solutions filling in empty slots in this classification scheme are also given. One of these new solutions has a physically reasonable equation of state and is a generalization of Buchdahl's n=1 polytrope model.