Revisiting Schwarzschild's constant density star in isotropic coordinates

Abstract

Herein we shall revisit the venerable 110-year-old topic of Schwarzschild's constant density star, emphasizing that for many (though not quite all) purposes it is much easier to analyze this spacetime in isotropic coordinates (\emph{versus} the more usually adopted Hilbert--Droste area coordinates). The relevant line element is particularly transparent, containing two simple rational functions of the radial coordinate, and the two physical parameters appearing in this line element are easily and readily interpretable in terms of the central density and central pressure of the star. Local properties in the stellar interior (such as the pressure profile) will be seen to be remarkably simple, though quasi-local properties like the Misner--Sharp mass are just a little bit trickier. Apart from its simplicity and clarity, the analysis is also of considerable pedagogical interest. For instance, there are a number of interesting special cases. Mathematically there is a perfectly good solution corresponding to a zero density star -- which can physically be interpreted as an explicit verification of the fact that pressure gravitates, even in the absence of mass-energy density. Additionally, there is a singular solution containing a naked singularity that satisfies all but one of the standard classical energy conditions. Furthermore you can even do both, combining zero density with a naked singularity -- so that pressure by itself can generate naked singularities -- at the cost of merely violating the dominant energy condition, the least physical of the standard energy conditions. We argue that many physically interesting features of Schwarzschild's star are very much under-appreciated.