Hamiltonian approach to relativistic star models

TL;DR

This paper introduces a Hamiltonian framework for static relativistic star models, classifies exact solutions via minisuperspace symmetries, and presents new solutions including a physically reasonable one generalizing Buchdahl's n=1 polytrope.

Contribution

It develops a Hamiltonian approach to relativistic stars, classifies solutions based on minisuperspace symmetries, and discovers new solutions with potential physical relevance.

Findings

Exact solutions linked to minisuperspace symmetries identified

Schwarzschild and Buchdahl solutions classified by Killing vectors/tensors

New physically reasonable solution generalizing Buchdahl's n=1 polytrope

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 2-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 integrations 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.