Generating exact polytropes in non-conservative unimodular geometries

Abstract

The trace-free Einstein equations contain one equation less than the complete field equations. In a static and spherically symmetric spacetime, the number of field equations is thus reduced to two. The equation of pressure isotropy of general relativity, however, is preserved thus showing that any known perfect fluid spacetime is a suitable candidate for the trace-free scenario. The extra freedom in imposing two constraints may now be exploited to include polytopes, something that is difficult in general relativity. The point here is that using any known exact solution one can find a polytropic star for various values of the polytropic index. One arrives at Tolman–Oppenheimer–Volkoff type equations and can study their solutions explicitly. Two examples of well-known stellar distributions that generate polytropes with physically reasonable behaviour are discussed. These models are regular, exhibit a sound speed that is never superluminal and are adiabatically stable in the sense of Chandrasekhar. We investigate a compactness measure confirming that our results are consistent with some observational data.