Building approximate stellar models in general relativity

Resumen

The research leading to this dissertation has been focused in the study of approximate ana- lytical stellar models in General Relativity, in particular using the scheme introduced by Cabe- zas et al. (2007) (CMMR) to build the interior and exterior spacetimes of an isolated rotating perfect fluid source surrounded by vacuum. It fills a gap in the literature since approximate interiors are mainly found following Hartle (1967); Hartle and Thorne (1968) and they involve numerical integrations. CMMR manages to find completely analytical expressions for the met- ric using, besides the slow rotation approximation, a post-Minkowskian one. 1. New uses of CMMR and theoretical applications We have expanded the use of CMMR in three ways. First, we provided a new model for a linear equation of state (EOS), where previous applications dealt with only with uniform en- ergy density and Newtonian polytropes (Martín et al., 2008). This EOS is versatile including the ones of exact solutions (Schwarzschild’s, Wahlquist’s) and also physically realistic compo- sitions, such as strange quark matter (SQM) using the simple MIT bag model (Witten, 1984) and more realistic SQM EOS like Dey et al. (1998) finding appropriate fittings for the parame- ters (Zdunik, 2000; Gondek-Rosinska et al., 2000). Second, we improved the matching to use Darmois-Israel conditions and checked the generality of our surface ansatz. Last, we have also built a global model with an interior made up from two layers with linear EOS but different parameters. This last solution gives the approximation scheme a lot of flexibility and possible uses, since most of the astrophysical compact sources can be modelled with between two and four layers of polytropic and SQM equations of state. This two kinds of compositions are now among the results already obtained with CMMR. The focus of the local research group is mainly theoretical and as such, part of our work has dealt with the analysis of the model properties and implications in the field of exact solutions of General Relativity. We have studied all the possible Petrov types of the interior, which are I, D and O, recovered several known results, including some by Fodor and Perjés (2000), extending the impossibility of uniform density sources to be of type D to non-asymptotically flat interiors. We also recovered an approximate Wahlquist metric, and found that the sound reason to use r 􏷟 to find its static limit (Whittaker’s solution) is that it is the only parameter whose vanishing makes twist vector zero. We recovered the result by Bradley et al. (2000) that Wahlquist cannot correspond to an isolated source, and also checked that the linear EOS interior cannot be matched to a Kerr exterior. These results are covered in Cuchí et al. (2013a,b). 2. Comparison with the AKM numerical code After getting satisfactory results analysing the correspondence between our metrics and some exact solutions, we decided to make some checks with stellar models obtained from numerical codes. These models are exact in the sense that no approximation is involved in their obtention. Our group, although used to symbolic computation programs for a long time, has no previous 1experience in this field so I made a brief visit at the Albert Einstein Institut at Gölm-Potsdam to learn about the latest codes. We decided to use the AKM code. The good results of the comparison with AKM encouraged us to program new routines to make the obtention of more approximate CMMR metrics automatically and improve the default graphic functions of Math- ematica to present them properly. The local hardware limitations were attenuated parallelizing some parts in a small cluster. The original aim of the comparison with numeric data was to check the global behaviour of the approximation, but also raised the question of how accurate our models can be in realistic scenarios. It turned out that we have relative errors around 10^-2 in the physical properties of objects with a simple MIT bag model equation of state, typical neutron star densities and masses up to one solar mass and spin rates of 350 Hz. For models with the solar parameters, it drops to 10^-6 . Errors are lower for the metric functions. This work, and the two-layer interior are in preparation for publication.